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A summary of natural hazard statistics for the year 2013 by the National Weather Service registered the fatalities, injuries and damages caused by severe weather \citep{NationalWeatherServiceOfficeofClimate2014}.
Among the different weather events, tornadoes were responsible for: 55 fatalities accounting for 12.33\% of the total coming in 4th place, 756 injuries for 27.33\% of the total - 2nd place, and \$3,642.18 million in property damage for 41.34\% of the total - 1st place.
These statistics highlight the potential impacts of tornadoes, which are one of the greatest weather-related threats to life and property in the U.S.
Part of these tornado-related fatalities are victims struck by airborne debris \citep{Bohonos1999}, which can also inflict major structural damage and increase the damage potential of the tornado as they are lofted along \citep{Marshall2002}.
Therefore, they are important in the study of tornadoes.

Continuous efforts have been led by the scientific community to improve the knowledge about tornadoes with the goal of increasing their predictability and to mitigate consequences.
Despite this, many aspects of tornado mechanics are still not completely understood \citep{Cheong2015}.
In this context, weather radars have proven to be an essential tool in tornado studies as their remote sensing capability allows retrieving  information in a way that would otherwise be extremely difficult and dangerous.
Furthermore, advances in weather radar technology have played a critical role by providing better measurements of the underlying phenomena.

The first meteorological application of a pulsed-Doppler radar was made in the 1950's, and since then it has become increasingly important for remote measurements of precipitation \citep{Doviak2006}.
However, it was not until the 1970's that tornadoes were observed with pulsed-Doppler weather radar \citep{Zrnic1975b}.
The discovery of the tornadic vortex signature (TVS) \citep{Brown1978}, which is a pattern of tornado formation before it is visible on the ground, suggested a potential for improved warning for severe weather. 
This eventually led to the nationwide deployment of a pulsed-Doppler weather radar (WSR-88D) network as a joint effort of the U.S. Departments of Commerce (DOC), Defense (DOD), and Transportation (DOT) --the NEXRAD \citep{Crum1993}.
The original WSR-88D network provided three main radar variables: radar reflectivity factor ($Z$), mean radial velocity ($\bar{v}_\mathrm{r}$), and spectrum width ($\sigma_v$).
By 2013, the WSR-88D network was upgraded to dual polarization, which provides new information about the scatterers within the radar resolution volume through an additional polarization channel \citep{Doviak2000}.
With the simultaneous transmission and reception of the H- and V-polarized waves, three additional polarimetric products were made available: differential reflectivity ($Z_\mathrm{DR}$), correlation coefficient ($\rho_\mathrm{HV}$), and differential phase ($\phi_\mathrm{DP}$).
Consequently, discrimination of meteorological and non-meteorological scatterers was possible, with potential application in tornado detection \citep{Ryzhkov2005}.

In light of the improved observation capabilities introduced by polarimetric radar measurements, the detection of tornadic debris signatures (TDS) was possible \citep{Ryzhkov2005}.
TDS are tornado-scale signatures with distinctive polarimetric characteristics collocated near a tornado vortex that are visible in radar observations after a tornado touches ground \citep{Ryzhkov2002}, that is related to the ejection and centrifuging of hydrometeors and debris by the cyclone \citep{Dowell2005}.
Since debris lofted by tornadoes can have polarimetric signatures that depend on their electrical size, shape, orientation and concentration \citep{Bodine2014b}, these signatures can be noticeably different from those of hydrometeors and can be used for polarimetric tornado detection \citep{Ryzhkov2005}.
Additionally, they are clear indicators of tornadoes when ground observation is limited or impossible (e.g., at night or during heavy rainfall) \citep{Kumjian2008}.
%A set of values typically found to be associated with TDS is suggested in \citet{Ryzhkov2005} as $Z > 45$ dBZ, $Z_\mathrm{DR} < 0.5$ dB, and $\rho_\mathrm{HV} < 0.8$, while \citet{Bodine2013} uses a more stringent set of values for $Z$ and $\rho_\mathrm{HV}$, with a set of thresholds defined as $Z < 43$ dBZ and $\rho_\mathrm{HV} < 0.82$, and another set of $Z < 51$ dBZ and $\rho_\mathrm{HV} < 0.72$.
A set of values typically found to be associated with TDS is suggested in \citet{Ryzhkov2005} and \citet{Bodine2013}, though it is still unknown exactly how the characteristics of different debris types affect different polarimetric radar measurements.

