In this chapter, the conclusions of this work are presented, and a few applications and recommendations for future work using the Bootstrap DPSD estimator are discussed.
The primary objective of this work was the development of the Bootstrap DPSD estimator, which is capable of obtaining accurate dual-polarimetric spectral density estimates from a single dwell.
A summary of the key aspects of each chapter is provided next.
The fundamentals of weather radar signal processing and spectral analysis were presented in Chapter 2, along with a brief description of the TDS and how important weather information could be masked in bulk radar estimates.
Typical non-polarimetric spectra were presented to demonstrate cases where the PSD alone would not be sufficient to determine the nature of the scatterers; and a hypothetical depiction of a spectral signature of tornadic debris was used as an example to demonstrate how the DPSDs could reveal additional spectral information that would aid in discriminating groups of scatterers of different nature moving at different radial velocities in a radar volume.
The advantages and disadvantages of the periodogram, Bartlett, Welch, and Daniell non-parametric PSD estimation methods were evaluated to be used for DPSD estimation.
While Bartlett, Welch, and Daniell PSD estimates can be used to obtain single-dwell DPSD estimates, the periodogram fails to yield useful single-dwell estimates.
However, it was shown that these degrade the frequency resolution considerably.
Conventional DPSD estimation methods average in frequency or use PSDs from multiple dwells that are averaged in range, azimuth, scan; however, the resolution in the averaged dimension can get significantly degraded.
This preliminary study helped in understanding the context of the problem, and showed the importance of having more accurate DPSD estimates for polarimetric spectral analyses.
In Chapter 3, the Bootstrap DPSD estimator was introduced.
First, the bootstrap resampling concept was presented, and the different block bootstrapping methods were evaluated taking into account their applicability to weather radar I/Q time-series signals.
Next, the design considerations of the Bootstrap DPSD estimator were presented.
In this context, a few indicators of the quality of the DPSD estimates, such as spectral leakage, statistical errors, and computational cost, were defined to establish comparison standards for selecting the optimal parameters in the design process of the Bootstrap DPSD estimator.
Additionally, signal pre-processing strategies such as signal extension, coherency correction, ratio of original samples and power correction, and post-processing strategies such as the bias correction were explained in detail.
In accordance with the quality indicators, performance analyses for the block bootstrap methods as well as for different configurations of the design strategies were conducted to obtain the optimal set of parameters that define the Bootstrap DPSD estimator.
Finally, a summary of the algorithm of the Bootstrap DPSD estimator was provided.
Chapter 4 began by describing the simulation methodology, and the calculation of the statistical errors.
Also, the relationship between bulk and spectral radar variables, and their differences in error dependence was discussed.
An analysis of error dependence on radar acquisition and signal parameters for the ideal estimator was presented, which helped simplify the complexity of subsequent performance analyses.  
Additionally, the effects in the spectral estimates of the interaction of two signals were briefly explained.
These preliminary analyses were important for a better interpretation of the results presented in the following chapter.
A statistical error performance of the DPSD estimators was then presented in single-signal single-dwell, single-signal multiple-dwell, and dual-signal analyses, providing an objective evaluation of the advantages and limitations of the Bootstrap DPSD estimator.
A few representative examples of the application of the Bootstrap DPSD estimator to real data and data obtained from a physical simulator were presented in Chapter 5.
The examples of real data focused more on the differences between the Bootstrap DPSD estimator and conventional DPSD estimators, and on the effects of averaging spectra in different dimensions, while the examples of simulated data focused on information that is masked by the bulk radar estimates, and that can be revealed with polarimetric spectral analyses.
These examples featured dual-polarimetric signatures of hydrometeor and non-hydrometeor scatterers in the same radar volume, which demonstrated the ability of the Bootstrap DPSD estimator to identify signals with different spectral characteristics.

