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\author{RYAN MICHAEL MAY}
\title{ESTIMATING AND MITIGATING ERRORS IN DUAL-POLARIZATION RADAR ATTENUATION CORRECTION}
\depositdate{2014}
\majorfield{SCHOOL OF METEOROLOGY}
\pdfinfo{
   /Author (Ryan Michael May)
   /Title (Estimating and Mitigating Errors in Dual-Polarization Radar Attenuation Correction)
   /CreationDate (D:20110403221170)
   /ModDate (D:20140125153500)
}

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\begin{document}
\memberone{Dr. Michael Biggerstaff, Chair}
\membertwo{Dr. Robert Palmer}
\memberthree{Dr. Ming Xue}
\memberfour{Dr. Tian-You Yu}
\memberfive{Dr. Louis Wicker}
\membersix{Dr. Mark Yeary}
\begin{preface}
\thispagestyle{empty}
This dissertation is dedicated to the love of my life, my wife Macie, without whose love and support this work would never have happened.
\prefacesection{Acknowledgements}
\setcounter{page}{4}
First and foremost, I want to acknowledge the support of my family and so many
friends. This dissertation has been (too) long in coming, and I would not have
made it to the finish line without your support.

I am grateful to my thesis advisor, Dr. Michael Biggerstaff, for all the long hours
of discussion that helped to shape the scope and direction of this work. His
mentorship has helped to make me a better writer and a better scientist.

I would also like to thank the members of my thesis committee, Drs. Robert
Palmer, Louis Wicker, Ming Xue, Mark Yeary, and Tian-You Yu. Your advice was
instrumental in helping me to chose a focused and manageable scope for my
thesis work. I am also grateful for your patience in dealing with the logistical
challenges that were associated with serving on my doctoral committee.

I would also like to express my thanks to Dr. Ted Mansell for performing the
numerical storm simulations that served as key input to this work.
%must leave a blank line next to avoid bug:

\tableofcontents
\listoftables
\listoffigures
\prefacesection{Abstract}
Correction for rain attenuation is an important data quality issue when using data collected by radars
operating at attenuating wavelengths, specifically C and X bands. Such issues are especially important
for quantitative use of the data, such as rainfall estimation, where a 3dB error in reflectivity factor can result
in more than 60\% error in the rainfall estimate. In this work, the errors
from several different attenuation correction techniques are examined. To test
the corrections, simulated time-series dual-polarization radar data are used. The basis for the 
simulations is the use of a discretized radar pulse, where each pulse element
generates the appropriately calculated stochastic value to give realistic radar
time series data. In addition to providing for
a sufficient number of elements to generate statistically meaningful data,
this discretized pulse model also
enables the simulation of spatial sampling aspects of the radar beam, allowing
for differential attenuation and phase shift across the radar beam.

These simulated data are used to quantify the performance of several rain attenuation correction algorithms: linear $\Phi_{DP}$, ZPHI, and Self-Consistent,
as well as a modified version of the Self-Consistent algorithm.
Using the simulated data and respective
truth fields, the performance of the algorithms is examined in detail across a variety of 
scattering and microphysics configurations, to study the impact of the assumptions made on the quality of algorithm performance. A wide array of radar
spatial sampling strategies are also examined to identify the impacts on
algorithm performance.
\end{preface}\chapter{Introduction}
\section{Motivation}
Radar-based Quantitative Precipitation Estimation (QPE) has been an active area
of research since the advent of radar meteorology
\citep{Marshall1947,Stout1968}. In fact, QPE is one of the primary missions of
the National Weather Service’s (NWS) Weather Surveillance Radar 1988-Doppler
(WSR-88D) network \citep{Crum1993,Klazura1993}. Traditional radar-based
estimation of precipitation relies on reflectivity (Z) only, which roughly
depends on the sum of the 6th power of the diameters of all scattering particles
(rain drops). This results in a non-unique relation between the volume of water
in the sampling volume of the radar pulse and the reflectivity estimated by the
radar---the relation depends on the actual distribution of droplet sizes
\citep{Stout1968}. Additionally, high reflectivity values are often associated
with the hail cores of severe storms, resulting in severe over-estimation of
rain in those areas. Non-meteorological echoes (e.g. ground clutter, insects,
birds, chaff, etc.) can also cause an automatic algorithm to estimate rain when
none is falling. Other problems include beam-blockage and attenuation, which 
causes rainfall underestimation due to the relation between Z and absolute
returned power.
Similarly, calibration of the radar is also important and is a non-trivial problem \citep{Atlas2002}.

To address these shortcomings, many radars now transmit and receive
electromagnetic radiation at two orthogonal polarizations (i.e., horizontal and
vertical). Since electromagnetic scattering is related to the size of the particle in the direction of wave polarization, having these two channels
gives much more information about the shape and orientation of the scatterers within the volume. For instance, liquid water drops are
increasingly deformed by aerodynamic stresses as their diameter increases. This
effect gives them a larger radar cross section (RCS) for horizontally polarized
waves than for vertically polarized. Thus, by comparing the amount of power
returned in the vertical and horizontal channels, $Z_{DR}$, one can estimate to what extent the raindrops are deformed from
spheres; this allows some inferences about the sizes of the drops to be made. 
Additionally, the non-spherical raindrops will result in a different effective index of refraction along the
radar propagation path. This difference results in a phase shift ($\phi_{DP}$)
between the channels; the range derivative of this parameter, $K_{DP}$, can be related to
rain amounts along the path \citep{Chandrasekar2001}. Using $K_{DP}$ and
$Z_{DR}$, regions with larger
water drops can be differentiated from those with only small (and mostly
spherical) drops. There are also applications for detecting ice crystals, hail,
and non-meteorological echoes
\citep[see for example][]{Straka2000,Jameson1983,Giuli1991,Hubbert1993}. 
Utilizing this increased information, dual-polarization radar data has been
shown to yield an improvement in radar-based QPE. This improvement, combined with
the ability to classify the radar echoes, has demonstrated sufficient gains in utility
that the WSR-88D network has been upgraded with dual-polarization capabilities
\citep{Ryzhkov2005a,Ryzhkov2005b,Doviak2000}.

As promising as they seems, there are significant
challenges remaining in using dual-polarization data for radar-based QPE. Careful
calibration of the radar is still important, and for dual polarization measurements, two separate channels need to be calibrated.
Also, the meteorological range of $Z_{DR}$ is much less than that of $Z$, so the relative calibration
of the two channels must be an order of magnitude more precise than the
absolute calibration of a single channel \citep{Ryzhkov2005c}.

\section{Attenuation Correction}
At a wavelength in S-band (nominally \SI{10}{\centi\meter}), attenuation is small and usually neglected; however,
 in heavy rain, one can still see attenuation up to \SI{.04}{db\per\kilo\meter} \citep{Bringi1990}. At
 wavelengths in C-band and X-band (nominally \SI{5}{\centi\meter} and \SI{3}{\centi\meter}, respectively),
 attenuation can be significant, up to \SI{0.3}{dB\per\kilo\meter} for C-band and \SI{2}{dB\per\kilo\meter} for X-band \citep{Bringi1990}. Therefore,
 data at these wavelengths must be corrected for attenuation before being used quantitatively. With only reflectivity data from a single channel available, however, 
trying to correct attenuation is unstable \citep{Hitschfeld1954,Hildebrand1978}. 

With the advent of dual-polarization radars, additional possibilities emerged for 
correcting attenuation of reflectivity. This also brought the new problem of 
correcting for \emph{differential} attenuation.
\cite{Bringi1990} laid the theoretical framework for such corrections by computing $K_{DP}$, attenuation ($A_H$),
and differential attenuation ($A_{DP}$) at S-, C-, and X-bands for a range of gamma DSD parameters. Linear regressions
through the resulting scatter of points showed promise in terms of producing linear relationships between $K_{DP}$ and both $A_H$ 
and $A_{DP}$, which could be used to estimate attenuation and correct for it. \cite{Aydin1989} instead proposed a
correction procedure for dual-polarization data that used an empirical relationship between ${A_H}/{Z_H}$ and
$Z_{DR}$.  \cite{Gorgucci1996} modified this procedure to use a more straightforward formulation of $A_H$ and
$A_{DP}$ in terms of $Z_H$ and $Z_{DR}$.  \cite{Gorgucci1998} performed a comparison between the methods
proposed by \cite{Hildebrand1978,Aydin1989,Bringi1990} at C-band. The corrections using only $Z$ and $Z_{DR}$
were found to perform much better than corrections using only $K_{DP}$. This is likely because $K_{DP}$,
as the range derivative of $\phi_{DP}$, is a difficult parameter to estimate robustly in all conditions
\citep{Gorgucci2000}. However, the ($Z_H$, $Z_{DR}$) method relies upon having a properly calibrated radar,
which is difficult to achieve and maintain \citep{Ryzhkov2005c}, especially operationally; conversely, calibration is not required to use $K_{DP}$.

\cite{Scarchilli1993}, working at C-band, introduced a simple method based on \cite{Bringi1990} that calculates
$A_H$ and $A_{DP}$ as linearly proportional to $K_{DP}$; this implies that the path integrated (differential) attenuation is linearly
related to the change in $\phi_{DP}$ up to that point.  The method then uses the corrected $Z_{DR}$ data to estimate backscatter 
differential phase ($\delta$), which can then be removed from the $\phi_{DP}$ data. This procedure is iterated until the differences in the 
$\phi_{DP}$ data between iterations became small.

\cite{Ryzhkov1995} applied the linear $A_H-\phi_{DP}$ and $A_{DP}-\phi_{DP}$ relations as well, but took the step of using an
empirical procedure to estimate the proportionality constants at S-band. These constants were estimated by looking at how a subset of the
$Z_H$ data (based on the corresponding $K_{DP}$ values) changed as a function of the total $\phi_{DP}$ change; the slope of this scatter
plot gave the appropriate $A_H-\phi_{DP}$. A similar procedure was performed for $Z_{DR}$ to estimate $A_{DP}$. This technique was applied
by \cite{Carey2000} at C-band to correct rainfall estimates in tropical convection. \cite{Carey2000} also noted the spread in published
coefficients at the time and attributed the variation to the dependence on a considerable amount of theoretical assumptions: temperature, drop size
distribution (DSD), and drop shape model. The study found that an advantage of empirical estimates was its ability to find coefficients
appropriate for the actual observed conditions.

\cite{Matrosov2002} extended this same approach to X-band, but slightly modified the
technique so that the drop shape factor, used to calculate the coefficient of proportionality between $A_H$ and $K_{DP}$,
is calculated from $Z_H$ and $Z_{DR}$ (initially corrected with a stock shape factor). The proportionality constants between $A_H$
or $A_{DP}$ and $K_{DP}$ are calculated from this shape factor. \cite{Anagnostou2006a} extended this slightly to
calculate the proportionality constant between $A_{DP}$ and $K_{DP}$ as well.  \cite{Anagnostou2006a} differs in that they find a ``best'' fit
shape factor for the whole case, where ``best'' was determined based on consistency of corrected data with raindrop spectra. These methods attempt to explicitly adjust for the dependence of the relationship between attenuation and $K_{DP}$ on the actual drop shapes. 

\cite{Smyth1998} proposed a scheme that used as a constraint the total path differential attenuation, estimated from the negative $Z_{DR}$ values
in a distant stratiform region; this constraint assumes that the intrinsic $Z_{DR}$ value for such a region is \SI{0}{dB}. This attenuation
($A_{DP}$) was distributed among all gates with sufficient $K_{DP}$ magnitude based on the value of $K_{DP}$ relative to the total
$\phi_{DP}$ change. The values of $K_{DP}$ and $A_{DP}$ were combined to retrieve the DSD, and from there the value of $A_H$. Having the 
values of both $A_H$ and $A_{DP}$ at all gates, corrected fields of $Z_H$ and $Z_{DR}$ were then generated. While the use of a region with
an intrinsic $Z_{DR}$ of \SI{0}{dB} can be robust for estimating the path integrated differential attenuation, automating the process
of identifying such regions is extremely difficult, making the technique difficult to implement operationally. Additionally, this procedure
offered no solution for scenarios without a distant stratiform region to act as a constraint.

\cite{Testud2000}, based on a ``rain-profiling'' method used for satellite-based
radar rainfall retrievals, used $\phi_{DP}$ as a constraint for correcting $Z_H$
for attenuation, using both $\phi_{DP}$ and $Z_H$ to estimate $A_H$ (hence being commonly referred to as the ZPHI method). The estimated
$A_H$ profile is then used to correct $Z_H$. Using the obtained specific attenuation, values for normalized intercept parameter ($N_0$)
were obtained, which were combined with specific attenuation to estimate rainfall. Separate sets of power law constants were given for the horizontal
and vertical polarizations, which could be used together to correct $Z_{DR}$ for differential attenuation. Alternatively, \cite{Testud2000} 
also provided a relation for calculating $A_{DP}$ from $A_H$ and $N_0$. \cite{Anagnostou2004}
applied this method to X-band, but the increased dependence of attenuation and $K_{DP}$
on microphysical parameters ($N_0$, etc.) led to the estimation over sets of range intervals within a radial. These intervals defined regions
over which $N_0$ was held fixed, and the procedure was iterated to find the optimal set of coefficients for the interval, increasing the
stability of the correction.  From these corrected data, rainfall estimates using $Z_H$, $Z_{DR}$, and $K_{DP}$ were found to be superior
to those using only $Z_H$ at X-band.

\cite{Bringi2001} extended the ZPHI approach with a ``self-consistent'' methodology that dynamically estimates the constant
of proportionality between $K_{DP}$ and $A_H$, $\alpha$. The procedure is iterative, using the radial profile of $Z_H$ and the change
in $\phi_{DP}$ over the radial to estimate a radial profile of $A_H$. This profile of $A_H$ is integrated using $\alpha$ to calculate an
estimated change in $\phi_{DP}$. The difference between the calculated and measured values is used to adjust $\alpha$ and the
procedure is iterated until convergence is achieved. Because the relationship between $A_{DP}$ and $K_{DP}$ is not as
linear at C-band, an estimate of the intrinsic $Z_{DR}$ at a far range is used to provide a constraint on the maximum $A_{DP}$, similar
to \cite{Smyth1998}. In this work, however, an empirical relationship is used to estimate an average $Z_{DR}$ value from the corrected
$Z_H$ data. \cite{Park2005a,Park2005b} extended this method to X-band in a straightforward fashion, only updating various coefficients and
empirical relations to ones appropriate for the wavelength. These updates came from a detailed evaluation of scattering simulations. \cite{Liu2006}
extends the approach slightly for $Z_H$ by using the Levenberg-Marquardt solver to improve convergence when solving for $\alpha$. \cite{Liu2006}
also modified the approach by running the same technique, using different coefficients, to correct $Z_V$ data and combining corrected $Z_H$
and $Z_V$ to obtain corrected $Z_{DR}$.

\cite{Gorgucci2006} corrected for attenuation and differential attenuation using a self-consistent approach at X-band as well, but with important
differences. Instead of extending ZPHI to adaptively estimate the coefficient relating $A_H$ and $\phi_{DP}$, \cite{Gorgucci2006} utilized the
full self-consistency of $Z_H$, $Z_{DR}$, $K_{DP}$, and ${A_H}$. Again, due to the challenges of working with $K_{DP}$, an estimated profile
of $\phi_{DP}$ was calculated from corrected estimates of $Z_H$, $Z_{DR}$, and $A_H$ (combined using a power-law). Differences between the
observed and calculated $\phi_{DP}$ profile were used to adjust a constant, which yielded a new estimate of $A_H$, and hence new profiles of
$Z_H$ and $Z_{DR}$. This procedure was iterated until the value of the constant converged; initial values were obtained from the procedure of
\cite{Bringi2001}. In this method, $A_{DP}$ was estimated using the same procedure (with different coefficients), but in a separate set of steps
and convergence.  \cite{Gorgucci2007} (working at C-band) modified this to simultaneously estimate $A_H$ and $A_{DP}$, making use of the
approximately linear relationship between the two variables. The unified approach was found to have reduced errors over the separated approach. The validation
of the technique was based on C-band profiles reconstructed from S-band data.

Beyond these main trees of approaches, there have been several novel techniques tried, with varying degrees of success.  \cite{Vulpiani2005} used
an artificial neural network to approximate the non-linear relationship for calculating attenuation from $Z_H$ and $Z_{DR}$. This algorithm worked
iteratively, like \cite{Hitschfeld1954}, running from the furthest range inward, constraining the total attenuation correction using $\phi_{DP}$, as
in \cite{Bringi2001}. This approach was demonstrated to have less error than the ZPHI approach, but training the neural network makes the
approach challenging from an operational implementation standpoint. \cite{Vulpiani2008} attempted to address the issue of spatial variability in
the coefficients relating attenuation to radar observables by using a Bayesian classifier. The classifier, using $Z_H$, $Z_{DR}$, $K_{DP}$, and
$\rho_{HV}$, identified regions as one of four classes (large drops or light, medium, or heavy rain); the results of the classification were used
to select an appropriate set of coefficients for estimating $A_H$ and $A_{DP}$ from $\phi_{DP}$. Using intrinsic values of $Z_{DR}$ at long ranges,
they demonstrated slightly reduced root mean square error in comparison with ZPHI.

\cite{Gourley2007} introduced a novel approach that advects cells between radar scans using a cross-correlation analysis. The decreases in
$Z_H$ and $Z_{DR}$ for individual cells were matched to increases in $\phi_{DP}$, which was used to retrieve the coefficients relating $\phi_{DP}$ to
$A_H$ and $A_{DP}$. The technique relied on the fact that the mean change in intrinsic $Z_H$ and $Z_{DR}$ over the whole radar coverage area
for several hours was approximately zero (i.e., no mean intensity changes).

A subject of much recent research are the so called ``hot spot'' regions, as identified by \cite{Ryzhkov1995}. These are regions of anomalously
high attenuation caused by the presence of anomalously large scatterers (compared with the sizes normally assumed for rain drop size
distributions). \cite{Tabary2009} and \cite{Borowska2011} attribute the cause of such areas to be wet ice particles, likely from melting hail. Because such
regions differ greatly from the surrounding areas in terms of the size distribution of scatterers, attenuation
from the areas are not properly corrected using mean profiles generated from most scattering simulations \citep{Carey2000}.  \cite{Gu2011}
proposed a correction method that identifies ``hot spot'' regions using $Z_H$ and $\rho_{HV}$; after identification, these regions were corrected for
attenuation using coefficients adjusted from their ``background'' values, as estimated in \cite{Ryzhkov1995}.

\section{Simulation}
Given all of the challenges and advances in radar-based QPE, one natural way to
evaluate the relative quality of different methods is through simulation. Recent
advances in numerical modeling have made it possible to simulate convective
storms at very fine scales over a broad range of environmental conditions
\citep[e.g.][]{Wicker1995,Lewellen1997,Jung2010}. Coupling a software radar simulator with
high-resolution numerical simulations, one can generate large sets of simulated
radar data that span a wide range of radar operating characteristics. These
simulated datasets can be used to provide truth data for objective evaluation of
algorithms.

Many approaches have been taken previously in simulating radar data, varying in
sophistication from simple time series simulation \citep{Zrnic1975} to moment
calculation \citep{Chandrasekar1987,Krajewski1993,Gosset2004} or full simulation
of radar returns from each pulse \citep{Capsoni1998,Capsoni2001,Muschinski1999}.
\cite{Zrnic1975} generated simulated time series radar data and Doppler spectra
using an assumed Gaussian distribution of velocities within the resolution
volume. \cite{Chandrasekar1987} looked at the variation of simulated
reflectivity values as a function of raindrop size distribution parameters.
Similarly, \cite{Krajewski1993} calculated values of reflectivity factor and
differential reflectivity using rainfall rates from a numerical model combined with an
assumed drop size distribution. Neither of these studies was concerned with
Doppler velocity or the impacts of scanning strategies. \cite{Jung2010} used
detailed scattering calculations to generate reflectivity, differential
reflectivity, differential propagation phase, and copolar cross-correlation data
from a numerical simulation. These data were used to look at the effects of
different numerical model microphysics schemes. \cite{Wood1997} evaluated the
effects of WSR-88D \citep{Crum1993} radar scanning strategies on the sampling of
mesocyclones and tornadoes. The effects of the scanning strategy were accounted
for by using an effective beamwidth for the radar, which was used to scan an
analytic vortex with a uniform reflectivity field. \cite{Gosset2004} used
analytic fields to calculate dual-polarization moments in order to analyze the
effect of non-uniform beam-filling on moment estimates. As mentioned previously, \cite{Zahiri2008} calculated polarimetric radar data from \SI{1}{\kilo\meter} 
numerical simulation output interpolated onto a polar grid; these simulated data included error by adding normally distributed noise terms. 
\cite{Berne2005} simulated reflectivity data using a sophisticated statistical model. Using stochastic, auto-regressive range-profiles of rain 
distribution parameters, they examined errors in the attenuation correction method of \cite{Hitschfeld1954}, as applied to satellite data. Beyond the
sophisticated statistical model, however, the radar simulation was simplistic, and did not include any beam-weighting or azimuthal gradients.

\cite{Capsoni1998} simulated the
pulse-to-pulse time series of radar data by combining the simulated returns from
individual representative hydrometeors within a radar volume (Monte Carlo
Sampling). This work was extended to generate polarimetric signatures by
\cite{Capsoni2001}. Due to the computational requirements of this approach, the
radar data were generated for only a single range gate only, and thus many
aspects of the scanning radar were not simulated. A similar approach was
presented by \cite{Cheong2008} for general use in generating simulated radar
signals (IQ data) at a single polarization from the output of numerical storm
models. \cite{Muschinski1999} used a different scheme for simulating time series
data for a wind profiler from a large eddy simulation. It relied on using a
fixed grid of scattering centers which return appropriate phase shifts;
when combined with a random initial phase, this procedure yielded realistic radar signals.

The work presented herein describes a radar simulator designed to simulate the returns from a
scanning Doppler radar on a pulse-to-pulse basis, based on the radar
configuration and scanning strategy used. This work extends that of
\cite{May2007}, which demonstrated that the simulator is capable of simulating
several radar data characteristics including: range resolution, azimuthal over-
and under-sampling, non-standard (anomalous) propagation, Rayleigh attenuation,
antenna sidelobes, velocity aliasing, and range aliasing. Starting with output
from a high-resolution numerical simulation, the simulator generates both
time-series and volume-averaged moments for H and V polarizations in the
linear basis. Time series data are generated using the method demonstrated by
\cite{Muschinski1999}. This approach is combined with that of \cite{Galati1995} to add appropriate correlation between the horizontal and vertical 
channels. These polarimetric capabilities will be demonstrated for a variety of wavelengths and scattering models.

\section{Application}
In developing methods for attenuation correction and radar-based QPE, validation is a frequent challenge.
Many validation methods rely on simple simulation, such as generating X-band profiles from S-band data. These have the benefit of removing some
DSD variability, but rely upon many assumptions about scattering processes in order to relate radar observables at one band to those at another. For attenuation
correction, another method is to compare co-located radars, with one operating
at a non-attenuating wavelength, such as S-band \citep{Anagnostou2006,Snyder2010}. This requires careful calibration of two radars and has errors
due to slightly different locations,
different observation paths, as well as differences in the intrinsic values due to resonance regime effects. Another validation approach, especially
for QPE, is to compare point measurements of rain from gauges (or full DSD measurements from disdrometers) to radar data, but this approach has problems due
to the inherent difference in scale between point measurements and the volume averaging involved in remote sensing. Additionally, at longer ranges
from the radar, the remotely sensed data can be more than a \si{\kilo\meter} above the point measurement, introducing significant variance into
the comparison.

Many of these previous shortcomings in validation can be addressed using a sophisticated radar simulation, such as that described previously.
In this work, errors inherent in attenuation correction at multiple wavelengths
(C- and X-band) are quantified and analyzed to identify
specific sources of these errors. In the process, several different techniques will be evaluated; of particular interest are the effects of temperature and
non-uniform beam-filling, including down-range impacts of differential attenuation across the radar beam.  Because the simulation includes full
propagation effects as well as a discretized radar pulse that can be filled with a heterogeneous field, the effects of non-uniform beam-filling are also taken into
account. Also, because the simulator contains all relevant truth data,
calculating errors in retrieved attenuation profiles is straightforward. Since the simulation is stochastic in nature, simulated data replicate the
variability seen in real data, including the effects of low signal to noise ratio; this implies that error estimates will reflect those of real world data.\chapter{Radar Simulator}
\section{Design}
Since the performance and accuracy of the simulation of radar data is critical
to the validity of the work that will follow, a large section of this work is
dedicated to a detailed description of the radar simulator used, which is an
extension of the work done by \cite{May2007}. The extensions can be summarized
as:
\begin{itemize}
    \item Inclusion of new scattering models for polarimetric variables
    \item Extension for model input which includes two-moment
        microphysics schemes
    \item Expansion to enable the generation of time series data in additional
        to simple volume average-based moments
    \item Generation of additional diagnostic fields to help identify regions where the pulse is non-uniformly filled.
\end{itemize}
The flow of the program is as follows (with the details covered in subsequent
 sections):
\begin{itemize}
    \item Read configuration
    \item Calculate radar variables from model input fields
    \item As the radar scans:
    \begin{itemize}
        \item Interpolate model fields to discretized radar resolution volume
        \item Generate a sample from the pulse
        \item As appropriate, generate a radial of data from the samples
    \end{itemize}
\end{itemize}

\subsection{Simulator configuration and input}
The behavior of the radar simulator is controlled by specifying radar
characteristics and scanning strategy (Table~\ref{tab:emul-params}). Note that
the antenna beamwidth, gain, and wavelength are treated independently to allow
for simulation of antennas with different classes of performance. The
actual form of the antenna pattern is fixed based on the analytic function
discussed later. It should also be noted that the polarization of the radar is
fixed to simultaneous horizontal and vertical transmission. This is an
implementation detail rather than a limitation, as adding transmit
polarization flexibility would be straightforward. The minimum detectable signal
is used as a threshold for the calculated moments and also as a noise power
value for the time series simulation. The pulse repetition time and pulse length
are given independently, but in reality they are usually constrained by the
duty cycle of the transmitter. Enforcement of a duty cycle is up to the user
when creating the configuration. The antenna pointing angles can be specified
for either full or sector plan-position indicator (PPI) scans, or range-height
indicator (RHI) scans. The simulator allows for overlapped sampling in both
azimuth (or elevation for RHI scans) and in range. In addition to the radar
configuration parameters, the drop shape model, canting angle distribution width (for T-matrix), and scattering model can be
specified at runtime. This simplifies the evaluation of the impacts of the use of 
these different models.

\begin{table}[ht]
    \centering
    \begin{tabular}{ | l | l | l |}
        \hline
        \textbf{Radar Parameters} & \textbf{Scanning Parameters} &
           \textbf{Simulation Parameters}\\
        \hline\hline
        Location & PRT & Input simulation\\
        Antenna beamwidth & Pulse length & Scattering model\\
        Antenna gain and sidelobes & Antenna rotation rate & Drop shape model\\
        Wavelength & Pulses per radial & Canting distribution width\\
        Transmit power & Radar gate spacing & PRNG seed\\
        Range to first gate & Scan fixed angle & \\
        Minimum detectable signal & Start and end angles & \\
        \hline
    \end{tabular}
    \caption{Simulator control parameters.}
    \label{tab:emul-params}
\end{table}

The input data to the radar simulator are three-dimensional gridded fields that
describe the state of the atmosphere. Wind components and the appropriate
variables describing the hydrometeor content (depending on the microphysics
scheme used) are required fields. Water vapor, temperature, and pressure are
also needed for calculating the atmospheric index of refraction, which is used
for determining the propagation path of the radar beam. Temperature is also used
in determining the dielectric constant for the hydrometeors.

\subsection{Calculation of Radar Variables}
\subsubsection{Scattering}
The simulator supports several different scattering assumptions: Rayleigh,
Rayleigh-Gans, Mie, and T-matrix \citep{Mishchenko1996,Chandrasekar2001,Doviak1993,Waterman1971}. For
Rayleigh-Gans and T-matrix scattering, there are options to use either spheres
or oblate spheroids as the assumed shape for hydrometeors. It should be noted
that using sphere as the shape for T-matrix or Rayleigh-Gans scattering produces
identical results to Mie and Rayleigh, respectively. When using oblate
spheroids for raindrops, by default the axis ratio ($r$) for the spheroids is calculated
using the polynomial form of \cite{Brandes2002}:
\begin{equation}
    r = \num{0.9951} + \num{0.0251} D - \num{0.03644} D^2
    + \num{5.303e-3}  D^3 - \num{2.492e-4} D^4
    \label{eq:brandes-shape-model}
\end{equation}
where $D$ is the equivolume diameter of the rain drop in \si{\milli\meter}. It
should be noted that the coefficient of the cubic term above, \num{5.303e-3}, is
not the value given by \cite{Brandes2002}. Both \cite{Brandes2002} and
\cite{Brandes2004} give a value of \num{5.030e-3}. However, this value results
in a curve that does not match the plot given in \cite{Brandes2004}, as shown in
Figure~\ref{fig:axis-ratio}; the value of \num{5.303e-3}, however, does produce a matching plot. 
\begin{figure}
    \includegraphics[width=2.6in,height=2.3in]{figures/Brandes2004-axis-ratio-plot.png}
    \includegraphics[scale=0.9]{figures/compare_axis_ratio_formulae}
    \caption{Left: As in \cite{Brandes2004} ``Raindrop axis ratio as a function
        of drop equivalent volume diameter derived from measurements. The curved
        solid line shows an empirical fit [Eq. (21)]; the dashed line shows the
        linear relation of Pruppacher and Beard (1970); and the dash–dot lines
        show the computational domain of Gorgucci et al. (2000)'' Right:
        Comparison of drop axis ratio relations using the coefficients published
        in \cite{Brandes2004} (dashed) and with the corrected value (solid).}
    \label{fig:axis-ratio}
\end{figure}
In addition to this relation, it is also possible to specify the use of the raindrop axis ratio relation introduced by
\cite{Pruppacher1970}:
\begin{equation}
	r =  \num{1.030} - \num{0.062}  D
\end{equation}
where $D$ is again in \si{\milli\meter}.

Another required parameter for performing the scattering calculations is the
refractive index ($m$) for water. The equations given by \cite{Ray1972} are used
to calculate the complex dielectric constant ($\epsilon_r + i\epsilon_i$) as a
function of temperature ($T$, \si{\celsius}) and wavelength ($\lambda$,
\si{\centi\meter}):
\begin{align}
    \epsilon_r &= \epsilon_\infty + \frac{(\epsilon_s - \epsilon_\infty)
        [1 + (\lambda_s / \lambda)^{(1 - \alpha)}\sin{(\alpha \pi / 2)}]}
        {1 + 2 (\lambda_s / \lambda)^{(1 - \alpha)} \sin{(\alpha \pi / 2)}
        + (\lambda_s / \lambda)^{2 (1 - \alpha)}}\\
    \epsilon_i &= \frac{(\epsilon_s - \epsilon_\infty)
        (\lambda_s / \lambda)^{(1 - \alpha)} \cos{(\alpha \pi / 2)}}
        {1 + 2 (\lambda_s / \lambda)^{(1 - \alpha)} \sin{(\alpha \pi / 2)}
        + (\lambda_s / \lambda)^{2 (1 - \alpha)}}
        + \frac{\sigma \lambda}{\num{18.8496e10}}
    \intertext{where $\epsilon_\infty$ is the high-frequency dielectric
    constant, $\epsilon_s$ is the static dielectric constant, $\lambda_s$ is the
    relaxation wavelength, $\alpha$ is a spread parameter, and $\sigma$ is a
    frequency-independent conductivity. These values are calculated as a
    function of temperature as:}
    \epsilon_s &= \num{78.54} [1.0 - \num{4.579e-3} (T - 25.0) \notag\\
        &+ \num{1.19e-5} (T - 25.0)^2 - \num{2.8e-8} (T - 25.0)^3]\\
    \epsilon_\infty &= \num{5.27137} + \num{0.0216474} T + \num{0.00131198} T^2\\
    \alpha &= \frac{\num{-16.8129}}{T + \num{273}} + \num{0.0609265}\\
    \lambda_s &= \num{0.00033836} \,\exp{\frac{\num{2513.98}}{T + 273}}\\
    \sigma &= \num{12.5664e8}
\end{align}
The index of refraction is found by taking the complex square root:
\begin{equation}
    m = \sqrt{\epsilon_r + i\epsilon_i}
\end{equation}
The use of these equations allow the simulator to capture the wavelength and
temperature dependence of the index of refraction, avoiding the use of lookup
tables with interpolation between fixed values for wavelength and temperature.
This permits calculation of scattering using the temperature at
each individual model grid point. Also, by having the wavelength dependence
captured in a functional form, the simulator can be used at arbitrary wavelengths and
not just be constrained to operating at general wavelengths appropriate to a
given band (such as \SI{3}{\centi\meter} for X-band).

All of the scattering calculations start from the scattering matrices
in the forward ($\bf{S}^f$) and backward($\bf{S}^b$) scattering directions,
given as:
\begin{align}
    \bf{S}^f(D) &= \begin{bmatrix} S_{HH}^f(D) & S_{HV}^f(D) \\
        S_{VH}^f(D) & S_{VV}^f(D) \end{bmatrix}\\
    \bf{S}^b(D) &= \begin{bmatrix} S_{HH}^b(D) & S_{HV}^b(D) \\
        S_{VH}^b(D) & S_{VV}^b(D) \end{bmatrix}
\end{align}
where $D$ is diameter and $H$ and $V$ in the subscript represent the
polarization for the incident and scattered wave. The calculation of the
scattering matrices depends on the scattering model used.

