**Processing to obtain Polarimetric Variables on the ORDA**

DO NOT DESTROY **--** version

July 2008

Dusan S. Zrnic, Valery M. Melnikov, and Igor Ivic

![](media/image1.png){width="4.830555555555556in"
height="2.651388888888889in"}

**Preamble**

This report is the last of the living and evolving document. It is the
guide to NWS and its contractor on how to integrate computations of dual
polarimetric variables into the architecture of the Open Radar Data
Acquisition (ORDA) system. Consideration is given to mitigation of range
and velocity ambiguities, to recovering loss of sensitivity, and
computation of polarimetric variables. Self sufficient appendices
further justify the recommendations and also present alternatives to
current processing schemes. The pertinent details (equations, flow
charts) are included and explained so that these could be programmed on
the system. We give some examples (or indicate by pointing to previous
reports where these are) of what we have tested on data and what still
remains to be verified as well as what issues remain to be resolved. The
technique of oversampling and whitening is purposely omitted because
there is no time to implement these on the initial system; these will be
part of subsequent upgrades. There is a separate document (in
preparation) describing the super resolution as proposed for "build 10"
(hardware/software modifications in 2008) which also applies to single
polarization radar.

This final version is appropriately named "Do Not Destroy". It contains
a heretofore not available section 2.2 wherein the thresholds for
improving detection are given. These should be updated at the end of
each volume scan.

Initially the intent of the authors, as implied by the name \"Destroy\",
was to replace (destroy) each version with an updated version. Meanwhile
the project dynamics and experience of the undiscovered "Philosopher -
author" destroyed this initial intent. The contract for dual
polarization upgrade has been issued hence a more stable and less
intimidating preamble was required. Therefore, some permanency became
desirable. We also realized that important aspects in dual polarization
upgrade (such as ground clutter issues exemplified with choice of H or V
channel for dictating clutter removal and adaptive recognition of ground
clutter, and recombination of supper resolution data) would fill too
much space. These deficiencies are one important reason the
"Philosopher" has not yet been discovered. Armed with new experience and
education the "wish to be Philosopher" decided to change the name and
help readers track the small changes (corrections) by adding on the
cover the months and year the latest changes have been made.

A new report will contain NSSL's recommendations on how to compute the
polarimetric variables, filter ground clutter, and deal with supper
resolution issues.

**TABLE OF CONTENTS**

> 1\.
> Introduction\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....
> \...\...\...1
>
> 2\. Mitigation of sensitivity loss in the dual polarization
> WSR-88D\...\...\...\...\...\...\...\...\.....4
>
> 2.1. The
> problem\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....4
>
> 2.2. The
> solution\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....5

a\) *Notation and
Computations\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....*
\...\...\...\...\...\...\...5

b\)
*Reflectivity\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....*\...\...\...\.....
\...\...\...6

c\) *Velocity and spectrum
width*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....
\...\...\...\...\...\....11

2.3. Computation of thresholds for various *M* and
*N~v~*/*N~h~*\...\...\...\...\...\...\...\...\...\...\...\....17

> 3\. Computation of polarimetric variables for the SHV
> scheme\...\...\...\...\...\...\...\...\...\...\....25
>
> 3.1. Standard method from time series
> data\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...25

a\) *Estimation of reflectivity
factor*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....25

b\) *Differential reflectivity
Z~DR~\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....*\...\...\...\...\...\.....26

c\) *Differential phase*
*Φ~DP~*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....26

d\) *Mean Doppler Velocity
v*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....27

e\) *Spectrum width
σ~v~*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....27

f\) *Correlation coefficient at zero lag
ρ~hv~(0)*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....27

3.2. Method based on
autocovariances\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....28

> 3.3. Spectral
> Computations\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....
> \...\...\...\...\...\....29

3.4 Estimation of the noise
powers\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...30

4\. Polarimetric measurements and mitigation of
ambiguities\...\...\...\...\...\...\...\...\...\...\...\.....31

> 4.1 Processing flow in the
> RDA\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....31

4,2 Polariometric
variables\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....34

4.2.1 Standard
VCP\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...34

4.2.2 Phase
coding\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....34

a\) *Surveillance
scan*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....35

b\) *Doppler
scan*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...35

c\) *Ground clutter filter and polarimetric
variables\...\...\...\...\...\...\...\...\...\...\.....*36

