When non-hydrometeor scatterers are dominant (e.g., in TDS), the desired weather information, which is given by the non-dominant scatterers, may be lost in bulk radar estimates.
It was mentioned previously that a major limitation with the conventional polarimetric spectral analysis techniques was the requirement of a large number of independent spectra in order to obtain accurate estimates.
Averaging spectra from different range gates, azimuths, dwells, or even by smoothing in the frequency domain, are different methods used in the past to estimate the DPSDs.
However, the associated resolution loss could obscure important details of the underlying phenomena, especially if it is evolving fast in time and in a small spatial extent.
The Bootstrap DPSD estimator provides a means to obtain better quality dual-polarimetric spectra based on a single dwell.
This chapter illustrates the performance of the Bootstrap DPSD estimator when applied to real and simulated data.
The first section presents examples of polarimetric spectral signatures observed in four different datasets obtained with different radars for two tornado cases, with the goal of illustrating the advantages of the Bootstrap DPSD estimator over conventional DPSD estimators, and over range-, azimuth-, scan-, or frequency-averaged DPSDs.
The next section shows examples of DPSDs obtained from a radar simulator, based on raindrops and debris in a tornadic vortex, to assess the impact of tornado dynamics on radar measurements from a spectral polarimetric perspective.

\section{Real Observations} 
The following analyses use I/Q time-series data collected with the KOUN, OU-PRIME, and PX-1000 radars during the 10 May 2010 Moore-Norman, OK, and the 20 May 2013 Moore, OK tornadoes.
A brief description of the radar systems is presented hereafter next and a summary of the characteristics of the radar systems is presented in Table \ref{tab:system}.
KOUN is an S-band polarimetric radar with a $0.9^\circ$ 3-dB beamwidth, range sampling of 250 m, and a peak transmit power of 750 kW; on these dates, it operated with a maximum unambiguous velocity of $27.5$ m/s.
OU-PRIME is a C-band polarimetric radar with a $0.45^\circ$ 3-dB beamwidth, a range sampling of up to 60 m, and a peak transmit power of 1 MW.
During the acquisition of the data, it operated with a range sampling of 125 m and a maximum unambiguous velocity of $16$ m/s.
PX-1000 is a transportable, X-band polarimetric radar, with a $1.8^\circ$ 3-dB beamwidth, a range sampling of up to 30 m, and two identical and independent solid state transmitters with total peak power of 200 W (at the time of data collection).
Data under analysis were collected with a maximum unambiguous velocity of $15.7$ m/s and a range sampling of up to $30$ m.
\begin{table}
\centering
\small
\begin{tabular}{|c|c|c|c|}\hline
\multicolumn{4}{|c|}{Transmitter}  \\ \cline{2-4}
						&	KOUN		&	OU-PRIME	&	PX-1000 \\ \cline{2-4}
Operating frequency			& 2705 MHz	& 5510 MHz		& 9550 MHz \\
Wavelength				& 11.1 cm		& 5.44 cm			& 3.14 cm \\	
Peak transmitter power		& 750 kW			& 1000 kW		& 0.2 kW SSPA*\\
Polarization				& STSR**	& STSR**	& STSR**\\
Pulse lengths				& 1.57 or 4.7 $\mu$s	& 0.4 to 2.0 $\mu$s	& 1 to 70 $\mu$s \\
\hline\hline
\multicolumn{4}{|c|}{Antenna}  \\ \cline{2-4}
						&	KOUN		&	OU-PRIME	&	PX-1000 \\ \cline{2-4}
Diameter					& 8.5 m			& 8.5 m			& 1.2 m \\
Beamwidth (-3 dB)			& 0.9$^\circ$		& 0.45$^\circ$		& 1.8$^\circ$ \\
Gain						& 44.5 dB			& 50 dB			& 38.5 dB \\
First SLL					& Better than -27 dB & Better than -27 dB	& Better than -26 dB \\
Cross-polar isolation		& Better than -35 dB & Better than -35 dB	& Better than -26 dB \\
Rotation rate				& 36$^\circ$/s max	& 30$^\circ$/s max	& 50$^\circ$/s max \\
\hline\hline
\multicolumn{4}{|c|}{Receiver}  \\ \cline{2-4}
						&	KOUN		&	OU-PRIME	&	PX-1000 \\ \cline{2-4}
%Minimum detectable signal	& -112 dBm		& -113 dBm		&
A/D converter bits			& 16 bit			& 16 bit			& 14 bit \\
Gate spacing				& 250 m			& 25-500 m		& 30 m \\ \hline
\end{tabular}
\begin{flushleft}
\scriptsize
* Solid State Power Amplifier. \\
** Simultaneous Transmission - Simultaneous Reception.
\end{flushleft}
\caption{Characteristics of the KOUN, OU-PRIME, and PX-1000 radar systems.}
\label{tab:system}
\end{table}

Detailed technical specifications on OU-PRIME can be found in \citet{Palmer2011}, and on PX-1000 in \citet{Cheong2013}.
Data were collected with KOUN and OU-PRIME radars for the 10 May 2010 case, and with KOUN and PX-1000 for the 20 May 2013 case.
The operational settings corresponding to the available data are summarized in Table \ref{tab:cases}.
\begin{table}
\centering
\small
\begin{tabular}{|c|c|c|}\hline
05/10/2010 data sets& KOUN				& OU-PRIME \\ \hline 
Time (UTC)		& 22:21:33-22:58:28 	& 22:00:47-22:59:12 \\
Scanning strategy	& PPI				& PPI \\
\multirow{2}{*}{Elevation angles ($^\circ$)}	& 0.5, 0.9, 1.4, 1.9, 2.4, 3.2, 4.0, 5.1, & 0.2, 1.0, 2.0, 3.0,  \\
									& 6.4, 8.0, 10.0, 12.5, 15.6, 19.5	& 4.0, 5.0, 6.5, 9.0 \\
Update time		& $\sim$4'20"				& $\sim$2'30" \\
Wavelength		& 11.1 cm				& 5.44 cm \\
PRF				& 1000 Hz			& 1176.5 Hz \\
Maximum unambiguous velocity & 27.5 m/s & 16 m/s \\
Range sampling 	& 250 m & 125 m \\
%Azimuthal resolution & 0.9$^\circ$ & 0.45$^\circ$ \\
\hline\hline
05/20/2013 data sets& KOUN				& PX-1000 \\ \hline 
Time (UTC)		& 19:46:29-20:42:31 	& 19:44:49-20:39:51 \\
Scanning strategy	& PPI				& PPI \\
\multirow{2}{*}{Elevation angles ($^\circ$)}	& 0.5, 1.0, 2.0, 3.0, 4.0,	& \multirow{2}{*}{2.6} \\
									&  5.0, 6.0, 8.0, 10.0 		& \\
Update time		& $\sim$4'20"				& $\sim$20" \\
Wavelength		& 11.1 cm				& 3.14 cm \\
PRF				& 1000 Hz			& 2000 Hz \\
Maximum unambiguous velocity & 27.5 m/s & 15.7 m/s \\
Range sampling 	& 250 m & 30 m \\
%Azimuthal resolution & 0.9$^\circ$ & 1.8$^\circ$ \\ 
\hline
\end{tabular}
\caption{Operational settings and acquisition parameters for the four cases under analysis.}
\label{tab:cases}
\end{table}