\paragraph{Rayleigh}
The Rayleigh approximation to the scattering from a dielectric sphere is the simplest, and the most familiar, method of calculating
radar observables.  Its inclusion within the simulator facilitates the use of the simulator as a teaching tool and also permits
evaluation of the impacts of making the assumption that Rayleigh scattering is the dominant effect. The Rayleigh approximation
is relatively accurate for scatterers whose size is small compared to the wavelength \citep{Doviak1993}. Under this assumption,
the effects of scattering are obtained by treating the sphere as a radiating dipole. This gives the following expressions for backscatter  ($\sigma_b$),
total scatter ($\sigma_s$), and absorption ($\sigma_a$) cross-sections:
\begin{align}
	\left| K\right|^2 &= \left| \frac{m^2 - 1}{m^2 + 2} \right|^2 \\
	\sigma_b &= \frac{\pi^5}{\lambda^4} \left| K \right|^2 D^6 \\
	\sigma_s &= \frac{2}{3} \sigma_b \\
	\sigma_a &= \frac{\pi^2}{\lambda} \Im{-K} D^3
\end{align}
$H$ and $V$ are omitted since Rayleigh scattering assumes spherical scatterers, and thus these scattering properties are
identical for the two polarizations. For consistency with the other scattering calculations, these efficiencies are converted to their
respective terms in the ($\bf{S}^f$) and backward($\bf{S}^b$) scattering matrices:
\begin{align}
S_{HH}^b(D) &= \frac{\sqrt{\sigma_b}}{4 \pi} \\
S_{VV}^b(D) &= S_{HH}^b(D) \\
\Re{S_{HH}^f(D)} &= \Re{S_{HH}^b(D)} \\
\Re{S_{VV}^f(D)} &= \Re{S_{VV}^b(D)} \\
\Im{S_{HH}^f(D)} &= \frac{\sigma_s + \sigma_a}{2 \lambda} \\
\Im{S_{VV}^f(D)} &= \Im{S_{HH}^f(D)}
\end{align}
The expressions for the forward scattering matrices, while not fully correct (for the real part of the complex scattering matrix), allow the extinction cross-section to 
be calculated in accordance with optical theorem
\citep{Ishimaru1991}. This allows all of the different scattering
models to use common code for obtaining relevant radar parameters from the scattering matrices in the forward and backward directions. The
real portion of the forward scattering matrix would be used to obtain a propagation phase; since the differential between H and V polarizations
is the quantity of interest, such a calculation is senseless with the Rayleigh approximation.

\paragraph{Rayleigh-Gans}
Rayleigh-Gans scattering extends Rayleigh scattering theory to non-\\spherical scatterers. This is achieved by adjusting the polarizability of the
scatterer based on the  orientation of the spheroid (prolate or oblate) and the polarization of the incident wave \citep{Chandrasekar2001}. 
This theory gives the backscattering matrix as:
\begin{align}
    \bf{S}^b(D) &= \frac{\pi^2 D^3 (m^2 - 1)}{6 \lambda^2}{\begin{bmatrix} \Lambda & 0 \\
        0 & \Lambda_z \end{bmatrix}}\\
%    \bf{S}^b(D) &= \begin{bmatrix} S_{HH}^b(D) & S_{HV}^b(D) \\
%        S_{VH}^b(D) & S_{VV}^b(D) \end{bmatrix} \\
	\Lambda_{x,y,z} &= \frac{1}{(m^2 - 1)\lambda_{x,y,z} + 1} \\
	\Lambda &= \Lambda_x = \Lambda_y \\
	\lambda_x &= \lambda_y = \frac{1 - \lambda_z}{2}
\end{align}
where $\lambda_{x,y,z}$ is the so-called depolarizing factor \citep{Chandrasekar2001} in the $x,y,z$ direction. These factors depend on the
shape of the scatterer, and are given for oblate spheroids and spheres as:
\begin{align}
	\lambda_z (sphere) &= \frac{1}{3} \\
	\lambda_z (oblate) &= \frac{1 + f^2}{f^2} \left(1 - \frac{1}{f}\tan^{-1}{f}\right) \\
	f^2 &= \frac{1}{e^2} - 1
\end{align}
where $e$ is the eccentricity of the spheroid. It can be shown that $\lambda_z$ for the sphere can be obtained by taking the limit
as $e$ goes to $\infty$. Using the value for a sphere, one obtains an identical expression as that given by unmodified Rayleigh theory.

Rayleigh-Gans theory gives the same matrix for the forward scatter direction as in the backscatter direction, with the exception of a flipped
sign on the $S_{HH}$ term. However, as before, it is modified so that the proper 
extinction coefficients are obtained using optical theorem \citep{Ishimaru1991}. In this case:
\begin{align}
	K_h &= \frac{m^2 - 1}{(m^2 - 1)\lambda_{x} + 1} \\
	\sigma_{sh} &= \frac{2 \pi^5}{3 \lambda^4} \left| K_h \right|^2 D^6 \\
	\sigma_{ah} &= \frac{\pi^2}{\lambda} \Im{-K_h} D^3 \\
	K_v &= \frac{m^2 - 1}{(m^2 - 1)\lambda_{z} + 1} \\
	\sigma_{sv} &= \frac{2 \pi^5}{3 \lambda^4} \left| K_h \right|^2 D^6 \\
	\sigma_{av} &= \frac{\pi^2}{\lambda} \Im{-K_h} D^3
\end{align}
These are used to adjust the forward scattering matrix as:
\begin{align}
	\Im{S_{HH}^f(D)} &= \frac{\sigma_{sh} + \sigma_{ah}}{2 \lambda} \\
	\Im{S_{VV}^f(D)} &= \frac{\sigma_{sv} + \sigma_{av}}{2 \lambda}
\end{align}
\paragraph{Mie}
The \cite{Mie1908} solution to the scattering problem is a relatively straightforward solution to the boundary value problem for the Laplace
equation \citep[see e.g][]{Chandrasekar2001}. This involves solving:
\begin{align}
	\epsilon_0 \nabla^2 \psi &= \vec{\nabla} \cdot \vec{P} \\
	\intertext{subject to the boundary conditions:}
	\hat{n}_1 \times \vec{E}^{in}_T + \hat{n}_2 \times (\vec{E}^i + \vec{E}^s) &= 0 \\
	\hat{n}_1 \times \vec{B}^{in}_T + \hat{n}_2 \times (\vec{B}^i + \vec{B}^s) &= 0 \\
	\intertext{where}
	\hat{n}_1 &= \hat{r} \\
	\hat{n}_2 &= -\hat{r}
\end{align}
where $\vec{E}$ and $\vec{B}$ are the electric and magnetic fields, respectively, the superscript $i$ and $s$ denote the incident and scattered
fields, respectively, and the superscript $in$ and subscript $T$ denote the total internal field.
The solution proceeds by expanding the scattered field in terms of vector spherical harmonics:
\begin{align}
	\vec{E}^s (k_0r, \theta, \phi) &= 2 \pi E_0 \sum^\infty_{n=1} (-\imath)^n \gamma^{1/2}_{1n} \vec{X}_{1n} \\
	\vec{X}_{1n} &= \left[\alpha_{o1n} \vec{M}_{o1n}(k_0r, \theta, \phi) + \imath \beta_{e1n} \vec{N}_{e1n}(k_0r, \theta, \phi) \right] \\
	\gamma^{1/2}_{mn} &= \sqrt{\frac{(2n + 1)(n - m)!\epsilon_m}{4 \pi (n + m)!}} \\
	\epsilon_m &= \begin{cases}
		2 & m > 0 \\
		1 & m = 0 \end{cases}
		\label{eq:mie_eq}
\end{align}
where $k_0$ is the free-space wavenumber, $k$ is the wavenumber within the sphere, and $\vec{M}$ and $\vec{N}$ are the vector spherical
wave functions (also called multipoles). The $Rg$ notation denote wave functions which are modified to be regular (finite) at the origin. The $o$
and $e$ subscripts denote represent whether the function is even or odd as a way of eliminating the use of complex harmonics. The expansion
coefficients, $\alpha_{o1n}$ and $\beta_{e1n}$, found using the boundary conditions, are:
\begin{align}
	\alpha_{o1n} &= \frac{
		\rho j_n(\rho) [\rho_0 j_n(\rho_0)]^\prime - \sqrt{\epsilon_r}\rho_0 j_n(\rho_0) [\rho j_n(\rho)]^\prime}{
		\sqrt{\epsilon_r} \rho_0 h_n^{(2)}(\rho_0) [\rho j_n(\rho)]^\prime - \rho j_n(\rho)[\rho_0 h_n^{(2)}(\rho_0)]^\prime} \\
	\beta_{e1n} &= \frac{
		\rho_0 j_n(\rho_0) [\rho j_n(\rho)]^\prime - \sqrt{\epsilon_r}\rho j_n(\rho) [\rho_0 j_n(\rho_0)]^\prime}{
		\sqrt{\epsilon_r} \rho j_n(\rho) [\rho_0 h_n^{(2)}(\rho_0)]^\prime - \rho_0 h_n^{(2)}(\rho_0)[\rho j_n(\rho)]^\prime} \\
	\rho_0 &= k_0 a \\
	\rho &= \rho_0 \sqrt{\epsilon_r} \\
	[\rho z_n(\rho)]^\prime &= \dfrac{d\rho z_n(\rho)}{d \rho}
	\label{eq:mie_coeffs}
\end{align}
where $j_n$ and $h_n^{(2)}$ are the spherical Bessel and (second kind) Hankel functions, respectively. Combining \eqref{eq:mie_eq} and
\eqref{eq:mie_coeffs} gives the following expressions for the scattering matrices \citep{Chandrasekar2001}:
\begin{align}
	\bf{S} &= \begin{bmatrix} \frac{i \cos{\phi_s}}{k_0} S_1(\theta_s) & \frac{-i \sin{\phi_s}}{k_0	} S_1(\theta_s) \\
 				\frac{i \sin{\phi_s}}{k_0} S_2(\theta_s) & \frac{i \cos{\phi_s}}{k_0} S_2(\theta_s) \end{bmatrix} \\
 	S_1(\theta_s) &= \sum_{n=1}^\infty \frac{2n + 1}{n(n+1)	}\left[
 			\alpha_{o1n}\frac{P_n^1(\cos{\theta_s})}{\sin{\theta_s}} + \beta_{e1n} \frac{dP_n^1(\cos{\theta_s})}{d\theta_s} \right] \\
 	S_2(\theta_s) &= \sum_{n=1}^\infty \frac{2n + 1}{n(n+1)	}\left[
 			\alpha_{o1n} \frac{dP_n^1(\cos{\theta_s})}{d\theta_s} + \beta_{e1n}\frac{P_n^1(\cos{\theta_s})}{\sin{\theta_s}} \right]
\end{align}
where $P_n^m(\cos{\theta})$ are the associated Legendre functions, and $\theta_s$ and $\phi_s$ define the scattering direction. For our
purposes, $\phi_s$ is \SI{0}{\degree} and $\theta_s$ is 0 or $\pi$ for forward scatter or backscatter, respectively. The actual implementation
is a Python-based port of Matlab code written by \cite{Matzler2002}.

\paragraph{T-Matrix}
The extended boundary condition, or T-Matrix, method was first proposed to solve electromagnetic scattering problems by \cite{Waterman1971}.
The approach is similar to that for Mie scattering, in that the incident and scattered fields are expanded into various multipole terms, solving for 
coefficients of vector spherical wave functions. However, the T-matrix method is applicable to non-spherical scatterers which do not have a simple
form in spherical coordinates. In such cases the boundary conditions do not lend themselves to a direct relation between the expansion coefficients
of the incident field and those of the scattered field. Due to the linearity of Maxwell's equations and the boundary conditions themselves,
the two sets of expansion coefficients can be related through a set of linear equations and matrix of coefficients.

In theory, this ``transition'' matrix (or T-matrix) is infinite in size; in practice, the procedure is iterated, increasing the number of terms until
the results converge. The radar simulation code relies on the FORTRAN implementation from \cite{Mishchenko1996}, which calculates the T-matrix for 
a fixed orientation; this has been wrapped in Python for convenience purposes. The reader is referred to \cite{Mishchenko1996} and the references therein
for the lengthy theoretical derivation.

\paragraph{Canting Angle Distribution}
For T-matrix scattering calculations, the simulator has the option of including a non-zero canting angle distribution width. While the mean
canting angle is assumed to be zero (a common assumption and a crucial one for the NEXRAD dual-polarization upgrade \citep{Doviak2000}), the
width of the distribution of angles can be changed to account for some random hydrometeor orientation due to turbulence. The simulation uses the axial
distribution \citep{Mardia1972}, which can be used to describe the distribution of a two-dimensional orientation angle in which one of the
components is fixed at \SI{90}{\degree}. Such a distribution is given by \citep{Chandrasekar2001}:
\begin{align}
	g_A(\theta) &= b(\kappa) \exp(-\kappa \cos^2{\theta}) \sin{\theta} \\
	b(\kappa) &= \frac{1}{2 \int_0^1 \exp(-\kappa t^2)\,dt}
	\label{eq:axial-distribution}
\end{align}
where $\kappa$ is the parameter that controls the width of the distribution. Figure~\ref{fig:angle-dist} shows the axial probability density function
for various values of $\kappa$.

\begin{figure}
	\centering
	\includegraphics{figures/angular-distribution}
	\caption{Axial probability density function for various values of the distribution width parameter, $\kappa$.}
	\label{fig:angle-dist}
\end{figure}

To include the effects of the orientation angle distribution in the calculation of radar observables, all expected values are converted from single
integrals over the drop size distribution to double integrals over both the DSD and the angular distribution. That is, integrals of the form:
\begin{align}
	A &= \int f(D) N(D)\,dD \\
	\intertext{become:}
	A &= \iint f(D,\theta) N(D) g_A(\theta)\,dD\,d\theta
    \label{eq:scatter-integrals}
\end{align}

\paragraph{Radar Observables}
By integrating various terms of these scattering matrices with respect to the drop size
distribution, at each model grid point the relevant radar parameters can be
calculated: reflectivity ($\eta$), backscatter differential phase shift ($\delta$),
wavenumber perturbation ($\Delta k$, due to propagation through the medium), and expected cross-correlation coefficient ($|\rho_{hv}|$)
\citep[chap. 3 and 4]{Chandrasekar2001}:
\begin{align}
    \eta_H &= 4 \pi \int_{0}^{D_{max}} |S_{HH}^b(D)|^2 N(D)\,dD\\
    \eta_V &= 4 \pi \int_{0}^{D_{max}} |S_{VV}^b(D)|^2 N(D)\,dD\\
    \delta &= arg(\int_{0}^{D_{max}} S_{HH}^{b*}(D) S_{VV}^b(D) N(D)\,dD)\\
    \Delta k_H &= \lambda \int_{0}^{D_{max}} S_{HH}^f(D) N(D)\,dD\\
    \Delta k_V &= \lambda \int_{0}^{D_{max}} S_{VV}^f(D) N(D)\,dD\\
    |\rho_{HV}| &= \frac{|\int_{0}^{D_{max}} S_{HH}^f(D) S_{VV}^{f*}(D) N(D)\,dD|}
    	{\sqrt{\int_{0}^{D_{max}} |S_{HH}^f(D)|^2 N(D)\,dD \int_{0}^{D_{max}} |S_{VV}^f(D)|^2 N(D)\,dD}}
\end{align}
While it is customary for such analytic forms to integrate from $0$ to $\infty$,
this is not realistic, as there is a maximum size, $D_{max}$, above which drops
will break up due to aerodynamic forces \citep{Magarvey1956}. A value of
\SI{1.0}{\centi\meter} is chosen here, which allows for a small number of large
drops (depending on the actual drop size distribution). This limit both
keeps the scattering calculation computationally tractable and eliminates 
contributions from non-realistic drops.

All of these calculations are performed for various values of temperature
and microphysics parameters and written into lookup tables. This allows the
calculations to be performed over the entire model domain much more rapidly
than by calculating the scattering matrices at each grid point, especially in
the case of T-matrix, which is the most computationally intensive of the
scattering models. It should also be noted that one significant limitation is that
the scattering calculations do not include mean canting at all--this includes
the orientation of the particle relative to the radar beam as the radar points away from horizontal.  To do this well
would require adding the angle of incidence of the radar beam relative to the
axis of symmetry of the particles as another dimension for the lookup tables.
This would make the memory requirements much larger as well as make the
interpolation process take longer. While this is a significant limitation for
simulating dual-polarization variables at high elevation angles, this is not
significant for simulating data at low elevations, such as those used in radar
QPE.

From the complex wavenumber perturbation, both the propagation phase shift ($K$) and
attenuation ($A$) per unit range can be calculated:
\begin{align}
    K_H &= 2 \Re{\Delta k_H}\\
    A_H &= 4 \Im{\Delta k_H}\\
    K_V &= 2 \Re{\Delta k_V}\\
    A_V &= 4 \Im{\Delta k_V}
    \label{eq:prop-calcs}
\end{align}
Each value has a factor of \num{2} that converts it to a quantity representing
the change over the two-way path. The attenuation terms have an additional
factor of \num{2} that converts the attenuation from representing amplitude loss
to representing a loss of power.

\paragraph{Drop Size Distributions}
Three different forms for drop-size distribution are permitted for calculating the
integrals of \eqref{eq:scatter-integrals} above: the \cite[MP,][]{Marshall1948}
exponential distribution, the modified gamma distribution \cite{Ulbrich1983}, and the gamma distribution as used by
\cite{Ziegler1985}. The choice of these distributions is motivated by the
numerical simulation output that has been used to date. In order to generate
the proper scattering fields, the drop size distribution needs to be calculated
at each model grid point, using the appropriate hydrometeor field. The MP
distribution is given by:
\begin{equation}
    N(D) = N_0 \exp(-\Lambda D)
\end{equation}
where $N_0$ is the intercept parameter and has a value of
\SI{8000}{\per\meter\cubed\per\milli\meter}, $D$ is the diameter in \si{\meter},
and $\Lambda$ is the slope parameter in \si{\per\meter}. $\Lambda$ can be calculated from the model
field rain concentration $q_r$:
\begin{equation}
    \Lambda = \sqrt[4]{\frac{\num{1000}\,\pi \rho_l N_0}{q_r}}
\end{equation}
where $\rho_l$ is the density of liquid water
(\SI{1000}{\kilogram\per\meter\cubed}) and the factor of \num{1000} converts
$N_0$ to \si{\meter\tothe{4}}.

The modified gamma distribution \cite{Ulbrich1983} is given by:
\begin{equation}
	N(D) = N_0 D^\mu \exp(-\Lambda D)
\end{equation}
where $\mu$ is the so-called shape parameter. Note that in this case that $N_0$ has units that depend on $\mu$ and are different
from those of the intecept parameter for the exponential distribution.

The gamma distribution of \cite{Ziegler1985}, henceforth called volume gamma, is
given as follows:
\begin{equation}
N(V) = \frac{N (\nu + 1)^{\nu + 1}}{V_0 \Gamma(\nu + 1)}
        \left(\frac{V}{V_0}\right)^{\nu}
        \exp\left[{-(\nu + 1) \frac{V}{V_0}}\right]
        \label{eq:volume-distribution}
\end{equation}
where $\nu$ is the shape parameter with a value of \num{-0.8}, $V_0$ is the
mean drop volume, $\Gamma$ is the Gamma function, and $N$ is the number
concentration, which is given by the numerical simulation. 
This can also be converted to terms of diameter instead of volume, making it more applicable to radar calculations:
\begin{equation}
    N(D) = \frac{3N (\nu + 1)^{\nu + 1}}{D_0 \Gamma(\nu + 1)}
        \left(\frac{D}{D_0}\right)^{3 \nu + 2}
        \exp\left[{-(\nu + 1) \left(\frac{D}{D_0}\right)^3}\right]
\end{equation}
where $D_0$ is the mean drop diameter, rather than volume.
The mean volume and diameter can be calculated from the simulation number concentration and rain concentration
fields as:
\begin{align}
    V_0 &= \frac{q_r}{\rho_l N} \\
    D_0 &= \sqrt[3]{\frac{6\,q_r}{\pi \rho_l N}}
\end{align}
Note that while this latter distribution is provided to maintain consistency with the numerical simulation input, its
use in the calculation of radar observables is problematic. Figure~\ref{fig:dsd-comparison} compares the volume gamma
distribution with a modified gamma distribution, an exponential distribution, and a Marshal-Palmer distribution, all having the same total
liquid water content, $q_r$. Additionally, with the exception of the Marshal-Palmer distribution, all have the same total number concentration.
The volume gamma distribution shows a significant excess in the
number of small drops, a lack of larger drops, and an overall character which does not match typical drop size distribution patterns. This has 
ramifications when calculating radar observables, especially $Z_{DR}$--the overabundance of small drops and lack of large drops would tend
towards \si{0}{dB} more than other distributions. This departure
from a typical distribution shape results from its original formulation in terms of volume, which causes the cube of diameter to appear
in the exponential; no set of coefficients for the modified gamma family of distributions for diameter can match this shape. The abundance of small
drops, tending towards infinity as drop volume approaches \num{0}, also creates problems when performing numerical integration of the
distribution.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/distribution-comparison.pdf}
	\caption{Comparison of drop size distributions as a function of diameter, $D$: 
		Marshall-Palmer, dotted black line; Exponential, black dashed line;
		Volume Gamma, black solid line; Modified Gamma with fixed shape 1.8102, red solid line.}
	\label{fig:dsd-comparison}
\end{figure}

In order to address these shortcomings but also stay as consistent as possible with the original DSD shape used by the model (more complex 
shape than a simple exponential distribution), several attempts
were made to fit the modified gamma distribution to the available parameters, $N$ and $q_r$, which can be expressed in terms of the zeroth
and third moments of the distribution, respectively. Starting from the expression for the moments of the modified gamma distribution:
\begin{equation}
	\int_{0}^{\infty} N_0 D^n e^{-\Lambda D} dD = N_0 \Gamma(n + 1) \Lambda^{-(n + 1)}
\end{equation}
and substituting for the zeroth and third moments yields:
\begin{align}
	N &= \overline{D^0} \\
	   &= N_0 \Gamma(1 + \mu) \Lambda^{-(\mu + 1)} \\
	q_r &= \frac{\rho_l \pi}{6} \overline{D^3} \\
	      &= \frac{\rho_l \pi N_0}{6} \Gamma(4 + \mu) \Lambda^{-(\mu + 4)}
\end{align}
By taking the ratio of $N$ and $q_r$ the complex $\Gamma$ functions (as well as $N_0$) can be eliminated:
\begin{equation}
	\frac{q_r}{N} = \frac{\rho_l \pi}{6} (\mu + 3)(\mu+2)(\mu+1) \Lambda^{-3}
	\label{eq:mu-poly-1}
\end{equation}
In order to reduce this expression further, a set of constraints is needed to be able to calculate a unique set of distribution parameters from the
model prognostic variables (this is an under-determined problem otherwise since there are three parameters but only two model variables).

One option for the constraint is the constrained gamma distribution of \cite{Zhang2001}. This work demonstrated, using empirical data, that
the value of shape of the distribution, $\mu$, largely can be predicted using the slope, $\Lambda$:
\begin{equation}
	\mu = \num{-0.016} \Lambda^2 + \num{1.213} \Lambda - \num{1.957}
	\label{eq:mu-lambda-relation}
\end{equation}
By substituting this into \eqref{eq:mu-poly-1} above, a sixth order polynomial in $\Lambda$ is obtained:
\begin{align}
	\num{-4.096e-6} \Lambda^6 + \num{9.31584e-4} \Lambda^5 - \num{.070592688} \Lambda ^4
         + \num{1.77976333} \Lambda^3 \notag\\
         +  \num{0.205717849} \Lambda^2 - \num{1.20627149} \Lambda
        - \num{0.0429204930} = \frac{6}{\rho_l \pi} \frac{q_r}{N} \Lambda^3
        \label{eq:constrained-gamma-poly}
\end{align}
Solving this polynomial is problematic, since it potentially has six roots for a given $q_r$ and $N$ pair; fortunately, in practice, there are only up
to two real, positive roots. This still leaves a problem of picking between one of two physically valid solutions. One possible strategy is
to pick the value closest to a moderate value of $\Lambda$; we choose \SI{10}{\per \milli \meter} for illustration here, but others options tested
include: maximum $\Lambda$, minimum $\Lambda$, closest to \SI{10}{\per \milli \meter}, and closest to $\mu$ of \num{2}, using \eqref{eq:mu-lambda-relation}. The value of \num{2} was chosen to reflect a representative value, chosen from the empirical data shown in \cite{Zhang2001}.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/z-distribution.pdf}
	\caption{Comparison of radar reflectivity values calculated at X-band for different drop size distributions, using the same number concentration
	and liquid water content taken from sample model output. These distributions are: exponential distribution, upper left;
	volume gamma, upper right; modified gamma with fixed shape, lower left; gamma constrained, lower right.}
	\label{fig:dsd-z-comparison}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/zdr-distribution.pdf}
	\caption{As in \ref{fig:dsd-z-comparison}, but for calculated $Z_{DR}$.}
	\label{fig:dsd-zdr-comparison}
\end{figure}

Figure~\ref{fig:dsd-z-comparison} shows the consequences of solving the sixth order polynomial to find the $\Lambda$ value. While the other
distributions (exponential, modified gamma with fixed shape, and volume gamma) show nearly identical distributions of radar reflectivity factor
(calculated using a sample set of model liquid water content and number concentration), the constrained gamma distribution shows instead two
distinct bands. It is likely that each of the two bands represent one of the families of roots being chosen from the two possible physical
solutions to \eqref{eq:constrained-gamma-poly}. This behavior is carried across all of the moments: $Z_{DR}$ (Figure~\ref{fig:dsd-zdr-comparison}),
$K_{DP}$ (Figure~\ref{fig:dsd-kdp-comparison}), $A_H$ (Figure~\ref{fig:dsd-atten-comparison}), and $A_D$ (Figure~\ref{fig:dsd-ad-comparison}).
This behavior of two distinct bands of behavior is present regardless of the method used to choose between the two solutions, though the details
of the bands do vary based on the method.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/kdp-distribution.pdf}
	\caption{As in \ref{fig:dsd-z-comparison}, but for calculated $K_{DP}$.}
	\label{fig:dsd-kdp-comparison}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/atten-distribution.pdf}
	\caption{As in \ref{fig:dsd-z-comparison}, but for calculated horizontal attenuation, $A_H$.}
	\label{fig:dsd-atten-comparison}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/ad-distribution.pdf}
	\caption{As in \ref{fig:dsd-z-comparison}, but for calculated differential attenuation, $A_D$.}
	\label{fig:dsd-ad-comparison}
\end{figure}

In light of the problems of using the $\Lambda-\mu$ relation of \cite{Zhang2001} to constrain the distribution, we chose instead to use the
conservation of the sixth moment of the diameter distribution as the necessary constraint to allow calculating all three parameters of the distribution.
This was chosen both because the combination of the zeroth, third, and sixth moments allows the math to be tenable and because conserving
the sixth moment of the distribution is approximately the same as conserving the equivalent radar reflectivity factor (under the Rayleigh
approximation). These three moments of the modified gamma distribution can be expressed as:
\begin{align}
	\overline{D^0} &= N_0 \Gamma(\mu + 1) \Lambda^{-(\mu + 1)} \\
	\overline{D^3} &= N_0 \Gamma(\mu + 4) \Lambda^{-(\mu + 4)} \\
	\overline{D^6} &= N_0 \Gamma(\mu + 7) \Lambda^{-(\mu + 7)}
\end{align}
In order to eliminate $N_0$ and $\Lambda$, these moments can be combined into a unitless ratio:
\begin{align}
	\frac{\overline{D^3}^2}{\overline{D^0}\,\overline{D^6}} &= \frac{\Gamma(\mu+4)\Gamma(\mu + 4)}{\Gamma(\mu + 1)\Gamma(\mu + 7)} \\
		&= \frac{(\mu + 3)(\mu + 2)(\mu+1)}{(\mu+6)(\mu+5)(\mu+4)}
		\label{eq:sixth-fit-poly}
\end{align}
In order to find a value for the left hand side of \eqref{eq:sixth-fit-poly}, we start
by noting the following relation between the moments of the diameter distribution and the moments of the volume distribution
\begin{equation}
	\overline{V^n} = {\left(\frac{\pi}{\num{6}}\right)}^n \overline{D^{3n}}
	\label{eq:volume-to-diameter}
\end{equation}
The relevant moments of the volume distribution from \eqref{eq:volume-distribution} above are:
\begin{align}
	\overline{V^0} &= N \\
	\overline{V^1} &= N V_0 \\
	\overline{V^2} &= N V_0^2 \frac{\nu + 2}{\nu + 1}
\end{align}
Combining these with \eqref{eq:volume-to-diameter} yields the desired left hand side of \eqref{eq:sixth-fit-poly}:
\begin{equation}
	\frac{\overline{D^3}^2}{\overline{D^0}\,\overline{D^6}} = \frac{{\left(\frac{\num{6}}{\pi}\right)}^2 (N V_0)^2}{N ({\left(\frac{\num{6}}{\pi}\right)}^2 N V_0^2 \frac{\nu + 2}{\nu + 1})}
\end{equation}
which simplifies to:
\begin{equation}
	\frac{\overline{D^3}^2}{\overline{D^0}\,\overline{D^6}} =  \frac{\nu + 1}{\nu + 2}
\end{equation}
Since $\nu$ is specified by the model (\num{-0.8} here), this reduces \eqref{eq:sixth-fit-poly} to a simple polynomial:
\begin{equation}
	\frac{\num{5}}{\num{6}} \mu^3 + \frac{\num{7}}{\num{2}} \mu^2 - \frac{\num{4}}{\num{3}} \mu - \num{14} = 0
\end{equation}

\begin{figure}
	\centering
	\includegraphics[]{figures/qr-sanity-check.pdf}
	\caption{Comparison of liquid water content calculated from the fit modified gamma distribution with that of the original
	model. All values lie along the 1-1 line, indicative that the procedure works to conserve water content.}
	\label{fig:qr-sanity-check}
\end{figure}

This polynomial, fortunately, has only a single real root, \num{1.81028329387715}. Using this value for
the shape parameter of the modified gamma distribution yields a distribution that matches the number concentration, liquid water content,
and Rayleigh approximation radar reflectivity factor of the distribution of the model. Figure~\ref{fig:qr-sanity-check} provides evidence that
this procedure works by comparing the liquid water content calculated from the fit distribution against that which was present in the original
model output. All values lie along the 1-1 correspondence line, indicating that value used to calculate the parameters of the distribution are
able to be retrieved again.

The improvement in the shape of the distribution obtained by using the modified gamma distribution is clear in Figure~\ref{fig:dsd-comparison}.
By using this modified gamma distribution, the number of drops as a function of diameter has a subjectively better shape (c.f. Figure 2 in
\cite{Ulbrich1983}); the abundance of small drops is addressed, as is the lack of larger drops. Looking at the calculated dual-pol parameters,
these changes to the distribution manifest in subtle ways. First, as shown in Figure~\ref{fig:dsd-zdr-comparison}, there is a \SI{0.5}{dB} difference
in peak $Z_{DR}$ values; this is likely the product of shifting liquid water mass from small spherical drops to larger, aspherical drops. Both
$A_H$ (Figure~\ref{fig:dsd-atten-comparison}) and $K_{DP}$ (Figure~\ref{fig:dsd-kdp-comparison}) have smaller values when using the modified
gamma distribution as well. $Z_H$ and $A_D$ are relatively unaffected by changes between these distributions. Overall, this points to a sensitivity of
the calculated moments to the drop size distribution; while not surprising, this does emphasize the importance of using a distribution that best
matches real world observations. In this case, reality much better reflects the modified gamma distribution for diameter rather than the volume
gamma distribution.

\subsubsection{Atmospheric refractive index}
To allow ray tracing of the radar pulse propagation path, the index of
refraction is calculated from the model temperature, $T$, water vapor pressure,
$e$, and air pressure, $p$, using the relation
\cite[chap.~1]{Bean1966}
\begin{equation}
    n=\left(\frac{C_d p}{T} + \frac{C_{w1} e}{T}
    + \frac{C_{w2}e}{T^2}\right) \times 10^{-6} + 1
\end{equation}
where $C_d$, $C_{w1}$, $C_{w2}$, have values \SI{0.776}{\kelvin\per\pascal},
\SI{0.716}{\kelvin\per\pascal}, and \SI{3.7e3}{\kelvin\squared\per\pascal},
respectively.

\subsubsection{Radial velocity}
To simplify calculations and reduce run-time memory requirements, the $u$, $v$,
and $w$ wind components are reduced to a radial velocity value based on the
known location of the radar relative to the model grid. The radial velocity,
$V_r$, is calculated by the projection of the total wind velocity vector onto
the radar beam:
\begin{equation}
    V_r=(u \sin{\theta} + v \cos{\theta}) \cos{\phi} + (w - w_t) \sin{\phi}
\end{equation}
where $u$ is the x-component of the wind, $v$ is the y-component of the wind,
$w$ is the z-component of the wind, $w_t$ is the average terminal fall speed for
the hydrometeors, $\theta$ is the azimuth angle measured clockwise from north,
and $\phi$ is the elevation angle. The average hydrometeor terminal fall speed
for the grid box is calculated as a backscatter cross-section-weighted average
given by:
\begin{equation}
    w_t = \eta^{-1} \int_0^{D_{max}} \sigma_b(D) V_t(D) N(D)\,dD
\end{equation}
where $\eta$ is the total reflectivity at the grid point, $\sigma_b(D)$ is the
backscatter cross-section, and $V_t(D)$ is the terminal fall speed as a function
of diameter, which is calculated using the fit of \cite{Brandes2002}:
\begin{equation}
    w_t(D) = \num{-0.1021} + \num{4.932} D - \num{0.9551} D^2
        + \num{0.07934} D^3 - \num{2.362e-3} D^4
\end{equation}
By using a backscatter cross-section-weighted mean, the terminal velocity better
reflects that which would be observed by the radar.
\subsection{Sampling of input fields}
To sample the virtual model atmosphere, the simulator calculates radar
observables along the path of individual pulses at the interval specified by the
pulse repetition time (PRT). This allows the input model fields, as well as the
state of the radar (such as antenna pointing angle) to change for individual
pulses. While the simulator is currently configured for a mechanically scanning
antenna, pulse-by-pulse calculation can be used to simulate measurements for a
phased-array radar as well. The pulse generated within the simulator defines the
volume of space that contributes to a sample taken along the radar beam. It is
bound in elevation and azimuth by a fixed multiple of the half-power beamwidth.
This multiple is chosen based on the number of sidelobes that are desired for
simulation in the antenna pattern. The pulse is bound in range by the specified
pulse length. This volume of space is subdivided into individual pulse elements,
such that, at the maximum range from the radar, the dimensions of each pulse
element are ten percent smaller than the model grid voxel at that range--ensuring that the pulse
element spacing is always finer than the model grid spacing in any direction.