4.2.3 Staggered
PRT\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....37

4.4. Filtering ground
clutter\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....37

4.5 Phase coding and computation of polarimetric
variables\...\...\...\...\...\...\...\...\...\.....38

> a\) *Assumptions and
> notation*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....39
>
> b\) *Algorithm*
> \...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....40
>
> 5\. Recommendations for initial implementation of polarimetric
> capability\...\...\...\...\....42
>
> 5.1
> Calibration\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....42
>
> 5.2 Determination of noise level and threshold of the
> variables\...\...\...\...\...\...\...\...\....42

5.3 Computations of polarimetric
variables\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....42

Appendix A

Probability of false alarm for the Legacy Threshold scheme of the Z
field\...\...\....44

Appendix B

Combinations of correlation functions for censoring radar data
\...\...\...\...\...\...\...\.....46

Appendix C

Computation of the noise
level\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...52

*Testing if noise is
white\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...*52

*Requirement for the correlation
coefficient*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....52

*Requirement for the differential
reflectivity*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....53

*Discussion*\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....54

**References**\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....55

**1. Introduction**

Four significant innovations are planed for the National network of the
WSR-88D radars. One is inclusion of schemes for mitigating
range/velocity (R/V) ambiguities, for which the phase coding scheme is
being tested in real time by the Radar Operation Center (ROC). Two is
super resolution. Three is addition of polarimetric capability. Four is
processing of oversampled signals in range. Specifics on how to
integrate these harmoniously are presented and various issues are
brought out. The processing of oversampled signals including the effects
of windowing is mentioned lightly because a separate report on the
subject has been prepared.

The weather radar users rightfully expect that the polarimetric WSR-88Ds
would produce significantly improved data and yield superior
quantitative precipitation estimates without compromising the capability
of the present radars. This consideration influenced our decision to
propose simultaneous transmission and reception of horizontally (H) and
vertically (V) polarized waves, the SHV mode (Doviak and Zrnic 1998). In
this mode all polarimetric variables are decoupled from Doppler
variables (velocity and spectrum width) so that in schemes for
mitigating range velocity ambiguities these can be treated the same as
reflectivity factor Z~h~. Thus the phased coding and staggered PRT apply
equally well to differential reflectivity Z~DR~, differential phase
Φ~DP~, and correlation coefficient ρ~hv~. Details on how to efficiently
compute Z~DR~, Φ~DP~, and ρ~hv~ depend on the specific path within a
technique. And the computing path can be data dependent in that some
require clutter filtering or separation of overlaid echo and some do
not. These details are brought out. In the alternate (switched) mode for
polarimetric measurements the Doppler variables are coupled with ρ~hv~.

One concern by meteorologists is the loss of sensitivity in the SHV
mode. It amounts to 3 dB per channel. This loss has been identified in
the report by Doviak and Zrnic (1998) who also proposed how to partially
mitigate its effects. The report by Melnikov and Zrnic (2004) suggests
several alternatives for mitigation and quantifies some of these.

In Section 2 are the recommendations for recapturing most of the loss in
sensitivity. These require combining the autocorrelations of the
Horizontal and Vertical signals at lag 1 for estimation of velocities.
This increases the sensitivity by about 1.5 to 2 dB. The gain comes at
low signal to noise ratios (SNRs) because the noises in the two channels
are uncorrelated. Further, at low SNR's the recommendation is to use the
powers in both H and V channels, the autocorrelation functions at lag 1
(both H channel and V channel), and the cross correlation function
between the H and V voltages. Linear combinations of the magnitudes of
these correlations and powers are tested against a threshold and at a
tolerable false alarm rate. This is done at SNR's which in the Baseline
system (ORDA build 8) would not pass the threshold; otherwise the
accepted Baseline thresholds are in place. This procedure is recommended
for all three spectral moments. Qualitative impressions of performance
exemplified in fields of reflectivity and Doppler velocity are
presented.

Although the desired evolutionary path of the WSR-88D over the next five
years is fairly well established (Zrnic and Zahrai 2003) the specifics
will depend on the priorities. These would have to balance the
capability of the signal processor, the time available to the
developers, the maturity of the techniques, and budgetary constraints.
At the moment, the maximum computational load of the ORDA is about 25 %.
If dual polarization is added this load would more than double, to over
50 %. Another doubling required for oversampling by a factor of only two
would saturate the system. Clearly the current ORDA does not have the
computational resources to accommodate dual polarization together with
oversampling with a smallest possible factor (two). Moreover, there are
demands for additional computations and data manipulations on the RDA
including placing some of the burden from the RPG to the RDA. This
presents a dilemma to the decision makers who face at least the
following four options. 1) Leave the processor as is and forgo
oversampling in the dual polarization mode until after the upgrade. 2)
Replace one board or processor chip on the board with a more powerful
one. 3) Replace the processor with the next generation processor which
according to the manufacturer should be ready by 2009. 4) Extract time
series data from the current ORDA into an additional powerful signal
processor as was done by Lincoln Laboratory for the TDWR radars.