Brief descriptions of the two weather events are presented next.
The first case took part in the second largest tornado outbreak documented in Oklahoma, which affected a large part of northern, central, and southern portions of the state.
According to \citet{NWSWFO2010}, during the late afternoon and early evening hours of May 10, 2010, thirteen different storms produced tornadoes, spawning a total of 36 tornadoes in the NWS Norman forecast area alone, and also producing significant structural damage over many areas with estimated losses in excess of \$595 million, three fatalities and over 450 injuries.
It is indicated that due to the potent combination of atmospherical conditions, the storms that developed quickly became tornadic after initiation, with typical storm motions of 50 to 60 mph.
Reports indicate that between 22:33 and 22:59 UTC, three to five tornadoes were occurring simultaneously every minute, which includes two EF-4 tornadoes (the Moore and Norman tornadoes), and two other EF-3 tornadoes.
Additionally, very large hail was reported in several locations with sizes up to 4.25" in diameter (softball size).
A detailed report of this event can be found in \citet{NWSWFO2010}.

The second case was part of another tornado outbreak, which took place in parts of the midwest and the Great Plains from May 18-20, 2013. 
According to \citet{NWSWFO2013}, where the most relevant aspects are summarized, during the afternoon and evening hours of May 20, 2013, several supercell thunderstorms developed along a dryline in central Oklahoma. 
It is indicated that one of these storms developed and rapidly intensified, producing a tornado to the west of Newcastle. 
The tornado, which touched down at 19:56 UTC, quickly became violent with an east-northeastward direction, across Moore and parts of south Oklahoma City causing catastrophic damage in these areas for about 40 minutes before finally dissipating.
The reported losses due to the tornado were over billions of dollars in damage, 24 fatalities, and over 200 injuries.
It was given a maximum rating of EF-5, making it the deadliest and most devastating tornado of the year in the United States.
In addition to the tornadoes, large hail and damaging winds caused significant damage in many areas.
A detailed report of this event can be found in \citet{NWSWFO2013}.

In-depth analyses based on weather radar observations of the 10 May 2010 case can be found in \citet{Bodine2011}, and of the 20 May 2013 case in \citet{Kurdzo2015}.
Hence, no discussions about the spatial and temporal evolution or characteristics of the storm will be given in this work.
The following sections will provide some examples of polarimetric spectral signatures estimated with the Bootstrap DPSD estimator that are not captured by the bulk polarimetric variables.