The pulse itself is propagated through the numerical grid using a ray-tracing
technique. For each range gate, the height of each horizontal slice of pulse
elements is determined separately by taking into account the atmospheric index
of refraction experienced by that particular ray element. This allows for
differential propagation across the vertical plane of the radar beam. Since any
part of the pulse that encounters the ground stops propagating, this allows for
partial beam blockage. This is more useful if terrain data are available rather
than the uniform ground level present in most model simulations. The change in
the height above ground, $\Delta h$, and change in range from the radar (along
the surface of the Earth), $\Delta r$, can be calculated from the incremental
change in range along the path, $\Delta s$, as
\begin{align}
    \Delta h &= \sqrt{h^2 + \Delta s^2
        + 2 h \Delta s \sqrt{1 - \frac{C^2}{n^2 h^2}}}\\
    \Delta r &= a \arcsin{(\frac{C \Delta s}{n h (h + \Delta h)})}\\
    C &= n_0 a \cos{\phi}
\end{align}
where $a$ is the radius of the earth, $h$ is the previous height of the element
above ground, $n$ is the index of refraction at height $h$, $n_0$ is the index
of refraction at the radar, and $\phi$ is the initial elevation angle of the
element \citep{Doviak1993}. The element's range from the radar along the surface
of the earth is then converted to standard two-dimensional Cartesian
coordinates, which are used to determine the location of the element on the
model grid.

The pulse volume is allowed to propagate through the environment as far as twice
the unambiguous range, $R_a$,
\begin{equation}
    R_a=\frac{c T_s}{2}
\end{equation}
where $T_s$ is the PRT and $c$ is the speed of light. Allowing the pulse to
propagate $2R_a$ from the radar means that after one PRT from the time the radar
is started there are two pulses propagating through the model field at any given
instant. When a sample is taken, the returns from both pulses are assigned
to the gate, producing the effects of range aliasing.

During propagation, each pulse element is assigned values of attenuation,
propagation phase shift, reflectivity, and backscatter phase shift for both
horizontal and vertical polarizations, as well as values for radial
velocity and co-polar cross-correlation coefficient. These values correspond to the grid point nearest to the element's
location in space. Nearest neighbor sampling is chosen over trilinear (or other
more sophisticated) interpolation to improve the computational efficiency of the
simulator. Since the pulse elements are generally much smaller than the grid
cells, this sampling method provides sufficient accuracy. The entire pulse
volume is stepped forward in range while keeping track of the total attenuation
and propagation phase shift along the path, for both polarizations. This running
total is kept for each pulse element, which allows for the calculation of
differential (in a geometric sense, not polarimetric) attenuation and phase
shift across the pulse.

\subsection{Sampling the pulse}
As the pulse is propagated through the model grid of the
simulated atmosphere, it is periodically sampled at an interval in range
dictated by the specified gate spacing. This allows for the gate and pulse
lengths to be independent. When a pulse sample is taken, two main processes
happen: time series data are generated and pulse averaged values are calculated.

\subsubsection{Time series generation}
Time series data are generated by combining the approaches used by \cite{Muschinski1999} and \cite{Galati1995,Torres2003a}.
The core of the procedure is to simulate radar returns for a volume of space by treating each element of the discretized pulse as a
``scattering center'' \cite{Muschinski1999} and assigning it a random, complex signal value.

The complex values from every pulse element are summed together and added to a
complex random noise value to generate a single IQ sample ($V$) for the range
gate for both horizontal and vertical polarization:
\begin{equation}
    V_{h,v} = A_N e^{-j\theta_N} + \sum_k A_{k(h,v)} 
\end{equation}
where $A_N$ is the noise amplitude, $\theta_N$ is the noise phase, and $A_k$ is the complex signal amplitude of the $k^{th}$ pulse
element. $A_N$ is calculated as the square root of a random value of noise
power, which is generated from an exponential distribution with an expected
value equal to the configured radar noise power (chosen to be the same as the
minimum detectable signal). $\theta_N$ is randomly generated uniformly between
$[0, 2\pi]$.

To control the correlation between the two polarizations, the value of
the co-polar cross-correlation coefficient for the pulse element is used. This value is used as the expected value for the cross-correlation between
the horizontal and vertical channels, and is used to combine two independent samples, $w_1$ and
$w_2$, from a white, complex Gaussian distribution \cite{Galati1995}:
\begin{align}
	A_{kh} &= \sqrt{P_{kh}} w_{1k} e^{-j\theta_k} \\
	A_{kv} &= \sqrt{P_{kv}} \left(\rho_{HVk} w_{1k} + \sqrt{1 - \rho_{HVk}^2} w_{2k} \right) e^{-j\theta_k} \\
    \theta_k &= \theta^n_k + \delta_k + \phi_k\\
    \theta^n_k &= \theta^{n-1}_k + \frac{4 \pi V_{rk} T_s}{\lambda}\\
    \theta^0_k &= 0
\end{align}
where $P_{kh}$ and $P_{kv}$ are the horizontal and vertical power, $\theta^n_k$ is the initial phase for the current pulse $n$, $\delta_k$ is
the backscatter phase shift, $\phi_k$ is the propagation phase shift, and the term
involving $V_{rk}$ (the radial velocity for the pulse element) represents the
Doppler phase shift (due to moving scatterers). $\delta_k$ and $\phi_k$
are those appropriate for the pulse element being calculated. The phase of $w_1$ and $w_2$ for each pulse element is kept constant.
This allows each individual pulse element to have a uniformly
distributed random initial phase and to track the total phase shift due to the
motion of scatterers \citep{Muschinski1999}. This approach for controlling the correlation of the horizontal and
vertical channels assumes that each
polarization has the same sample-time auto-correlation function and that the cross-correlation function is just
the cross-correlation coefficient times the autocorrelation.

The horizontal and vertical powers are calculated as:
\begin{equation}
    P_k = \frac{P_t g^2 \lambda^2 f^4(\theta_k,\phi_k) |W(r_0, r)|^2 \eta_k}
        {(4 \pi)^3 l_k^2(r) r^2} \Delta r \sin\phi_k \,\Delta\phi \,\Delta\theta
    \label{eq:pulse-power}
\end{equation}
where $P_t$ is the transmitted power, $g$ is the system gain, $\lambda$ is the
wavelength, $r$ is range from the radar, $l_k^2$ is the two-way attenuation
factor, $f^2$ is the normalized, one-way antenna pattern, $\theta_k$ is the
azimuth angle relative to the beam center, $\phi_k$ is the elevation angle
relative to beam center, $\eta_k$ is the reflectivity (backscattering cross
section per unit volume), and $W$ is the range weighting function. $\Delta r$,
$\Delta \phi$, and $\Delta \theta$ represent the spacing between pulse elements
in range, elevation, and azimuth, respectively. This expression comes from
making a discrete approximation to the integrand given by \cite{Doviak1993},
where the power, $P$, for a sample taken at range $r_0$ is given by:
\begin{align}
    P(r_0) &= \iiint\limits_{V}{\eta(r) I(r_0, r) \,dV}\\
    \intertext{where}
    I(r_0,r) &= \frac{P_t g^2 \lambda^2 f^4(\theta,\phi) |W(r_0, r)|^2}
        {(4 \pi)^3 l^2(r) r^4}\\
    dV &= r^2 \,dr sin(\phi) \,d\phi \,d\theta
\end{align}

All random values are generated using the Mersenne Twister pseudorandom number
generator \citep{Matsumoto1998}. This generator is chosen because it produces
high-quality pseudorandom numbers with good run time performance and has readily
available source code.

The simulator assumes a Gaussian range weighting function and a normalized
antenna pattern with the following form \citep[chap. 3]{Doviak1993}
\begin{equation}
    f^2(\theta) = \left(\frac{8 J_2((\pi D_a \sin{\theta})/\lambda)}
        {((\pi D_a \sin{\theta})/\lambda)^2}\right)^2
    \label{eq:antenna-pattern}
\end{equation}
where $J_2$ is the second-order Bessel function of the first kind, $\theta$ is
the angular offset from boresight and $D_a$ is the diameter of the antenna.
$D_a$ for \eqref{eq:antenna-pattern} above can be calculated from the half-power
beamwidth, $\theta_1$, as
\begin{equation}
    D_a = \frac{1.27 \lambda}{\theta_1}
\end{equation}
where $\lambda$ is the wavelength. \cite{Doviak1993} state that
\eqref{eq:antenna-pattern} describes the antenna pattern for the first few
sidelobes quite well for a parabolic antenna. However, it is limited in that it
gives sidelobes of a fixed level and location (c.f. Figure~
\ref{fig:antenna-pattern}), prohibiting configuration of sidelobes with
arbitrary magnitude.
\begin{figure}
	\centering
	\includegraphics{figures/antenna_pattern}
	\caption{Simulator antenna pattern for a one-degree half-power beamwidth radar.}
	\label{fig:antenna-pattern}
\end{figure}

\subsubsection{Pulse averaged values}
Pulse averaged values are a way to generate a single data value for the pulse
to represent a radar observable. These averages include the antenna and range
weighting functions, as well as reflectivity weighting where appropriate. These
are useful for inter-comparison with the time series data, as they are not
contaminated with noise or other artifacts that occur with IQ-based estimates.
Consequently, they are useful for generating moments for use as truth fields
representing the best case for radar observations. They are also useful if
time-series data generation is not desired and the simpler averages give
sufficient realism for the application.

Several values are calculated currently: power ($\overline{P}$, both horizontal
and vertical), power-weighted average radial velocity ($\overline{V_r}$),
power-weighted radial velocity variance ($\sigma_{vr}$), power-weighted average
$\overline{K_{DP}}$, and power-weighted differential back\-scatter ph\-ase
($\overline{\delta_{hv}}$). The power-weighting for the latter two variables
only include antenna and range weighting. $\overline{K_{DP}}$ is treated
differently because it is a propagation-based variable. $\overline{\delta_{hv}}$
is so weighted because its use is not intended as true moment data, but rather
as a diagnostic tool to see what areas might be impacted by its presence. Thus,
only a representative average is needed. The power-weightings are performed over
all of the pulse elements as follows, where $P_k$ is as in
\eqref{eq:pulse-power}:
\begin{align}
    \overline{P} &= \sum_{k}P_k\\
    \overline{V_r} &= \frac{\sum_{k}{P_k V_{rk}}}{\overline{P}}\\
    \sigma_{vr} &= \frac{\sum_{k}{P_k V_{rk}^2}}{\overline{P}} - \overline{V_r}^2\\
    \overline{K_{DP}} &= \frac{\sum_{k}{f^4(\theta_k, \phi_k) |W(r_0, r)|^2 (K_{Hk} - K_{Vk})}}
        {\sum_{k}{f^4(\theta_k, \phi_k) |W(r_0, r)|^2}}\\
    \overline{\delta_{hv}} &= \frac{\sum_{k}{f^4(\theta_k, \phi_k) |W(r_0, r)|^2 (\delta_{Hk} - \delta_{Vk})}}
        {\sum_{k}{f^4(\theta_k, \phi_k) |W(r_0, r)|^2}}
\end{align}
where quantities with the $k$ subscript represent those for the $k^{th}$ pulse
element.

\subsection{Moment calculation}
The sampling of model data is repeated for the number of pulses that are to be
averaged for a radial of data, as specified by the scanning strategy. Moment
data (horizontal and vertical power, Doppler velocity, Doppler spectrum width,
specific differential phase, and backscatter differential phase) are then
generated at each range gate along the radial. Power, specific differential
phase, and backscatter differential phase are calculated as the average of all
their respective samples for the specified number of pulses at that range gate.
Radial velocity is calculated as the power-weighted average of all velocity
samples (one per pulse) at that range gate. To simulate velocity aliasing, this
average is restricted to a value within the Nyquist interval and is given by:
\begin{align}
    V_a &= V_r + 2 n V_{NYQ}
    \intertext{where}
    n &=
    \begin{cases}
        0  & \text{for}~ |V_r| <= V_{NYQ}\\
        \frac{V_{NYQ} - V_r}{2 V_{NYQ}} - 1 & \text{for}~ V_r > V_{NYQ}\\
        1 - \frac{V_{NYQ} + V_r}{2 V_{NYQ}} & \text{for}~ V_r < -V_{NYQ}
    \end{cases}
\end{align}
where $V_a$ is the aliased velocity value, $V_r$ is the original (unaliased)
radial velocity, $V_{NYQ}$ is the Nyquist (or aliasing) velocity, and $n$, an
integer, is the number of Nyquist intervals by which the $V_a$ differs from
$V_r$. One advantage of simulated data is that the unaliased Doppler velocity is
known and can be output as well. This is useful for testing dealiasing
algorithms or when one wishes to eliminate velocity aliasing from the problem
under consideration.

Spectrum width is calculated as the square root of the power-weighted average of
the variance for each sample, which is the variance of all velocity values
within the pulse. Initial attempts at simulating spectrum width used only the
variance of the individual velocity samples that were themselves an average over
the entire pulse. That approach yielded unreasonably low spectrum width values.
By taking into account the variance of all velocity values within all pulses,
the spectrum width reflects the effects of antenna rotation and wind
shear across the radar beam; however, the sub-grid scale atmospheric turbulence
is neglected. Moreover, since this method does not use a true power spectrum at
each range gate, this simulated spectrum width does not take into account a
limited Nyquist interval. This option exists as a holdover from an earlier
version and it is recommended to use spectrum width values generated from
time-series data as they will be much more representative of real-world
behavior.

In addition to the moments above, equivalent reflectivity factor ($Z_e$) is
calculated from the average power using:
\begin{equation}
    Z_e = \frac{2^{10} \ln{2} \lambda^2 r^2 \overline{P_r}}
        {\pi^3 P_t g^2 \theta_1^2 c \tau |K_W|^2}
\end{equation}
where $\tau$ is the pulse duration.

To estimate moments from time series data, the standard autocorrelation-based
methods are used. The
single polarization moments, power ($\hat{P}$), mean Doppler velocity
($\hat{V}_r$), and Doppler spectrum width ($\hat{\sigma}_v$) are calculated as
\cite[chaps. 5 and 6]{Chandrasekar2001}:
\begin{align}
    \hat{R}(l) &= \sum_{n=0}^{N-l-1} V(n + l) V^*(n) \\
    \hat{P} &= \hat{R}(0) \\
    \hat{V_r} &= \frac{-\lambda}{4 \pi \,T_s} \arg{\hat{R}(1)} \\
    \hat{\sigma}_v &= \frac{\lambda}{2 \pi \,T_s \sqrt{2}}
        \left[\ln{\left|\frac{\hat{R}(0)}{\hat{R}(1)}\right|}\right]^{\frac{1}{2}}
\end{align}
where $\hat{R}(l)$ is the autocorrelation at lag $l$. The dual-polarization
moments are estimated as \cite[chaps. 5 and 6]{Chandrasekar2001}:
\begin{align}
    \hat{R}_{v,h}(l) &= \sum_{n=0}^{N-l-1} V_{v}(n+l) V_{h}^*(n) \\
    \hat{Z}_{DR} & = \frac{\hat{P}_h - \hat{N}_h}{\hat{P}_v - \hat{N}_v} \\
    |\hat{\rho}_{co}| &= \frac{|\hat{R}_{v,h}(0)|}{\sqrt{\hat{P}_h\hat{P}_v}} \\
    \hat{\phi}_{DP} &= \arg{\hat{R}_{v,h}(0)}
\end{align}
where $\hat{R}_{v,h}(l)$ is the co-polar cross-correlation and $\hat{N}_h$ and
$\hat{N}_v$ are noise power estimates for the horizontal and vertical channels,
respectively. The simulator's configured noise power is used for these values,
which represents a best case scenario. In addition, $K_{DP}$ is calculated as:
\begin{equation}
    \hat{K}_{DP}(r) = \frac{\hat{\phi}_{DP}(r + \Delta r)
        - \hat{\phi}_{DP}(r - \Delta r)}{2\,\Delta r}
\end{equation}
which is a centered-difference approximation to the $1^{st}$ derivative of
$\hat{\phi}_{DP}$. This approach is chosen purely for ease of implementation to
demonstrate the generation of $\hat{K}_{DP}$ data. Such estimates are not used
in practice (and will not be used in future work) because the discrete
difference amplifies the noise in the original data.

\section{Demonstration}
Simulated data were generated for different radar characteristics to illustrate
the simulator's capabilities to simulate dual-polarization radar data as a
function of radar wavelength. The source simulation used as input is a COMMAS model simulation of multi-cell convection.
The COMMAS model is a storm-scale model that includes
prognostic equations for momentum, pressure, potential temperature, and
turbulent kinetic energy \citep{Wicker1995}. This particular simulation was run
on a grid with a uniform spacing of \SI{100}{\meter} in x, y, and z, initialized
using a series of buoyancy perturbations. The microphysics are a modified
version of those described by \cite{Ziegler1985}, but include only liquid-phase
hydrometeors. The simulated data here rely on a single time step
\SI{3600}{\second} from the start of the simulation. A plot of the winds and
rain concentration from \SI{350}{\meter} AGL is shown in Figure~
\ref{fig:commas-output}.

\begin{figure}
    \centering
    \includegraphics[scale=0.8]{figures/commas_wz_3600_qr_wind_vectors}
    \caption{Rain water content, shaded contours, and horizontal wind, vectors,
    for the COMMAS simulation used to produce the sample datasets.}
    \label{fig:commas-output}
\end{figure}

Note that all radar simulations that follow were run using the same random
number seed, \num{4346}. This reduces variations due to different
realizations of the sequence of random numbers in the Monte Carlo simulation. In
fact, if the same number of random numbers are used (affected by number of
pulses and number of sub elements in a pulse), there are no differences in the simulated output due to the
Monte Carlo simulation. Table~\ref{tab:common-params}
lists the common parameters used to define the radar and scanning strategy for the examples that will be shown.
\begin{table}
    \centering
    \begin{tabular}{ | l | l | }
        \hline
        Antenna gain & \SI{45.5}{dB} \\
        Peak power & \SI{250}{\kilo\watt} \\
        First range gate & \SI{500}{\meter} \\
        Noise power & \SI{-113}{dBm} \\
        Elevation & \SI{0.5}{\degree} \\
        PRT & \SI{0.667}{\milli\second} \\
        Rotation Rate & \SI{20}{\degree\per\second} \\
        Pulses per radial & \num{75} \\
        Gate length & \SI{125}{\meter} \\
        Antenna Limits & Main-lobe only \\
        \hline
    \end{tabular}
    \caption{Radar and scanning parameters common to the example simulations.}
    \label{tab:common-params}
\end{table}
The capabilities highlighted here focus on broad scale changes to radar capabilities: wavelength, beamwidth, antenna pattern, and radial size.
The parameters varying across these examples are outlined in Table~\ref{tab:common-params}.
\begin{table}
	\centering
	\begin{tabular}{ | l | l | l | l | l | l | l | }
		\hline
		Experiment & C & CBW & CRW & S & X \\
		Wavelength &  \SI{5}{\centi\meter} & \SI{5}{\centi\meter} & \SI{5}{\centi\meter} & \SI{10}{\centi\meter} &
			\SI{3.21}{\centi\meter} \\
		Beamwidth & \SI{1.0}{\degree} & \SI{2.0}{\degree} & \SI{2.0}{\degree} & \SI{1.0}{\degree} & \SI{1.0}{\degree} \\
		Radial Spacing & \SI{1.0}{\degree} & \SI{1.0}{\degree} & \SI{2.0}{\degree} & \SI{1.0}{\degree} & \SI{1.0}{\degree} \\
		\hline
	\end{tabular}
	\caption{Parameters of note for the example simulations}
	\label{tab:varying-params}
\end{table}
The following sections examine the results from these simulations and highlight features that capture effects observed in real-world data.

\subsection{Wavelength}
The experiments S, C, and X highlight the differences at these different respective wavelengths. The base configuration is for a typical radar with a \SI{1.0}{\degree}
two-way \SI{3}{dB} beamwidth, transmitting \SI{250}{\kilo\watt} of peak power.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/single_S_1_MainLobe_1.png}
	\caption{Plan Position Indicators (PPI) of the basic moment data for the base S experiment: (a) Horizontal Reflectivity Factor ($Z_H$)
		(b) horizontal Doppler velocity ($V_{RH}$) (c) horizontal Doppler spectrum width ($\sigma_{vH}$)
		(d) horizontal attenuation ($A_H$) }
	\label{fig:s-base}
\end{figure}
\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/dual_S_1_MainLobe_1.png}
	\caption{Plan Position Indicators (PPI) of the dual-polarization moment data for the S experiment: (a) Differential Reflectivity Factor ($Z_{DR}$)
		(b) differential phase ($\Phi_{DP}$) (c) co-polar cross-correlation coefficient ($\rho_{HV}$)
		(d) differential attenuation ($A_D$) }
	\label{fig:s-dual}
\end{figure}

At S-band, the horizontal reflectivity factor ($Z_H$) in Figure~\ref{fig:s-base} shows a 
mutli-cell storm complex. The
Doppler velocity field shows two distinct areas of surface divergence, evidenced by
the couplets of in-bound, and then out-bound, velocities as one moves down the radial
away from the radar. Looking at the PPI of horizontal attenuation, there is a peak in
total attenuation of about \SI{2.5}{dB}. Turning to the dual-polarization moments in 
Figure~\ref{fig:s-dual}, the $Z_{DR}$ field agrees well with that of $Z_H$, with higher
values of $Z_{DR}$, indicative of more oblate drops, co-located with higher reflectivity 
values. For most of the area of weather signal, the values of $Z_{DR}$ range from 
\SI{0}{dB} to \SI{2.5}{dB}. At the far range from the radar (along \SI{0}{\degree} 
azimuth), there are some negative values of $Z_{DR}$; these are not representative of
intrinsic values from the precipitation particles, but rather are caused by differential
attenuation. This is validated by panel (d) in Figure~\ref{fig:s-dual}, where the $A_D$
field shows a peak value of approximately \SI{0.5}{dB} of attenuation. Along with the
$Z_{DR}$ field, the $\rho_{HV}$ and $\Phi_{DP}$ fields look as typical for weather. The
$\rho_{HV}$ values are very close to \num{1.0} for the weather, with small departures, 
as low as \num{0.99}. The $\Phi_{DP}$ field shows an overall monotonic increase with
range from the radar.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/single_C_1_MainLobe_1.png}
	\caption{As in Figure~\ref{fig:s-base}, but for the C experiment.}
	\label{fig:c-base}
\end{figure}
\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/dual_C_1_MainLobe_1.png}
	\caption{As in Figure~\ref{fig:s-dual}, but for the C experiment.}
	\label{fig:c-dual}
\end{figure}

Figure~\ref{fig:c-base}, for comparison, shows the same fields as
Figure~\ref{fig:s-base}, but for a radar operating at C-band. Since the wind field in
this multi-cell complex is relatively weak, the Doppler fields show no significant
differences from those at S-band. However, the reflectivity field at C-band shows a
pronounced decrease in intensity with range when compared to that for S-band. This is an
effect of attenuation, which panel (d) in Figure~\ref{fig:c-base} shows is over
\SI{20}{dB}. For the dual polarization moments, shown in Figure~\ref{fig:c-dual}, the
most prominent difference is the presence of differential attenuation, which is over
\SI{5}{dB} for a significant portion of the data around \SI{0}{\degree} azimuth. This
greatly degrades the quality of the $Z_{DR}$ data, which now drops to negative values
beyond \SI{10}{\kilo\meter} range. The $\rho_{hv}$ field also shows some differences,
with values around \num{0.98} within the weather. Much of this decrease is caused by
noise biasing the $\rho_{hv}$ estimates where attenuation has decreased the signal
intensity. The PPI of $\Phi_{DP}$  also shows the impact of changing wavelength, where
the peak values have doubled from \SI{100}{\degree} at S-band to \SI{200}{\degree} at
C-band, which is consistent with scattering theory.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/single_X_1_MainLobe_1.png}
	\caption{As in Figure~\ref{fig:s-base}, but for the X experiment.}
	\label{fig:x-base}
\end{figure}
\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/dual_X_1_MainLobe_1.png}
	\caption{As in Figure~\ref{fig:s-dual}, but for the X experiment.}
	\label{fig:x-dual}
\end{figure}

At X-band, the single polarization moments (Figure~\ref{fig:x-base}) show very large
differences from S-band due to the effects of attenuation. The PPI of $Z_H$ shows a
swath of data along \SI{0}{\degree} azimuth where the signal is completely extinguished;
the calculated attenuation in this region is up to \SI{100}{dB}. As a result, the
Doppler velocity and spectrum width moment data in this same region are unavailable,
with only noise being detected. In Figure~\ref{fig:x-dual}, the dual-polarization moment
data also show the effects of increased attenuation. The field of $Z_{DR}$ is almost
entirely negative, owing to differential attenution, which has a peak value of
\SI{16}{dB}. The field of $\Phi_{DP}$, where the signal has not been extinguished, shows
a peak of \SI{250}{\degree}.

\subsection{Antenna}
To demonstrate the impacts of changing antenna configuration, the configuration
described as CBW in Table~\ref{tab:varying-params} changes the configured width of
the mainlobe to \SI{2}{\degree}, while keeping the azimuthal sampling interval of the
data at \SI{1}{\degree}.

Figure~\ref{fig:c-bw-base} shows the impact of this configuration change on the
single polarization data, which is to smear the data across radials. This effect is
most visible at the edges of the mutli-cell complex, where the change in beamwidth
causes the apparent azimuthal extent of the complex to increase. One can also see
areas of weak reflectivity in the original data (Figure~\ref{fig:c-base}) that are
filled with larger values in the data taken with a larger beamwidth. The
dual-polarization moments (Figure~\ref{fig:c-bw-dual}) show similar effects with
regard to smearing. The $\rho_{hv}$ data also show values with decreased magnitude
as a result of the larger sampling volume.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/single_C_2_MainLobe_1.png}
	\caption{As in Figure~\ref{fig:s-base}, but for the CBW experiment.}
	\label{fig:c-bw-base}
\end{figure}
\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/dual_C_2_MainLobe_1.png}
	\caption{As in Figure~\ref{fig:s-dual}, but for the CBW experiment.}
	\label{fig:c-bw-dual}
\end{figure}

\subsection{Sampling}
The last of the demonstration cases considered here examines changing the azimuthal
sampling interval, independent of beamwidth, as specified by the CBW experiment. This
experiment modifies the sampling width by just doubling the number of samples used to
calculate the data for a radial, from \num{75} to \num{150}; this changes the azimuthal
sampling interval from \SI{1.0}{\degree} to \SI{2.0}{\degree}. For this configuration,
the \SI{2}{\degree} beamwidth from the CBW experiment has been kept, so that these data
have matching sampling interval and beamwidth.
Figure~\ref{fig:c-rw-base} shows the single-polarization moment data for this
configuration, while Figure ~\ref{fig:c-rw-dual} shows the dual-polarization data.
Compared to the data for the CBW experiment shown in the previous section, these data
generally look blurred, with features showing much less distinction. This makes sense
since each radial in this data is the combination of two radials from the CBW data.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/single_C_2_MainLobe_2.png}
	\caption{As in Figure~\ref{fig:s-base}, but for the CRW experiment}
	\label{fig:c-rw-base}
\end{figure}
\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/dual_C_2_MainLobe_2.png}
	\caption{As in Figure~\ref{fig:s-dual}, but for the CRW experiment}
	\label{fig:c-rw-dual}
\end{figure}

\subsection{Time Series}
As an example of time series data generation capabilities,
Figure~\ref{fig:ts_general} shows the IQ data and power spectral density (PSD),
for both the horizontally and vertically polarized channels, from an area
around the main updraft of the storm for the S experiment. This region is
chosen to give strong returns and moderate velocity. The PSD is calculated from
the IQ data using the periodogram \citep{Harris1978} with a Hanning window and data
padded to \num{128} samples. The PSD shows a distinct peak around
\SI{7}{\meter\per\second} with a shape around that peak that generally resembles
a Gaussian distribution. The horizontal and vertical channels are slightly
different in this case, with the vertical channel having slightly less power at
all velocities, due to the intrinsic $Z_{DR}$. The fact that this occurs at all velocities is a
consequence of the power from the main signal leaking into
the other frequencies. This effect is not seen in Figure~\ref{fig:ts_noise}, which shows a similar plot
for a region that does not contain any weather signal. Instead, the H and V channels show little correlation in the
PSD, and there is no systematic bias between the channels.
\begin{figure}
    \centering
    \includegraphics[scale=0.95]{figures/tsplot_general}
    \caption{Plots of time series information for the S-band example.  These data are from the main
        core of the storm at \SI{355.5}{\degree} azimuth and \SI{25.5}{\kilo\meter}
        range. (a) Horizontal IQ data, (b) Vertical IQ data, (c) Power spectra for
        horizontal (blue) and vertical (green) channels.}
    \label{fig:ts_general}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.95]{figures/tsplot_noise}
    \caption{As in Figure~\ref{fig:ts_general}, but at \SI{342.5}{\degree} azimuth, corresponding to an area devoid of weather echo,
    showing data for pure noise.}
    \label{fig:ts_noise}
\end{figure}\chapter{Attenuation Correction}
While significant research effort has been put into radar-based QPE, especially
using dual-polarization radar data, there are still significant problems.
Attenuation is an important problem in the use of dual-polarization radar data
at any wavelength for QPE. Even at S-band, where attenuation of a single
channel may not be significant, differential attenuation can significantly bias
the calculation of differential reflectivity. When biased differential
reflectivity values are fed into a power-law-based QPE algorithm, this bias is
exacerbated. With the increasing use of radar systems at attenuating wavelengths
(such as C- and X-band), the problem of correcting data for attenuation has been
an area of active research.

\section{Techniques}
The basis for most radar rain attenuation correction schemes is in estimating the 
attenuation using other radar observables, such as reflectivity factor or, for
dual-polarization radars, differential propagation phase. In this work, we examine
several correction algorithms: Linear, ZPHI, Self-Consistent, and Modified Self-Consistent.

\subsection{Linear}
The Linear method, put forth by \cite{Bringi1990}, is the most straight-forward of the 
attenuation-correction methods. This method uses a direct relationship between specific
differential phase, $K_{DP}$, and both specific horizontal attenuation, $A_H$, and the
specific differential attenuation, $A_D$:
\begin{align}
	A_H &= \gamma_H K_{DP}^{\beta_H} \label{eq:linear-kdp-h}\\
	A_{DP} &= \gamma_D K_{DP}^{\beta_D}\label{eq:linear-kdp-diff}
\end{align}
The coefficients in these relations are found by calculating the radar observables, 
combining theoretical scattering calculations and measured drop size distributions.
For this algorithm, $A_H$ and $A_{DP}$ are regressed against $K_{DP}$,
resulting in values for $\beta_H$ and $\beta_D$ that are
close to \num{1.0}, though not exact. By prescribing a value of \num{1.0} for the exponents in these
relations, not only are the relations simplified, but the need to calculate $K_{DP}$
is entirely removed. Then simple integration of \eqref{eq:linear-kdp-h} and
\eqref{eq:linear-kdp-diff} above results in the following set of equations:
\begin{align}
	a_H(r) &= \gamma_H \Phi_{DP}(r)\label{eq:linear-phi-h}\\
	a_D(r) &= \gamma_D \Phi_{DP}(r)\label{eq:linear-phi-diff}
\end{align}
Thus, \eqref{eq:linear-phi-h} (and \eqref{eq:linear-phi-diff}) above describes a 
completely linear relation between
the path integrated (differential) attenuation, $a_H$ (and $a_D$) at a range, 
$r$, and the measured differential phase, $\Phi_{DP}$, at that range. By eliminating the use of $K_{DP}$, the problem of estimating
a derivative from noisy data is avoided. For this technique, there is one free parameter in each equation, $\gamma$, that relates
attenuation (or differential attenuation) to $\Phi_{DP}$; this parameter is empirically determined by a power-law regression on theoretical scattering 
calculations performed using drop size distribution data.

This technique has several shortcomings. First, the exponent in the best fit curve is not necessarily \num{1.0}; by prescribing
such a value, errors are necessarily introduced (albeit in the name of simplicity
and operational robustness of the technique). Second, while $K_{DP}$ calculation
is avoided, with all its inherent challenges, the $\Phi_{DP}$ data can still be contaminated by effects of non-uniform beam-filling and
differential back-scatter phase. Quality control must be performed on the data to ensure that such effects do not result in anomalous
attenuation correction. Finally, the relationship between attenuation and $\Phi_{DP}$ is sensitive to the assumptions made in the scattering
calculations. If the actual atmospheric conditions and processes differ from those 
assumed (e.g. temperature or drop shapes), the Linear technique will be unable
to compensate for these differences and thus yield less accurate results.