Nevertheless, in recent years, research on TDS has shown promising results in different applications.
\citet{Schultz2012} and \citet{Schultz2012a} explore the use of TDS as a tool for forecasters to improve the warning decision-making process and to assess the potential threat of a tornado in a near-real time operational environment, while \citet{Scharfenberg2005} reported TDS were used to enhance the confidence of tornado detection.
TDS may also provide information about the damage potential and the intensity of a tornado \citep{Bodine2013}.
Several studies \citep{Ryzhkov2005, Bluestein2007, Bodine2011} have shown evidence of negative $Z_\mathrm{DR}$ values in TDS from different tornado cases, suggesting a possible common alignment of the debris within the tornadic vortex, or a scattering in the Mie regime due to large debris.
In \citet{Bodine2014a}, a statistical analysis of TDS is performed using dual-frequency observations in an attempt to better characterize debris scattering and polarimetric signatures.
Logically, important scientific questions about the dynamics of a tornado, and the relationship between them and the radar measurements arise: 
How do tornado dynamics affect debris centrifuging? 
How does the centrifuging of tornadic debris vary for debris with different characteristics? 
How do the size and concentration of lofted debris relate to tornado intensity and damage severity and extent? 
What causes negative $Z_\mathrm{DR}$ in tornadoes?
The answers to these questions could provide a better understanding of tornadoes, which would potentially aid in the nowcasting and detection of imminent threats, as well as to provide better near real-time estimation of the tornado intensity and damage severity and extent.
However, any attempt to exploit the advantages of the TDS detection will come with its challenges.

While TDS can be helpful in the detection of tornadoes \citep{Scharfenberg2005, Schultz2012, Schultz2012a}, there are clear limitations when assessing tornado intensity through radar observations, and thus it is still a topic of much ongoing research.
One such limitation is due to the effects of centrifuging, which causes hydrometeors and debris to move at slightly different velocities within a tornado vortex.
In addition, Doppler radars measure the scatterers' motion rather than the actual wind speed and as such, significant biases can be introduced in wind radar measurements \citep{Dowell2005}.
This can occur whether the signal scattered by debris is stronger than that of hydrometeors or vice versa.
Moreover, TDS are more likely to be seen in observations due to the debris signal power dominating in a particular radar resolution volume \citep{Ryzhkov2005}, thus measurements of mean radial velocity in volumes with TDS are more prone to centrifuging bias.
Hypothetically, this error in measurement can be corrected if the velocity of the debris is known and can be separated from the velocity of hydrometeors which passively trace the wind. 
Since the radar variables are computed by averaging the contribution of all scatterers within a radar resolution volume, an alternative way to retrieve the velocities must be employed. This can be achieved through spectral analysis.

The power spectral density (PSD) of weather radar signals is the power-weighted distribution of the radial velocities in a radar resolution volume \citep{Doviak2006}. 
In a PSD, the power return of scatterers moving at different Doppler velocities within a given resolution volume is shown.
\citet{Janssen1985} found that approximately 75\% of the spectra observed with a phased array system from precipitation at close ranges were Gaussian-shaped.
For the other 25\% of the cases with non-Gaussian spectra, spectral analysis can provide important information about the distribution of radial velocities in the radar resolution volume \citep{Yu2009}.
%Typical weather signals have Gaussian-like spectra \citep{Zrnic1975a} with wide spectrum width, while other type of signals observed with weather radar, such as ground clutter, also have Gaussian-like spectra but with narrow spectrum width and centered at zero velocity \citep{Doviak2006}.
When assuming a Gaussian distribution, however, spectral analysis is only useful when there is a multimodal signal composition, with each scatterer group having its own velocity in the resolution volume; i.e., more than one peak in the PSD.
In the case where the weather signal is mixed with ground clutter, the spectra will show the distribution of both signals as a function of their Doppler velocities with two peaks corresponding to each type of scatterer, where the ground clutter signal is at zero and the weather signal can be moving at any Doppler velocity \citep{Doviak2006}.
Provided that the weather signal does not have a zero mean velocity, the ground clutter can be  filtered from the spectra without significantly corrupting the weather signal \citep{Torres1999}.
In the cases where the signals overlap in the spectra, techniques have been developed to mitigate the clutter influence and reconstruct the weather signal to provide better radar estimates \citep{Cao2012, Torres2014}.
However, it is difficult to discriminate the nature of the non-static scatterers contained in the spectra since the peaks in a PSD do not contain any information other than the power returns and their velocity towards the radar.
Without further information about the distribution of scatterers, any attempt to classify them would be difficult. 
%The most common method to estimate the PSD is with the periodogram estimator \cite{StoicaBook}. 

Additional spectral information can be gathered from the dual polarimetric spectral densities (DPSD).
The DPSD depicts the polarimetric characteristics of scatterers moving at different Doppler velocities within a radar resolution volume. 
These tools for polarimetric spectral analysis provide three additional spectral variables: the spectral differential reflectivity ($sZ_\mathrm{DR}$), the spectral correlation coefficient ($s\rho_\mathrm{HV}$), and the spectral differential phase ($s\phi_\mathrm{DP}$).
Although a universal definition of the spectral polarimetric variables has not been established, the term DPSD used herein can refer to any one spectral polarimetric variable or a set of them.
Provided that the polarimetric characteristics of the scatterers are known, it is hypothesized that discrimination of scatterer type in the spectra would be possible by correlating the DPSD values to known polarimetric characteristics of different scatterer types, though identification and characterization of all scatterers in the volume would not be possible. 