\subsection{Case 1: 10 May 2010, OU-PRIME and KOUN}
To make a fair comparison between the datasets corresponding to different radars, it is important that the spatial and temporal characteristics of the event are matched as closely as possible.
In this example, the KOUN set corresponding to 22:22:38 UTC with an elevation angle of $1.36^\circ$, and the OU-PRIME set corresponding to 22:23:22 with an elevation angle of $0.96^\circ$ were selected, which are approximately one minute apart.
The PPIs for the selected scans for each radar are shown in Figure \ref{fig:PPI_1}.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/chap5/KOUN_PPI_1.eps}
(a)
\includegraphics[width=1\linewidth]{figures/chap5/OUPRIME_PPI_1.eps}
(b)
\caption{PPI plots corresponding to (a) KOUN data at 22:22:38 UTC, and elevation $\phi=1.36^\circ$ with azimuth $\theta=30^\circ$ highlighted, and (b) OU-PRIME data at 22:23:22 UTC, and elevation $\phi=0.96^\circ$ with azimuth $\theta=25^\circ$ highlighted: $\mathrm{SNR}_\mathrm{H}$ (top left), $\bar{v}_r$ (top right), $Z_\mathrm{DR}$ (bottom left), and $\rho_\mathrm{HV}$ (bottom right). Grid lines are $30^\circ$ and 10 km apart.}
\label{fig:PPI_1}
\end{figure}
KOUN data were grouped into $2.0^\circ$ radials with a $0.5^\circ$ azimuthal spacing, yielding approximately 79 pulses per radial.
OU-PRIME data were grouped into $1.0^\circ$ radials with a $0.5^\circ$ azimuthal spacing, yielding approximately 70 pulses per radial.
Range-Doppler plots are useful for spectral analyses, as they illustrate spectral variables as a function of range and radial velocity, with the intensity representing the particular spectral variable. 
Each row in the $y$-axis of the range-Doppler plots represents the spectrum for a given range gate, and the $x$-axis represents the radial velocity.
With the aid of range-Doppler plots, it is possible to observe the spatial distribution and radar-relative motion of the scatterers in a particular radial; and with the DPSDs, it is also possible to detect any significantly different scatterer signatures for given radar resolution volumes. 
First, the differences between the different DPSD estimators will be illustrated.
By computing the DPSD estimates, distinct spectral signatures can be found near azimuth $\theta = 30^\circ$ in KOUN data, and $\theta=25^\circ$ in OU-PRIME data, as depicted in Figures \ref{fig:dpsd_1}-\ref{fig:dpsd_1_periodogram}.
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/KOUN_1_DPSD.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/OUPRIME_DPSD.eps}\\
(b)
\caption{Range-Doppler plots of single-dwell Bootstrap DPSD estimates corresponding to: (a) KOUN data at 22:22:38 UTC, elevation $\phi=1.36^\circ$, and azimuth $\theta=30^\circ$, and (b)  OU-PRIME data at 22:23:22 UTC, elevation $\phi=0.96^\circ$, and azimuth $\theta=25^\circ$ (after velocity dealiasing).}
\label{fig:dpsd_1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/KOUN_1_DPSD_daniell.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/OUPRIME_DPSD_daniell.eps}\\
(b)
\caption{Same as Figure \ref{fig:dpsd_1}, but for Daniell DPSD estimates.}
\label{fig:dpsd_1_daniell}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/KOUN_1_DPSD_periodogram.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/OUPRIME_DPSD_periodogram.eps}\\
(b)
\caption{Same as Figure \ref{fig:dpsd_1}, but for periodogram DPSD estimates.}
\label{fig:dpsd_1_periodogram}
\end{figure}
The DPSDs were estimated with a Blackman window (-58 dB SLL), with no zero-padding for the DFT.
Furthermore, a 20-dB SNR threshold is used to censor low-SNR spectral coefficients.
For a single dwell (i.e., $K=1$), both Bootstrap and Daniell DPSD estimators provide qualitatively meaningful results, while the periodogram estimator fails to yield useful results.
For an easier interpretation, the velocities were manually de-aliased (i.e., $x$-axis from 0 to +2$v_\mathrm{a}$) with information about the storm motion, illustrating the peak of the stronger signal moving at similar radial velocities from both radar measurements.
Additionally, estimates with $K=3$ averaged spectra from adjacent radials are included for clarity.
Each spectrum corresponds to a coverage of approximately $4^\circ$ for the KOUN dataset and $2.5^\circ$ for the OU-PRIME dataset (Figures \ref{fig:dpsd_3}-\ref{fig:dpsd_3_periodogram}).
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/KOUN_1_DPSD_K_3.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/OUPRIME_DPSD_K_3.eps}\\
(b)
\caption{Range-Doppler plots of $K=3$ averaged Bootstrap DPSD estimates corresponding to: (a) KOUN data at 22:22:38 UTC, elevation $\phi=1.36^\circ$, and azimuth $\theta=30^\circ$, and (b)  OU-PRIME data at 22:23:22 UTC, elevation $\phi=0.96^\circ$, and azimuth $\theta=25^\circ$.}
\label{fig:dpsd_3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/KOUN_1_DPSD_daniell_K_3.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/OUPRIME_DPSD_daniell_K_3.eps}\\
(b)
\caption{Same as Figure \ref{fig:dpsd_3}, but for Daniell DPSD estimates.}
\label{fig:dpsd_3_daniell}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/KOUN_1_DPSD_periodogram_K_3.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/OUPRIME_DPSD_periodogram_K_3.eps}\\
(b)
\caption{Same as Figure \ref{fig:dpsd_3}, but for periodogram DPSD estimates.}
\label{fig:dpsd_3_periodogram}
\end{figure}
It can be observed that the spectral signatures are similar in both cases (using $K=1$ and $K=3$), although the single-dwell estimates are noisier.
As it has been mentioned in previous chapters, it is desirable to minimize the averaging in any dimension to avoid masking spectral features that may be present.
Upon a quick examination, the Daniell estimates are qualitatively similar to the Bootstrap estimates as previously shown.
However, the Daniell estimator smoothes the spectra in frequency, and it also does not take into account additional signal information obtainable with bootstrapping.
Hence, the  $s\rho_\mathrm{HV}$ tends to be higher for higher SNR spectral coefficients and lower where the spectral coefficients are more affected by the smearing.
Moreover, it can be demonstrated that the quality of the Daniell estimates are significantly degraded in cases with lower frequency resolution; i.e., for lower number of samples per dwell.
By arbitrarily reducing the azimuthal extent of a radial to $0.5^\circ$, and keeping an azimuthal spacing of $0.5^\circ$ for the KOUN dataset at the same azimuth of $\theta = 25^\circ$, the number of pulses per radial is approximately $20$.
Figure \ref{fig:dpsd_lessdeg} shows the estimated DPSDs, where the spectral broadening due to averaging in frequency of the Daniell estimates is more evident.
Additionally, the region around 31.5 km in range is continuously high in $s\rho_\mathrm{HV}$, which could be interpreted as one signal with a wide spectrum width, depicted in Figure \ref{fig:dpsd_lessdeg_line}.
\begin{figure}
\centering
\begin{minipage}[c]{0.495\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap5/KOUN_lessdeg.eps}\\
(a)\\
\end{minipage}
\begin{minipage}[c]{0.495\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap5/KOUN_lessdeg_daniell.eps}\\
(b)\\
\end{minipage}
\includegraphics[width=1.0\linewidth]{figures/chap5/b_d_comp.pdf}\\
(c)\\
\caption{Range-Doppler plots of single-dwell DPSD estimates corresponding to KOUN data at 22:22:38 UTC, elevation $\phi=1.36^\circ$, azimuth $\theta=30^\circ$ and $0.5^\circ$ radials with an average of $20$ pulses per radial, for (a) Bootstrap estimator, (b) Daniell estimator, and (c) zoomed comparison of both estimates.}
\label{fig:dpsd_lessdeg}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap5/lessdeg.eps}
\caption{Line spectra at $31.5$ km for the range-Doppler plots of Figure \ref{fig:dpsd_lessdeg}. Bootstrap (blue) and Daniell (green) estimates illustrating the effects of degraded frequency resolution.}
\label{fig:dpsd_lessdeg_line}
\end{figure}