\subsection{ZPHI}
To address shortcomings in the Linear $\Phi_{DP}$ algorithm, \cite{Testud2001} proposed the ZPHI technique, based on an algorithm to
correct for attenuation in satellite profiles. The core relation in this technique is between reflectivity and attenuation,
as described by \cite{Hitschfeld1954}. The \cite{Hitschfeld1954} algorithm is improved upon by
providing a constraint for the total path attenuation. For satellites, this is provided by the ground; for terrestrial radar applications, this
constraint is provided by estimating the total path attenuation from the total change in $\Phi_{DP}$ along the path. The full technique
is described by the following set of equations:
\begin{align}
	I(r, r_0) &= \num{0.46}b\int_r^{r_0}Z_a^b(s)\,ds \\
	A(r_0) &= \frac{Z_a^b(r_0)}{I(r_1,r_0)} \lbrace 10^{\num{0.1}b\gamma\Delta\Phi} - 1\rbrace \\
	A(r) &= \frac{Z_a^b(r)}{I(r_1,r_0) + \lbrace 10^{\num{0.1}b\gamma\Delta\Phi} - 1\rbrace I(r, r_0)}
	  \times \lbrace 10^{\num{0.1}b\gamma\Delta\Phi} - 1\rbrace
	\label{eq:zphi}
\end{align}
Essentially, the technique works by calculating the total attenuation from the total 
path change in $\Phi_{DP}$, and distributing the total attenuation along the path
based on the (attenuated) reflectivity values. It should be noted that the process of relating the total path-integrated attenuation to $\Phi_{DP}$
implicitly makes use of the same assumptions as the Linear $\Phi_{DP}$ algorithm.
To correct for differential attenuation, the horizontal and vertical reflectivities can be corrected for attenuation separately; then corrected
differential reflectivity can be calculated from the individual reflectivities.

This technique has several strengths versus the Linear $\Phi_{DP}$ algorithm. By relying on the relationship between reflectivity and attenuation,
the ZPHI algorithm is much less sensitive to assumptions made about the scattering process. This dependence is captured by $b$, which is
a free parameter that is empirically determined from scattering calculations.
Also, by using $\Phi_{DP}$ only to calculate the
total path attenuation, the problems of non-uniform beam-filling and differential backscatter phase are limited; only the total change in $\Phi_{DP}$
is needed, nominally reducing the requirements to two good values of $\Phi_{DP}$.

However, the technique is not immune to the problems of invalid scattering 
assumptions. The $\gamma$ above is the same as that for the linear $\Phi_{DP}$ 
algorithm; the same sensitivity to assumptions is present in estimating the total 
attenuation from
$\Phi_{DP}$.  Thus, depending on actual atmospheric conditions, the total attenuation may be under- or over-estimated.

\subsection{Self-Consistent}
The self-consistent attenuation correction algorithm was introduced by \cite{Bringi2001} as an improvement on ZPHI. As the name suggests,
this method relies on the internal consistency of the data, in particular attenuation and $\Phi_{DP}$. This method iteratively performs the ZPHI
correction to try to find an optimal value for the $\gamma$ coefficient, attempting to address the sensitivity to scattering processes.
The ``optimal'' value of $\gamma$ is found by minimizing the mean absolute 
difference between the true profile of $\Phi_{DP}$ and a calculated profile
of $\Phi_{DP}$; this profile is calculated from the algorithm's estimated profile
of attenuation using an inverted form of the Linear $\Phi_{DP}$ relation with
the value of $\gamma$ being tested. The procedure is explicitly defined as:
\begin{align}
	\phi_{DP}^c(r;\gamma) &= 2 \int_{r_0}^r \frac{A_h(s;\gamma)}{\gamma}\,ds \\
	\gamma_{min} &\leq \gamma \leq \gamma_{max} \\
	Error &= \sum_{j=1}^N \left| \Phi_{DP}^{filt}(r_j) - \Phi_{DP}^c(r_j;\gamma) \right|
	\label{eq:zphi-sc}
\end{align}

The advantage of this technique is its ability to address the sensitivity of the
algorithms to assumptions about scattering processes; however,
only the $\gamma$ coefficient (relating attenuation to $\Phi_{DP}$) is optimized. The $b$ coefficient (relating attenuation and
reflectivity) remains fixed; while this coefficient is much less sensitive to scattering assumptions, this is a limitation to the adaptiveness of the
technique. Additionally, by using the linear $\Phi_{DP}$ relation, there is still the assumption that the power in the power law relation between
$K_{DP}$ and attenuation is \num{1.0}; such an assumption is a limitation to the attenuation model prescribed by the equations and cannot
be addressed by the adaptive procedure. Also, while this technique is robust in theory, in our testing we have observed that the technique can
have convergence problems, especially on rays with limited data. Such problems are not limited to individual rays, unfortunately, but extend
across small regions. For such rays, the procedure is not truly converging, but is stopping at the bounds prescribed by the algorithm's optimization procedure.

\subsection{Modified Self-Consistent}
In the course of examining the behavior and performance of the self-consistent
method, some artifacts were observed. One problem was that for rays with
little data coverage (and small total attenuation and $\Phi_{DP}$), the errors
were drastically larger. The poor performance for these rays comes as a result
of the optimization procedure not being able to converge to a value within
the specified bounds for $\gamma$; consequently, an anomalously large or
small value for $\gamma$ results. The second problem observed was that
the algorithm showed an oscillating pattern as a function of azimuth,
alternating between slightly greater and slightly less than true attenuation for
several rays at a time. Figure~\ref{fig:gamma-dist} shows the values
of the optimized $\gamma$ as a function of azimuth. Comparing the
estimated values to true values calculated from simulated data, the rays with
bad (non-converged) values are clearly visible. One can also see changes in
the converged value of $\gamma_h$ that do not correspond to changes
in the true optimal value of $\gamma_h$.

\begin{figure}
	\centering
	\includegraphics{figures/gamma_plot}
	\caption{Distribution of $\gamma_h$ values as a function of
	azimuth. Values obtained from optimization in the self-consistent algorithm
	are in blue, while true values are in green. The median of the optimized values
	is given by the dashed black line.}
	\label{fig:gamma-dist}
\end{figure}

To address these shortcomings and increase the robustness of the
Self-Consistent algorithm, we introduce a Modified Self-Consistent algorithm.
In this version, the optimal values of $\gamma_h$ from all rays is collected and
the median $\gamma_h$ value is calculated. This median is used then as the
optimal value
of $\gamma_h$ for all rays. As an example, the median for the $\gamma_h$ 
values
given in Figure~\ref{fig:gamma-dist} is plotted as a horizontal line. While this
value is clearly greater than the true value in places (and less in others), it
has the beneficial behavior of removing outlier values as well as producing
smoother results. Figure~\ref{fig:sc-compare} compares the estimates
of specific attenuation for horizontal attenuation (at C band) for both the
modified and un-modified Self-Consistent algorithms. The modified method
shows a tighter clustering of values, due to the reduction in outlier points. It also
appears that the modified procedure introduces a slight negative bias, 
with more points below the one-to-one correspondence line.

\begin{figure}
	\centering
	\includegraphics[scale=0.95]{figures/sc-scatter}
	\caption{Scatter plots of estimated specific attenuation for self-consistent
	(left) and modified Self-Consistent (right) versus true specific attenuation
	values.}
	\label{fig:sc-compare}
\end{figure}

The performance of this algorithm versus that of the unmodified
Self-Consistent method will be discussed further in later chapters.

\section{Finding Coefficients}
All of the algorithms described previously rely on free parameters to relate attenuation to radar observables (i.e., reflectivity and $\Phi_{DP}$).
The original works (\cite{Bringi1990}, \cite{Testud2000}, and \cite{Bringi2001}) determined these coefficients empirically using regression on
calculated radar observables. However, there are two obstacles to the use of these values here. First, the scattering calculations used to generate
these parameters were based on datasets of drop size distributions; because these data would not match the prescribed drop size distribution
that underlies the model simulation, their use could introduce errors in the attenuation correction. While certainly such errors exist in
real world applications of these techniques, the goal here is to provide a best-case scenario for these techniques. As such, differences in
assumptions between the algorithms and the radar data simulation are minimized where possible. The second challenge is that the original
works have differing, or even undocumented, assumptions regarding their scattering calculations; it is difficult to create a set of radar
simulation parameters that is known to completely match those assumptions
made by several different works.

The challenges with using exiting published coefficients necessitates the 
development of a procedure for calculating a new set of regression parameters. This procedure has the benefit of creating parameters
that represent a best case scenario for algorithm performance; since the assumptions used in generation of the parameters can be matched
exactly in the process of radar data simulation, resulting errors are inherent to the attenuation correction algorithm itself. Having such coefficients
with precisely known assumptions also allows for systematic testing of how violating these assumptions affects algorithm performance; this is the
subject of the next chapter.

\subsection{Scattering Calculations}
To generate empirical fits between radar observables (i.e., $K_{DP}$ and reflectivity) and (differential) attenuation values, we start with a set of
scattering matrices calculated using the T-matrix method, as outlined in the previous chapter. The relevant assumptions for the
scattering calculations are given in Table~\ref{tab:regression-parms}. Using these assumptions, the scattering matrices are calculated across
a range of diameters from \num{0.01} to \SI{8}{\milli\meter}.

\begin{table}[ht]
    \centering
    \begin{tabular}{ | l | l | l |}
        \hline
        Wavelength & \SI{5.5}{\centi\meter}, \SI{3.21}{\centi\meter} \\
        Temperature & \SI{283}{\kelvin} \\
        Canting Angle Distribution Width & \num{10} \\
        Shape Model & \cite{Brandes2004} \\
        \hline
    \end{tabular}
    \caption{Parameters used in scattering calculations for regression coefficients}
    \label{tab:regression-parms}
\end{table}

To turn the scattering matrices into $K_{DP}$, reflectivity, attenuation, and differential attenuation, we use equations \eqref{eq:scatter-integrals}
and \eqref{eq:prop-calcs}, just as in the radar simulation. The model simulation grid, which
gives a set of rain water concentration ($q_r$) and number concentration ($N_0$) values, functions as a set of relevant DSD data. Even when the grid is thresholded to contain measurable
rain ($q_r > \SI{0.5}{\kilo\gram\per\meter\cubed}$ and $N_0 > \SI{100}{\per\meter\cubed})$, there are still too many points to work with;
as a result, we arbitrarily subset those points by a factor of \num{100}. In Figure~\ref{fig:regress-data} the resulting subset of the data is shown.
These data represent a wide spread of number concentration values, with the higher rainwater content values clustered around a number
concentration of \SI{10000}{\per\meter\cubed}.
\begin{figure}
	\centering
	\includegraphics[scale=0.70]{figures/regress-data}
	\caption{Scatter plot of rain water parameters used for regressions. Rain water content (\si{\gram\per\meter\cubed}) is plotted along
	               the x-axis and number concentration (\si{\per\meter\cubed}) is plotted logarithmically along the y-axis.}
	\label{fig:regress-data}
\end{figure}
To turn these two parameters into actual drop size distributions, a modified gamma distribution, as described previously, is prescribed. Using
the model-appropriate fixed shape parameter of \num{1.81}, we have sufficient information to generate a unique drop size distribution for
each rain content-number concentration pair.

\subsection{Regressions}
For the attenuation correction techniques described previously, the following set of empirical relations are required:
\begin{align}
	A_H &= \gamma_H K_{DP}\\
	A_V &= \gamma_V K_{DP}\\
	A_H &= a_H Z_H ^ {b_H}\\
	A_V &= a_V Z_V ^ {b_V}
	\label{eq:atten-relations}
\end{align}
$\gamma_{H,V}$ and $b_{H,V}$ are the empirical coefficients that need to be determined through regressions on sets of
$A_{H,V}$, $Z_{H,V}$, and $K_{DP}$ provided by the scattering calculations ($a_{H,V}$ are unused in the algorithms). $\gamma_{H,V}$
can be found in a straightforward fashion using linear regression, while the $b_{H,V}$ can be found by using logarithms to transform the power
law regression into a linear regression problem. Figure~\ref{fig:za-h-unweighted} shows the results of the regression (for both C- and X-bands)
between reflectivity factor ($Z_H$) and specific attenuation ($A_H$) at horizontal polarization. Qualitatively, the resulting fit does not
appear to capture well the relationship between $Z_H$ and $A_H$, especially at C-band; the increasing error with increasing
values of $A_H$ confirms this. Figure~\ref{fig:za-v-unweighted} shows similar problems with the fit for vertical polarization.
Figures~\ref{fig:ka-h-unweighted} and \ref{fig:ka-v-unweighted} show the same analysis, but for the regression between specific differential
phase ($K_{DP}$) and specific attenuation at horizontal and vertical polarizations, respectively. The results here are much less problematic;
overall the curves seem to fit the raw values relatively well, though for horizontal polarization the slope of the line looks, qualitatively, to be
too small.

\begin{figure}
	\centering
	\includegraphics[scale=0.7]{figures/za_H_unweighted}
	\caption{Results of applying regression on scattering data for C-band (left) and X-band (right). The top panels show the raw calculated
		specific attenuation ($A_H$) values (red), as well as those calculated from the regression (blue), as a function of
		reflectivity factor ($Z_H$), for horizontal
		polarization. The bottom panels show the resulting error as a function of the true specific attenuation values.}
	\label{fig:za-h-unweighted}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.7]{figures/za_V_unweighted}
	\caption{As in Figure~\ref{fig:za-h-unweighted}, but for vertical polarization.}
	\label{fig:za-v-unweighted}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.7]{figures/ka_H_unweighted}
	\caption{Results of applying regression on scattering data for C-band (left) and X-band (right). The top panels show, for
	    horizontal polarization, the raw calculated
		specific attenuation ($A_H$) values (red), as well as those calculated from the regression (blue), as a function of
		specific differential phase ($K_{DP}$). The bottom panels show the resulting error as a function of the true specific attenuation values.}
	\label{fig:ka-h-unweighted}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.7]{figures/ka_V_unweighted}
	\caption{As in Figure~\ref{fig:ka-h-unweighted}, but for vertical polarization.}
	\label{fig:ka-v-unweighted}
\end{figure}

There are two main reasons for the poor regression fits. First, the set of points used in the regression is dominated
by points corresponding to low attenuation values; this is a direct consequence of using the model grid's precipitation fields as the source
of DSD data, as most of the model field that has precipitation is only filled with light rain. Combined with the mathematics of linear regression,
the curve that minimizes the RMS error fits, for the most part, the numerous points at low attenuation values. The errors from the lack of fit
at high attenuation value are more than countered by the fact that the line fits many points at low values. This particular problem affects both
the regressions using $K_{DP}$ and $Z_{H,V}$.

The second problem, which only affects the regressions on $Z_{H,V}$, come from the reformulation of the power law fit as a linear regression.
The logarithmic transformation of the data (to make it a linear problem), changes the measurement of errors. The deviations of the regression
curve from the data (i.e., subtraction between points) in logarithmic space, actually represent ratios in the space of the original points being
fit with a power law. Effectively, the regression is minimizing the RMS of relative error rather than that of absolute error. This has the effect
of reducing the error contribution of points at higher attenuation values. This effect explains why the regressions for $Z_{H,V}$ were,
qualitatively, less good than those for $K_{DP}$.

One method to try to improve these fits would be to resample the model DSD fields to remove the over-represented low attenuation values.
To do so objectively would not be simple; more importantly, this would not address all of the problems for the power law
regressions. Instead, we have chosen to use weighted linear regression, using the square of the attenuation value as the weights. Doing so
makes the larger attenuation values have a greater impact on the regression, countering both the problems of numerous low values and the
logarithmic scaling.

\begin{figure}
	\centering
	\includegraphics[scale=0.7]{figures/za_H_weighted}
	\caption{As in Figure~\ref{fig:za-h-unweighted}, but for weighted regression.}
	\label{fig:za-h-weighted}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.7]{figures/za_V_weighted}
	\caption{As in Figure~\ref{fig:za-v-unweighted}, but for weighted regression.}
	\label{fig:za-v-weighted}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.7]{figures/ka_H_weighted}
	\caption{As in Figure~\ref{fig:ka-h-unweighted}, but for weighted regression.}
	\label{fig:ka-h-weighted}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[scale=0.7]{figures/ka_V_weighted}
	\caption{As in Figure~\ref{fig:ka-v-unweighted}, but for weighted regression.}
	\label{fig:ka-v-weighted}
\end{figure}

Figure~\ref{fig:za-h-weighted} shows the results when weighting is applied in the regression between specific attenuation and
reflectivity factor at horizontal polarization. Qualitatively, the resulting curves do a much
better job of capturing the relationship between $A_H$ and $Z_H$, especially at C-band. Quantitatively, the errors for high attenuation values
are greatly reduced, without greatly increasing the errors for small values of $A_H$.
The results for vertical polarization (Figure~\ref{fig:za-v-weighted}) show similar improvements.
Figures~\ref{fig:ka-h-weighted} and \ref{fig:ka-v-weighted} do not show the same 
clear gains for the regression between $K_{DP}$ and $A_H$, though there
do appear to be slight improvements. Overall, the effect of weighting in the regression has been to create fits to the data that
provide useful estimates for the parameters needed in \eqref{eq:atten-relations} above; the weighted regressions appear to capture well the relationships between
the radar observables. The final values for these parameters, which will
be used in the later chapters, are presented in Table~\ref{tab:regress-results}.

\begin{table}[ht]
    \centering
    \begin{tabular}{ | l | l | l |}
        \hline
        & \textbf{C-band} & \textbf{X-band}\\
        \hline\hline
        $\beta_H$ &  \num{0.770618335} & \num{0.62143} \\
        $\beta_V$ &  \num{0.812085167} & \num{0.681336195} \\
        $\gamma_H$ & \num{0.10014759039795704} & \num{0.3315601537701545} \\
        $\gamma_V$ & \num{0.07342783867155196} & \num{0.27888686163888265} \\
        \hline
    \end{tabular}
    \caption{Final parameter values from regression.}
    \label{tab:regress-results}
\end{table}\chapter{Modeling Errors in Attenuation Correction}
The goal here is to examine and quantify the sensitivity of the previously
described algorithms to the assumptions made in calculating the coefficients
used in those techniques. Thus, it explores the impact of errors in modeling the
relationships between the radar variables.  Such sensitivity of radar parameters
has previously been demonstrated by \cite{Aydin1989}, \cite{Bringi1990},
and \cite{Carey2000}, among others. The work here, carries the
study of such sensitivity further by using fully-simulated radar data as well as
full algorithm implementations.

To perform this study, the radar data simulation procedure
described previously is used to generate data using different
sets of assumptions. For each of these sets, the attenuation correction
algorithms are run to generate estimates of the attenuation field, which
are compared against the true attenuation field, as calculated by the
radar simulation. These comparisons are done for both C- and X-band, which
permits exploration of any differences between the two bands.

\begin{table}
    \centering
    \begin{tabular}{ | l | l | }
        \hline
        Antenna gain & \SI{45.5}{dB} \\
        Antenna \SI{3}{dB} Beamwidth & \SI{0.25}{\degree} \\
        Antenna Limits & Main-lobe only \\
        Peak power & \SI{250}{\kilo\watt} \\
        First range gate & \SI{500}{\meter} \\
        Noise power & \SI{-113}{dBm} \\
        Elevation & \SI{0.5}{\degree} \\
        PRT & \SI{0.667}{\milli\second} \\
        Rotation Rate & \SI{15}{\degree\per\second} \\
        Pulses per radial & \num{25} \\
        Gate length & \SI{100}{\meter} \\
        \hline
    \end{tabular}
    \caption{Radar and scanning parameters common to the simulations}
    \label{tab:modelling-common-params}
\end{table}

Table~\ref{tab:modelling-common-params} lists the parameters common
across all of the experiments involved in studying the effects of modeling
errors. To limit the impact of radar sampling geometry on the results,
the radar and scanning configuration is chosen to produce very high-resolution
data, higher than usually is encountered in real data; the effects
of sampling will be explored later. These parameters give radials with a spacing
of \SI{0.25}{\degree}, matching the \SI{3}{dB} beamwidth.

\begin{table}
    \centering
    \begin{tabular}{ | c | c | c | c | c | }
        \hline
        Experiment & Wavelength & Canting Width & Temperature & Shape Model \\
        \hline
        Control & \SI{5.5}{\centi\meter}, \SI{3.21}{\centi\meter} & \num{10} & \SI{283}{\kelvin} & Brandes \\
        Canting & \SI{5.5}{\centi\meter}, \SI{3.21}{\centi\meter} & \num{20} & \SI{283}{\kelvin} & Brandes \\
        Shape & \SI{5.5}{\centi\meter}, \SI{3.21}{\centi\meter} & \num{10} & \SI{283}{\kelvin} & Pruppacher \\
        Temperature & \SI{5.5}{\centi\meter}, \SI{3.21}{\centi\meter} & \num{10} & Grid (\SI{295}{\kelvin}) & Brandes \\
        Wavelength & \SI{5.0}{\centi\meter}, \SI{3.0}{\centi\meter} & \num{10} & \SI{283}{\kelvin} & Brandes \\
        Combined & \SI{5.0}{\centi\meter}, \SI{3.0}{\centi\meter} & \num{20} & Grid (\SI{295}{\kelvin}) & Pruppacher \\
        \hline
    \end{tabular}
    \caption{Parameters differing between different experiments}
    \label{tab:modelling-exp-params}
\end{table}

Table~\ref{tab:modelling-exp-params} lists the parameters for the various
experiments that are examined here. The Control experiment serves as the
baseline of performance for the algorithms, where the assumptions used
in simulating the radar data match those used to find the coefficients
for the algorithms; as such, it also represents a best case
for performance. The other experiments, Canting, Shape, Temperature, and 
Wavelength, are chosen to systematically test how algorithm performance
varies as assumptions involving these aspects of scattering are varied. The
final experiment, Combined, is essentially the opposite of the Control
experiment, in that all of the assumptions are violated; as such, it might be
considered to represent, theoretically, a worst case scenario for algorithm
performance.

\begin{figure}
    \centering
    \includegraphics[scale=0.47]{figures/C-band-calc-attenuation}
    \caption{ Top Left: Specific attenuation as a function of reflectivity factor
    for horizontal polarization. Top Right: Specific attenuation at horizontal
    polarization as a function specific differential phase.
    Bottom Left: Specific attenuation as a function of reflectivity factor
    for vertical polarization.
    Bottom Right: Specific attenuation at vertical polarization as a function
    of specific differential phase. All calculations assume a C-band wavelength.}
    \label{fig:C-Attenuation-calcs}
\end{figure}

To frame the results from the experiments,
Figures~\ref{fig:C-Attenuation-calcs} and \ref{fig:X-Attenuation-calcs} show
plots of attenuation as a function of both reflectivity factor and specific
differential phase, for both horizontal and vertical polarizations. These
plots give some expectation of the results of the experiments, while 
making a simplifying assumption of a \cite{Marshall1947} drop size
distribution; this implies that they do not reflect completely the results
from the full simulation (using a modified gamma distribution). For
reference, the curve corresponding to the regression coefficients
(for the correction algorithms) is also plotted.

In Figure~\ref{fig:C-Attenuation-calcs}, it is apparent that at C-band the
relationship between reflectivity factor and attenuation, for both
polarizations, is relatively immune to the assumptions made; the biggest
change comes from switching the wavelength. It is also seems from these plots
that the regression coefficient between reflectivity and attenuation captures
the various relationships relatively well. The relationship between specific
differential phase and attenuation, however, varies much more with changing
assumptions. Again, the greatest impact seems to come from switching
wavelength, though the effect is even greater than for reflectivity factor. It should 
also be noted that the effect of changing wavelength was of the opposite sign
to the effects of changing every other assumption. Consequently, the 
curve for the Combined experiment lies very close to that of Control, a result
of the errors for changing the various assumptions canceling out.
The regression
coefficient curve in this case is well below the others, as well. This
hints that calculating attenuation using differential phase may
result in underestimation; or it may just be a product of using the
\cite{Marshall1947} drop size distribution here, while the coefficient was
optimized for a modified gamma distribution. The results of the individual
experiments answer this question.

At X-band, in Figure~\ref{fig:X-Attenuation-calcs}, the relationship between
reflectivity factor and attenuation seems to be more sensitive to the
assumptions made than at C-band. Also, the regression coefficient curve in this
case seems to be noticeably different than the other curves; again, this could
be caused by the difference in the drop size distribution used. Looking at the
relations between attenuation and specific differential phase, it is
apparent that the assumptions made greatly impact the result. As before,
for the case of the Combined experiment the errors seem to cancel out,
leaving the Control and Combined curves very close. For horizontal
polarization, the regression curve is right along the Control (and Combined);
at vertical polarization the regression curve is not as good a match for Control. The
results from the experiments with the full simulation are needed to
see whether this is an artifact of the drop size distribution difference.

\begin{figure}
    \centering
    \includegraphics[scale=0.47]{figures/X-band-calc-attenuation}
    \caption{As in Figure~\ref{fig:C-Attenuation-calcs}, but for X-band.}
    \label{fig:X-Attenuation-calcs}
\end{figure}

\FloatBarrier
\section{Control}
As mentioned previously, the Control experiment is designed to represent a best
case for algorithm performance, where the parameters used to simulate the
radar data match the assumptions made when calculating the coefficients for the
attenuation correction algorithms. The parameters include fairly routine assumptions, such
as fixed temperature (\SI{283}{\kelvin}) and the \cite{Brandes2002} shape
model (as described in \eqref{eq:brandes-shape-model}). Given the matching
assumptions and the high resolution data, any errors here reflect fundamental
limitations of the algorithms due to the assumptions made about power law
relationships between the variables, as well as any errors introduced by the
radar's sampling process.

\FloatBarrier
\subsection{C band}
Figure~\ref{fig:C_Control_Attenuation_H} shows the fields of horizontal
attenuation that result from running the various algorithms, as well as the
true field of attenuation. Overall, all algorithms show good agreement. The
greatest values of attenuation are found at the far range due north from the
radar. Here, the Linear and ZPHI algorithms have noticably higher values
compared to the truth field; the Modified Self-Consistent (MSC) algorithm,
on the other hand,  underestimates the attenuation values. The
Self-Consistent (SC) algorithm appears to capture the true attenuation
almost perfectly.

These observations are largely corroborated by
Figure~\ref{fig:C_Control_Attenuation_Difference_H}, which shows the output
from the algorithms with the true attenuation field subtracted, aiding in
highlighting the areas that differ. The differences for Linear and ZPHI are in the
range of \SI{2}{dB} too high in the core, while MSC only shows a slight low
bias of approximately \SI{0.5}{dB}. SC shows fluctuations of some rays
with low values and some with high, with the magnitude of the differences on
the order of \SI{1}{dB}; overall, the output field from SC appears unbiased. As 
mentioned before, the radial-to-radial variation for SC is a consequence of the
independent optimization for $\gamma$ that is applied to each radial, and is the
impetus for the development of the MSC. In this case, MSC does not show
the oscillatory behavior, but still manages to show an overall field that appears
unbiased with regards to estimates of attenuation.

Figures~\ref{fig:C_Control_Attenuation_V} and
\ref{fig:C_Control_Attenuation_Difference_V} show similar analyses for
vertical polarization. Here, all of the algorithms, overall, demonstrate accuracy
in estimating the amount of attenuation. The Linear and ZPHI algorithms yield an overall overestimate of attenuation, up to \SI{1}{dB}. MSC, conversely,
shows an overall slight underestimation of attenuation, on the order of about
\SI{0.5}{dB}. SC displays changes on a per radial basis; while the mean
of its field appears that it could be comparable to the others, the individual
radials show errors larger than the other algorithms.

2D histograms of the specific attenuation at horizontal and vertical
polarizations,
shown in Figures~\ref{fig:C_Control_Specific_Attenuation_H_scatter} and
\ref{fig:C_Control_Specific_Attenuation_V_scatter}, respectively, help to
examine the algorithm performance better by focusing on the accuracy
on a gate by gate basis. At horizontal polarization, the bias in the Linear
algorithm is evidenced by the ribbon of largest values being located above
one-to-one line. This is also true for ZPHI, though in this case, the algorithm
shows two distinct maximal bands in specific attenuation, and the one-to-one
line goes between them; the band with the highest counts sits
above the line, indicating a slight high bias. SC and MSC also demonstrate
this two-banded structure; however, for these algorithms, especially MSC,
the band of highest point counts lies along the one-to-one line. It should
also be mentioned that all of the ZPHI-based algorithms (ZPHI, SC, MSC) show
a universal underestimation of higher values of specific attenuation, though 
these values are less common. At vertical polarization, all of the histograms
tighten up, showing less overall spread. All of the algorithms have greater
concentrations of points along the one-to-one line, reinforcing the notion
that all of the algorithms have decreased errors for vertical polarization. These
results are reflected in Table~\ref{tab:C-stats-Control}, which shows that
SC and MSC have the lowest biases at horizontal polarization, while all algorithms
show lower values of mean squared-error (MSE) at vertical polarization.

Based on these results, it is also instructive to look at the results for
differential attenuation, since it is more directly related to biases in
$Z_{DR}$. Figures~\ref{fig:C_Control_Differential_Attenuation} and
\ref{fig:C_Control_Differential_Attenuation_Difference} show PPIs of
differential attenuation and its difference from the true value, respectively.
Here the Linear algorithm shows a clear high bias, of approximately \SI{1.2}{dB},
which would be a significant bias for $Z_{DR}$. ZPHI shows a decreased bias
with respect to this, though the errors still approach \SI{1.0}{dB}. The SC
algorithm again shows an overall unbiased field, with significant
spatial variability in the errors. MSC appears to smooth this out well, though
it is still showing errors of approximately \SI{0.5}{dB} outside of the core, where
there are slightly negative errors. The 2D histogram of specific differential
attenuation,
Figure~\ref{fig:C_Control_Specific_Differential_Attenuation_scatter}, shows
these same patterns of biases, including the negative biases of SC and MSC
at higher values. Again, the calculated bias and MSE values
(Table~\ref{tab:C-stats-Control}) favor the results of the SC and MSC
algorithms.

Overall, all of the algorithms perform relatively well when the coefficients
used agree with the nature of the data. Linear begins to show some problems
correcting differential attenuation, as the resulting bias is high relative to the magnitude
of $Z_{DR}$. It should also be noted that, given its better performance at
vertical polarization, the problem stems from its bias at estimating horizontal
attenuation. The Control experiment also demonstrates that the optimization
procedure for SC does well at reducing the bias from the ZPHI algorithm,
and subsequently the MSC does well at smoothing out the results from
SC.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Attenuation_H}
    \caption{Plan Position Indicators (PPIs) of attenuation for horizontal
    polarization at C-band for the Control experiment from various sources:
    (a) True
    field calculated from model (b) Linear algorithm (c) ZPHI algorithm
    (d) Self-Consistent algorithm (e) Modified Self-Consistent algorithm.
    Range rings are plotted every \SI{10}{\kilo\meter} from the radar.}
    \label{fig:C_Control_Attenuation_H}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Attenuation_Difference_H}
    \caption{Plan Position Indicators (PPIs) for the Control experiment of the
    difference between the
    true horizontal attenuation values (from the model) at C-band
    and those calculated by algorithms: (a) Linear algorithm (b) ZPHI algorithm
    (c) Self-Consistent algorithm (d) Modified Self-Consistent algorithm.
    Range rings are plotted every \SI{10}{\kilo\meter} from the radar.}
    \label{fig:C_Control_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:C_Control_Attenuation_V}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_Difference_H}, but for
    vertical polarization.}
    \label{fig:C_Control_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Specific_Attenuation_H_scatter}
    \caption{2D histograms for the C-band Control experiment of true specific
    horizontal attenuation (x-axis) and calculated specific horizontal
    attenuation (y-axis) for: (a) Linear algorithm (b) ZPHI algorithm
    (c) Self-Consistent algorithm (d) Modified Self-Consistent algorithm.}
    \label{fig:C_Control_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Specific_Attenuation_V_scatter}
    \caption{As in Figure~\ref{fig:C_Control_Specific_Attenuation_H_scatter}
    but for vertical polarization.}
    \label{fig:C_Control_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H} but for
    differential attenuation.}
    \label{fig:C_Control_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_Difference_H} but for
    differential attenuation.}
    \label{fig:C_Control_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Control_Specific_Differential_Attenuation_scatter}
    \caption{As in Figure~\ref{fig:C_Control_Specific_Attenuation_H_scatter}
    but for specific differential attenuation.}
    \label{fig:C_Control_Specific_Differential_Attenuation_scatter}
\end{figure}

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0310 & 0.0047 & 0.9612\\
        ZPHI & 0.0281 & 0.0070 & 0.9233\\
        SC & 0.0074 & 0.0070 & 0.9173\\
        MSC & -0.0009 & 0.0059 & 0.9305\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0095 & 0.0020 & 0.9639\\
        ZPHI & 0.0071 & 0.0033 & 0.9240\\
        SC & -0.0027 & 0.0042 & 0.9061\\
        MSC & -0.0087 & 0.0034 & 0.9257\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0410 & 0.0023 & 0.9021\\
        ZPHI & 0.0287 & 0.0019 & 0.8291\\
        SC & 0.0091 & 0.0012 & 0.8315\\
        MSC & 0.0064 & 0.0009 & 0.8896\\
        \hline
    \end{tabular}
    \caption{Bias, mean squared-error, and $r^2$ for the specific attenuation
    results for the Control experiment at C-band.}
    \label{tab:C-stats-Control}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, the pattern of horizontal attenuation looks similar to that at C-band,
as shown by the PPIs in Figure~\ref{fig:X_Control_Attenuation_H}. One
difference is that the core of highest attenuation values (now peaking around
\SI{100}{dB}) is in the area of missing data for the algorithms, since the
actual data are completely attenuated here; the signal is well below the noise.
As a result, the algorithms have no chance of matching truth here. For the
ZPHI algorithms, it is also clear that they are not able to match the high values
calculated by the Linear algorithm. From
Figure~\ref{fig:X_Control_Attenuation_Difference_H} it is apparent, based on
the differences when the true attenuation values are subtracted, that the
Linear algorithm is doing much better than the other algorithms in this area
along the outside of the core. This is likely caused by ZPHI's
reliance on reflectivity data, which while above the noise threshold, are still heavily
contaminated by the amount of attenuation incurred. Even
outside these regions, the Linear algorithm appears to be doing the best
job, though the other algorithms demonstrate more reasonable performance
outside the heavily attenuated area. Once again, the radial-to-radial variations
are clear in the differences for SC, which are eliminated by the MSC algorithm.