From a broader perspective, this work can be summarized as follows.
The bulk radar estimates depict useful information about the characteristics and motion of meteorological phenomena, though these measurements are susceptible to biases when the signal of interest is contaminated by other types of scatterers in the radar volume.
The dual-polarimetric spectral densities may unveil additional information for groups of scatterers moving at different Doppler velocities, which can potentially aid in the characterization of distinct scatterer types.
Identification of different polarimetric spectral signatures is important for many applications. 
For example, by discriminating hydrometeors in spectra, it should be possible to obtain more accurate wind velocity measurements, which is very important for tornado intensity and damage potential estimation.
Previous estimation methods required averaging $K\geq 20$ spectra to get DPSD estimates of desirable error levels, which could be obtained from different range gates, radials, scans.
A smaller number of spectra could be averaged, though the quality of the estimates is usually insufficient for quantitative spectral analyses.
Good resolution is required in all dimensions in order to capture important features of meteorological phenomena that evolve relatively fast in time, in a small spatial extent, and with scatterers moving at different velocities within the radar volume (e.g., tornadoes).
The Bootstrap DPSD estimator was introduced as a means to compute the DPSDs from a single dwell with minimal resolution loss.
It employs the bootstrap resampling concept, which is a useful method to measure statistical properties of estimators when the available sample size is small.
Briefly, the estimator pre-processes and then bootstraps the conditioned I/Q time-series signals to obtain I/Q pseudo-realizations, which are in turn used to obtain bootstrapped PSD estimates.
The DPSDs are then computed by averaging the bootstrapped estimates, and a bias correction is applied to obtain the final estimates.
The pre- and post-processing strategies, as well as the appropriate selection of parameters are at the core in the design of the Bootstrap DPSD estimator.
The Bootstrap DPSD estimator shows superior error statistics when compared to conventional DPSD estimators for single-dwell as well as for multiple-dwell estimates, and it was shown that it meets the performance of the ideal estimator with about half the number of averaged independent spectra.
However, the Bootstrap DPSD estimator has a particular limitation in that, by attempting to correct the signal coherency, it could introduce an incoherence that adds a minimal bias into the spectral correlation coefficient estimates.
Further analyses of the impacts of this limitation are needed.
Additionally, the Bootstrap DPSD estimator shows better performance than the conventional DPSD estimators when discriminating polarimetric signatures of signals corresponding to different groups of scatterers moving at different radial velocities in the radar volume.
However, the ideal estimator still outperforms the single-dwell Bootstrap DPSD estimator in the dual-signal analysis.
Though a multiple-dwell dual-signal analysis was not conducted, it is expected that the Bootstrap DPSD estimator will have superior performance compared to  conventional and the ideal DPSD estimators.  
The potential of the Bootstrap DPSD estimator was demonstrated with a few representative examples using data from real tornado cases and from a physically-based simulator.
It was illustrated in the examples how polarimetric spectral analyses can unveil additional information obscured by bulk estimates.
It is expected that spectral analysis can provide more insight to better understand tornado dynamics and their connection to weather radar measurements.
However, to validate the observations from a physical point of view and to answer the scientific questions that motivated this work, more in-depth analyses are required.
Such studies are beyond the scope of this work and a few recommendations for future work are proposed next.

%One important application for the polarimetric spectral analysis is debris centrifuging bias correction.
By identifying the spectral coefficients that correspond to hydrometeors using the DPSDs, it is possible to obtain better estimates of the weather components, and to consequently improve the bulk radar estimates for the signal of interest. 
A simple spectral-based algorithm would entail thresholding the spectra for components with high SNR (e.g., $s\mathrm{SNR}_\mathrm{H}>20$ and $s\mathrm{SNR}_\mathrm{V}>20$ dB), and to perform a Gaussian fit on the PSD, though algorithms of higher complexity can be implemented.
Based on the PSD, multiple Gaussian fits could determine the number of signals in the spectra, their power distribution, mean radial velocities and spectrum widths.
The polarimetric characteristics of each signal could be discriminated with the DPSDs, by analyzing the distribution of $sZ_\mathrm{DR}$ and $s\rho_\mathrm{HV}$ that correspond to the spectral coefficients of the fitted curves.
Such additional information of the scatterers in a radar resolution volume could be exploited for many different applications.
Velocity bias correction should be possible by filtering out the signals that are more likely to be of non-hydrometeor nature, and by recalculating the spectra (if more than one signal is tagged as weather after filtering), or by simply using the fitted curve of the weather signal (if only one signal is left after filtering).
Another important application is a spectral-based hydrometeor classification algorithm (HCA).
Since evidence of unimodal spectra with different polarimetric characteristics has been found, it is possible to extend the HCA to work with spectral coefficients, as to provide a distribution of hydrometeors for a given radar resolution volume.
Moreover, as it was suggested in an example in Chapter 5, the contribution from range-folded echoes may be mitigated from bulk radar estimates, provided that their spectral signature is different.
Additionally, the Bootstrap DPSD estimator can be used to improve already existing analysis techniques.
The performance of previous DSD retrieval algorithms using spectral analysis may be improved by using Bootstrap DPSD estimates.
Also, clutter mitigation techniques can now be based on the spectral polarimetric variables to more accurately discriminate moving clutter and zero-mean radial velocity weather signals.

Polarimetric spectral signatures that are tagged as coming from non-hydrometeors can be useful in other applications.
From a qualitative point of view, the Bootstrap DPSD estimator is an important tool of analysis in quest to answer some of the science questions that motivated this work.
As it was presented in Chapter 5, identification of non-hydrometeor scatterers may provide more insight into debris ejection and loading dynamics, as well as for common alignment of debris in tornadic flows.
Such hypotheses can be validated by cross-referencing the polarimetric spectral signatures obtained with the Bootstrap DPSD, with visual outputs and statistical studies of the positions, velocities and orientations of the debris particles in the physically-simulated scene.
Additionally, the connection between different physical and electrical debris characteristics with radar variables can be tested and characterized.
This includes the dependence of spectral differential reflectivity, spectral correlation coefficient, and differential velocity with different debris characteristics, including size, shape, material, concentration, orientation, etc.
Also, hypotheses about the differences in radar data (both bulk and spectral estimates) for different radar wavelengths can also be validated with the aid of the physically-based simulator.

While the bootstrap term derives from the phrase: ``to pull oneself up from one's bootstrap", the Bootstrap DPSD estimator is not magical. 
The general idea of the estimator is that it rearranges its limited, yet available data, in a way that additional information (which is otherwise lost with other estimators) can be extracted.
It is expected that weather radar researchers could use this work to gain some insight into the interpretation of polarimetric spectral signatures.
Additionally, further work to improve the current state of the Bootstrap DPSD estimator, and the understanding of spectral polarimetry, in general, is encouraged.