Moving on to a more interpretative analysis of the estimates, typical features such as unimodal weather signal, bimodal weather, and ground clutter are readily apparent in the range-Doppler plots.
Examples of these typical DPSDs are shown in Figure \ref{fig:dpsd_typical}.
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/wx_dpsd.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/wxgc_dpsd.eps}\\
(b)
\caption{Plots of Bootstrap DPSD estimates corresponding to KOUN data at 22:22:38 UTC, elevation $\phi=1.36^\circ$, and azimuth $\theta=30^\circ$: (a) weather signature at 9.5 km, and (b) weather and ground clutter signature at 4.25 km.}
\label{fig:dpsd_typical}
\end{figure}
Other interesting signatures are the unimodal signal with markedly different polarimetric characteristics for different radial velocities spanning from approximately 20 km to 40 km in range in the OU-PRIME observation, the bimodal signal spanning from approximately 20 km to over 50 km in range in the KOUN observation, and a multimodal signal composed of weather, ground clutter, and an isolated peak at $\sim$8 km in the KOUN observation.
The DPSDs of these signatures are shown in Figure \ref{fig:dpsd_other}.
\begin{figure}
\begin{minipage}[c]{0.495\linewidth}
\centering
\includegraphics[width=1.12\linewidth]{figures/chap5/unimodal2_dpsd.eps}\\
(a)\\
\end{minipage}
\begin{minipage}[c]{0.495\linewidth}
\centering
\includegraphics[width=1.12\linewidth]{figures/chap5/bimodal_dpsd.eps}\\
(b)\\
\end{minipage}
\centering
\includegraphics[width=0.6\linewidth]{figures/chap5/multimodal_dpsd.eps}\\
(c)\\
\caption{Plots of Bootstrap DPSD estimates corresponding to: (a) unimodal spectra with different polarimetric characteristics at 36 km, (OU-PRIME) (b) bimodal spectra at 32.25 km (KOUN), and (c) multimodal spectra at 9 km (KOUN).}
\label{fig:dpsd_other}
\end{figure}
While it is difficult to provide a precise explanation for these observations, some hypotheses can be elaborated. %fig:dpsd_3
The unimodal signal with differences in $sZ_\mathrm{DR}$ (Figure \ref{fig:dpsd_3}b, between approximately 20 and 42.5 km in range) may be attributed to smaller raindrops being centrifuged or size sorting, while the decreasing values in range would possibly be due to attenuation. 
However, collocated with these $sZ_\mathrm{DR}$ values are really low $s\rho_\mathrm{HV}$ values, which might be an indication of non-meteorological scatterers.
An example of the spectra at a range gate of 36 km is depicted in Figure \ref{fig:dpsd_other}a.
Additionally, since OU-PRIME operates at C-band, it might be more susceptible to non-Rayleigh scattering from scatterers relatively larger than hydrometeors.
Also, another indication that supports the hypothesis of non-meteorological scatterers lies farther ahead in range, where a wide distribution of lower $s\mathrm{SNR}$ with negative $sZ_\mathrm{DR}$ and low $s\rho_\mathrm{HV}$ is surrounding the weather signal.
The KOUN spectral data also shows a weaker signal of low $sZ_\mathrm{DR}$ that is not apparent in the OU-PRIME spectral data.
This weaker signal is observed within a bimodal spectrum (Figure \ref{fig:dpsd_other}b), along with a stronger weather signal with varying $sZ_\mathrm{DR}$ values, and relatively high $s\rho_\mathrm{HV}$ values.
Although the weaker signal has high $s\rho_\mathrm{HV}$ values in the KOUN data, this may be due to the fact that these presumably non-hydrometeor scatterers might be small in size such that the scattering is mostly homogeneous.
Another plausible explanation is that the weaker signal corresponds to a range-folded echo, as evidence of range folding was observed in the vicinity of the radial under analysis in later PPI scans for the same dataset.
If such were the case, the range-folded echoes could be mitigated with polarimetric spectral analysis, showing another important potential application of the Bootstrap DPSD estimator.
Finally, a signal with multiple peaks in the spectrum (Figure \ref{fig:dpsd_other}c) can be observed in range gates near 8 km in the KOUN range-Doppler plots.
The peak at approximately 2 m/s corresponds to weather, with $sZ_\mathrm{DR}$ values close to 0 (small raindrops) and high $s\rho_\mathrm{HV}$.
A wider signal with negative $sZ_\mathrm{DR}$ and varying values of $s\rho_\mathrm{HV}$ can be seen with a mean radial velocity of approximately 14 m/s. The polarimetric characteristics of this spectral signature are similar to those of debris.
And also, an isolated peak of a scatterer with high $sZ_\mathrm{DR}$ and high $s\rho_\mathrm{HV}$ with a motion of over 20 m/s can be seen in the spectra.
This unknown scatterer could be a single large piece of debris being lofted in the air, and a hypothesis for the velocity being higher than the surrounding wind could be explained if the object was ejected radially in the direction of the radar beam with a higher tangential velocity.
Alternatively, since ground clutter contamination is still affecting at this elevation angle and range, the echo could correspond to a vehicle.
It is important to note that these characteristics can be appreciated even with single dwell DPSDs, showing the potential of the Bootstrap DPSD estimator in different spectral analysis applications.
 
\subsection{Case 2: 20 May 2013, KOUN and PX-1000}
A PPI of the selected elevation angle and time is shown in Figure \ref{fig:PPI_2}.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/chap5/KOUN_PPI_2.eps}
(a)
\includegraphics[width=1\linewidth]{figures/chap5/PX1000_PPI.eps}
(b)
\caption{PPI plots corresponding to (a) KOUN data at 20:23:59 UTC, and elevation $\phi=2.99^\circ$ with azimuth $\theta=27^\circ$ highlighted, and (b) PX-1000 data at 20:23:56 UTC, and elevation $\phi=2.6^\circ$ with azimuth $\theta=21^\circ$ highlighted: $\mathrm{SNR}_\mathrm{H}$ (top left), $\bar{v}_r$ (top right), $Z_\mathrm{DR}$ (bottom left), and $\rho_\mathrm{HV}$ (bottom right). Grid lines are $30^\circ$ and 10 km apart.}
\label{fig:PPI_2}
\end{figure}
For the PX-1000 dataset, the update time is 20 seconds, the range gate spacing is 30 m with a resolution of 112 m, while the azimuthal spacing is $0.5^\circ$ with a resolution of $1.8^\circ$.
Since range is oversampled, it is possible to compute range-averaged DPSDs without significantly compromising the amount of information.
Additionally, with faster update times, it would also be reasonable to obtain dwell-averaged DPSDs, as long as the underlying phenomenon is not evolving too rapidly.
A comparison of DPSDs computed by averaging in different dimensions is possible with this dataset, which will be discussed next.
The single-dwell Bootstrap DPSD estimate with $K=1$ (first column, Figure \ref{fig:dpsd_dim_avg}) is compared to those that resulted from the Daniell estimator with a three-point moving average (second column, Figure \ref{fig:dpsd_dim_avg}), and by averaging three gates in range (third column, Figure \ref{fig:dpsd_dim_avg}), three radials in azimuth (fourth column, Figure \ref{fig:dpsd_dim_avg}), and three scans in time (last column, Figure \ref{fig:dpsd_dim_avg}).
%, and three spectral coefficients in frequency (Figure \ref{fig:dpsd_frequency}).
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap5/dim_avg_title.pdf}\\
\begin{minipage}[c]{0.495\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap5/dim_avg_snrh.pdf}\\
(a)\\
\end{minipage}
\begin{minipage}[c]{0.495\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap5/dim_avg_snrv.pdf}\\
(b)\\
\end{minipage}
\begin{minipage}[c]{0.495\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap5/dim_avg_szdr.pdf}\\
(c)\\
\end{minipage}
\begin{minipage}[c]{0.495\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap5/dim_avg_srhv.pdf}\\
(d)\\
\end{minipage}
\includegraphics[width=1.0\linewidth]{figures/chap5/cbar.pdf}\\
\caption{Comparison of different averaging methods. PX-1000 data at 20:23:56 UTC, elevation $\phi=2.6^\circ$ and azimuth $\theta=21^\circ$, estimated with the Bootstrap DPSD estimator using a single dwell (first column), with the Daniell three-point single-dwell estimator (second column), range averaging estimator (third column), azimuth averaging estimator (fourth column), and scan-to-scan averaging estimator (fifth column), for: (a) $s\mathrm{SNR}_\mathrm{H}$, (b) $s\mathrm{SNR}_\mathrm{V}$, (c) $sZ_\mathrm{DR}$, and (d) $s\rho_\mathrm{HV}$.}
\label{fig:dpsd_dim_avg}
\end{figure}