At vertical polarization, similar behavior is exhibited by the algorithms' base
results (Figure~\ref{fig:X_Control_Attenuation_V}) and the
differences (Figure~\ref{fig:X_Control_Attenuation_Difference_V}),
though the errors (outside of the heavy attenuation zone) are
reduced in magnitude. 	Looking at the 2D histograms of specific attenuation
for both horizontal and vertical polarization in
Figures~\ref{fig:X_Control_Specific_Attenuation_H_scatter} and
\ref{fig:X_Control_Specific_Attenuation_V_scatter}, respectively, it is
clear that the Linear
algorithm matches the true attenuation values very well, with only slight
underestimation at the highest values. For the ZPHI-based algorithms,
again the two-banded structure is evident, with one band on either side
of the one-to-one line.

The computed bias and MSE values for this experiment
(Table~\ref{tab:X-stats-Control}) agree with these observations. While the
Linear algorithm does not have the lowest bias, it is comparable to that of the
other algorithms. The MSE for the Linear algorithm, is the lowest of all algorithms.
It should also be noted that the difference in $r^2$ values between the Linear
and ZPHI-based algorithms reflects the two-band structure present in the
specific attenuation results.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Control_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_Difference_H} but
    for X-band.}
    \label{fig:X_Control_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Control_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H} but for X-band.}
    \label{fig:X_Control_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Control_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_Control_Attenuation_H} but for
    vertical polarization.}
    \label{fig:X_Control_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Control_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_Control_Attenuation_Difference_H} but for
    vertical polarization.}
    \label{fig:X_Control_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Control_Specific_Attenuation_H_scatter}
    \caption{As in Figure~\ref{fig:C_Control_Specific_Attenuation_H_scatter}
    but for X-band.}
    \label{fig:X_Control_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Control_Specific_Attenuation_V_scatter}
    \caption{As in Figure~\ref{fig:X_Control_Specific_Attenuation_H_scatter}
    but for vertical polarization.}
    \label{fig:X_Control_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Control_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_Control_Attenuation_H} but for
    differential attenuation.}
    \label{fig:X_Control_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Control_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_Control_Attenuation_Difference_H} but for
    differential attenuation.}
    \label{fig:X_Control_Differential_Attenuation_Difference}
\end{figure}

For differential attenuation, the results in
Figure~\ref{fig:X_Control_Differential_Attenuation} look very similar to
those for the individual polarizations. In the differences from the true field 
(Figure~\ref{fig:X_Control_Differential_Attenuation_Difference}), the zone 
where the ZPHI algorithms
have problems has increased slightly in size; it now starts at
\SI{20}{\kilo\meter} range. Outside of this area, all of the algorithms look
comparable, exhibiting a high bias in the estimated attenuation, with
peak errors around \SI{1.2}{dB}. In the 2D histograms for specific
differential attenuation in
Figure~\ref{fig:X_Control_Specific_Differential_Attenuation_scatter},
these high biases are clearly evident, with all of the algorithms having
most of the values above the one-to-line.

Overall, for the Control experiment, the Linear algorithm shows the most
success estimating attenuation at X-band, as its sole reliance on differential phase
makes it more robust than the ZPHI-based algorithms, at least in this case.
It should also be noticed
that the performance of these latter algorithms were all very similar. This is
unsurprising since SC and MSC do not need to optimize away from the
base value; this value is already optimal for the assumptions used
to simulate the data.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Control_Specific_Differential_Attenuation_scatter}
    \caption{As in Figure~\ref{fig:X_Control_Specific_Attenuation_H_scatter}
    but for specific differential attenuation.}
    \label{fig:X_Control_Specific_Differential_Attenuation_scatter}
\end{figure}

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0237 & 0.0722 & 0.9708\\
        ZPHI & -0.0327 & 0.2279 & 0.9058\\
        SC & 0.0151 & 0.2200 & 0.9082\\
        MSC & 0.0079 & 0.1985 & 0.9157\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & -0.0004 & 0.0500 & 0.9712\\
        ZPHI & -0.0414 & 0.1607 & 0.9071\\
        SC & 0.0036 & 0.1533 & 0.9098\\
        MSC & -0.0115 & 0.1418 & 0.9160\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0384 & 0.0044 & 0.9561\\
        ZPHI & 0.0063 & 0.0120 & 0.8136\\
        SC & 0.0106 & 0.0118 & 0.8197\\
        MSC & 0.0216 & 0.0117 & 0.8201\\
        \hline
    \end{tabular}
    \caption{Bias, mean squared-error, and $r^2$ for the specific attenuation
    results for the Control experiment at X-band.}
    \label{tab:X-stats-Control}
\end{table}

\FloatBarrier
\section{Canting}
The Canting experiment modifies the Control experiment by changing the
prescribed width of the canting angle distribution ($\kappa$ in
\eqref{eq:axial-distribution}) for the simulation from
\num{10} to \num{20}, changing the standard deviation of canting angles
from \SI{13.2}{\degree} to \SI{9.2}{\degree}. There still is no mean canting,
only a narrower range of
orientations distributed around \SI{0}{\degree}. The effect of this decreased
spread is to make the scatterers appear less isotropic to the radar, slightly
increasing the magnitude of the dual-polarization signatures. Previously,
\cite{Aydin1989}, with an assumed Gaussian
distribution of canting angles, found a \SI{6}{\percent} error in $K_{DP}$
resulted when increasing the standard deviation of canting angle from
\SI{0}{\degree} to \SI{10}{\degree}. 

\FloatBarrier
\subsection{C band}
In the Canting experiment, the horizontal attenuation estimates
(Figure~\ref{fig:C_Canting_Attenuation_H}) show mark\-edly higher values
for the Linear and ZPHI algorithms, while SC and MSC show practically
identical output compared to the Control experiment. The same is also true
at vertical polarization (Figure~\ref{fig:C_Canting_Attenuation_V}). With the
narrower distribution of angles, the intrinsic values of $K_{DP}$, and hence
$\Phi_{DP}$ increase, causing the Linear and ZPHI algorithms to
estimate more total path attenuation. The SC and MSC algorithms,
however, by virtue of the optimization procedure, adjust their
internal $\gamma$ coefficient to account for the change in canting.

The impacts of this change are quite evident in the difference fields for
horizontal (Figure~\ref{fig:C_Canting_Attenuation_Difference_H}) and
vertical (Figure~\ref{fig:C_Canting_Attenuation_Difference_V}) polarizations.
For the Linear and ZPHI algorithms, the errors increase to \SI{3}{dB},
gaining around \SI{1}{dB} in error from a simple change in canting;
SC and MSC, conversely, show the same pattern of errors as in Control.
These errors are also
reflected in the histograms of specific attenuation at horizontal and vertical 
polarizations (Figures~\ref{fig:C_Canting_Specific_Attenuation_H_scatter} and
\ref{fig:C_Canting_Specific_Attenuation_V_scatter}, respectively). Here
the points for the Linear and ZPHI algorithms shift upwards, indicating a more pronounced positive bias. In contrast, the SC and MSC algorithms
show qualitatively the same spread of points as in the Control experiment.
The computed bias
values (Table~\ref{tab:C-stats-Canting}) for the SC and MSC algorithms are much
lower for both polarizations; this decrease in bias comes with comparable (SC),
or even slightly better (MSC), values for MSE.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H} but for the
    Canting experiment.}
    \label{fig:C_Canting_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H} but for the
    Canting experiment.}
    \label{fig:C_Canting_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_Canting_Attenuation_H} but for
    vertical polarization.}
    \label{fig:C_Canting_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_Canting_Attenuation_Difference_H} but for
    vertical polarization.}
    \label{fig:C_Canting_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Specific_Attenuation_H_scatter}
    \caption{As in Figure~\ref{fig:C_Control_Specific_Attenuation_H_scatter}
    but for the Canting experiment.}
    \label{fig:C_Canting_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Specific_Attenuation_V_scatter}
    \caption{As in Figure~\ref{fig:C_Canting_Specific_Attenuation_H_scatter}
    but for vertical polarization.}
    \label{fig:C_Canting_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_Canting_Attenuation_H} but for
    differential attenuation.}
    \label{fig:C_Canting_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_Canting_Attenuation_Difference_H} but for
    differential attenuation.}
    \label{fig:C_Canting_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Canting_Specific_Differential_Attenuation_scatter}
    \caption{As in Figure~\ref{fig:C_Canting_Specific_Attenuation_H_scatter}
    but for specific differential attenuation.}
    \label{fig:C_Canting_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, Figure~\ref{fig:C_Canting_Differential_Attenuation}
shows an increase in attenuation as well, though this time the increase is for
all algorithms, as well as the truth field. This increase in differential attenuation
was likely concealed by the larger range of values at the individual polarizations.
The differences from truth (
Figure~\ref{fig:C_Canting_Differential_Attenuation_Difference}), show
that this increase in differential attenuation was well captured by all the
algorithms. SC and MSC show only minor changes; Linear and ZPHI show
slight increases in errors, on order of \SI{0.2}{dB}, maintaining their
overall positive bias. It should be noted, though,
that this is not a negligible bias for $Z_{DR}$. The histogram of specific
differential attenuation agrees with this, showing no shifts in the distributions
of points. However, there are appreciable increases in the spread of points away
from the one-to-one ratio line for the Linear and ZPHI algorithms; this
indicates an overall reduction in accuracy for these algorithms. 

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0465 & 0.0068 & 0.9612\\
        ZPHI & 0.0432 & 0.0094 & 0.9136\\
        SC & 0.0082 & 0.0071 & 0.9165\\
        MSC & -0.0009 & 0.0060 & 0.9306\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0243 & 0.0032 & 0.9642\\
        ZPHI & 0.0214 & 0.0041 & 0.9173\\
        SC & -0.0024 & 0.0042 & 0.9037\\
        MSC & -0.0096 & 0.0034 & 0.9254\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0430 & 0.0026 & 0.9035\\
        ZPHI & 0.0304 & 0.0024 & 0.8030\\
        SC & 0.0098 & 0.0013 & 0.8336\\
        MSC & 0.0077 & 0.0010 & 0.8916\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Canting experiment.}
    \label{tab:C-stats-Canting}
\end{table}

\FloatBarrier
\subsection{X band}
The results at X-band mirror those for C-band, following from the
same increase in differential phase due to the decrease in spread of
canting angles. At both horizontal, Figure~\ref{fig:X_Canting_Attenuation_H},
and vertical, Figure~\ref{fig:X_Canting_Attenuation_V}, polarizations, the
Linear and ZPHI algorithms show an increase in the calculated attenuation, while
the SC and MSC algorithms show no perceptible changes; more importantly, the
true attenuation does not change perceptibly. The difference fields in
Figure~\ref{fig:X_Canting_Attenuation_Difference_H} clarify this, showing that
both Linear and ZPHI have a positive bias in their estimates of horizontal
attenuation, with errors on the order of \SI{3}{dB}.
This is a marked change from the Control experiment where both of these
algorithms had results that were closer to being unbiased. There is an especially
large change for the Linear algorithm due north from the radar, where the change
is approximately \SI{5}{dB}. This same pattern is evident for vertical polarization
in Figure~\ref{fig:X_Canting_Attenuation_Difference_V}, though the change for 
the Linear algorithm is more stark given the minimal errors that were present in its
output for the Control experiment. The histograms for specific attenuation
at horizontal, Figure~\ref{fig:X_Canting_Specific_Attenuation_H_scatter}, and
vertical, Figure~\ref{fig:X_Canting_Specific_Attenuation_V_scatter}, polarizations
show a
shift in points upwards for the Linear and ZPHI algorithms, agreeing with the
observed increase in bias. The histograms for the SC and MSC show no
qualitative changes, which is also in agreement with the PPIs of attenuation; the
computed bias values (Table~\ref{tab:X-stats-Canting}) change much more for
the Linear and ZPHI algorithms than for the SC and MSC algorithms.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Canting_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Canting_Attenuation_H} but for
    X-band.}
    \label{fig:X_Canting_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Canting_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Canting_Attenuation_Difference_H} but for
    X-band.}
    \label{fig:X_Canting_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Canting_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_Canting_Attenuation_H} but for
    vertical polarization.}
    \label{fig:X_Canting_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Canting_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_Canting_Attenuation_Difference_H} but for
    vertical polarization.}
    \label{fig:X_Canting_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Canting_Specific_Attenuation_H_scatter}
    \caption{As in Figure~\ref{fig:C_Canting_Specific_Attenuation_H_scatter}
    but for X-band.}
     \label{fig:X_Canting_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Canting_Specific_Attenuation_V_scatter}
    \caption{As in Figure~\ref{fig:X_Canting_Specific_Attenuation_H_scatter}
    but for vertical polarization.}
    \label{fig:X_Canting_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Canting_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_Canting_Attenuation_H} but for
    differential attenuation.}
    \label{fig:X_Canting_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Canting_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_Canting_Attenuation_Difference_H} but for
    differential attenuation.}
    \label{fig:X_Canting_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Canting_Specific_Differential_Attenuation_scatter}
    \caption{As in Figure~\ref{fig:X_Canting_Specific_Attenuation_H_scatter}
    but for specific differential attenuation.}
    \label{fig:X_Canting_Specific_Differential_Attenuation_scatter}
\end{figure}

The results for differential attenuation also agree with those at C-band. While
the actual differential attenuation values in
Figure~\ref{fig:X_Canting_Differential_Attenuation} show moderately larger
value across the board, the difference from the true values in
Figure~\ref{fig:X_Canting_Differential_Attenuation_Difference} show no
appreciable difference from the results for the Control experiment. The results
for ZPHI show some changes in the highest core of attenuation, but this is on the
fringes of the fully-extinguished signal, so values here are likely sensitive
to the signal-to-noise ratio. This change does produce a difference in the
histogram of specific differential attenuation, in
Figure~\ref{fig:X_Canting_Specific_Differential_Attenuation_scatter}. Here
the ZPHI algorithm has noticeably shifted many points from well below the
one-to-one line to above, and the core bands have shifted upwards as well;
this is likely a reflection of moving some of the anomalously low points closer to
their true values. The effect of this change is an increase in the net bias
for the ZPHI algorithm (Table~\ref{tab:X-stats-Canting}); this increase is the
largest of any of the algorithms.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0960 & 0.0862 & 0.9709\\
        ZPHI & 0.0330 & 0.1882 & 0.9200\\
        SC & 0.0185 & 0.2225 & 0.9080\\
        MSC & 0.0096 & 0.1977 & 0.9167\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0716 & 0.0597 & 0.9713\\
        ZPHI & 0.0250 & 0.1284 & 0.9219\\
        SC & 0.0066 & 0.1530 & 0.9091\\
        MSC & -0.0102 & 0.1392 & 0.9166\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0404 & 0.0049 & 0.9564\\
        ZPHI & 0.0182 & 0.0120 & 0.8322\\
        SC & 0.0107 & 0.0129 & 0.8219\\
        MSC & 0.0220 & 0.0129 & 0.8216\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Canting experiment.}
    \label{tab:X-stats-Canting}
\end{table}

\FloatBarrier
\section{Shape}
The Shape experiment modifies the Control experiment by changing the prescribed
shape model (i.e., the relationship between drop axis ratio and size) from the
\cite{Brandes2002} polynomial model to the \cite{Pruppacher1970} model.
Given that both models are empirical fits to data, neither one represents a
guaranteed match to natural processes. Instead, this experiment is designed
to highlight the sensitivity of the algorithms (through the coefficients) to the
chosen drop shape model by examining how performance changes when the
assumption is violated. 

\FloatBarrier
\subsection{C band}
In the Shape experiment, the results for horizontal attenuation in
Figure~\ref{fig:C_Shape_Attenuation_H} for the Linear and ZPHI algorithms
show very pronounced increases from the values in the Control experiment. 
These values are well excess of those shown by the true attenuation field. The 
SC algorithm shows only a minor increase over the results from the Control
experiment, while the MSC algorithm actually shows a decrease. For vertical
polarization, in Figure~\ref{fig:C_Shape_Attenuation_V}, the results are similar.
For the difference from the true values of horizontal attenuation,
Figure~\ref{fig:C_Shape_Attenuation_Difference_H} shows that the errors
for Linear and ZPHI have increased to almost \SI{6}{dB} (from initial errors
around \SI{1}{dB}). For SC, while the errors of individual rays have changed,
the overall error field remains unbiased, with
a slight increase in the amount of negative departure for some rays. The MSC
algorithm now shows a negative bias, with the peak of errors now approaching
\SI{1}{dB}, in contrast with errors very close to \SI{0}{dB} for the Control
experiment. For vertical
polarization (Figure~\ref{fig:C_Shape_Attenuation_Difference_V}), the
results are qualitatively identical, though the magnitudes of the errors are
slightly smaller.

Looking at the histograms of specific attenuation for horizontal and vertical 
polarizations, in
Figures~\ref{fig:C_Shape_Specific_Attenuation_H_scatter} and
\ref{fig:C_Shape_Specific_Attenuation_V_scatter}, respectively, the
pronounced bias for the Linear algorithm is clearly visible. For the ZPHI algorithm,
not only is the increase in bias visible, but the points are much more spread
out. The SC algorithm does not show any significant changes, though there is
a small region of points that now extend above the main core of points; this
was not present in the Control experiment. For the MSC algorithm, the curve
of points now slopes slightly further down, agreeing with the
observation of the increase in negative bias.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H} but for the
    Shape experiment.}
    \label{fig:C_Shape_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_Difference_H} but for
    the Shape experiment.}
    \label{fig:C_Shape_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_Shape_Attenuation_H} but for
    vertical polarization.}
    \label{fig:C_Shape_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_Shape_Attenuation_Difference_H} but for
    vertical polarization.}
    \label{fig:C_Shape_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Specific_Attenuation_H_scatter}
    \caption{As in Figure~\ref{fig:C_Control_Specific_Attenuation_H_scatter},
    but for the Shape experiment.}
    \label{fig:C_Shape_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Specific_Attenuation_V_scatter}
    \caption{As in Figure~\ref{fig:C_Shape_Specific_Attenuation_H_scatter},
    but for vertical polarization.}
    \label{fig:C_Shape_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_Shape_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:C_Shape_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_Shape_Attenuation_Difference_H}, but
    for differential attenuation.}
    \label{fig:C_Shape_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Shape_Specific_Differential_Attenuation_scatter}
    \caption{As in Figure~\ref{fig:C_Shape_Specific_Attenuation_H_scatter}
    but for specific differential attenuation.}
    \label{fig:C_Shape_Specific_Differential_Attenuation_scatter}
\end{figure}

The results at horizontal and vertical polarizations lead to a similar pattern
of results for differential attenuation. The PPIs of the raw data in
Figure~\ref{fig:C_Shape_Differential_Attenuation} show the same large increase
in values for the Linear and ZPHI algorithms, as well as a slight decrease in
differential attenuation calculated by the MSC algorithm. Looking at the
differences from truth in
Figure~\ref{fig:C_Shape_Differential_Attenuation_Difference}, it is clear that
the Linear algorithm has a large area where the bias in differential attenuation
is well over \SI{2}{dB}. For ZPHI, the errors are less but have increased to
\SI{1.5}{dB}. While the SC algorithm's results are mostly unchanged from
the Control experiment, the MSC algorithm now has a large negative bias
around \SI{1}{dB}. The histogram of specific differential attenuation in
Figure~\ref{fig:C_Shape_Specific_Differential_Attenuation_scatter} is
consistent with the PPIs, and matches what was seen for specific attenuation
for the individual polarizations, including the large increase in the spread of
points for the ZPHI algorithm.

Overall, changing the shape model from the \cite{Brandes2002} model to the
\cite{Pruppacher1970} model drastically increased the errors for the algorithms
using fixed coefficients at C-band. Since the \cite{Pruppacher1970} model
yields more oblate drops, this change increases the magnitude of the polarimetric
variables; consequently, this increases the magnitude of the $\Phi_{DP}$
data without an increase in the reflectivity field. Together, this leads the
Linear and ZPHI algorithms to increased estimations of attenuation. The SC
algorithm, as in the Canting experiment, is able to compensate for this through
its optimization procedure. The MSC also benefits from this optimization,
but the changes made to filter the optimized coefficients have introduced
a significant negative bias.

The computed bias and MSE values (Table~\ref{tab:C-stats-Shape}) show that
the SC and MSC algorithms are clearly the best performers, having the lowest
values for both columns. Between the two algorithms, neither demonstrates
consistently better performance
across all polarizations. For $r^2$ values, while the SC and MSC are not able to
produce values higher than those of Linear, the differences are decreased over
those in the Control experiment.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.1191 & 0.0251 & 0.9560\\
        ZPHI & 0.1171 & 0.0379 & 0.8265\\
        SC & 0.0170 & 0.0079 & 0.9083\\
        MSC & -0.0088 & 0.0070 & 0.9279\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0819 & 0.0128 & 0.9621\\
        ZPHI & 0.0803 & 0.0167 & 0.8616\\
        SC & 0.0070 & 0.0040 & 0.9055\\
        MSC & -0.0074 & 0.0034 & 0.9273\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0683 & 0.0056 & 0.8986\\
        ZPHI & 0.0535 & 0.0082 & 0.5224\\
        SC & 0.0087 & 0.0012 & 0.8437\\
        MSC & -0.0082 & 0.0014 & 0.8586\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Shape experiment.}
    \label{tab:C-stats-Shape}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band the results are similar to those at C-band. The PPIs of both horizontal
and vertical attenuation, Figures~\ref{fig:X_Shape_Attenuation_H} and
\ref{fig:X_Shape_Attenuation_V}, respectively, show very large increases in
the estimated attenuation for the Linear and ZPHI algorithms. Again, this
comes from the increased differential phase that results from using the
\cite{Pruppacher1970} shape model. Looking at the differences from
the true values in Figures~\ref{fig:X_Shape_Attenuation_Difference_H}
and \ref{fig:X_Shape_Attenuation_Difference_V}, the errors for the Linear
algorithm have increased to well over \SI{10}{dB}, while ZPHI has errors
peaking at \SI{10}{dB}. The SC and MSC see increases in errors as well,
though these increases are much less drastic; the MSC algorithm increases
to around \SI{3}{dB} (from around \SI{1}{dB}). The SC algorithm's estimate
of attenuation, overall, shows no net increase; however,
there are now some individual areas that have errors up to \SI{5}{dB}. Indeed,
it was this problem with some rays showing larger errors that motivated the
development of MSC. The histograms of specific attenuation values, for
both polarizations in
Figures~\ref{fig:X_Shape_Specific_Attenuation_H_scatter} and
\ref{fig:X_Shape_Specific_Attenuation_V_scatter}, match those at C-band.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Shape_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Shape_Attenuation_H}, but for X-band.}
    \label{fig:X_Shape_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Shape_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Shape_Attenuation_Difference_H}, but
    for X-band.}
    \label{fig:X_Shape_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Shape_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_Shape_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:X_Shape_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Shape_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_Shape_Attenuation_Difference_H}, but
    for vertical polarization.}
    \label{fig:X_Shape_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Shape_Specific_Attenuation_H_scatter}
    \caption{As in Figure~\ref{fig:C_Shape_Specific_Attenuation_H_scatter},
    but for X-band.}
    \label{fig:X_Shape_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Shape_Specific_Attenuation_V_scatter}
    \caption{As in Figure~\ref{fig:X_Shape_Specific_Attenuation_V_scatter},
    but for vertical polarization.}
    \label{fig:X_Shape_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Shape_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_Shape_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:X_Shape_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Shape_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_Shape_Attenuation_Difference_H}, but
    for differential attenuation.}
    \label{fig:X_Shape_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Shape_Specific_Differential_Attenuation_scatter}
    \caption{As in Figure~\ref{fig:X_Shape_Specific_Attenuation_H_scatter},
    but for specific differential attenuation.}
    \label{fig:X_Shape_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, Figure~\ref{fig:X_Shape_Differential_Attenuation}
shows that for the Shape experiment, differential attenuation increases
for all, including the true values. Looking at the difference from truth 
(Figure~\ref{fig:X_Shape_Differential_Attenuation_Difference}), the errors
for Linear and ZPHI increase as well; for Linear the errors are now up to
\SI{2}{dB} (up from \SI{1}{dB}), and ZPHI has errors up to \SI{1.5}{dB}, which
is an increase from around \SI{0.5}{dB} for the Control experiment. This is
discounting the area where extinction of the signal is causing problems.
As at the individual polarizations, SC shows an increase in errors for some
rays, but no systematic increase, while MSC shows an overall increase in its
positive bias, though of much less magnitude than Linear and ZPHI.
Figure~\ref{fig:X_Shape_Specific_Differential_Attenuation_scatter} shows
the histogram of specific differential phase, which matches well those
from horizontal and vertical polarizations.

In terms of bias and MSE, the SC and MSC algorithms stand out from the
other algorithms (Table~\ref{tab:X-stats-Shape}).  Between the two algorithms,
in this experiment, the results from the MSC algorithm appear slightly better,
as they yield a smaller MSE while having comparable biases.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.4642 & 0.4232 & 0.9667\\
        ZPHI & 0.3904 & 0.9875 & 0.7668\\
        SC & 0.0334 & 0.2623 & 0.8933\\
        MSC & 0.0360 & 0.2282 & 0.9066\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.4093 & 0.3313 & 0.9678\\
        ZPHI & 0.3619 & 0.9809 & 0.7046\\
        SC & 0.0232 & 0.1687 & 0.8981\\
        MSC & 0.0090 & 0.1530 & 0.9078\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0841 & 0.0116 & 0.9537\\
        ZPHI & 0.1683 & 0.3639 & 0.3179\\
        SC & 0.0116 & 0.0172 & 0.7968\\
        MSC & 0.0259 & 0.0169 & 0.8008\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Shape experiment.}
    \label{tab:X-stats-Shape}
\end{table}

\FloatBarrier
\section{Temperature}
The Temperature experiment is designed to explore the sensitivity of the
algorithms to changes in temperature; such sensitivity arises from the use
of a fixed temperature when calculating the regression coefficients. In this
experiment, instead of prescribing a fixed temperature for simulating the radar
data, the temperature from the model grid is used.
Figure~\ref{fig:model-temperature} shows the actual field of temperatures
that is used; the field of temperatures is generally around \SI{295}{\kelvin},
though there is some spatial variability. However, this variability is smaller
than the magnitude of the change from the fixed temperature of
\SI{283}{\kelvin} that was used to generate the regression coefficients.

Temperature sensitivity of the relationship between attenuation and
differential phase was previously mentioned by \cite{Bringi1990}, and its impact
on attenuation correction was briefly described by \cite{Carey2000}
and \cite{Aydin1989}.  The aim
here is to quantify this effect within the context of actual algorithms run
on (simulated) data.

\begin{figure}
	\centering
	\includegraphics[scale=0.87]{figures/commas_wz_3600_temperature.pdf}
	\caption{Temperature near the surface of the model grid used to
	simulate radar data.}
	\label{fig:model-temperature}
\end{figure}

\FloatBarrier
\subsection{C band}
For the Temperature experiment at C-band, the PPIs of attenuation for 
horizontal,
Figure~\ref{fig:C_Temperature_Attenuation_H}, and vertical,
Figure~\ref{fig:C_Temperature_Attenuation_V}, show clear decreases
in attenuation in both SC and MSC algorithms, as well as the true attenuation,
that result from the increase in temperature. Subtracting out the true values,
the corresponding differences in
Figures~\ref{fig:C_Temperature_Attenuation_Difference_H} and
\ref{fig:C_Temperature_Attenuation_Difference_V} show that the
decrease in SC and MSC capture well the real change in intrinsic attenuation,
as they show no significant increase in errors; MSC shows a small increase in
its negative bias, on order of \SI{0.1}{dB}, while SC shows some rays that
increase the magnitude of their errors by a similar amount, with no systematic
changes. The attenuation estimates from Linear and ZPHI, however, now
show almost \SI{5}{dB} of error. The histograms of specific attenuation
for both polarizations, in
Figures~\ref{fig:C_Temperature_Specific_Attenuation_H_scatter} and
\ref{fig:C_Temperature_Specific_Attenuation_V_scatter}, show clearly
the biases of Linear and ZPHI, as the bands of most points shift noticeably
upwards above the one-to-one ratio line. For SC and MSC, the change
between the Control and Temperature experiments is to drastically shorten
the maximum range of values, but there is no observable change in terms
of shifts relative to the one-to-one line; there is a clear negative bias at
higher values of specific attenuation, though this is consistent between the
experiments.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H}, but for the
    Temperature experiment.}
    \label{fig:C_Temperature_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_Difference_H}, but
    for the Temperature experiment.}
    \label{fig:C_Temperature_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_Temperature_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:C_Temperature_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_Temperature_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:C_Temperature_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Specific_Attenuation_H_scatter}
    \caption{As in Figure~\ref{fig:C_Control_Specific_Attenuation_H_scatter},
    but for the Temperature experiment.}
    \label{fig:C_Temperature_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_Temperature_Specific_Attenuation_H_scatter}, but for
    vertical polarization.}
    \label{fig:C_Temperature_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_Temperature_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:C_Temperature_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_Temperature_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:C_Temperature_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Temperature_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_Temperature_Specific_Attenuation_H_scatter},
    but for specific differential attenuation.}
    \label{fig:C_Temperature_Specific_Differential_Attenuation_scatter}
\end{figure}

Looking at differential attenuation now, the PPIs for the Temperature
experiment in Figure~\ref{fig:C_Temperature_Differential_Attenuation}
do not show significant changes. The true values exhibit only a small decrease,
which from this figure appears to be matched by all algorithms, with the
exception of a slight increase in the differential attenuation estimated by
the Linear algorithm. Consequently, this causes the Linear algorithm to show
the greatest increase in errors when comparing the differences from truth
in Figure~\ref{fig:C_Temperature_Differential_Attenuation_Difference} to
the results from the Control experiment. The Linear algorithm's error has
increased from around \SI{1.2}{dB} to \SI{1.5}{dB}. The ZPHI algorithm's
errors, in comparison with the Control experiment also show an increase.
Conversely, the SC algorithm shows almost no change in errors, while the MSC
shows only a minor increase in its positive bias by around \SI{0.1}{dB}. In
Figure~\ref{fig:C_Temperature_Specific_Differential_Attenuation_scatter}, the 
histograms of specific differential attenuation largely match these minor changes,
with a few notable exceptions. For the Linear algorithm, the band of points
shifts upwards, giving the positive bias. Conversely, the band for ZPHI develops a negative
bias that increases with increasing values of specific differential attenuation;
this is countered by an increase spread of points above the
one-to-one line, resulting in the net increase in positive bias seen in the
PPIs of integrated attenuation. The SC and MSC algorithms also show the
same increase in negative bias at larger values.

Overall, again the fixed coefficient algorithms at C-band show significant
increases in errors in estimating attenuation for individual polarizations.
The adaptive algorithms, SC and MSC, are able to adjust well for these changes
with only minor increases in errors. It is worth noting that while the
errors for the polarizations greatly increase for Linear and ZPHI, the increase
in errors for differential attenuation is not nearly as great. This is because
the changes in intrinsic horizontal and vertical attenuation were close in
magnitude, resulting in only small changes to the differential attenuation.
In this experiment, the difference between the algorithms is quite pronounced.
The biases for the SC and MSC algorithms  (Table~\ref{tab:C-stats-Temperature})
are almost an order of magnitude less
than those of the Linear and ZPHI algorithms; this is achieved while also having
a smaller MSE.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.1009 & 0.0191 & 0.9554\\
        ZPHI & 0.0968 & 0.0230 & 0.8453\\
        SC & 0.0087 & 0.0062 & 0.8974\\
        MSC & -0.0009 & 0.0050 & 0.9235\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0727 & 0.0111 & 0.9617\\
        ZPHI & 0.0689 & 0.0112 & 0.8652\\
        SC & -0.0022 & 0.0036 & 0.8695\\
        MSC & -0.0135 & 0.0028 & 0.9175\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0540 & 0.0037 & 0.8869\\
        ZPHI & 0.0392 & 0.0043 & 0.6153\\
        SC & 0.0081 & 0.0013 & 0.8099\\
        MSC & 0.0118 & 0.0010 & 0.8850\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Temperature experiment.}
    \label{tab:C-stats-Temperature}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, increasing the temperature increases the intrinsic attenuation
at both polarizations, seen in
Figures~\ref{fig:X_Temperature_Attenuation_H} and 
\ref{fig:X_Temperature_Attenuation_V},
as opposed to decreases as observed at C-band. The corresponding
differences from the intrinsic value, in
Figures~\ref{fig:X_Temperature_Attenuation_Difference_H} and
\ref{fig:X_Temperature_Attenuation_Difference_V}, show a significant
increase in the errors for the Linear algorithm, with pronounced negative
bias around \SI{5}{dB}. All of the ZPHI-based algorithms, conversely, do
not show significant increases in errors. ZPHI and MSC do show increases
in magnitude to about \SI{1}{dB}, but these increases occur in both positive and
negative directions, leaving their respective fields unbiased. These
observations match the histograms for specific attenuation at horizontal
and vertical polarizations, shown in
Figures~\ref{fig:X_Temperature_Specific_Attenuation_H_scatter} and
\ref{fig:X_Temperature_Specific_Attenuation_V_scatter}, respectively.
The Linear algorithm shows an increased negative bias for higher attenuation
values, versus the results from the Control experiment. This agrees with
not only the large negative value for the path through the core of reflectivity,
but also with the fact that the negative bias vanishes for smaller attenuation
values along the edges of the storms. The ZPHI-based algorithms show increases
in the spread of points, but no significant shifts in the bands of points, with
the ZPHI and MSC algorithms showing the greatest spread. The computed values
of bias and MSE (Table~\ref{tab:X-stats-Temperature}) agree with this
observation. Here, the values for the Linear algorithm show the greatest change
from the Control experiment, with the other algorithms' error statistics
remaining much more stable.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Temperature_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Temperature_Attenuation_H}, but for
    X-band.}
    \label{fig:X_Temperature_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Temperature_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Temperature_Attenuation_Difference_H},
    but for X-band.}
    \label{fig:X_Temperature_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Temperature_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_Temperature_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:X_Temperature_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Temperature_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_Temperature_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:X_Temperature_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Temperature_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_Temperature_Specific_Attenuation_H_scatter}, but
    for X-band.}
    \label{fig:X_Temperature_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Temperature_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_Temperature_Specific_Attenuation_H_scatter}, but
    for vertical polarization.}
    \label{fig:X_Temperature_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Temperature_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_Temperature_Attenuation_H}, but
    for differential attenuation.}
    \label{fig:X_Temperature_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Temperature_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_Temperature_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_Temperature_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Temperature_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_Temperature_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:X_Temperature_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the PPIs in
Figure~\ref{fig:X_Temperature_Attenuation_Difference_H} show only small
increase across the board, with the exception of a small decrease for the
Linear algorithm. The differences in
Figure~\ref{fig:X_Temperature_Differential_Attenuation_Difference} show
that this results in an increased negative bias for the Linear algorithm,
similar to what was observed for the individual polarizations. The
ZPHI-based algorithms remain largely unchanged in regards to errors, with
only a small decrease in the errors for the ZPHI algorithm. The histograms
of specific differential phase in
Figure~\ref{fig:X_Temperature_Specific_Differential_Attenuation_scatter}
show that the increase in negative bias, as well as ZPHI's decreased errors,
are a result of a negative bias for larger values of attenuation.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & -0.0239 & 0.1073 & 0.9682\\
        ZPHI & -0.0773 & 0.3614 & 0.8760\\
        SC & 0.0174 & 0.2654 & 0.9053\\
        MSC & 0.0178 & 0.2546 & 0.9081\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & -0.0363 & 0.0770 & 0.9687\\
        ZPHI & -0.0767 & 0.2473 & 0.8821\\
        SC & 0.0064 & 0.1798 & 0.9101\\
        MSC & -0.0043 & 0.1772 & 0.9114\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0215 & 0.0039 & 0.9565\\
        ZPHI & -0.0067 & 0.0166 & 0.7636\\
        SC & 0.0086 & 0.0148 & 0.7909\\
        MSC & 0.0184 & 0.0151 & 0.7834\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Temperature experiment.}
    \label{tab:X-stats-Temperature}
\end{table}

\section{Wavelength}
The Wavelength experiment examines the impact of changing the wavelength
while staying within the same band. While attenuation is clearly a function of
wavelength, evidenced by the definitive changes going from S- to C- to X-band,
within these bands changes in attenuation are often neglected. This experiment
aims to quantify the effect of this assumption by changing the wavelength
from the Control values, \SI{5.5}{\centi\meter} and \SI{3.21}{\centi\meter}
for C- and X-bands, to \SI{5.0}{\centi\meter} and \SI{3.0}{\centi\meter},
respectively. These represent changes of only \SI{10}{\percent}.