The signal processing on the upgraded WSR-88D will be greatly influenced
by available hardware. A good choice for the oversampling factor is five
but at the moment it is not known if this will be feasible. There are
also many other details which can influence the overall architecture of
signal processing in the dual polarization mode. Such is the possibility
to instantaneously (radial by radial) decide if clutter or other
artifacts are present in the data and to remove these. Ultimately
spectral processing coupled with decision making in the RDA would
comprise these advanced concepts. Herein we stop short on these topics.
Rather we present recommendations for implementations at the time of the
upgrade (year 2009).

Another issue concerns determination of the noise powers *N~h~* and
*N~v~*, in the two receivers; these must be estimated. The preferred way
to estimate these is on each radial of data. This could be done in the
spectral domain by identifying the white noise contribution using a
technique like in the GMAP clutter filter. Or it could be in the time
domain by locating the delays where signal is absent. Nonetheless, such
advanced techniques might not be ready for implementation by the time of
the upgrade.

In Section 3 we list three ways for computing polarimetric variables.
The standard way is a likely candidate for surveillance scans and for
any scans in which the noise powers are estimated on each radial. If
there is uncertainty in the noise powers, estimates from covariances are
preferred because these are not biased by white noise. Computations of
the polarimetric variables from complex spectra offer adaptive
filtering. These computations are required in cases of ground clutter
filtering regardless of what kind of range velocity mitigation scheme is
operating (i.e., phase coding, staggered PRT, or else). Further, such
computations should be done on the weak trip signal from phased coded
schemes.

Section 4 provides information on the data flow and major computations
in the dual polarization radar. The interplay between clutter filtering
and the mitigation of ambiguities is also addressed so that the readers
learn the musts of the technique. Abbreviated functional description of
how to process the phase coded signals is given to make a point about
spectral processing. The description is sufficient for those
knowledgeable to be able to incorporate it into the complete algorithm.
Thus, this section does indicate what needs to be done as well as how
and where to do it. It is left to the professionals to incorporate these
suggestions into the system.

Section 5 contains the recommendation for processing that should be
implemented at the time of deployment. Its essence is to retain all the
existing capabilities of the single polarization system (apart from a 3
dB loss of power per channel). Thus, it does not capitalize on the
immense potency of dual polarization which is left for subsequent
upgrades.

Throughout the report some equations are numbered in bold font and
bracketed within square brackets, \[ \], to highlight their importance.

**2. Mitigation of Sensitivity loss in the dual polarization**

**WSR-88D**

The issue of sensitivity in the proposed dual polarization mode for the
WSR-88D is discussed in this section as well as the proposed scheme to
gain back this loss.

**2.1 The Problem**

In the simultaneous mode of H, V transmission and reception the
transmitted power is split into two channels (Fig 2.1). Hence, on
reception, the signal to noise ratio in each channel is one half of what
it would be if all the power were in a single channel (see the reports
by Doviak and Zrnic 1998 and Melnikov and Zrnic 2004). Thus, there is a
3 dB loss in sensitivity compared to single channel (legacy) WSR-88D.
This loss has two effects on the spectral moments. One is the increase
in the errors of estimates. The other is that more data are below
display and processing thresholds and hence are lost.

![](media/image2.png){width="6.0in" height="3.600787401574803in"}

Fig. 2.1 Block diagram of the SHV dual polarization system.

Herein is the recommendation on how to mitigate some of the effects of
sensitivity loss. For the same radar dwell time and at low SNRs (\~ less
than 5 dB), there would be an increase in standard deviation of
estimates if the baseline processing is used. More sophisticated
processing in the spectral or autocovariance domain based on maximum
likelihood principles could reduce the errors. Also coherent summations
of the H and V samples could recover the loss (Melnikov and Zrnic 2004).
But, the additional calculations required by either method are not
justified. There are other more important computations that the system
ought to do at the time of first deployment. Hence, we recommend that
the benefits of more sophisticated processing at low SNRs be
established. Then, if warranted, such processing could be incorporated
into the system.

In the baseline system the default thresholds for displaying and
subsequent processing reflectivity and velocity fields are as follows.
For reflectivity the SNR threshold is 2 dB and data with significant
returns satisfy

10log~10~(P-N) \> 10log~10~(N)+2 dB, (2.1)

where P is the power estimate at a range location (time delay) and N is
the noise estimate obtained from calibrations at end of volume scans.
The false alarm due to noise for the threshold defined with this
equation and *M*=17 samples for reflectivity estimates is 1.17×10^-6^.
It was obtained from theoretical analysis (Appendix A) as well as
simulations. For velocity estimates, the SNR threshold is 3.5 dB.