%\begin{figure}
%\centering
%\includegraphics[width=.65\linewidth]{figures/chap5/PX1000_DPSD.eps}
%\caption{Range-Doppler plots of PX-1000 data at 20:23:56 UTC, elevation $\phi=2.6^\circ$ and azimuth $\theta=21^\circ$, estimated with the Bootstrap DPSD estimator using a single dwell (no averaging, $K=1$).}
%\label{fig:dpsd_single}
%\end{figure}
With range averaging, some information in range may be obscured, though if range is oversampled, the impact on the quality of DPSD estimates should be minimal.
%\begin{figure}
%\centering
%\includegraphics[width=.65\linewidth]{figures/chap5/PX1000_DPSD_daniell_range.eps}
%\caption{Same as Figure \ref{fig:dpsd_single}, but using the Daniell (three-point moving average) estimator with range averaging ($K=3$).}
%\label{fig:dpsd_range}
%\end{figure}
The azimuth-averaged DPSDs exhibit spectral leakage, mainly due to discontinuities in azimuth during the data acquisition.
Spectral products are vulnerable to these artifacts, and in such a case, averaging in another dimension would be more appropriate.
%\begin{figure}
%\centering
%\includegraphics[width=.65\linewidth]{figures/chap5/PX1000_DPSD_daniell_azimuth.eps}
%\caption{Same as Figure \ref{fig:dpsd_range}, but with azimuthal averaging ($K=3$).}
%\label{fig:dpsd_azimuth}
%\end{figure}
Additionally, scan-to-scan-averaging does not reduce the resolution in range or azimuth, but it is sensitive to the time between scans.
By averaging three dwells that are 20 seconds apart, it yields an effective observation time of approximately 1 minute.
If the phenomena is evolving fast in time, extraneous information may be gathered by averaging in time; e.g., the signature with high $s\rho_\mathrm{HV}$ with mean velocities close to 12.5 m/s is absent from all other DPSD estimates.
%\begin{figure}
%\centering
%\includegraphics[width=.65\linewidth]{figures/chap5/PX1000_DPSD_daniell_time.eps}
%\caption{Same as Figure \ref{fig:dpsd_range}, with scan-to-scan averaging ($K=3$)}
%\label{fig:dpsd_time}
%\end{figure}
%Averaging in frequency does not require spectra from additional range gates or scans, but it ``smears" the spectral information, as depicted in Figure \ref{fig:dpsd_frequency}.
%As previously mentioned, this is especially important in cases where the number of samples per dwell is small, where the frequency resolution is already poor.
%By averaging in frequency, this low resolution can degrade even further and the ability to discriminate features in spectra could be severely hindered.
%\begin{figure}
%\centering
%\includegraphics[width=.65\linewidth]{figures/chap5/PX1000_DPSD_frequency.eps}
%\caption{Same as Figure \ref{fig:dpsd_single}, but using frequency averaging ($K=1$).}
%\label{fig:dpsd_frequency}
%\end{figure}

Similar spectral signatures as those found in previous examples can also be observed in the KOUN dataset for the 20 May 2013 case.
Using a single spectrum for DPSD estimation ($K=1$) and at an azimuth of $\theta=27^\circ$, a unimodal spectra with marked polarimetric characteristics can be observed (Figure \ref{fig:KOUN_2_DPSD}).
\begin{figure}
\centering
\includegraphics[width=.9\linewidth]{figures/chap5/KOUN_2_DPSD_new.eps}
\caption{Range-Doppler plots of KOUN data at 20:23:59 UTC, elevation $\phi=2.99^\circ$ and azimuth $\theta=27^\circ$, estimated with the Bootstrap DPSD single-dwell estimator ($K=1$).}
\label{fig:KOUN_2_DPSD}
\end{figure}
The weaker signal is collocated with low $sZ_\mathrm{DR}$, while the stronger signal has small $sZ_\mathrm{DR}$.
The $s\rho_\mathrm{HV}$ behaves differently for both signals as their power ratio varies in range.
From ranges 32 to 34 km, the dominant signal has high $sZ_\mathrm{DR}$, and as the dominant signal power weakens for higher positive radial velocity (i.e., the right edge of the spectral signature), the mixing is stronger such that none of the signals are dominant, thus exhibiting a lower $s\rho_\mathrm{HV}$ (Figure \ref{fig:r33km}a).
From ranges 25 to 31 km, a similar effect is observed but the dominant signal is weaker for lower radial velocities; i.e., left side of the spectrum (Figure \ref{fig:r33km}b).
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/r33km.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/r27km.eps}\\
(b)
\caption{Plots of Bootstrap DPSD estimates corresponding to KOUN data at 20:23:59 UTC, elevation $\phi=2.99^\circ$, and azimuth $\theta=27^\circ$. Signatures at: (a) 33.5 km, and (b) 27.75 km.}
\label{fig:r33km}
\end{figure}
From ranges 22 to 24 km, the low $sZ_\mathrm{DR}$ signal is wider and less dominant, while the high $sZ_\mathrm{DR}$ signal is narrower  and likely moving at about the same mean radial velocity as the less dominant signal; these marked differences in the unimodal spectra are more evident in $sZ_\mathrm{DR}$ (Figure \ref{fig:r22km}a).
From ranges 19 to 21 km, an overall decrease in $s\rho_\mathrm{HV}$ is evident, which could be explained by no signal being completely dominant over the other in the spectra (Figure \ref{fig:r22km}b).
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/r22km.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/r20km.eps}\\
(b)
\caption{Same as Figure \ref{fig:r33km} but for signatures at: (a) 22.5 km, and (b) 20 km.}
\label{fig:r22km}
\end{figure}
And finally, from ranges 18 km and below, the power of the high $sZ_\mathrm{DR}$ signal starts weakening as the wider, low $sZ_\mathrm{DR}$ signal starts exhibiting higher $s\rho_\mathrm{HV}$ (Figure \ref{fig:r17km}a).
Another spectral signature featured in this radial is the helicopter at about 7 km in range (Figure \ref{fig:r17km}b).
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/r17km.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/r7km.eps}\\
(b)
\caption{Same as Figure \ref{fig:r33km} but for signatures at: (a) 17.75 km, and (b) 7.5 km.}
\label{fig:r17km}
\end{figure}
It is known that this particular scatterer is a helicopter because it exhibits (in PPI plots, not shown) spatially isolated, unusually high returned power, rapid movement as a function of time, mean radial velocities that do not follow the surrounding winds, and non-homogeneous scattering (low $\rho_\mathrm{HV}$).
One last example of a spectral signature of a TDS range gate is shown in Figure \ref{fig:tds_dpsd}.
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/db_line_data.eps}
\caption{Plots of Bootstrap DPSD estimates corresponding to KOUN data at 20:14:16 UTC, elevation $\phi=1.98^\circ$, azimuth $\theta=-29^\circ$, and range $r=9.5$ km. The dashed black lines in $s\hat{Z}_\mathrm{DR}$ and $s\hat{\rho}_\mathrm{HV}$ indicate the values of the bulk estimates.}
\label{fig:tds_dpsd}
\end{figure}
The range gate corresponds to KOUN data at 20:14:16 UTC, elevation $\phi=1.98^\circ$, azimuth $\theta=-29^\circ$, and range 9.5 km, with a $\hat{Z}_\mathrm{DR}$ of -2.83 dB and a $\hat{\rho}_\mathrm{HV}$ of 0.50, which is consistent with a typical TDS.
The spectra is multimodal and it exhibits different spectral polarimetric signatures collocated with different peaks in radial velocity, though it is unlikely that they are associated with typical weather signals due to the difference in polarimetric characteristics.
Additionally, an isolated peak at 15 m/s can be observed with $s\hat{Z}_\mathrm{DR} < -10$ dB, which may be debris that is being ejected from the vortex.