\FloatBarrier
\subsection{C band}
In the Wavelength experiment, the PPIs of attenuation at horizontal and
vertical polarization, in
Figures~\ref{fig:C_Wavelength_Attenuation_H} and
\ref{fig:C_Wavelength_Attenuation_V}, respectively, exhibit large increases
in the amount of attenuation. The change in the intrinsic attenuation due to
the decrease in wavelength appears to be captured well by the SC and MSC
algorithms. The Linear and ZPHI algorithms do increase as well, though
the magnitude of their increase is much less than the intrinsic values. This is
reflected in Figures~\ref{fig:C_Wavelength_Attenuation_Difference_H}
and \ref{fig:C_Wavelength_Attenuation_Difference_V}, which show the
differences from truth at horizontal and vertical polarizations, respectively.
For both polarizations, the Linear and ZPHI algorithms show significant
negative biases of around \SI{5}{dB}; this is a large change from the smaller
\SI{1}{dB} positive bias shown in the Control experiment. Conversely, the SC
and MSC algorithms show no significant biases, nor any significant changes
from the Control experiment. The histograms of specific attenuation at these
two polarizations, in
Figures~\ref{fig:C_Wavelength_Specific_Attenuation_H_scatter} and
\ref{fig:C_Wavelength_Specific_Attenuation_V_scatter}, show large shifts
down in the main band(s) of points, compared to the Control experiment
for the Linear and ZPHI algorithms; this is a reflection of the negative bias
in the Wavelength experiment. SC and MSC, show no
perceptible shift, but only an expansion of the set of points out to larger
values, due to the overall increase in attenuation with lesser wavelength.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H}, but for the
    Wavelength experiment.}
    \label{fig:C_Wavelength_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_Difference_H}, but
    for the Wavelength experiment.}
    \label{fig:C_Wavelength_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_Wavelength_Attenuation_H} but for
    vertical polarization.}
    \label{fig:C_Wavelength_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_Wavelength_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:C_Wavelength_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_Control_Specific_Attenuation_H_scatter}, but for
    the Wavelength experiment.}
    \label{fig:C_Wavelength_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_Wavelength_Specific_Attenuation_H_scatter}, but
    for vertical polarization.}
    \label{fig:C_Wavelength_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_Wavelength_Attenuation_H}, but
    for differential attenuation.}
    \label{fig:C_Wavelength_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_Wavelength_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:C_Wavelength_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Wavelength_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_Wavelength_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:C_Wavelength_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the same large increase in attenuation is
observable for all PPIs in
Figure~\ref{fig:C_Wavelength_Differential_Attenuation} in comparison with
the Control experiment. In contrast to the results for the individual
polarizations, here the Linear and ZPHI match much better
the intrinsic results as well as the performance of the SC and MSC algorithms.
The difference from the true values, shown in
Figure~\ref{fig:C_Wavelength_Differential_Attenuation_Difference},
show that with the exception of SC, all of the algorithms exhibit a pattern
of a small positive bias along the edges of the reflectivity structure (smaller
intrinsic differential attenuation), with a small negative bias in the middle
of the reflectivity (larger intrinsic differential attenuation). For MSC, this
represents an increase in the negative bias by about \SI{0.5}{dB}. For
the Linear and ZPHI algorithms this is a much greater change; before these
algorithms had positive biases of approximately \SI{1}{dB}, but now show
negative biases. The SC algorithm shows no change in its overall bias, but
individual rays show increases in errors in both directions. The
histograms of specific differential attenuation,
Figures~\ref{fig:C_Wavelength_Specific_Differential_Attenuation_scatter}
reflect these changes in bias. The ZPHI and Linear algorithms show
large downward shifts in the main bands of points; however, since the
results for the Control experiment had a pronounced positive bias, the
downward shift brings the results closer to the one-to-one ratio line, at least
for lower attenuation values. The plot for the Linear algorithm shows a
significant downward tilt at higher values of specific differential attenuation,
which results in the negative bias in the core of reflectivity. For ZPHI,
there is no noticeable curve to the points, but rather there is an overall
increase in the number of points
below the one-to-one line. The SC and MSC algorithms show no appreciable
shift, but only an expansion to larger values.

The stark
contrast in performance between the SC and MSC algorithms is likely
a reflection of the limits of how attenuation is modelled in the ZPHI
algorithm. In the SC approach, when the $\gamma$ coefficient can vary
on a per ray basis, the algorithm can compensate for the change between
the high and low attenuation regimes better; this comes at the cost of
more radial-to-radial variation. In the MSC approach, having a fixed coefficient
across the field limits the ability of the algorithm to adjust between regimes,
and instead the results show regions with both positive and negative biases.

Overall, the change in wavelength within C-band has a large impact on the
results. Again, the SC and MSC algorithms' optimization procedure
largely compensates for this effect, yielding lower biases and MSE values
(Table~\ref{tab:C-stats-Wavelength}). Linear and ZPHI, on the other hand
show large changes, with especially large errors at calculating the attenuation
for individual polarizations, which are reflected in the much greater biases for
specific attenuation.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & -0.0629 & 0.0171 & 0.9596\\
        ZPHI & -0.0641 & 0.0259 & 0.9002\\
        SC & 0.0076 & 0.0123 & 0.9304\\
        MSC & 0.0013 & 0.0117 & 0.9351\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & -0.0704 & 0.0125 & 0.9627\\
        ZPHI & -0.0701 & 0.0191 & 0.8922\\
        SC & -0.0020 & 0.0074 & 0.9244\\
        MSC & -0.0063 & 0.0069 & 0.9318\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0124 & 0.0015 & 0.9129\\
        ZPHI & 0.0010 & 0.0019 & 0.8474\\
        SC & 0.0099 & 0.0018 & 0.8652\\
        MSC & 0.0068 & 0.0014 & 0.8895\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Wavelength experiment.}
    \label{tab:C-stats-Wavelength}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, the results of changing the wavelength are similar to those
observed at C-band. The PPIs of horizontal attenuation in
Figure~\ref{fig:X_Wavelength_Attenuation_H} show a large increase
in the intrinsic attenuation. The algorithms show some changes from the
results in the Control experiment, but in these images it is difficult to tell the
exact nature due to the area of completely extinguished signal. Results at
vertical polarization, Figure~\ref{fig:X_Wavelength_Attenuation_V}, are
similar. The corresponding PPIs of the difference from truth, shown in
Figures~\ref{fig:X_Wavelength_Attenuation_Difference_H} and
\ref{fig:X_Wavelength_Attenuation_Difference_V}, show large changes for
the Linear and ZPHI algorithms and minimal changes in the errors for the SC
and MSC algorithms. The Linear algorithm fares very badly in changing the
wavelength at X-band, with errors approaching \SI{8}{dB} at both
polarizations before the
signal completely extinguishes; this is an exceedingly large change in performance
given that this algorithm shows the least errors (nearly \SI{0}{dB}) of all
the algorithms in the Control experiment. ZPHI also shows a large increase in
errors, with errors approaching \SI{5}{dB} (discounting the fringes around the
areas of extinguished signal). SC and MSC show no significant changes
in the errors in these same areas, in comparison with the Control experiment.
The histograms for specific attenuation (in
Figures~\ref{fig:X_Wavelength_Specific_Attenuation_H_scatter} and
\ref{fig:X_Wavelength_Specific_Attenuation_V_scatter} for horizontal
and vertical polarizations, respectively), show that for the Linear algorithm,
the band of points has rotated to the right of the one-to-one ratio line.
ZPHI develops a large area of points below the one-to-one line in
comparison with the Control experiment results. These effects cause both the
bias and MSE for the algorithm (Table~\ref{tab:X-stats-Wavelength}) to increase
greatly from their respective values in the results for the Control experiment;
these results
are also in agreement with the observed changes in the PPIs of attenuation. For 
the SC and MSC algorithms,
there are no significant changes other than the expanded range due to larger
values of attenuation; there are no changes relative to the one-to-one line.
There are, however, modest changes to the computed bias and MSE values
for the algorithms (Table~\ref{tab:X-stats-Wavelength}).

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Wavelength_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Wavelength_Attenuation_H}, but for
    X-band.}
    \label{fig:X_Wavelength_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Wavelength_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Wavelength_Attenuation_Difference_H},
    but for X-band.}
    \label{fig:X_Wavelength_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Wavelength_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_Wavelength_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:X_Wavelength_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Wavelength_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_Wavelength_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:X_Wavelength_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Wavelength_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_Wavelength_Specific_Attenuation_H_scatter}, but for
    X-band.}
    \label{fig:X_Wavelength_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Wavelength_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_Wavelength_Specific_Attenuation_H_scatter} but for
    vertical polarization.}
    \label{fig:X_Wavelength_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Wavelength_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_Wavelength_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:X_Wavelength_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Wavelength_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_Wavelength_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_Wavelength_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Wavelength_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_Wavelength_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:X_Wavelength_Specific_Differential_Attenuation_scatter}
\end{figure}

In comparison with the results at the individual polarizations, the results
for differential attenuation are much less drastic. The PPI of the intrinsic
attenuation in Figure~\ref{fig:X_Wavelength_Differential_Attenuation}
shows that the Wavelength experiment has slightly increased intrinsic
differential attenuation in comparison with the Control experiment. The
differences from the true values, in
Figure~\ref{fig:C_Wavelength_Differential_Attenuation_Difference}, show
that the only algorithm that experiences any significant change from the
Control experiment is the Linear algorithm. The Linear algorithm, as in the Wavelength
experiment at C-band, exhibits a decrease in  errors in estimating differential
attenuation in comparison with results for the
Control experiment, with peak errors decreasing from \SI{1.0}{dB} to
\SI{0.5}{dB}. The histogram of specific differential attenuation
(Figure~\ref{fig:X_Wavelength_Specific_Differential_Attenuation_scatter})
shows that this decrease is the result from a small shift in the band of points,
such that more of the band lies along the one-to-one ratio line, with a small
negative bias at the highest values of specific differential attenuation.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & -0.1385 & 0.1323 & 0.9726\\
        ZPHI & -0.1923 & 0.5343 & 0.8485\\
        SC & 0.0003 & 0.2619 & 0.9178\\
        MSC & -0.0077 & 0.2398 & 0.9233\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & -0.1521 & 0.1064 & 0.9727\\
        ZPHI & -0.1931 & 0.4143 & 0.8389\\
        SC & -0.0081 & 0.1994 & 0.9123\\
        MSC & -0.0267 & 0.1900 & 0.9161\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0220 & 0.0040 & 0.9585\\
        ZPHI & -0.0168 & 0.0211 & 0.7163\\
        SC & 0.0009 & 0.0167 & 0.7992\\
        MSC & 0.0109 & 0.0163 & 0.7970\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Wavelength experiment.}
    \label{tab:X-stats-Wavelength}
\end{table}

\section{Combined}
Based on the all of the previous experiments, the Combined experiment
examines a kind of ``worst-case'' scenario; whereas the previous experiments
systematically probe the effects of violating individual assumptions, this
experiment simultaneously violates all of the assumptions. By combining
all of the effects, the experiment seeks to identify how errors may increase
further with more incorrect assumptions. In considering this a ``worst-case'',
this implicitly assumes that the errors stack together; as will be
shown, it is just as possible that errors are of opposite sign and
cancel.

\FloatBarrier
\subsection{C band}
In the Combined experiment, the base field of horizontal attenuation,
in Figure~\ref{fig:C_Combined_Attenuation_H}, shows large increases
across the board. Visually, the increase in intrinsic attenuation seems to be
best matched by the SC algorithm. The Linear and ZPHI algorithms show large
increases, much higher than the intrinsic values, while the MSC algorithm
does not appear to increase enough. These results are mirrored at vertical
polarization, shown in Figure~\ref{fig:C_Combined_Attenuation_V}. These
observations are confirmed in the differences from the intrinsic values for
horizontal and vertical polarizations, shown in
Figures~\ref{fig:C_Combined_Attenuation_Difference_H} and
\ref{fig:C_Combined_Attenuation_Difference_V}, respectively. Linear shows
the greatest amount of error, around \SI{5}{dB}. ZPHI follows
closely with errors of the same magnitude but showing less total area with
that magnitude. The results from MSC, unlike the others, show the development
of a pronounced negative bias, around \SI{2}{dB} at both polarizations, in the
core of the reflectivity structure. Since SC does not show any such bias, this
is likely a result of a fixed coefficient across the domain being unable to
match well the intrinsic relationships between radar observables and
attenuation. The histograms of specific attenuation for both polarizations,
in Figures~\ref{fig:C_Combined_Specific_Attenuation_H_scatter} and
\ref{fig:C_Combined_Specific_Attenuation_V_scatter}, show the upward shift
of the points for both Linear and ZPHI, which causes the observed positive
bias. These histograms also show a slight difference between the SC and MSC algorithms;
while the histogram of points for SC is balanced around the one-to-one
ratio line, the MSC algorithm has more points below, as well as a band of
points curving away from this line at higher values of specific attenuation.
All of these effects contribute to the observed negative bias for MSC. This
behavior is not evident, however, in the bias and MSE values for the experiment
(Table~\ref{tab:C-stats-Combined}), which are not significantly changed from
the values for MSC in the Control experiment. The bias values for the
ZPHI and Linear algorithms are much larger.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_H}, but for the
    Combined experiment.}
    \label{fig:C_Combined_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Control_Attenuation_Difference_H},
    but for the Combined experiment.}
    \label{fig:C_Combined_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_Combined_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:C_Combined_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_Combined_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:C_Combined_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_Control_Specific_Attenuation_H_scatter}, but for the
    Combined experiment.}
    \label{fig:C_Combined_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_Combined_Specific_Attenuation_H_scatter}, but for
    vertical polarization.}
    \label{fig:C_Combined_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_Combined_Attenuation_H} but for
    differential attenuation.}
    \label{fig:C_Combined_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_Combined_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:C_Combined_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/C_Combined_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_Combined_Specific_Attenuation_H_scatter}, but for
    specific differential attenuation.}
    \label{fig:C_Combined_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the PPIs across the board show an increase in
magnitude in Figure~\ref{fig:C_Combined_Differential_Attenuation}. The
relative size of the increase matches those from the individual polarizations,
with SC visually matching the intrinsic values best, Linear and ZPHI being too high,
and MSC being not high enough. The differences from truth in
Figure~\ref{fig:C_Combined_Differential_Attenuation_Difference} quantify
these errors. Here, the MSC algorithm shows the biggest errors as well
as the biggest change; the pronounced negative bias of approximately
\SI{1.3}{dB} represents a large change from the almost \SI{0}{dB} error
observed in the same region in the Control experiment. Again, this is different
from the SC algorithm, where the only change is slight increases to the
magnitude of errors in individual rays but no change in overall bias. The Linear
algorithm also shows an increase in its positive bias, changing from
approximately \SI{1}{dB} in the Control experiment to \SI{1.5}{dB} here.
The ZPHI shows a two-fold change in its behavior. Along the edges of the
reflectivity structure, where attenuation is less, the positive bias
increases from around \SI{0.5}{dB} to approximately \SI{1.0}{dB}. In
the core of the storm, the bias switches from slightly positive
to slightly negative. This behavior is reflected in the histogram of
specific differential attenuation shown in
Figure~\ref{fig:C_Combined_Specific_Differential_Attenuation_scatter}.
In comparison with the same histogram for the Control experiment, the ZPHI
algorithm here shows a much larger region of points above the one-to-one
line for small attenuation values, while an increase in negative bias for
larger values. For the Linear algorithm, the histogram shows the shift
in points corresponding to the increased positive bias. Conversely,
the MSC algorithm shows an increase in the area of points below the line,
a change not observed in the plot for SC; this corresponds to the increasing
negative bias for MSC in the Combined experiment.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.1335 & 0.0302 & 0.9418\\
        ZPHI & 0.1311 & 0.0478 & 0.8375\\
        SC & 0.0221 & 0.0132 & 0.9128\\
        MSC & -0.0104 & 0.0143 & 0.9202\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.1050 & 0.0187 & 0.9512\\
        ZPHI & 0.1027 & 0.0259 & 0.8402\\
        SC & 0.0107 & 0.0063 & 0.9085\\
        MSC & -0.0093 & 0.0065 & 0.9221\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0512 & 0.0044 & 0.8924\\
        ZPHI & 0.0375 & 0.0063 & 0.7293\\
        SC & 0.0119 & 0.0025 & 0.8605\\
        MSC & -0.0088 & 0.0029 & 0.8465\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Combined experiment.}
    \label{tab:C-stats-Combined}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, the Combined experiment shows, in
Figures~\ref{fig:X_Combined_Attenuation_H} and
\ref{fig:X_Combined_Attenuation_V}, a pronounced increase in attenuation
over the Control results for both polarizations. For the Linear algorithm, the
differences from true attenuation for the Combined experiment
(Figures~\ref{fig:X_Combined_Attenuation_Difference_H} and
\ref{fig:X_Combined_Attenuation_Difference_V} for horizontal and vertical
polarizations, respectively) exhibit a very large increase; here the errors have
increased to a peak magnitude of \SI{10}{dB}, up from around \SI{1}{dB} for
the Control experiment. The ZPHI algorithm results also show a large, although
lesser, increase, with errors now approaching almost \SI{7.5}{dB}. The SC
and MSC algorithms show smaller increases, with errors near \SI{0}{dB} for
the Control experiment now approaching \SI{2}{dB}. These large increases
in error are reflected in the histograms of specific attenuation for both
polarizations in Figures~\ref{fig:X_Combined_Specific_Attenuation_H_scatter}
and \ref{fig:X_Combined_Specific_Attenuation_V_scatter}; here the
results for Linear and ZPHI both show large shifts in points upwards above
the one-to-one ratio line, consistent with the large positive biases
exhibited. The computed bias values for these two algorithms in this experiment 
are over ten times as large as those for the Control experiment
(Table~\ref{tab:X-stats-Combined}).
Another feature of these histograms is a bulge in some points
upwards in the results for the SC algorithm. These are a reflection of
degraded results for some of the rays. MSC, through its use of smoothing
the optimized coefficients, is able to avoid such rays.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Combined_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_Combined_Attenuation_H}, but for
    X-band.}
    \label{fig:X_Combined_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Combined_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_Combined_Attenuation_Difference_H},
    but for X-band.}
    \label{fig:X_Combined_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/X_Combined_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_Combined_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:X_Combined_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Combined_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_Combined_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:X_Combined_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Combined_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_Combined_Specific_Attenuation_H_scatter}, but for
    X-band.}
    \label{fig:X_Combined_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Combined_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_Combined_Specific_Attenuation_H_scatter}, but for
    vertical polarization.}
    \label{fig:X_Combined_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Combined_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_Combined_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:X_Combined_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Combined_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_Combined_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_Combined_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/X_Combined_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_Combined_Specific_Attenuation_H_scatter}, but for
    specific differential attenuation.}
    \label{fig:X_Combined_Specific_Differential_Attenuation_scatter}
\end{figure}

Figure~\ref{fig:X_Combined_Differential_Attenuation} shows that the
results for differential attenuation are similar, with an increase in differential
attenuation across the board, both intrinsic and calculated. The PPIs of the
differences from the true values, in
Figure~\ref{fig:X_Combined_Differential_Attenuation_Difference}, show that SC,
having the lowest overall errors, does the best job of capturing these changes.
The MSC and ZPHI algorithms are close, but they show increases in errors along 
the edges of the storms; here the errors have increased to over \SI{1.2}{dB},
up from less than \SI{1}{dB} in the Control experiment. Linear shows errors
up to \SI{1.5}{dB}, though it should be noted that the other algorithms do not produce
usable data in these regions. The histograms of specific differential attenuation,
given in
Figure~\ref{fig:X_Combined_Specific_Differential_Attenuation_scatter},
show pronounced shifts in the bands for the Linear and ZPHI algorithms, which
correspond to the increases in positive bias of the estimates. The SC and
MSC algorithms do not show such pronounced shifts, though they do appear
to have increased numbers of points above the one-to-one line.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.3420 & 0.2691 & 0.9653\\
        ZPHI & 0.2355 & 0.4268 & 0.8988\\
        SC & -0.0041 & 0.4510 & 0.8867\\
        MSC & -0.0144 & 0.4748 & 0.8811\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.3115 & 0.2088 & 0.9659\\
        ZPHI & 0.2453 & 0.3956 & 0.8777\\
        SC & 0.0033 & 0.2802 & 0.8933\\
        MSC & -0.0330 & 0.3185 & 0.8815\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0499 & 0.0082 & 0.9565\\
        ZPHI & 0.0419 & 0.0287 & 0.7810\\
        SC & -0.0011 & 0.0298 & 0.7549\\
        MSC & 0.0081 & 0.0296 & 0.7458\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Combined experiment.}
    \label{tab:X-stats-Combined}
\end{table}

\section{Conclusion}
This chapter examines the results of running several attenuation correction
techniques, Linear, ZPHI, Self-Consistent, and Modified Self-Consistent
using simulated radar data. The focus here is on the impacts of the assumptions
made in developing these techniques and how the violation of these
assumptions impacts algorithm performance. The baseline of performance is
established using the Control experiment, where all assumptions made in
obtaining the coefficients used in the algorithms are matched in the
simulation procedure. At C-Band, Self-Consistent and Modified Self-Consistent
prove most successful, thought Linear and ZPHI algorithms yielded useful
results provided small biases are acceptable for a given application. At X-band,
the Linear algorithm actually shows the least error, though the rest of the
algorithms produced results that were only slightly worse. Overall, the
algorithms all worked quite well in the Control experiment.

In the other experiments, when these assumptions were systematically
violated, the Linear and ZPHI techniques, because they rely upon fixed
coefficients, yielded much higher errors than the Self-Consistent and
Modified Self-Consistent algorithms, at both X- and C-band. In almost all cases
these latter algorithms were able to deliver reasonable results. While the
biases for individual polarizations from these algorithms could still be in the
range of 
\SI{1}{dB} to \SI{2}{dB}, this was much more usable than the errors for the
Linear and ZPHI algorithms in most cases.

Differential attenuation proved challenging for the algorithms in most cases,
which is attributed to a collection of causes. Primarily, the challenge for
correcting differential attenuation is that $Z_{DR}$ is smaller, and thus its
acceptable bias is an order of magnitude less than that for horizontal
(or vertical) reflectivity. The other problem is that while the algorithms
demonstrate very good results for calculating attenuation for individual
polarizations, using the difference between the results exacerbates any errors.

In terms of sensitivity to assumptions, changing any one of them (canting,
shape model, temperature, or wavelength), yielded pronounced changes, both
in terms of the resulting attenuation field and in the results of the algorithms.
This implies that successful use of these techniques depends upon carefully
making assumptions that match nature. For physical processes
(i.e., canting and shape model), this implies the need for good data and
parameterizations of the microphysical processes involved. For something
like temperature, which varies greatly, this implies a greater challenge.
For fixed coefficient techniques, though, the biases introduced are significant, so making
use of any temperature data available could yield significant gains in reducing
the bias in reflectivity due to under- or over-correction for attenuation.

Most
important of all effects, was the sensitivity to wavelength within the
band. Changing the wavelength by only \SI{10}{\percent} introduced at least
\SI{5}{dB} of error in the attenuation estimates by the Linear and ZPHI
algorithms; this runs counter to the common wisdom that one can use
coefficients for these algorithms that are appropriate to a given band. It is
possible, then, to gain algorithm performance by simply tuning
coefficients to the true operating wavelength of a given radar system. While
the Self-Consistent algorithms were able to largely compensate for all of the
violated assumptions, including wavelength, the best algorithm performance
occurred for the Control experiment. Therefore, It is important to all algorithms
to take care in making accurate assumptions; using the true wavelength of the
radar, rather than some general waveband, is one of the simplest ways to
eliminate errors.
\chapter{Sampling Errors in Attenuation Correction}
Whereas the previous chapter focuses on the assumptions underlying the
physical model used for scattering calculations, here the focus is on the impact
of radar operating parameters on attenuation correction, specifically those
parameters that impact the spatial sampling of the data. This includes hardware
effects, such as the beamwidth or antenna sidelobes, as well as the data spacing.

One source of error arising from spatial sampling is integration within the
resolution volume. The regressions underlying the attenuation correction
coefficients fit radar observables calculated from a single drop size
distribution; however, radar resolution volumes can contain scatterers
spread across many different distributions. How integration of these scattering
processes transforms into the actual measured radar data has profound implications 
for the performance of attenuation correction techniques. Another potential
impact of spatial sampling is non-uniform beam-filling (NBF). NBF specifically
refers to the idea that gradients in precipitation amount and/or type
across the radar beam can cause artifacts in data, especially $\Phi_{DP}$.
Since $\Phi_{DP}$ is a critical variable in the process of correcting for
attenuation, NBF poses a significant problem for robust attenuation correction.

As in the previous chapter, the impacts of spatial sampling are tested here with
simulated data created using the previously described simulation procedure.
Sets of data are generated, for both C- and X-band, using radar and scanning
configurations that explore different aspects of radar spatial sampling. The
results of the different attenuation correction algorithms (Linear, ZPHI, SC,
and MSC) are compared to the truth field, calculated using the same
spatial sampling, to see how algorithm performance changes.

The base radar and simulation configuration is given in
Table~\ref{tab:spatial-common-params}. The parameters here match
closely those used previously, with the exception that the antenna is
rotating more quickly at \SI{20}{\degree\per\second}; this value was chosen
to more naturally achieve the larger radial spacing used here without modifying
the PRT. The parameters of the scattering model (canting width, wavelength,
temperature, and shape model) are selected to match those used to generate
the regression coefficients used in the correction algorithms, as was done
in the Control experiment in the modeling chapter. This gives a best case
scenario for algorithm performance and helps to ensure that any resulting
errors are attributable to spatial sampling.

\begin{table}
    \centering
    \begin{tabular}{ | l | l | }
        \hline
        Antenna gain & \SI{45.5}{dB} \\
        Peak power & \SI{250}{\kilo\watt} \\
        First range gate & \SI{500}{\meter} \\
        Noise power & \SI{-113}{dBm} \\
        Elevation & \SI{0.5}{\degree} \\
        PRT & \SI{0.667}{\milli\second} \\
        Canting Width & \num{10} \\
        Wavelength & \SI{5.5}{\centi\meter}, \SI{3.21}{\centi\meter} \\
        Temperature & \SI{283}{\kelvin} \\
        Shape Model & Brandes \\
        Rotation Rate & \SI{20}{\degree\per\second} \\
        \hline
    \end{tabular}
    \caption{Radar and scanning parameters common to the simulations}
    \label{tab:spatial-common-params}
\end{table}

\begin{table}
    \centering
    \begin{tabular}{ | c | c | c | c | c | }
        \hline
        Experiment & Radial Spacing & Beamwidth & Antenna Limits & Gate Width \\
        \hline
        Control & \SI{1}{\degree} & \SI{1}{\degree} & Main-lobe only & \SI{125}{\meter} \\
        Sidelobe & \SI{1}{\degree} & \SI{1}{\degree} & \num{1} Side-lobe & \SI{125}{\meter} \\
        Beamwidth & \SI{1}{\degree} & \SI{2}{\degree} & Main-lobe only & \SI{125}{\meter} \\
        Radial Width & \SI{2}{\degree} & \SI{2}{\degree} & Main-lobe only & \SI{125}{\meter} \\
        Range Resolution & \SI{1}{\degree} & \SI{1}{\degree} & Main-lobe only & \SI{250}{\meter} \\
        \hline
    \end{tabular}
    \caption{Parameters differing between different experiments}
    \label{tab:spatial-exp-params}
\end{table}

The individual experiments are listed in
Table~\ref{tab:spatial-exp-params}. The Control experiment here
establishes baseline performance with a common radar configuration,
\SI{1}{\degree} beamwidth with matched radial sampling. The other
experiments systematically modify this configuration to explore the effects
of sidelobes, increased beamwidth, radial sampling interval, and range
resolution. Radial spacing is controlled by modifying the number of pulses
used to create a radial of data; \num{75} and \num{150} pulses are used to
create \SI{1}{\degree} and \SI{2}{\degree} radials, respectively. Also,
pulse width is matched to gate width, such that a \SI{0.8}{\micro\second}
pulse is used with the \SI{125}{\meter} gate spacing and a
\SI{1.6}{\micro\second} pulse is used with the \SI{250}{\meter} gate spacing.

\section{Control}
The Control experiment establishes the reference point for the other
experiments, with a scanning strategy that yields \SI{1.0}{\degree} radial
spacing, matched to the \SI{3}{dB} beamwidth. This differs
from the Control experiment in the previous chapter by using both larger
gates (\SI{125}{\meter} versus \SI{100}{\meter}), as well as larger
beamwidth and radial spacing (\SI{1.0}{\degree} versus \SI{0.25}{\degree}).
These parameters are selected to form a baseline around a more reasonable
(in comparison with standard real-world operations) spatial 
sampling than that of the modeling study, which is configured specifically to 
minimize the effects of spatial sampling.

\FloatBarrier
\subsection{C band}
At C-band, the PPIs of horizontal attenuation for the Control experiment,
shown in Figure~\ref{fig:C_sp_Control_Attenuation_H}, show that the Linear
and ZPHI algorithms produce the highest estimates for horizontal attenuation,
higher than the intrinsic values. The SC and MSC algorithms are much closer
to the intrinsic values, though they each are slightly higher and lower,
respectively, than the intrinsic values. When the intrinsic attenuation values
are subtracted from the algorithm results, in
Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H}, the positive
bias for the Linear and ZPHI algorithms exhibits maxima in the core
of the reflectivity, with a value around \SI{2}{dB}. The SC algorithm overall
seems to be unbiased, but individual rays appear to have errors of
approximately \SI{0.5}{dB} of either sign. The MSC algorithm, on the
other hand, does well to smooth out these errors and produces a
field that seems to be unbiased and has only small errors.
Histograms of horizontal specific attenuation, in
Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, show how
these errors stack up on a gate-by-gate basis. Both the ZPHI and Linear
algorithms show that the bands of the highest numbers of points are slightly
above the one-to-one line, which causes the observed positive bias. Conversely,
both the SC and MSC algorithms have their bands of the
greatest numbers of points right along the one-to-one line. The results for
MSC also show a tighter clustering of points, which is why the algorithm
exhibits less errors than the SC algorithm.

At vertical polarization, the results are similar. The PPIs of vertical attenuation
in Figure~\ref{fig:C_sp_Control_Attenuation_V} show that Linear and ZPHI
again have the highest values, though visually they appear much closer to
the intrinsic values. The MSC and SC algorithms are, at least visually, difficult
to distinguish from the intrinsic results. The differences from the intrinsic
values, shown in Figure~\ref{fig:C_sp_Combined_Attenuation_Difference_V},
show that indeed the differences for Linear and ZPHI are less; only a positive
bias of up to \SI{1}{dB} is observed here. SC, on the other hand, appears to
have developed a slight negative bias; while errors still change from radial to
radial, the magnitude of negative errors (approaching \SI{1}{dB}) are greater
than the positive errors (less than \SI{0.5}{dB}). MSC, again, smooths out
these errors and produces a field with no apparent bias and small errors.
The histograms of vertical specific attenuation, in
Figure~\ref{fig:C_sp_Control_Specific_Attenuation_V_scatter},
explain the sources for these behaviors. For the Linear and ZPHI algorithms,
the bands of most points sits just above the one-to-one line, yielding a small
positive bias. For the SC algorithm, while the main band of points is along
the one-to-one line, there are still significant numbers of points below this
line, yielding the small negative bias. The MSC algorithm, on the other hand,
has a tighter grouping of points and thus does not have the significant
area of points below the line. This observed behavior yields the best bias
and MSE values (Table~\ref{tab:C-stats-Control-sp}) while still giving an
$r^2$ value that is comparable to that of the Linear algorithm.