Previous work in DPSD have used different methods for their estimation as well as other spectral variables.
Some works have dealt with classification of scatterers to filter out non-meteorological targets.
In \citet{Bachmann2007a}, different biological scatterers were discriminated using Daniell's method to estimate the PSD, which is a three-point moving average filter on the power spectra estimate, and averaging over several adjacent range gates.
Additionally, an enhanced version of the velocity-azimuth display (VAD), the spectral VAD was explored, showing promising results for separating bird and insect flight speeds. 
In \citet{Moisseev2009}, a filter for adaptive clutter and noise suppression based on the spectral polarimetric variables was presented. 
This work used additional spectral variables such as textures of spectral differential reflectivity and spectral differential phase.
The spectra were averaged in range to reduce the variance of spectral estimates.
\citet{Alku2015} expanded on \citet{Moisseev2009} by proposing the classification of a wide variety non-meteorological echoes based on thresholds set by the DPSD values and strategies to mitigate such signal contaminants. 
The spectra were estimated by averaging over four range gates.
In a similar application, \citet{Unal2009} studied the use of the spectral linear depolarization ratio and found improved measurements of atmospheric returns. The DPSD was estimated using time averaging of up to 40 spectra, for an observation time of ~1 minute.
Other works used DPSD estimates to study the microphysics and dynamics of weather events.
\citet{Yanovsky2011} studied the retrieval of microphysical and turbulence characteristics of rain using phenomenological models and through the analysis of the spectral differential reflectivity and spectral linear depolarization ratio.
Here, the DPSD were estimated with spectra averaged in time over an observation time of ~15 minutes. 
\citet{Dufournet2010} and \citet{Spek2008} assessed the retrieval of ice and mixed-phase cloud microphysics characteristics using models for ice particles with spectral differential reflectivity, spectral linear depolarization ratio and spectral correlation coefficient. The spectra were averaged in time to obtain the DPSD using ~10 observations. 
\citet{Moisseev2006} and \citet{Moisseev2007}, retrieved the drop-size distribution (DSD) through the spectral differential reflectivity using the average of independent simulated spectra.
\citet{Yu2013} analyzed a hailstorm at a low elevation angle using spectral polarimetry to verify shear-induced size sorting mechanisms, which was further validated with simulations. 
Finally, there is only one work in the literature that studies the quality of the DPSD estimates.
In \citet{Yu2012}, a much needed, thorough analysis of the statistical quality of the spectral polarimetric variables was performed, yielding the number of independent observations needed to ensure the errors of the polarimetric variables are within the desired limits.
It is important to note that all of these studies have used, in one way or another, the averaging of the  PSD to obtain the DPSD estimate.
In this study, focus is directed to the spectral differential reflectivity and the spectral correlation coefficient, while the spectral differential phase is not analyzed because it may not convey as much information to discriminate hydrometeors from debris.

While the idea of analyzing the spectral signatures of TDS --whether it is for a better characterization of the scattering properties of debris or to discriminate debris from hydrometeors in the spectra-- might seem simple at first,  there are many constraints that prevent an adequate analysis.
The main limitation for a polarimetric spectral analysis for tornado observations is the large amount of independent observations needed to achieve the required accuracy and precision, shown in \citet{Yu2012} as  approximately 20 independent observations to achieve optimal statistical quality. 
Currently, the operational WSR-88D can provide one scan for an elevation angle every few minutes \citep{Crum1993}. 
It is essential to average more than one spectra to get useful DPSD estimates.
Furthermore, it is easy to show that with one independent observation the estimator fails to yield useful results for the spectral correlation coefficient.
To account for this limitation, additional data may be obtained from adjacent gates in radial, range, different dwells, or even by smoothing the spectral estimates, but they ultimately end up reducing the spatial, time, or frequency resolution, and thus far, an optimal solution to this problem has not been proposed in the previous literature.
Since tornadoes are events that evolve fast in time, in a relatively small spatial extent, with different scatterers contained within the tornadic vortex \citep{Bodine2013}, it is in our interest to preserve the best resolution possible in all dimensions.

To overcome these constraints, a new estimator of the DPSD is developed.
In this work, the Bootstrap DPSD estimator is presented, which accounts for the aforementioned limitations and computes the DPSD using one dwell with minimal resolution loss.
In Chapter \ref{chap_2}, the concepts of polarimetric radar I/Q time-series signal processing, TDS, spectral analysis and estimation of spectral variables are presented. Also, an assessment of the advantages and limitations of currently available methods of PSD and DPSD estimation is included.
A complete description of the fundamental concepts and considerations taken in the design of the Bootstrap DPSD estimator is presented in Chapter \ref{chap_3}.
Thorough analysis of the quality of the Bootstrap DPSD estimator under different scenarios by comparing the ideal and conventional methods are presented in Chapter \ref{chap_4}.
Chapter \ref{chap_5} shows the results of using the Bootstrap DPSD estimator on data sets of real weather events, and data obtained from simulations of hydrometeors and debris in a tornado vortex from a physical polarimetric radar time-series simulator \citep{Cheong2015}.
Finally, concluding remarks and recommendations for future work are presented in Chapter \ref{chap_6}.