In summary, a few examples of the Bootstrap DPSD estimator applied to real data have been shown.
Additionally, it has been demonstrated that the Bootstrap DPSD estimates show more details compared to the conventional estimators.
Also, it was possible to identify spectral signatures using single-dwell estimates, in contrast to averaging in range, azimuth, scan, or frequency, where the resolution is degraded.
The following section presents results of the Bootstrap DPSD estimator applied to data from a physically based simulator.

\section{Simulated Observations}
To generate realistic I/Q weather radar signals in a tornadic environment, a polarimetric radar time-series simulator \citep{Cheong2015}, SimRadar, is used (Figure \ref{fig:simradar}).
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/chap5/simradar.png}
\caption{User interface of the polarimetric radar time-series simulator (SimRadar).}
\label{fig:simradar}
\end{figure}
Briefly, this physical simulator synthesizes radar returns based on the position, velocity, orientation, and radar cross section (RCS) of scatterers, which include both raindrops and debris.
It ingests large-eddy-simulation (LES) models for tornado-vortex wind fields and trajectories of the scatterers.
The tumbling behavior of non-spherical objects and the drag forces are calculated based on a 6DOF model \citep{Maruyama2012, Richards2012} and LES-meteorological-background wind vectors.
For the RCS of hydrometeors, the electrical scattering model of an oblate raindrop is used \citep{Bringi2001}, and for debris, the RCS measurements of a leaf at different orientations are ingested into the radar simulator. 
The body of the leaf is modeled by a rectangular plate of 6-cm wide by 8-cm long; the stem is modeled as a cylinder of 12-cm long attached to the body.
The DSD of the raindrops is modeled with a Marshall-Palmer distribution.
Finally, the contribution of each scatterer is weighted by its radial and angular position relative to the radar beam. 
To produce the simulated I/Q time-series signals, contributions from all scatterers are coherently summed after updating the physical properties of the scene at each time step.
Detailed technical specifications on the simulation framework can be found in \citet{Cheong2015}.
Thanks to the versatility of the simulator, many scenarios with different radar acquisition and simulation scene parameters can be generated, making it an incredibly powerful and useful tool to tackle the scientific questions regarding tornado dynamics and interaction with debris. 
While the goal here is not to answer these scientific questions, a few examples of the Bootstrap DPSD estimates of simulated data are presented next to illustrate the potential utility of spectral-analysis tools in the study of tornado dynamics.

\subsection{Simulation Case 1: Weather Only}
The radar parameters (Table \ref{tab:sim1}) are similar to typical WSR-88D values, though the PRT is arbitrarily modified to ensure there is no aliasing in velocity, and the range resolution is 30 m with a gate spacing of 15 m.
The scene is populated with $1.024\times 10^6$ raindrops with 5 different diameters for the weather-only case, and $1.024\times 10^6$ leaves for the debris-only case.
In both cases, there are about 240 scatterers per cell in both cases.
The rest of the simulation parameters are listed in Table \ref{tab:sim1}.
\begin{table}
\centering
\begin{tabular}{|c|c|} \hline
\multicolumn{2}{|c|}{Radar parameters}	\\ \hline
PRT				&$0.2$ ms		\\
Wavelength		&$10$ cm			\\	
Peak transmit power	& 50 kW			\\
Antenna gain		& 50 dBi			\\
Beamwidth		& $1.0^\circ$		\\
Pulse width		& 0.2 $\mu$s		\\
Range resolution	& 30 m			\\
Gate spacing		& 15 m			\\
Samples per dwell	& 100			\\
Azimuthal sampling	& $0.5^\circ$		\\
Maximum unambiguous velocity	& 82.5 m/s	\\ \hline
\end{tabular}