For the PPIs of results, at both polarizations neither the raw results nor the
differences from the intrinsic values display any significant deviation from
the Control experiment in the modeling study. While the change in resolution
is clearly visible, the structure of the errors is unchanged between the
two experiments. However, the histograms of specific attenuation do
show an interesting change between the experiments. In the modeling
study, which had a higher resolution, the results for all of the ZPHI-based
algorithms showed a pronounced structure with two bands along which most
points lay; one band was situated above the one-to-one line and one below. In 
this Control experiment, with \SI{1.0}{degree} radial spacing, the two bands
have collapsed to a single band that follows the one-to-one line. Also, all
algorithms show tighter clustering of points in the lower-resolution data. This
effect extends to the computed $r^2$ values, which have increased for all of the
ZPHI algorithms versus the values for the modeling study.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Attenuation_H}
    \caption{Plan Position Indicators (PPIs) of attenuation for horizontal
    attenuation at C-band for the Control experiment from various sources:
    (a) True
    field calculated from model (b) Linear algorithm (c) ZPHI algorithm
    (d) Self-Consistent algorithm (e) Modified Self-Consistent algorithm.
    Range rings are plotted every \SI{10}{\kilo\meter} from the radar.}
    \label{fig:C_sp_Control_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Attenuation_Difference_H}
    \caption{Plan Position Indicators (PPIs) for the Control experiment of the
    difference between the
    true horizontal attenuation values (from the model) at C-band
    and those calculated by algorithms: (a) Linear algorithm (b) ZPHI algorithm
    (c) Self-Consistent algorithm (d) Modified Self-Consistent algorithm.
    Range rings are plotted every \SI{10}{\kilo\meter} from the radar.}
    \label{fig:C_sp_Control_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:C_sp_Control_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:C_sp_Control_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Specific_Attenuation_H_scatter}
    \caption{2D histograms for the C-band Control experiment of true specific
    horizontal attenuation (x-axis) and calculated specific horizontal
    attenuation (y-axis) for: (a) Linear algorithm (b) ZPHI algorithm
    (c) Self-Consistent algorithm (d) Modified Self-Consistent algorithm.}
    \label{fig:C_sp_Control_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, but for
    vertical polarization.}
    \label{fig:C_sp_Control_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:C_sp_Control_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:C_sp_Control_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Control_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, but for
    specific differential attenuation.}
    \label{fig:C_sp_Control_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the PPIs of the base values, in
Figure~\ref{fig:C_sp_Combined_Differential_Attenuation}, show that both
Linear and ZPHI have values that are much larger, while SC and MSC appear to
be closer, but too large and too small, respectively.
The PPIs of the differences from intrinsic values, in
Figure~\ref{fig:C_sp_Control_Differential_Attenuation}, show that the Linear
algorithm has the most error, with a bias approaching \SI{1.25}{dB}. ZPHI
has a slightly smaller positive bias, only approaching \SI{1.0}{dB}. SC
shows errors fluctuating significantly from radial to radial,
with errors of \SI{0.5}{dB} in either direction. While the MSC algorithm smooths
out this behavior, this
smoothing yields a significant negative bias in the middle of the field, with
a magnitude approaching \SI{0.5}{dB}; this is a change from the behavior for
the individual polarizations. Histograms of the specific
differential attenuation values
(Figure~\ref{fig:C_sp_Control_Specific_Differential_Attenuation_scatter})
show that both Linear and ZPHI's points are
shifted above the one-to-one line, causing the observed positive biases.
The core band of points for the SC algorithm lies along the
one-to-one line, though having significant spread. The MSC algorithm,
conversely, has tilted away from the one-to-one line at higher
attenuation values, causing the negative bias in the core of the reflectivity
structure. This agrees with the computed bias values in
Table~\ref{tab:C-stats-Control-sp}, which show that SC and MSC have the lowest
values, as well as the lowest values for MSE. For the first time, the MSC
has the best values of $r^2$; the spatial smoothing removes artifacts that
had caused MSC's variance to increase, and hence decrease the $r^2$ value.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0314 & 0.0030 & 0.9789\\
        ZPHI & 0.0286 & 0.0036 & 0.9629\\
        SC & 0.0059 & 0.0035 & 0.9573\\
        MSC & -0.0025 & 0.0026 & 0.9705\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0101 & 0.0011 & 0.9820\\
        ZPHI & 0.0079 & 0.0014 & 0.9665\\
        SC & -0.0031 & 0.0021 & 0.9493\\
        MSC & -0.0055 & 0.0014 & 0.9686\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0407 & 0.0020 & 0.9380\\
        ZPHI & 0.0285 & 0.0014 & 0.8904\\
        SC & 0.0079 & 0.0007 & 0.8911\\
        MSC & -0.0014 & 0.0005 & 0.9453\\
\hline
    \end{tabular}
    \caption{Bias, mean squared-error, and $r^2$ for the specific attenuation
    results for the Control experiment at C-band.}
    \label{tab:C-stats-Control-sp}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, due to the extinguished signal in the core of the reflectivity, it is
more difficult to evaluate the performance of the algorithms using the
PPIs of horizontal attenuation in Figure~\ref{fig:X_sp_Control_Attenuation_H}.
The differences from the intrinsic values shown in
Figure~\ref{fig:X_sp_Control_Attenuation_Difference_H} show that all of
the algorithms have similar performance, with positive biases (ignoring the
regions of low-signal for the ZPHI algorithms) around \SI{2}{dB}.
ZPHI appears to be slightly better than the rest, while MSC slightly worse.
The histograms of specific attenuation, in
Figure~\ref{fig:X_sp_Control_Specific_Attenuation_H_scatter},
show some very subtle results behind these biases. For the Linear and ZPHI
algorithms, the main bands of points lie along the one-to-one line. However,
the spread of points is uneven, with more points above the one-to-one line.
For the ZPHI and SC algorithms, the spread is more even, but at lower
attenuation values, the core of very many points lies above the line, causing
the small bias. While these may seem like small effects, it is important to
keep in mind that the observed biases are a few \si{dB} on top of calculated
values on order \SI{60}{dB}. A small bias in specific attenuation values
in the range of \SI{1}{dB\per\kilo\meter} is easily sufficient to yield the
observed biases.

The results for vertical polarization are similar, with PPIs of the base attenuation
values shown in Figure~\ref{fig:X_sp_Control_Attenuation_V}. The differences
from the intrinsic values, shown in
Figure~\ref{fig:X_sp_Control_Attenuation_Difference_V}, show that all
algorithms have an overall positive bias. Here, ZPHI performs best, with the
errors having a magnitude around \SI{2}{dB}. The other algorithms have larger
errors that approach \SI{4}{dB}. Looking at the histograms of
vertical specific attenuation in
Figure~\ref{fig:X_sp_Control_Specific_Attenuation_V_scatter}, the cause
of the differences is subtle. The ZPHI algorithm exhibits slightly less spread
in the points, lessening the overall positive bias.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_Control_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_H}, but for
    X-band.}
    \label{fig:X_sp_Control_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Control_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H},
    but for X-band.}
    \label{fig:X_sp_Control_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_Control_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_sp_Control_Attenuation_H}, but
    for vertical polarization.}
    \label{fig:X_sp_Control_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Control_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_sp_Control_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:X_sp_Control_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Control_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, but for
    X-band.}
    \label{fig:X_sp_Control_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Control_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_Control_Specific_Attenuation_H_scatter}, but
    for vertical polarization.}
    \label{fig:X_sp_Control_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Control_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_sp_Control_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:X_sp_Control_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Control_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_sp_Control_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_sp_Control_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Control_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_Control_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:X_sp_Control_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, it is challenging to evaluate algorithm
performance from the base PPIs in
Figure~\ref{fig:X_sp_Control_Differential_Attenuation}. The differences
from the intrinsic values, in
Figure~\ref{fig:X_sp_Control_Differential_Attenuation_Difference}, make
the picture clear that ZPHI and SC perform the best of all here. Both algorithms
have biases less than \SI{1}{dB} which, while significant, are much better than
the \SI{1.25}{dB} positive bias for the Linear algorithm. MSC, on the other hand,
demonstrates a breakdown in performance, with errors over \SI{2}{dB}. Such
performance is unexpected given the performance of SC and ZPHI and is likely
due to a poor choice in ``smoothed'' coefficient. The histograms of specific
differential attenuation in
Figure~\ref{fig:X_sp_Control_Specific_Differential_Attenuation_scatter} show
clear shifts for all of the algorithms, which yield the observed biases. Here,
the MSC algorithm also shows a large spread of points above the line,
which creates the exceptionally bad algorithm performance. This is reflected
in the computed bias values for MSC
(Table~\ref{tab:X-stats-Control-sp}), which are the highest for the horizontal
polarization and differential cases.

Unlike at C-band, the X-band Control experiment here at a azimuthal spacing of
\SI{1.0}{\degree} displays some interesting differences from the Control
experiment of the modeling study at \SI{0.25}{\degree} azimuthal spacing. The
histograms display similar changes as at C-band, with the bands of points
clustering more tightly and the multiple bands for the ZPHI-based algorithms
collapsing down to a single band. However, the attenuation results for the
individual polarizations all show larger errors here than for the higher
resolution data. It is interesting to note that the increase in errors is not reflected 
in the differential attenuation results, with the exception of the anomalous
performance of the MSC algorithm. That the change in errors with the change
in spacing from \SI{0.25}{\degree} to \SI{1.0}{\degree} is only observed
at X-band, and not C-band, is likely a consequence of the much higher
attenuation values observed at X-band.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0530 & 0.0501 & 0.9789\\
        ZPHI & -0.0034 & 0.1266 & 0.9453\\
        SC & 0.0365 & 0.1244 & 0.9468\\
        MSC & 0.0991 & 0.1105 & 0.9526\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0254 & 0.0335 & 0.9793\\
        ZPHI & -0.0131 & 0.0789 & 0.9518\\
        SC & 0.0260 & 0.0774 & 0.9529\\
        MSC & 0.0152 & 0.0670 & 0.9580\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0394 & 0.0031 & 0.9752\\
        ZPHI & 0.0098 & 0.0087 & 0.8600\\
        SC & 0.0098 & 0.0085 & 0.8637\\
        MSC & 0.1006 & 0.0364 & 0.7057\\
\hline
    \end{tabular}
    \caption{Bias, mean squared-error, and $r^2$ for the specific attenuation
    results for the Control experiment at X-band.}
    \label{tab:X-stats-Control-sp}
\end{table}

\section{Sidelobe}
The purpose of the Sidelobe experiment is to test the effects of antenna
sidelobes on the accuracy of attenuation correction techniques. This is
achieved by modifying the Control experiment to include the first antenna
sidelobes, which have peaks around \SI{25}{dB} below the mainlobe level.
This experiment tests the impact of the smearing effect of sidelobes on
the accuracy of the correction techniques without modifying the inherent
azimuthal resolution of the radar antenna.

\FloatBarrier
\subsection{C band}
At C-band, the PPIs of horizontal attenuation in
Figure~\ref{fig:C_sp_Sidelobe_Attenuation_H} show minimal differences
from those in the Control experiment. The only observable differences
are a couple of anomalous rays in the ZPHI algorithms and a slight increase
in the estimated attenuation for the MSC algorithm. The PPIs with the intrinsic
values subtracted, shown in
Figure~\ref{fig:C_sp_Sidelobe_Attenuation_Difference_H}, demonstrate
that the change for MSC is very minor,
on the order of \SI{0.1}{dB}. At vertical polarization, the PPIs of the
attenuation values (in
Figure~\ref{fig:C_sp_Sidelobe_Attenuation_V}) show no visual differences from 
Control. Even the differences from the
intrinsic values in Figure~\ref{fig:C_sp_Sidelobe_Attenuation_Difference_V}
show no discernable difference from the Control experiment, outside of the
extra spurious ray in the ZPHI-based algorithms.

Consequently, the histograms
of specific attenuation for both polarizations, in
Figures~\ref{fig:C_sp_Sidelobe_Specific_Attenuation_H_scatter} and
\ref{fig:C_sp_Sidelobe_Specific_Attenuation_V_scatter}, show almost no
change between the Control and Sidelobe experiment. The exception is that
the MSC algorithm, for horizontal specific attenuation, shows a slight upward
shift for the Sidelobe experiment, bringing its core band of values into
better alignment with the one-to-one line. This corresponds to a slight change
in the bias computed for horizontal polarization
(Table~\ref{tab:C-stats-Sidelobe}), from \SI{-0.0025}{dB\per\kilo\meter}
to \SI{0.0015}{dB\per\kilo\meter}. This change is a consequence of
a better-matching coefficient being selected as the median, possibly a
consequence of the bad rays. This highlights that despite goals to the
contrary, the selection of the coefficient in the MSC algorithm can be
sensitive to the data being processed. As opposed to in the modeling study,
this sensitivity is likely more pronounced here because of the
decreased number of radials of data.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_H}, but for
    the Sidelobe experiment.}
    \label{fig:C_sp_Sidelobe_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H},
    but for the Sidelobe experiment.}
    \label{fig:C_sp_Sidelobe_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_sp_Sidelobe_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:C_sp_Sidelobe_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_sp_Sidelobe_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:C_sp_Sidelobe_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, but for
    the Sidelobe experiment.}
    \label{fig:C_sp_Sidelobe_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Sidelobe_Specific_Attenuation_H_scatter}, but for
    vertical polarization.}
    \label{fig:C_sp_Sidelobe_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_sp_Sidelobe_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:C_sp_Sidelobe_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_sp_Sidelobe_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:C_sp_Sidelobe_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Sidelobe_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Sidelobe_Specific_Attenuation_H_scatter},
    but for differential attenuation.}
    \label{fig:C_sp_Sidelobe_Specific_Differential_Attenuation_scatter}
\end{figure}

Given the lack of changes for the individual polarizations, it is unsurprising
that PPIs of both the base differential attenuation fields and the differences
from the intrinsic values, shown in
Figures~\ref{fig:C_sp_Sidelobe_Differential_Attenuation} and
\ref{fig:C_sp_Sidelobe_Differential_Attenuation_Difference}, respectively,
exhibit no significant changes. The only exception is the MSC algorithm,
which shows a decrease in its bias in estimating differential attenuation, a
consequence of the change observed at horizontal polarization. This is
reflected in the histograms of specific differential attenuation in
Figure~\ref{fig:C_sp_Sidelobe_Specific_Differential_Attenuation_scatter},
which only change for MSC, showing the same shift to better agreement
with the one-to-one line.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0314 & 0.0030 & 0.9789\\
        ZPHI & 0.0286 & 0.0036 & 0.9630\\
        SC & 0.0059 & 0.0035 & 0.9573\\
        MSC & 0.0015 & 0.0025 & 0.9706\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0101 & 0.0011 & 0.9820\\
        ZPHI & 0.0079 & 0.0014 & 0.9665\\
        SC & -0.0032 & 0.0021 & 0.9493\\
        MSC & -0.0055 & 0.0014 & 0.9687\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0407 & 0.0020 & 0.9380\\
        ZPHI & 0.0285 & 0.0014 & 0.8903\\
        SC & 0.0080 & 0.0007 & 0.8910\\
        MSC & 0.0053 & 0.0005 & 0.9530\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Sidelobe experiment.}
    \label{tab:C-stats-Sidelobe}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, the changes from the Control experiment are still minimal, though
greater than at C-band. The PPIs of the base results for both horizontal
and vertical polarizations, shown in
Figures~\ref{fig:X_sp_Sidelobe_Attenuation_H} and
\ref{fig:X_sp_Sidelobe_Attenuation_V}, respectively, exhibit no qualitative
changes. For the Linear algorithm, the results look completely
identical. The differences from the intrinsic values of horizontal attenuation,
in Figure~\ref{fig:X_sp_Sidelobe_Attenuation_Difference_H}, show some
small fluctuations, likely due to small changes in the data near the low-signal
region. The MSC algorithm shows some slightly larger changes, due
to the aforementioned sensitivity. The PPIs of differences for vertical
polarization, in Figure~\ref{fig:X_sp_Sidelobe_Attenuation_Difference_V},
show some small changes as well, but much fewer than observed at
horizontal polarization. The histograms of specific attenuation for both
horizontal and vertical polarizations, in
Figures~\ref{fig:X_sp_Sidelobe_Specific_Attenuation_H_scatter} and
\ref{fig:X_sp_Sidelobe_Specific_Attenuation_V_scatter}, respectively,
show that the only meaningful change from the Control experiment
is for the MSC algorithm; here, the values shift closer to the one-to-one
line, in agreement with the observed decrease in bias in the difference PPI.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_Sidelobe_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Sidelobe_Attenuation_H}, but for
    X-band.}
    \label{fig:X_sp_Sidelobe_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Sidelobe_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Sidelobe_Attenuation_Difference_H},
    but for X-band.}
    \label{fig:X_sp_Sidelobe_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_Sidelobe_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_sp_Sidelobe_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:X_sp_Sidelobe_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Sidelobe_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_sp_Sidelobe_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:X_sp_Sidelobe_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Sidelobe_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Sidelobe_Specific_Attenuation_H_scatter}, but for
    X-band.}
    \label{fig:X_sp_Sidelobe_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Sidelobe_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_Sidelobe_Specific_Attenuation_H_scatter}, but
    for vertical polarization.}
    \label{fig:X_sp_Sidelobe_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Sidelobe_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_sp_Sidelobe_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:X_sp_Sidelobe_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Sidelobe_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_sp_Sidelobe_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_sp_Sidelobe_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Sidelobe_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_Sidelobe_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:X_sp_Sidelobe_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the results reflect the lack of change for the
individual polarizations. The PPIs of differential attenuation in
Figure~\ref{fig:X_sp_Sidelobe_Differential_Attenuation} show no
qualitative changes for the Linear, ZPHI, and SC algorithms. However, the MSC
algorithm does show a noticeable increase in value, a consequence of its 
increase in calculated horizontal attenuation for the Sidelobe experiment.
These changes do help to resolve the anomalously high differences from
intrinsic values that were observed for the MSC algorithm in the Control
experiment, yielding much improved computed values for the bias in
Table~\ref{tab:X-stats-Sidelobe}.
Figure~\ref{fig:X_sp_Sidelobe_Differential_Attenuation_Difference} shows
that these differences have decreased from over \SI{2}{dB} to around
\SI{1.25}{dB}. The histograms of specific differential attenuation in
Figure~\ref{fig:X_sp_Sidelobe_Specific_Differential_Attenuation_scatter}
show that the values for MSC are now much more tightly clustered around
the one-to-one line, in agreement with the observed improvement in
performance.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0531 & 0.0501 & 0.9789\\
        ZPHI & -0.0029 & 0.1206 & 0.9482\\
        SC & 0.0372 & 0.1187 & 0.9495\\
        MSC & 0.0545 & 0.0986 & 0.9562\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0254 & 0.0336 & 0.9793\\
        ZPHI & -0.0162 & 0.0898 & 0.9444\\
        SC & 0.0230 & 0.0884 & 0.9452\\
        MSC & 0.0167 & 0.0760 & 0.9516\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0394 & 0.0031 & 0.9752\\
        ZPHI & 0.0067 & 0.0103 & 0.8333\\
        SC & 0.0060 & 0.0100 & 0.8412\\
        MSC & 0.0388 & 0.0102 & 0.8509\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Sidelobe experiment.}
    \label{tab:X-stats-Sidelobe}
\end{table}

\section{Beamwidth}
The Beamwidth experiment examines the impact of the antenna's
\SI{3}{dB} beamwidth, and hence azimuthal resolution, on the accuracy of
attenuation corrections. Here, the azimuthal spacing is kept at
\SI{1.0}{\degree}, so the data sampling interval is no longer matched to
the resolution. This experiment, combined with the Radial Width experiment,
helps to separate the two effects at play when determining azimuthal smearing
in data.

\FloatBarrier
\subsection{C band}
In the Beamwidth experiment, the results for horizontal attenuation, in
Figure~\ref{fig:C_sp_Beamwidth_Attenuation_H}, show only minor
differences from those in the Control experiment. Based on the PPIs of
the difference from intrinsic values, in
Figure~\ref{fig:C_sp_Beamwidth_Attenuation_Difference_H}, the only
substantive change at horizontal polarization is that for the MSC algorithm
the negative bias in the core of the reflectivity is gone; however,
this has been replaced by positive differences of a similar magnitude on either
side. Overall this likely represents a shift in the optimized coefficient obtained by
the algorithm. The histograms of horizontal specific attenuation in
Figure~\ref{fig:C_sp_Beamwidth_Specific_Attenuation_H_scatter} show
no significant changes, other than a slight shift upward for the band of
points for the MSC algorithm. This brings the points into slightly better
alignment with the one-to-one line, reflecting the shift in bias from
\SI{-0.0025}{dB\per\kilo\meter} to \SI{0.0065}{dB\per\kilo\meter}
(Table~\ref{tab:C-stats-Beamwidth}).

At vertical polarization, there is even less change,
compared to the Control results, with no visible differences in the PPIs shown in
Figure~\ref{fig:C_sp_Beamwidth_Attenuation_V}. The differences from
the intrinsic values, in
Figure~\ref{fig:C_sp_Beamwidth_Attenuation_Difference_V}, do not show 
much change either; the exception is that the pronounced negatively
biased rays in the results for the SC algorithm show reduced errors.
The histograms of vertical specific attenuation
(Figure~\ref{fig:C_sp_Beamwidth_Specific_Attenuation_V_scatter}) do not 
show any qualitative changes from the results in the Control experiment;
the reduction in errors for the few rays of the SC algorithm's results is too small 
to be clear in the histogram.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_H}, but for
    the Beamwidth experiment.}
    \label{fig:C_sp_Beamwidth_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H},
    but for the Beamwidth experiment.}
    \label{fig:C_sp_Beamwidth_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_sp_Beamwidth_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:C_sp_Beamwidth_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_sp_Beamwidth_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:C_sp_Beamwidth_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, but
    for the Beamwidth experiment.}
    \label{fig:C_sp_Beamwidth_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Beamwidth_Specific_Attenuation_H_scatter}, but
    for vertical polarization.}
    \label{fig:C_sp_Beamwidth_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_sp_Beamwidth_Attenuation_H}, but
    for differential attenuation.}
    \label{fig:C_sp_Beamwidth_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_sp_Beamwidth_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:C_sp_Beamwidth_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Beamwidth_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Beamwidth_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:C_sp_Beamwidth_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the PPIs in
Figure~\ref{fig:C_sp_Beamwidth_Differential_Attenuation} show that the
only change is a substantial increase in the differential attenuation calculated 
using the MSC algorithm, a direct consequence of the increase in its estimated
horizontal attenuation. The PPIs of the difference from the intrinsic values,
in Figure~\ref{fig:C_sp_Beamwidth_Differential_Attenuation_Difference}, show
that this increase has turned a pronounced negative bias within the core of
the reflectivity structure into an overall positive bias around \SI{0.5}{dB}; this 
positive bias is still less than that of the ZPHI algorithm. The
histograms of specific differential attenuation in
Figure~\ref{fig:C_sp_Beamwidth_Specific_Differential_Attenuation_scatter}
show that only MSC shows a substantive change, with an upward shift in
its band of points; this is in agreement with the increase in bias shown in
Table~\ref{tab:C-stats-Beamwidth}.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0322 & 0.0029 & 0.9808\\
        ZPHI & 0.0293 & 0.0035 & 0.9631\\
        SC & 0.0049 & 0.0032 & 0.9582\\
        MSC & 0.0065 & 0.0022 & 0.9712\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0115 & 0.0011 & 0.9829\\
        ZPHI & 0.0091 & 0.0013 & 0.9668\\
        SC & -0.0037 & 0.0020 & 0.9514\\
        MSC & -0.0063 & 0.0013 & 0.9694\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0403 & 0.0019 & 0.9456\\
        ZPHI & 0.0278 & 0.0014 & 0.8880\\
        SC & 0.0072 & 0.0007 & 0.8975\\
        MSC & 0.0148 & 0.0006 & 0.9478\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Beamwidth experiment.}
    \label{tab:C-stats-Beamwidth}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, the effect of changing beamwidth is much more pronounced,
evidenced by the PPIs of horizontal attenuation shown in
Figure~\ref{fig:X_sp_Beamwidth_Attenuation_H}. Here, all the
ZPHI-based algorithms show increases in the amount of calculated
attenuation, versus their results from the Control experiment. Similar increases
are observed for vertical polarization, shown in
Figure~\ref{fig:X_sp_Beamwidth_Attenuation_V}. Taking the differences
from the intrinsic attenuation, shown in
Figures~\ref{fig:X_sp_Beamwidth_Attenuation_Difference_H} and
\ref{fig:X_sp_Beamwidth_Attenuation_Difference_V}, it is clear that
all of the algorithms, including Linear, have increased positive biases
in their calculated attenuation fields at both polarizations. The histograms
of specific attenuation for horizontal and vertical polarization, shown in
Figures~\ref{fig:X_sp_Beamwidth_Specific_Attenuation_H_scatter} and
\ref{fig:X_sp_Beamwidth_Specific_Attenuation_V_scatter}, respectively,
show that these increases in bias are not due to any shift in the main bands
of points. Rather, it is due to an increase in the spread of points above
the one-to-one ratio line. These are reflected by increases in not only the
computed biases, but in the MSE values as well
(Table~\ref{tab:X-stats-Beamwidth}).

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_Beamwidth_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Beamwidth_Attenuation_H}, but
    for X-band.}
    \label{fig:X_sp_Beamwidth_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Beamwidth_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Beamwidth_Attenuation_Difference_H},
    but for X-band.}
    \label{fig:X_sp_Beamwidth_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_Beamwidth_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_sp_Beamwidth_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:X_sp_Beamwidth_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Beamwidth_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_sp_Beamwidth_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:X_sp_Beamwidth_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Beamwidth_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Beamwidth_Specific_Attenuation_H_scatter}, but
    for X-band.}
    \label{fig:X_sp_Beamwidth_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Beamwidth_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_Beamwidth_Specific_Attenuation_H_scatter}, but for
    vertical polarization.}
    \label{fig:X_sp_Beamwidth_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Beamwidth_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_sp_Beamwidth_Attenuation_H}, but
    for differential attenuation.}
    \label{fig:X_sp_Beamwidth_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Beamwidth_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_sp_Beamwidth_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_sp_Beamwidth_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Beamwidth_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_Beamwidth_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:X_sp_Beamwidth_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the changes are more minor. The PPIs of
differential attenuation, in
Figure~\ref{fig:X_sp_Beamwidth_Differential_Attenuation}, show that the
most significant changes come from filling in area around the low-signal
area. All of the ZPHI-based algorithms show significant changes here,
especially the MSC algorithm; outside this region, the Beamwidth
experiment results for differential attenuation are consistent with those
for the Control experiment. The differences from the intrinsic values,
shown in
Figure~\ref{fig:X_sp_Beamwidth_Differential_Attenuation_Difference},
confirm these observations. Outside of the areas that were discussed,
the only other changes observed here are that a few radials from the output
of the SC algorithm changed their relative bias. Their proximity to
the low signal region makes such data suspect. The
histograms for specific differential attenuation, in
Figure~\ref{fig:X_sp_Beamwidth_Specific_Differential_Attenuation_scatter},
show that there are no major changes to the character of the results for
differential attenuation in the Beamwidth experiment. None of the main bands
of points show any shift; there are changes in outlier points, but these are not systematic. The fact that the individual polarizations show
significant changes, while differential attenuation remains relatively unchanged
between the two experiments, indicates that the changes experienced
by the two polarizations are very consistent.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.1009 & 0.0974 & 0.9596\\
        ZPHI & 0.0454 & 0.1231 & 0.9415\\
        SC & 0.0687 & 0.1277 & 0.9413\\
        MSC & 0.2072 & 0.3123 & 0.8805\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0730 & 0.0762 & 0.9537\\
        ZPHI & 0.0303 & 0.0921 & 0.9389\\
        SC & 0.0574 & 0.0974 & 0.9371\\
        MSC & 0.0909 & 0.1060 & 0.9334\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0407 & 0.0030 & 0.9776\\
        ZPHI & 0.0129 & 0.0135 & 0.7606\\
        SC & 0.0075 & 0.0115 & 0.7927\\
        MSC & 0.1524 & 0.1543 & 0.3618\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Beamwidth experiment.}
    \label{tab:X-stats-Beamwidth}
\end{table}

\section{Radial Width}
The Radial Width experiment is the counterpart to the previous Beamwidth
experiment. The beamwidth from the previous experiment is maintained
at \SI{2.0}{\degree}, while the radial spacing is increased to match. Effectively,
two radials from the Beamwidth experiment will be combined into one in the
Radial Width experiment. Comparisons between the two will reflect the impacts
of this averaging on the correction techniques. Also, comparing against the
Control baseline, which has sampling matched between beamwidth and
radial spacing at \SI{1.0}{\degree}, will explore how the overall change
in resolution volume impacts the corrections.

\FloatBarrier
\subsection{C band}
For the Radial Width experiment, the PPIs of attenuation at horizontal and
vertical polarizations, in Figures~\ref{fig:C_sp_RadialWidth_Attenuation_H}
and \ref{fig:C_sp_RadialWidth_Attenuation_V}, respectively, show no
significant changes in comparison with the results of the Beamwidth
experiment, in regards to the amount of attenuation; clearly different
is the coarseness of the data, which is expected based on the change to the
radial spacing. The PPIs of the difference between the algorithm calculation
and the intrinsic values, in
Figures~\ref{fig:C_sp_RadialWidth_Attenuation_Difference_H} and
\ref{fig:C_sp_RadialWidth_Attenuation_Difference_V}, confirm the lack
of significant differences. The MSC algorithm does show some small changes
from small negative errors to small positive errors, but no significant
changes to the magnitude of errors. This is likely another reflection of the
sensitivity of MSC. The histograms of specific attenuation, in
Figures~\ref{fig:C_sp_RadialWidth_Specific_Attenuation_H_scatter} and
\ref{fig:C_sp_RadialWidth_Specific_Attenuation_V_scatter}, show no 
significant changes from the results in the Beamwidth experiment (other than
an expected decrease in the total number of points). This is also reflected in the
lack of significant changes to the bias and MSE values in
Table~\ref{tab:C-stats-Radial Width}.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_H}, but for
    the Radial Width experiment.}
    \label{fig:C_sp_RadialWidth_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H},
    but for the Radial Width experiment.}
    \label{fig:C_sp_RadialWidth_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_sp_RadialWidth_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:C_sp_RadialWidth_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_sp_RadialWidth_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:C_sp_RadialWidth_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, but for
    the Radial Width experiment.}
    \label{fig:C_sp_RadialWidth_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_RadialWidth_Specific_Attenuation_H_scatter}, but
    for vertical polarization.}
    \label{fig:C_sp_RadialWidth_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_sp_RadialWidth_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:C_sp_RadialWidth_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_sp_RadialWidth_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:C_sp_RadialWidth_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RadialWidth_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_RadialWidth_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:C_sp_RadialWidth_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the results are similar. The PPIs of the base
attenuation fields in
Figure~\ref{fig:C_sp_RadialWidth_Differential_Attenuation} show no
significant changes, except that the MSC algorithm appears to have produced
less total differential attenuation. This is confirmed in
Figure~\ref{fig:C_sp_RadialWidth_Differential_Attenuation_Difference},
which shows that the differences from the intrinsic values of differential
attenuation have changed from an overall, small positive bias to a negative
bias, around \SI{0.25}{dB} in the core of the reflectivity structure, as well as 
small positive biases on either side. The errors for the other algorithms show no 
significant changes with respect to the Beamwidth experiment. The 
histograms of specific differential attenuation, in
Figure~\ref{fig:C_sp_RadialWidth_Specific_Differential_Attenuation_scatter},
show that the core band of points for the MSC algorithm has shifted down,
putting the band of points along the one-to-one line for small values; for higher
values, the band now curves below the one-to-one line. This yields
the observed lessened bias for smaller attenuation along the edge of the
reflectivity structure and pronounced negative bias for the higher values of
attenuation in the middle of the structure. Overall, this value dependent bias
reflects a failure of the algorithm to effectively capture the relationship
between radar observables and attenuation.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0334 & 0.0030 & 0.9806\\
        ZPHI & 0.0306 & 0.0036 & 0.9622\\
        SC & 0.0071 & 0.0031 & 0.9584\\
        MSC & 0.0035 & 0.0022 & 0.9716\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0121 & 0.0011 & 0.9832\\
        ZPHI & 0.0099 & 0.0013 & 0.9665\\
        SC & -0.0019 & 0.0018 & 0.9527\\
        MSC & -0.0007 & 0.0012 & 0.9693\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0398 & 0.0019 & 0.9479\\
        ZPHI & 0.0274 & 0.0014 & 0.8851\\
        SC & 0.0069 & 0.0006 & 0.8937\\
        MSC & 0.0002 & 0.0004 & 0.9461\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Radial Width experiment.}
    \label{tab:C-stats-Radial Width}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, the PPIs of horizontal and vertical attenuation, in
Figures~\ref{fig:X_sp_RadialWidth_Attenuation_H} and
\ref{fig:X_sp_RadialWidth_Attenuation_V}, respectively, show no
observable changes in magnitude between the Beamwidth and Radial Width
experiments. The differences from the intrinsic values, shown in
Figures~\ref{fig:X_sp_RadialWidth_Attenuation_Difference_H} and
\ref{fig:X_sp_RadialWidth_Attenuation_Difference_V}, confirm that there
are no significant changes in the structure of the errors of the algorithms. The
histograms of specific attenuation, in
Figures~\ref{fig:X_sp_RadialWidth_Specific_Attenuation_H_scatter} and
\ref{fig:X_sp_RadialWidth_Specific_Attenuation_V_scatter}, also exhibit no
shifts or tilts that would indicate a change in performance. It is worth 
noting
that the histograms for the Radial Width experiment show significantly fewer
points at the higher values of specific attenuation; this is a reflection of how
the larger azimuthal spacing smooths the data and reduces peaks. Also,
at both polarizations, the ZPHI algorithms develop a collection of outlier points
around a value of true specific attenuation of \SI{1.5}{dB\per\kilo\meter};
these points contribute to an increase in the bias and MSE
(Table~\ref{tab:X-stats-Radial Width}) but are otherwise insignificant compared
to the core band of points.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_RadialWidth_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_RadialWidth_Attenuation_H}, but for
    X-band.}
    \label{fig:X_sp_RadialWidth_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RadialWidth_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_RadialWidth_Attenuation_Difference_H},
    but for X-band.}
    \label{fig:X_sp_RadialWidth_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_RadialWidth_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_sp_RadialWidth_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:X_sp_RadialWidth_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RadialWidth_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_sp_RadialWidth_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:X_sp_RadialWidth_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RadialWidth_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_RadialWidth_Specific_Attenuation_H_scatter}, but
    for X-band.}
    \label{fig:X_sp_RadialWidth_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RadialWidth_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_RadialWidth_Specific_Attenuation_H_scatter}, but
    for vertical polarization.}
    \label{fig:X_sp_RadialWidth_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RadialWidth_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_sp_RadialWidth_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:X_sp_RadialWidth_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RadialWidth_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_sp_RadialWidth_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_sp_RadialWidth_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RadialWidth_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_RadialWidth_Specific_Attenuation_H_scatter}, but
    for specific differential attenuation.}
    \label{fig:X_sp_RadialWidth_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the results are similarly stable between the
Beamwidth and Radial Width experiments. The PPIs of the base fields, in
Figure~\ref{fig:X_sp_RadialWidth_Differential_Attenuation}, show no
significant changes in the algorithms, outside of the low-signal region which
is expected to be dependent on the sensitivity of the radar. The differences
from the intrinsic values, in
Figure~\ref{fig:X_sp_RadialWidth_Differential_Attenuation_Difference}, show
that the errors have not changed outside of the low-signal region. The 
anomalous regions in the MSC algorithm have had the magnitude of errors
greatly reduced, though the values are still anomalously high. The
histograms of the points, in
Figure~\ref{fig:X_sp_RadialWidth_Specific_Differential_Attenuation_scatter},
also show no significant changes to the core bands of points. For the
ZPHI-based algorithms, many of the outlier points are better clustered around
the one-to-one line, especially for the MSC algorithm. This is another
consequence of the smoothing effect of the larger radial spacing.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.1331 & 0.1477 & 0.9379\\
        ZPHI & 0.0743 & 0.1567 & 0.9216\\
        SC & 0.0910 & 0.1521 & 0.9264\\
        MSC & 0.1697 & 0.2799 & 0.8775\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0969 & 0.1074 & 0.9340\\
        ZPHI & 0.0514 & 0.1131 & 0.9215\\
        SC & 0.0750 & 0.1187 & 0.9199\\
        MSC & 0.1016 & 0.1394 & 0.9095\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0416 & 0.0030 & 0.9782\\
        ZPHI & 0.0127 & 0.0169 & 0.6911\\
        SC & 0.0046 & 0.0106 & 0.7978\\
        MSC & 0.0779 & 0.0694 & 0.4196\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Radial Width experiment.}
    \label{tab:X-stats-Radial Width}
\end{table}

\section{Range Resolution}
While all previous experiments explored aspects of azimuthal sampling, the
Range Resolution experiment looks at how the resolution in the alternate
direction, along the range, impacts the attenuation corrections. Given that
all of the algorithms rely on range integrals of various forms, range
resolution and spacing is a very relevant consideration for the operation of
attenuation correction techniques.