\begin{tabular}{|c|c|c|}\hline
\multicolumn{3}{|c|}{Simulation domain}									\\ \hline
Scatterer type				& Raindrops	& Debris (Leaf)					\\
Number of scatterers		& $1.024\times 10^6$ & $1.024\times 10^6$			\\
Density					& \multicolumn{2}{|c|}{$\sim 470$ particles per cell}	\\
DSD						& \multicolumn{2}{|c|}{Marshall-Palmer}			\\ 
Number of diameter classes	& \multicolumn{2}{|c|}{5 (1, 2, 3, 4, 5 mm)}			\\
LES model				& \multicolumn{2}{|c|}{Suction vortices}			\\ 
Size of simulation domain	&  \multicolumn{2}{|c|}{864.77 m x 738.81 m x 217.02 m} \\
\hline
\end{tabular}
\caption{Radar acquisition and simulation domain parameters.}
\label{tab:sim1}
\end{table}
The PPI plots of the simulation with weather-only scenes are shown in Figure \ref{fig:sim_ppi_1}.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/chap5/SIM_WX_PPI.eps}
\caption{PPI plots of simulated weather-only data at elevation $\phi=3.00^\circ$: $\mathrm{SNR}_\mathrm{H}$ (top left), $\bar{v}_\mathrm{r}$ (top right), $Z_\mathrm{DR}$ (bottom left), and $\rho_\mathrm{HV}$ (bottom right).}
\label{fig:sim_ppi_1}
\end{figure}
Briefly, the weather-only case shows the tornado vortex structure in the SNR field and a TVS in the velocity field.
Also, the $Z_\mathrm{DR}$ plot shows that moderately-sized raindrops with $Z_\mathrm{DR}$ of approximately 2 dB are moving at a distance of $\sim$200 m in radius and with smaller sized raindrops in the core of the vortex, resembling what appears to be a ring in the SNR and $Z_\mathrm{DR}$ fields.
The $\rho_\mathrm{HV}$ is also consistently high, as expected, since the simulation domain is only populated with raindrops.
Spectral analysis shows unimodal weather spectra for most parts of the domain.
As an example, a range-Doppler plot of a radial corresponding to an azimuth of $\theta = -10^\circ$ depicts this signature (Figure \ref{fig:sim_rd_wx}).
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/wx_rd_-10.eps}
\caption{Range-Doppler plots of simulated weather data at elevation $\phi=3.00^\circ$ and azimuth $\theta = -10^\circ$.}
\label{fig:sim_rd_wx}
\end{figure}
Moreover, some signatures of possible size sorting due to the drag forces can be observed as larger drops (larger $sZ_\mathrm{DR}$) are collocated with smaller radial velocities than smaller raindrops (smaller $sZ_\mathrm{DR}$).
Again, the correlation coefficient, indicates high values owing to the homogeneity of the scatterers in motion.
The DPSDs of a particular range gate exhibiting this behavior are shown in Figure \ref{fig:sim_dpsd_wx}.
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/wx_sim_dpsd.eps}
\caption{Bootstrap DPSD estimates of simulated weather data at elevation $\phi=3.00^\circ$, azimuth $\theta = -10^\circ$ and $r=1.925$ km.}
\label{fig:sim_dpsd_wx}
\end{figure}

\subsection{Simulation Case 2: Debris Only}
A more interesting case is the one with the debris signals only (Figure \ref{fig:sim_ppi_db}), where a ring can be appreciated in the SNR and $Z_\mathrm{DR}$ fields, while $\rho_\mathrm{HV}$ exhibits a low value region collocated near the core of the tornado vortex.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/chap5/SIM_DB_PPI.eps}
\caption{PPI plots of simulated debris-only data at elevation $\phi=3.00^\circ$: $\mathrm{SNR}_\mathrm{H}$ (top left), $\bar{v}_\mathrm{r}$ (top right), $Z_\mathrm{DR}$ (bottom left), and $\rho_\mathrm{HV}$ (bottom right).}
\label{fig:sim_ppi_db}
\end{figure}
The low to negative values of $Z_\mathrm{DR}$ could be explained by some degree of common alignment of the leaves, which is also supported by the fact that the $\rho_\mathrm{HV}$ in the annular region is also high (i.e., common alignment results in more homogeneous scattering).
A range-Doppler plot of a radial across the core (azimuth $\theta = 1.5 ^\circ$, Figure \ref{fig:sim_rd_db}) shows multimodal spectra in power, as well as a wide range of $sZ_\mathrm{DR}$ values, while the $s\rho_\mathrm{HV}$ suggests centrifuging of debris.
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/db_rd_1_50.eps}
\caption{Range-Doppler plots of simulated debris data at elevation $\phi=3.00^\circ$ and azimuth $\theta = 1.5^\circ$.}
\label{fig:sim_rd_db}
\end{figure}
With the radar beam pointing approximately perpendicular to the vortex-relative wind direction, a reasonable assumption is that most leaves that are carried by the wind will have a mean radial velocity close to zero.
This observation is clearer between ranges $1.85$ and $2.05$ km, and between $2.25$ and $2.4$ km.
Additionally, the high $s\rho_\mathrm{HV}$ values collocated with the mean radial velocities suggest higher homogeneity of the scatterers, which in turn supports the hypothesis of commonly aligned debris.
Finally, another interesting signature observed in these range-Doppler plots is how a few of the non-dominant peaks in the spectra exhibit higher motion towards (away from) the radar for the southern (northern) region of the vortex, as if these debris were being ejected radially outwards from the core (Figures \ref{fig:sim_db_dpsd}a and \ref{fig:sim_db_dpsd}b).
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/chap5/db_line_1_940.eps}\\
(a)\\
\includegraphics[width=0.72\linewidth]{figures/chap5/db_line_2_360.eps}\\
(b)
\caption{Bootstrap DPSD estimates simulated debris data at elevation $\phi=3.00^\circ$ and azimuth $\theta=1.5^\circ$ at: (a) 1.94 km, and (b) 2.36 km.}
\label{fig:sim_db_dpsd}
\end{figure}
They are also collocated with low $s\rho_\mathrm{HV}$ values, suggesting these leaves are tumbling rapidly and are not commonly aligned with the wind motion.

\subsection{Simulation Case 3: Weather and Debris Mix}
The next simulated scenario is composed of weather and debris (Figure \ref{fig:sim_wd_ppi}), each with a count of $1.024 \times 16^6$, totaling $2.048 \times 10^6$ particles with a density of approximately 470 particles per radar resolution volume, and with the same radar acquisition parameters.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/chap5/SIM_WD_PPI.eps}
\caption{PPI plots of simulated weather and debris data at elevation $\phi=3.00^\circ$: $\mathrm{SNR}_\mathrm{H}$ (top left), $\bar{v}_\mathrm{r}$ (top right), $Z_\mathrm{DR}$ (bottom left), and $\rho_\mathrm{HV}$ (bottom right).}
\label{fig:sim_wd_ppi}
\end{figure}
It is possible to analyze each signal and the composite signal separately, with the goal of testing hypotheses regarding velocity biasing.
In a raindrops-only scenario, the differential velocity (i.e., the difference between the velocities estimated from the H and V channels) is expected to be minimal, provided that the raindrops are small, since the velocities estimated from each channel should be approximately equal.
This hypothesis is tested with the weather-only signal (Figure \ref{fig:sim_wx_dv}), and the results are as expected.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/chap5/difv_wx.eps}
\caption{PPI plot of differential velocity for weather only at elevation $\phi=3.00^\circ$.}
\label{fig:sim_wx_dv}
\end{figure}
Following a similar logic, the differential velocity for the composite weather-and-debris signal is expected to have a higher dispersion since the radial velocities estimated from either channel are more vulnerable to biases owing to the possibility that one signal could be more dominant than the other for a given polarization, and vice versa.
The differential velocity of the weather-and-debris scenario is shown in Figure \ref{fig:sim_wd_dv}.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/chap5/difv_wd.eps}
\caption{PPI plot of differential velocity for weather and debris at elevation $\phi=3.00^\circ$.}
\label{fig:sim_wd_dv}
\end{figure}
Furthermore, the weather and weather-and-debris cases can be compared closely as they correspond to the same simulation scene, allowing the subtraction of the true wind motion from the composite case in order to assess the biases introduced by the presence of debris.
This difference in velocity (Figure \ref{fig:sim_wx_delta}), i.e., the mean velocity of weather and debris minus the mean velocity of only weather, provides an insight into the measurement bias in the mean radial velocities.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/chap5/difv_1.eps}
\caption{PPI plot of difference in mean radial velocities between weather and debris, and weather at elevation $\phi=3.00^\circ$.}
\label{fig:sim_wx_delta}
\end{figure}
A difference close to zero can be interpreted as the wind velocity in the weather-and-debris case following the true wind motion accurately, while a negative difference means the wind motion is underestimated, and a positive difference represents overestimation.
In a similar analysis, the difference in velocity between the debris-only case and the composite case is compared in Figure \ref{fig:sim_db_delta}, where the values close to zero indicate that the radar is likely measuring debris motion rather than wind motion.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/chap5/difv_2.eps}
\caption{PPI plot of difference in mean radial velocities between weather and debris, and debris at elevation $\phi=3.00^\circ$.}
\label{fig:sim_db_delta}
\end{figure}
Upon closer inspection of the northwestern quadrant where the wind motion seems to be underestimated, spectral analysis shows that the spectra is mostly dominated by debris ($\theta = 0.5^\circ$, Figure \ref{fig:sim_all}a), as the signatures in power are very similar to those of the debris-only case (Figure \ref{fig:sim_all}b).
\begin{figure}
\centering
\scriptsize
\includegraphics[width=0.7\linewidth]{figures/chap5/wd_rd_10.eps}\\
(a)\\
\includegraphics[width=0.7\linewidth]{figures/chap5/db_rd_10.eps}\\
(b)\\
\includegraphics[width=0.7\linewidth]{figures/chap5/wx_rd_10.eps}\\
(c)\\
\caption{Range-Doppler plots of Bootstrap DPSD estimates at elevation $\phi=3.00^\circ$, and azimuth $\theta = 0.5^\circ$, for: (a) weather and debris, (b) debris only, and (c) weather only.}
\label{fig:sim_all}
\end{figure}
Clearly, the biases in the mean radial velocities are due to the high-power returns from debris.
However, even when the weather signal is dominated to some degree by debris, some signatures can be observed in the spectral correlation coefficient.
As it was mentioned in Section \ref{sec:dualsignal}, depending on the difference in power of the signals that are mixed in the composite signal, certain features remain possible to be identified.
By comparing the range-Doppler plots of the three cases (weather, debris, and weather and debris), it is evident that a line of low $s\rho_\mathrm{HV}$ is embedded in the weather-and-debris spectral signature, and it is collocated mostly with the peak of the weather spectral signature (Figure \ref{fig:sim_all}).
Additionally, the decrease in $s\rho_\mathrm{HV}$ can be attributed to the individual signals being comparable in power (i.e., within 20 dB of each other), such that none of them are dominant in that particular spectral coefficient, hence decreasing the homogeneity of the scattered signal.
Finally, focus is directed to the northeastern quadrant, where the velocity is overestimated.
At a radial in the $\theta = 7.5^\circ$ direction, the mixing of both signals is more widespread, and at farther ranges, the debris spectral signatures start diverging more rapidly from the mean wind motion (Figure \ref{fig:sim_ejection}).
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/chap5/wd_rd_ejection.eps}
\caption{Range-Doppler plots of Bootstrap DPSD estimates for weather and debris at elevation $\phi=3.00^\circ$, and azimuth $\theta = 7.5^\circ$.}
\label{fig:sim_ejection}
\end{figure}
This might be an indication of debris ejection, since the fastest moving object appears to have a radial velocity close to the maximum  radial velocity observed tangential to the vortex.
The bulk radar estimates mask the velocities of all underlying groups of scatterers that are depicted in the DPSDs.
Through visual analysis and statistical study of the simulation scene, it would be possible to determine the positions, velocities and orientations of the scatterers, in order to validate the results of the polarimetric spectral analysis.
However, such study is beyond the scope of this work.

\section{Summary}
The advantages of using the Bootstrap DPSD estimator over conventional DPSD estimators were shown for single- and multiple-dwell cases.
Also, it has been demonstrated that spectral signatures and qualitative features of the spectra can be identified with a single dwell estimate, without the resolution losses associated with range-, azimuth-, scan-, or frequency-averaged DPSDs.
Polarimetric spectral analysis can unveil additional information masked in bulk estimates, and a few representative spectral signatures have been presented as examples to illustrate this.
While a few interpretations for the observations have been provided, a validation of these is beyond the scope of this work.
It is deemed appropriate to perform more in-depth studies with the simulator and the Bootstrap DPSD estimator to confirm the hypotheses discussed herein. 
Such work would entail exploiting many of the advantages provided by such a powerful simulation tool in its many possible configurations, in conjunction with the dual-polarimetric spectral analysis capabilities provided by the Bootstrap DPSD estimator.
It is believed that the results of such studies will provide a better understanding of tornado dynamics and its connection to weather radar measurements, as well as to answer the scientific questions that motivated this work.
In the next chapter, the conclusion for this work and discussions for future work will be presented.