\FloatBarrier
\subsection{C band}
The PPIs of the base attenuation fields for the Range Resolution experiment
are given in Figures~\ref{fig:C_sp_RangeResolution_Attenuation_H}
and \ref{fig:C_sp_RangeResolution_Attenuation_V}, for horizontal and
vertical polarizations, respectively. These fields exhibit no significant differences
from the results for the Control experiment. This consistency with the
Control experiment also extends to the differences from the intrinsic values, 
shown in Figures~\ref{fig:C_sp_RangeResolution_Attenuation_Difference_H}
and \ref{fig:C_sp_RangeResolution_Attenuation_Difference_V}. This lack
of change is reflected in the histograms for specific attenuation, shown in
Figures~\ref{fig:C_sp_RangeResolution_Specific_Attenuation_H_scatter}
and \ref{fig:C_sp_RangeResolution_Specific_Attenuation_V_scatter},
for horizontal and polarizations, respectively. The only significant change is that
all algorithms show fewer points with high values of specific attenuation, a
consequence of the smoothing due to the larger pulse and gate spacing. 
However, the
histograms have no observable changes with regards to the accuracy of the
correction algorithms. This is reflected in the computed biases and MSE values
as well (Table~\ref{tab:C-stats-Range Resolution}) which show only minor changes
from the values for the Control experiment.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_H}, but for the
    Range Resolution experiment.}
    \label{fig:C_sp_RangeResolution_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H},
    but for the Range Resolution experiment.}
    \label{fig:C_sp_RangeResolution_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_sp_RangeResolution_Attenuation_H}, but
    for vertical polarization.}
    \label{fig:C_sp_RangeResolution_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Attenuation_Difference_V}
    \caption{As in
    Figure~\ref{fig:C_sp_RangeResolution_Attenuation_Difference_H}, but
    for vertical polarization.}
    \label{fig:C_sp_RangeResolution_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, but for
    the Range Resolution experiment.}
    \label{fig:C_sp_RangeResolution_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_RangeResolution_Specific_Attenuation_H_scatter},
    but for vertical polarization.}
    \label{fig:C_sp_RangeResolution_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_sp_RangeResolution_Attenuation_H}, but
    for differential attenuation.}
    \label{fig:C_sp_RangeResolution_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Differential_Attenuation_Difference}
    \caption{As in
    Figure~\ref{fig:C_sp_RangeResolution_Attenuation_Difference_H}, but
    for differential attenuation.}
    \label{fig:C_sp_RangeResolution_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_RangeResolution_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_RangeResolution_Specific_Attenuation_H_scatter},
    but for specific differential attenuation.}
    \label{fig:C_sp_RangeResolution_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the results largely follow those for the individual
polarizations. The PPIs of the base differential attenuation results, in
Figure~\ref{fig:C_sp_RangeResolution_Differential_Attenuation}, show that
the MSC algorithm is the only one that changes magnitude versus the results
from the Control experiment. The differences from the intrinsic values,
in Figure~\ref{fig:C_sp_RangeResolution_Differential_Attenuation_Difference},
show that the effect of this change is to reduce the negative bias in the
core of the reflectivity structure from \SI{0.5}{dB} to \SI{0.25}{dB}. The
remainder of the algorithms show no change in the structure of their errors
versus the Control experiment. The histograms of specific differential
attenuation, in Figure~\ref{fig:C_sp_RangeResolution_Specific_Differential_Attenuation_scatter},
show that the change in the results for the MSC algorithm result from
a band at higher values being pulled into the main cluster rather than
extending below the one-to-one line. The other algorithms show only a reduction 
of high values, similar to that observed for the individual polarizations.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0313 & 0.0063 & 0.9356\\
        ZPHI & 0.0284 & 0.0038 & 0.9590\\
        SC & 0.0021 & 0.0035 & 0.9548\\
        MSC & -0.0054 & 0.0027 & 0.9690\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0101 & 0.0029 & 0.9368\\
        ZPHI & 0.0078 & 0.0014 & 0.9642\\
        SC & -0.0065 & 0.0022 & 0.9467\\
        MSC & -0.0114 & 0.0016 & 0.9677\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0392 & 0.0023 & 0.8609\\
        ZPHI & 0.0280 & 0.0015 & 0.8741\\
        SC & 0.0073 & 0.0007 & 0.8822\\
        MSC & 0.0034 & 0.0005 & 0.9445\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Range Resolution experiment.}
    \label{tab:C-stats-Range Resolution}
\end{table}

\FloatBarrier
\subsection{X band}
The horizontal attenuation results for X-band in the Range Resolution
experiment, in
Figure~\ref{fig:X_sp_RangeResolution_Attenuation_H}, show
larger attenuation values for all of the ZPHI-based algorithms.  The differences
from the intrinsic values, shown in
Figure~\ref{fig:X_sp_RangeResolution_Attenuation_Difference_H}, show
that these increases are constrained, at least for ZPHI and SC, to the region
of low-signal. The areas that are expected to be more stable show no changes
versus the Control experiment. The MSC algorithm is the lone exception
and shows an overall
decrease in its positive bias, from \SI{5}{dB} to around \SI{3}{dB}. Given that
this change is not reflected by the ZPHI and SC algorithms, it is a
consequence of a different coefficient being obtained by the optimization
process of the algorithm. The Linear algorithm exhibits the development of an
area with negative bias near the radar, with values moving from near
\SI{0}{dB} to around \SI{1}{dB}. The histograms for horizontal specific
attenuation, in
Figure~\ref{fig:X_sp_RangeResolution_Specific_Attenuation_H_scatter},
show no significant changes from the Control experiment, at least in regards
to biases. The MSE value for the Linear algorithm
(Table~\ref{tab:X-stats-Range Resolution}) do reflect a large increase over the
results of the Control experiment and are likely related to the increase in errors
observed earlier; none of the other algorithms exhibit this increase. 
All of the algorithms do reflect a shrinking of the overall range of
values, a reflection of the smoothing process due to the increases in pulse
width and gate size in the Range Resolution experiment.

For vertical polarization, the results are similar. The base fields of attenuation
in Figure~\ref{fig:X_sp_RangeResolution_Attenuation_V} show increases
in estimated attenuation in the region of low-signal. Outside of these areas,
the only significant changes to the structure of the errors shown in
Figure~\ref{fig:X_sp_RangeResolution_Attenuation_Difference_V} are for
the Linear algorithm near the radar, where a small negative bias around
\SI{1}{dB} develops. The histograms of vertical specific attenuation in
Figure~\ref{fig:X_sp_RangeResolution_Specific_Attenuation_V_scatter} show
no significant changes other than the loss of points at higher values of
specific attenuation.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_RangeResolution_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_RangeResolution_Attenuation_H},
    but for X-band.}
    \label{fig:X_sp_RangeResolution_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RangeResolution_Attenuation_Difference_H}
    \caption{As in
    Figure~\ref{fig:C_sp_RangeResolution_Attenuation_Difference_H}, but
    for X-band.}
    \label{fig:X_sp_RangeResolution_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_RangeResolution_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_sp_RangeResolution_Attenuation_H},
    but for vertical polarization.}
    \label{fig:X_sp_RangeResolution_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RangeResolution_Attenuation_Difference_V}
    \caption{As in
    Figure~\ref{fig:X_sp_RangeResolution_Attenuation_Difference_H}, but
    for vertical polarization.}
    \label{fig:X_sp_RangeResolution_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RangeResolution_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_RangeResolution_Specific_Attenuation_H_scatter},
    but for X-band.}
    \label{fig:X_sp_RangeResolution_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RangeResolution_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_RangeResolution_Specific_Attenuation_H_scatter},
    but for vertical polarization.}
    \label{fig:X_sp_RangeResolution_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RangeResolution_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_sp_RangeResolution_Attenuation_H},
    but for differential attenuation.}
    \label{fig:X_sp_RangeResolution_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RangeResolution_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_sp_RadialWidth_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_sp_RangeResolution_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_RangeResolution_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_RangeResolution_Specific_Attenuation_H_scatter},
    but for specific differential attenuation.}
    \label{fig:X_sp_RangeResolution_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the results are largely the same as at the individual
polarizations. The base field of differential attenuation, in
Figure~\ref{fig:X_sp_RangeResolution_Differential_Attenuation}, shows
no significant changes in magnitude, and this is also reflected by no
significant changes in the errors from intrinsic values, shown in
Figure~\ref{fig:X_sp_RangeResolution_Differential_Attenuation_Difference}.
The one exception to the lack of changes is that the modifications to the
range resolution has helped improve the results for MSC around the area
of low signal; this is likely just a consequence of the technique's sensitivity.
The histograms of specific differential attenuation, in
Figure~\ref{fig:X_sp_RangeResolution_Specific_Differential_Attenuation_scatter},
show some notable differences. For the ZPHI-based algorithms, the points
are better clustered in the Range Resolution experiment, versus the Control
experiment; the MSC algorithms points have especially lost some spread, an
outcome that drives the observed improvement to the previously anomalous
results. The Linear algorithm shows a greater spread in points.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.0507 & 0.1286 & 0.9425\\
        ZPHI & -0.0048 & 0.1114 & 0.9504\\
        SC & 0.0258 & 0.1136 & 0.9500\\
        MSC & 0.0264 & 0.0966 & 0.9560\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.0237 & 0.0900 & 0.9424\\
        ZPHI & -0.0190 & 0.0859 & 0.9455\\
        SC & 0.0114 & 0.0869 & 0.9448\\
        MSC & 0.0037 & 0.0751 & 0.9513\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0378 & 0.0056 & 0.9295\\
        ZPHI & 0.0085 & 0.0092 & 0.8442\\
        SC & 0.0051 & 0.0105 & 0.8192\\
        MSC & 0.0187 & 0.0099 & 0.8321\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Range Resolution experiment.}
    \label{tab:X-stats-Range Resolution}
\end{table}

\section{Combined}
The Combined experiment is separate from the others, as it does not involve
exploring sampling directly. Instead, this experiment, like in the previous
modeling study, looks at the impact of the violation of assumptions made
about scattering. Here the radar simulation is performed with the
baseline sampling parameters (\SI{1.0}{\degree} beamwidth and radial
spacing, \SI{125}{\meter} gate spacing) so that sampling effects are also
included. For the violated assumptions, this case is designed to be reasonable
and representative, so only the shape model and temperature assumptions
are violated; this uses the \cite{Pruppacher1970} shape model (as opposed
to \cite{Brandes2002}) and the model simulation temperatures (approximately
\SI{295}{\kelvin} as opposed to \SI{283}{\kelvin}). These aspects are chosen
because they are both important and challenging to address (from the
standpoint of natural variability) when obtaining coefficients for the algorithms.
Overall, this experiment is designed to reflect a real-world set of sampling
intervals along with a reasonable set of modeling errors.

\FloatBarrier
\subsection{C band}
The base attenuation fields for the Combined experiment at C-band are
presented in Figures~\ref{fig:C_sp_Combined_Attenuation_H} and
Figure~\ref{fig:C_sp_Combined_Attenuation_V}, for horizontal and
vertical polarizations, respectively; these PPIs show the impact of the
modifications to the scattering model. The intrinsic attenuation has
decreased for both polarizations, and this decrease is matched fairly well
by the SC and MSC algorithms. The Linear and ZPHI algorithms, however, show
a significant increase between the Control and Combined experiments. With
the intrinsic values subtracted, the PPIs in
Figures~\ref{fig:C_sp_Combined_Attenuation_Difference_H} and
\ref{fig:C_sp_Combined_Attenuation_Difference_V} show that this change
greatly increases the biases for the Linear and ZPHI algorithms, though Linear
has a larger area of peak errors. At horizontal polarization, this changes the
errors from around \SI{2}{dB} to \SI{10}{dB}; for vertical the change is
slightly smaller, from \SI{1}{dB} to almost \SI{8}{dB}. Conversely, the SC
and MSC algorithms only exhibit minors changes in the structure of their
errors between the experiments. The histograms of specific attenuation
for horizontal,
Figure~\ref{fig:C_sp_Combined_Specific_Attenuation_H_scatter}, and for
vertical, Figure~\ref{fig:C_sp_Combined_Specific_Attenuation_V_scatter},
polarizations show very large upward shifts in the points for both the
Linear and ZPHI algorithms, causing the positive bias. These correspond to
greater than an order of magnitude increase in the computed values of
bias and MSE (Table~\ref{tab:C-stats-Combined-sp}). An additional problem
for the ZPHI algorithm is that the distribution of points changes from being
tightly clustered around the one-to-one line to being very spread out; this
poor fit represents a very poor match of ZPHI's internal scattering model
to the data. The points from the SC and MSC algorithms, conversely, show
no significant shifts between the Control and Combined experiments.
While the computed bias values increase a little, the MSE values stay the same;
more importantly, the bias values for SC and MSC are an order of magnitude
smaller than those of the Linear and ZPHI algorithms
(Table~\ref{tab:C-stats-Combined-sp}). This is
evidence that the optimization of a single coefficient is sufficient to correct
the problems observed by ZPHI.

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_H}, but for
    the Combined experiment.}
    \label{fig:C_sp_Combined_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Control_Attenuation_Difference_H},
    but for the Combined experiment.}
    \label{fig:C_sp_Combined_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Attenuation_V}
    \caption{As in Figure~\ref{fig:C_sp_Combined_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:C_sp_Combined_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:C_sp_Combined_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:C_sp_Combined_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Control_Specific_Attenuation_H_scatter}, but
    for the Combined experiment.}
    \label{fig:C_sp_Combined_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Combined_Specific_Attenuation_H_scatter}, but for
    vertical polarization.}
    \label{fig:C_sp_Combined_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:C_Combined_Attenuation_H},
    but for differential attenuation.}
    \label{fig:C_sp_Combined_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:C_sp_Combined_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:C_sp_Combined_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/C_Combined_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Combined_Specific_Attenuation_H_scatter},
    but for specific differential attenuation.}
    \label{fig:C_sp_Combined_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the PPIs in
Figure~\ref{fig:C_sp_Combined_Differential_Attenuation} show that the
Linear and ZPHI algorithms have the most trouble with the changes between
the Control and Combined experiments. The differences from the intrinsic
values, in
Figure~\ref{fig:C_sp_Combined_Differential_Attenuation_Difference}, show
that for these two algorithms, their positive biases have increased from around
\SI{1}{dB} to well over \SI{2}{dB}. As in the individual polarizations, the
histograms of the points, in
Figure~\ref{fig:C_sp_Combined_Specific_Differential_Attenuation_scatter},
show that the Linear and ZPHI algorithms experience large shifts in the points
between the Control and Combined experiments. The SC and MSC algorithms
experience no such shift.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.1911 & 0.0564 & 0.9661\\
        ZPHI & 0.1877 & 0.0823 & 0.7022\\
        SC & 0.0172 & 0.0043 & 0.9294\\
        MSC & 0.0035 & 0.0028 & 0.9629\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.1445 & 0.0339 & 0.9759\\
        ZPHI & 0.1419 & 0.0416 & 0.7687\\
        SC & 0.0079 & 0.0019 & 0.9240\\
        MSC & -0.0028 & 0.0012 & 0.9644\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0843 & 0.0079 & 0.9179\\
        ZPHI & 0.0653 & 0.0141 & 0.2917\\
        SC & 0.0062 & 0.0009 & 0.8761\\
        MSC & 0.0022 & 0.0007 & 0.9411\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:C-stats-Control}, but for
    the Combined experiment.}
    \label{tab:C-stats-Combined-sp}
\end{table}

\FloatBarrier
\subsection{X band}
At X-band, the PPIs of horizontal and vertical attenuation, in
Figures~\ref{fig:X_sp_Combined_Attenuation_H} and
\ref{fig:X_sp_Combined_Attenuation_V}, show that the effect of the 
Combined experiment is to increase the intrinsic attenuation. Unlike C-band,
the SC and MSC algorithms appear to decrease, while the Linear
and ZPHI algorithms increase. The differences from intrinsic values, in
Figures~\ref{fig:X_sp_Combined_Attenuation_Difference_H} and
\ref{fig:X_sp_Combined_Attenuation_Difference_V}, show that the net
result is to increase errors across the board. The
SC and MSC algorithms have smaller changes, increases on the order of
\SI{1}{dB}. For the Linear and ZPHI algorithms, this change is from
\SI{1}{dB} to well-over \SI{10}{dB} at both polarizations. This is
reflected in the histograms of specific attenuation, shown in
Figures~\ref{fig:X_sp_Combined_Specific_Attenuation_H_scatter} and
\ref{fig:X_sp_Combined_Specific_Attenuation_V_scatter}, which show
large shifts for both the Linear and ZPHI algorithms. The Linear algorithm
exhibits a shift of its entire band of points above the one-to-one ratio line,
which results in the extremely large positive bias, approaching
\SI{0.5}{dB\per\kilo\meter} (Table~\ref{tab:X-stats-Combined-sp}).
The effect on ZPHI is
slightly smaller, in that the shift only occurs for smaller attenuation values;
the larger values still lie along the one-to-one line. This still results
in much larger computed values for bias and MSE, with the bias approaching
\SI{0.4}{dB\per\kilo\meter}. For ZPHI, the spread of points
also changes, from having a few outliers below the one-to-one line, to having
many more outliers, almost all of which are above that line.

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_Combined_Attenuation_H}
    \caption{As in Figure~\ref{fig:C_sp_Combined_Attenuation_H}, but for
    X-band.}
    \label{fig:X_sp_Combined_Attenuation_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Combined_Attenuation_Difference_H}
    \caption{As in Figure~\ref{fig:C_sp_Combined_Attenuation_Difference_H},
    but for X-band.}
    \label{fig:X_sp_Combined_Attenuation_Difference_H}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.91]{figures/spatial/X_Combined_Attenuation_V}
    \caption{As in Figure~\ref{fig:X_sp_Combined_Attenuation_H}, but for
    vertical polarization.}
    \label{fig:X_sp_Combined_Attenuation_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Combined_Attenuation_Difference_V}
    \caption{As in Figure~\ref{fig:X_sp_Combined_Attenuation_Difference_H},
    but for vertical polarization.}
    \label{fig:X_sp_Combined_Attenuation_Difference_V}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Combined_Specific_Attenuation_H_scatter}
    \caption{As in
    Figure~\ref{fig:C_sp_Combined_Specific_Attenuation_H_scatter}, but
    for X-band.}
    \label{fig:X_sp_Combined_Specific_Attenuation_H_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Combined_Specific_Attenuation_V_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_Combined_Specific_Attenuation_H_scatter}, but
    for vertical polarization.}
    \label{fig:X_sp_Combined_Specific_Attenuation_V_scatter}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Combined_Differential_Attenuation}
    \caption{As in Figure~\ref{fig:X_sp_Combined_Attenuation_H}, but for
    differential attenuation.}
    \label{fig:X_sp_Combined_Differential_Attenuation}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Combined_Differential_Attenuation_Difference}
    \caption{As in Figure~\ref{fig:X_sp_Combined_Attenuation_Difference_H},
    but for differential attenuation.}
    \label{fig:X_sp_Combined_Differential_Attenuation_Difference}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.92]{figures/spatial/X_Combined_Specific_Differential_Attenuation_scatter}
    \caption{As in
    Figure~\ref{fig:X_sp_Combined_Specific_Attenuation_H_scatter}, but for
    specific differential attenuation.}
    \label{fig:X_sp_Combined_Specific_Differential_Attenuation_scatter}
\end{figure}

For differential attenuation, the PPIs in
Figure~\ref{fig:X_sp_Combined_Differential_Attenuation} show increases
across the board, both for the intrinsic values and the algorithms. The
differences from the intrinsic values, in
Figure~\ref{fig:X_sp_Combined_Differential_Attenuation_Difference},
show that Linear and ZPHI have the largest increases in errors, as was the case for
the individual polarizations. The Linear algorithm's positive bias increases from
approximately \SI{1}{dB} to over \SI{2}{dB}. The ZPHI algorithm, outside
of the low-signal region, has its positive bias increase from around \SI{0.5}{dB}
to over \SI{1}{dB}. Conversely, the SC algorithm exhibits no
significant changes in errors between the Control and Combined experiments,
while the MSC algorithm reduces its errors, as some of its anomalously bad
results are corrected. These changes are reflected by the histograms
of specific attenuation, in
Figure~\ref{fig:X_sp_Combined_Specific_Differential_Attenuation_scatter},
which show a large shift for the results of the Linear algorithm above the
one-to-one ratio line. The ZPHI
algorithm shows an increase in the spread of its points, with the spread
being biased above the one-to-one line.

\begin{table}
    \centering
    \begin{tabular}{| c | c | c | c |}
        \hline
        Algorithm & Bias (\si{dB\per \kilo\meter}) & MSE (\si{dB\squared \per \kilo\meter \squared}) & $r^2$ \\
        \hline
        \hline
        \multicolumn{4}{|c|}{Horizontal}\\
        \hline
        Linear & 0.4650 & 0.3601 & 0.9675\\
        ZPHI & 0.3973 & 0.7653 & 0.8163\\
        SC & 0.0551 & 0.1962 & 0.9300\\
        MSC & 0.0690 & 0.1902 & 0.9322\\
        \hline
        \multicolumn{4}{|c|}{Vertical}\\
        \hline
        Linear & 0.4151 & 0.2852 & 0.9677\\
        ZPHI & 0.3743 & 0.8375 & 0.7451\\
        SC & 0.0455 & 0.1177 & 0.9376\\
        MSC & 0.0374 & 0.1245 & 0.9343\\
        \hline
        \multicolumn{4}{|c|}{Differential}\\
        \hline
        Linear & 0.0705 & 0.0076 & 0.9709\\
        ZPHI & 0.1441 & 0.2471 & 0.3742\\
        SC & 0.0129 & 0.0170 & 0.8062\\
        MSC & 0.0252 & 0.0195 & 0.7740\\
\hline
    \end{tabular}
    \caption{As in Table~\ref{tab:X-stats-Control}, but for
    the Combined experiment.}
    \label{tab:X-stats-Combined-sp}
\end{table}

\FloatBarrier
\section{Conclusion}
This chapter examined the effects of running several different attenuation
correction algorithms, Linear, ZPHI, Self-Consistent, and Modified Self-Consistent
on radar data with different spatial sampling characteristics. These configurations
explored azimu\-thal effects, including sidelobes, mainlobe beamwidth, and radial
spacing, as well as the range resolution effect of changing pulse width and
gate spacing. Remarkably, the algorithms overall showed very little sensitivity
to these changes. The initial Control experiment, which started with matched
beamwidth and radial spacing of \SI{1.0}{\degree} showed no significant
differences from the modeling study. The lack of sensitivity to sampling
continued through the remainder of the experiments that only tested 
aspects of spatial sampling.

One effect revealed by this study is that the Modified Self-Consistent
algorithm can be sensitive to the data. By virtue of its single optimized value,
any spuriously obtained coefficient can bias all of the results. The method
of choosing the median value originally was chosen to try to smooth the results
from the Self-Consistent algorithm
in a way that is less sensitive to outliers. However, these results reveal that
when few radials of data are available, as occurred when moving from the
\SI{0.25}{\degree} spacing of the modeling study to the \SI{1.0}{\degree}
(or even \SI{2.0}{\degree}) used here, the ranking of coefficients can be
sensitive to bad radials, degrading algorithm performance. When the algorithm
performs well, the results are indeed azimuthally more smooth and
more robust than Self-Consistent itself; this lends credence to the concept of
smoothing coefficients from the Self-Consistent algorithm. However, the choice
of median is not sufficiently robust for operational applications.

The Combined experiment, which used both a \SI{1.0}{\degree} spacing as well
as violating the assumed temperature and drop shape model, demonstrates
the sensitivity of the fixed-coefficient algorithms (Linear and ZPHI) to the
validity of the assumed scattering model for standard resolution data (lower
than that of the modeling study); these methods had up to \SI{10}{dB} of 
errors with the multiple violated assumptions. This highlights the need to use 
more sophisticated methods, like Self-Consistent, that can automatically tune 
coefficients to yield the best results for the data at hand. In the results shown
here, these algorithms (SC and MSC) resulted in an order of magnitude less
bias in estimates of specific attenuation at the individual polarizations.\chapter{Conclusion}
This work examines the practical aspects of performing correction for rain-induced
attenuation in dual-polarization radar data. Given the large biases in reflectivity
data that are caused by attenuation at C- and X-band, correction for this effect
is important for many radar applications, including QPE.

\section{Simulator}
This study of attenuation is conducted through the use of simulated radar data.
The simulation procedure builds on the work of \cite{May2007}, and extends it
to produce dual-polarization time series data. The dual-polarization aspect of the
simulation is accomplished by the inclusion of full T-matrix based scattering
calculations \citep{Mishchenko1993}, as well as supporting the use of
the output from two-moment model microphysics schemes \citep{Ziegler1985}.
These calculations form the foundation for calculating radar intrinsic variables,
like reflectivity factor, specific attenuation, and specific differential phase shift. 
By feeding these starting variables into the path integration procedure of
\cite{May2007}, the path-integrated attenuation and differential phase are
properly simulated, including differential effects across the radar antenna.
The time-series simulation is based upon a combination of the work of
\cite{Muschinski1999} and \cite{Galati1995}; together, these form the
basis of using the radar scattering parameters to generate stochastic time
series data. This stochastic data allows the simulation to accurately capture
the errors in the radar data due to both intrinsic variability and the procedure
of moment estimation; this ensures that errors identified using simulated data
represent those experienced by real-world radar systems.

The use of simulated data is vital to this study. Previous studies of the accuracy
of radar attenuation algorithms, such as \cite{Carey2000} and 
\cite{Gorgucci1998}, rely upon indirect comparisons, either to rain gauge data
(to examine changes to biases in QPE) or to data from a separate,
non-attenuated, radar system.  The benefit of using simulated data is that
the true, non-attenuated reflectivity fields are known and can serve as a direct
truth field in any analysis.

\section{Algorithms}
The attenuation correction techniques examined consist of two broad
families: Linear \citep{Bringi1990} and ZPHI \citep{Testud2000}. The ZPHI
family, in additition to the ZPHI algorithm itself, also includes the Self-Consistent
algorithm \citep{Bringi2001}. This work also tests a proposed Modified Self-
Consistent technique, which attempts to address shortcomings in the
robustness of the standard Self-Consistent algorithm.

All of the aforementioned techniques require empirically determined
coefficients that represent the relations between radar observables and
attenuation. To maximize the potential algorithm performance and because
different algorithms used differing assumptions for coefficients, this work
calculates its own set of ``optimal'' coefficients. This ensures the
coefficients accurately reflect the known assumptions used in the simulation
of data, as well as reflect the drop size distribution used in the underlying
numerical weather simulation. 

\section{Modeling Errors}
The first study conducted examined the sensitivity of the algorithms to 
assumptions made in calculating the coefficients, such as wavelength,
temperature, shape model, etc. The baseline of performance is established
using a Control experiment, where the radar data are simulated using a set
of assumptions that match those used to obtain the coefficients for the
algorithms. Overall, all algorithms worked well at both C- and X-bands in the
Control experiment.

In the other experiments, when these assumptions were systematically
violated, the Linear and ZPHI techniques, because they rely upon fixed
coefficients, yielded much higher errors than the Self-Consistent and
Modified Self-Consistent algorithms, at both X- and C-band. In almost all cases
these latter algorithms were able to deliver reasonable results. While the
biases for individual polarizations from these algorithms could still be in the
range of 
\SI{1}{dB} to \SI{2}{dB}, this was much more usable than the errors for the
Linear and ZPHI algorithms in most cases.

In terms of sensitivity to assumptions, changing any one of them (canting,
shape model, temperature, or wavelength), yielded pronounced changes, both
in terms of the resulting attenuation field and in the results of the algorithms.
This implies that successful use of these techniques depends upon carefully
making assumptions that match nature. For physical processes
(i.e., canting and shape model), this implies the need for good data and
parameterizations of the microphysical processes involved. For something
like temperature, which varies greatly, this implies a greater challenge.
For fixed coefficient techniques, though, the biases introduced are significant, so making
use of any temperature data available could yield significant gains in reducing
the bias in reflectivity due to under- or over-correction for attenuation.

Most important of all effects was the sensitivity to wavelength within the
band. Changing the wavelength by only \SI{10}{\percent} introduced at least
\SI{5}{dB} of error in the attenuation estimates by the Linear and ZPHI
algorithms; this runs counter to the common wisdom that one can use
coefficients for these algorithms that are appropriate to a given band. It is
possible, therefore to gain algorithm performance by simply tuning
coefficients to the true operating wavelength of a given radar system. While
the Self-Consistent algorithms were able to largely compensate for all of the
violated assumptions, including wavelength, the best algorithm performance
occurred for the Control experiment. Therefore, It is important to all algorithms
to take care in making accurate assumptions; using the true wavelength of the
radar, rather than some general waveband, is one of the simplest ways to
eliminate errors.

\section{Spatial Errors}
The second study examined the effects of the spatial sampling 
characteristics of the radar data on algorithm performance. This includes azimuthal effects (sidelobes, mainlobe beam\-width, and radial
spacing), as well as the range resolution effect of changing pulse width and
gate spacing. Remarkably, the algorithms overall showed very little sensitivity
to these changes. The initial Control experiment, which started with matched
beamwidth and radial spacing of \SI{1.0}{\degree} showed no significant
differences from the modeling study. The lack of sensitivity to sampling
continued through the remainder of the experiments that only tested 
aspects of spatial sampling.

One effect that was revealed by this study is that the Modified Self-Consistent
algorithm can be sensitive to the data. By virtue of its single optimized value,
any spuriously obtained coefficient can bias all of the results. While the
Modified Self-Consistent technique does at times show improved, smoother
results, the sensitivity is a problem to its robustness. Further work is necessary
to adjust the technique and find a method that better smooths its coefficients
to produce results that are consistently better than the regular Self-Consistent
technique.

The spatial study also performed an experiment that combined standard
spatial sampling (\SI{1.0}{\degree} azimuthal spacing) with violating the 
assumed temperature and drop shape model. This experiment demonstrates
the sensitivity of the fixed-coefficient algorithms (Linear and ZPHI) to the
validity of the assumed scattering model for standard resolution data (lower
than that of the modeling study); these methods had up to \SI{10}{dB} of 
errors with the multiple violated assumptions. This highlights the need to use 
more sophisticated
methods, like Self-Consistent, that can automatically tune coefficients to
yield the best results for the data at hand; these algorithms, in the cases
studied here, result in an order of magnitude less
bias in estimates of specific attenuation.

\OUsinglespace
\bibliographystyle{ametsoc.bst}
\bibliography{Dissertation}

\end{document}