**2.2 The solution**

The recommended solution is explained in this sub section.

a\) *Notation and Computations*

Let *R~h~*(*T~s~*) and *R~v~*(*T~s~*) be the autocorrelations of the
horizontally and vertically polarized returns at lag *T~s~* (where
*T~s~* is the pulse repetition time). These autocorrelations are
computed from the time series data or from the spectra.

From time series *R~h~*(*T~s~*) is obtained as

![](media/image3.wmf) (2.2)

where *M* is the number of samples, index *n* refers to time (*nT~s~*),
and *H* is the complex return at a specific range location.

From the spectrum *S~h~*(*k*)

![](media/image4.wmf) (2.3)

Warning! The DFT in (2.3) produces circular autocorrelation which
contains the product of the first and last term of the sequence
*H*(*n*). This product is subtracted from the sum to produce a result
that is exactly equal to (2.2). Otherwise errors in the estimate of the
autocorrelation would increase. There are other ways to obtain correct
value of *R~h~*(*T~s~*) such as padding at least one zero either at the
beginning or end of the sequence. Formulas identical to (2.2 and 2.3)
apply to the autocorrelations of the *V* signals.

On the legacy WSR-88D the computation (2.2) is used. On the ORDA if
autocorrelations are computed from the spectra (2.3) should be the
choice.

One more correlation function is pertinent to the proposed thresholding
scheme. It is the cross correlation

![](media/image5.wmf) (2.4)

which will be abbreviated with *R~hv~*.

Depending on the spectrum width, differential reflectivity, and
scatterer type any one of the three correlation functions could have the
largest magnitude. An optimum threshold scheme can be constructed from a
combination of the correlation functions and other polarimetric
variables. In such scheme the conditional probability of threshold
crossing by noise would be fixed i.e., p(X\>T│noise) = C while the
conditional probability of detecting the signal plus noise would be
maximized, i.e., p(X\>T│signal+noise). X is the vector with elements
which are compared element by element to the vector threshold T. Results
of this type of optimization will be presented in a subsequent report.
Nonetheless, the pertinent probability density and simple considerations
for our choice are described in Appendix B.

Meanwhile we proceed with the following intuitive and simple approach.
We take the sum combinations of the powers and magnitudes of correlation
functions and compare these to a suitable set of thresholds.

b\) *Reflectivity*

We considered the following approach to threshold the reflectivity
field:

at SNR~h~ \>2 dB (2.5a)

\...\...\...\...\...\...\...\...\.....use 2 dB threshold (i.e., given by
Eq. 2.1)

at -1 dB \< SNR~h~ \< 2 dB\...

\...\...\...\...\...\...\...\...\.....use ![](media/image6.wmf)
**\[2.5b\]**

where *T* is the threshold in power units and the multiplicative
coefficients *α, β,* and *γ* are determined to maximize the probability
of detection at tolerable probabilities of false alarm. As of this
writing we have not determined the best choice of the multiplicative
coefficients, but have used a value of 1 for each. The bold equation
number and square brackets emphasize that this and similar expression
should be implemented in the RDA.

Concerning the partition into regions of SNR~h~ \> 2 dB and -1 dB \<
SNR~h~ \< 2 dB, we have determined that there is miniscule difference in
performance if \[2.5b\] is used exclusively in the full range of SNRs
(i.e., some signals with SNR \< -1 dB will pass the detection criterion
on the strength of their coherencies); for one case, in the whole PPI we
noticed four points that differed (by repetitively switching between the
two fields and observing the enlarged display).

Evaluation of the "best" tested sum \[2.5b\] was performed using a set
of dual-polarization time series data. This set was collected at the
long PRT (unambiguous range 460 km), and the number of time samples *M*
was 17, the same as in the first surveillance cut of VCP11 for
reflectivity measurements; the elevation was 0.48^o^. To investigate the
effect of the 3 dB power loss, noise power was doubled in each channel
by simply adding the noise samples as:

![](media/image7.wmf). (2.6)

The noise power in the H channel is *N~h~*, and *N* is the average noise
in the two channels. Each noise sample is generated in MATLAB. Threshold
for each approach was set so that the probability of false alarm (PFA)
is 10^-5^, and the results are summarized in Appendix B. Herein, we
present the pertinent reflectivity fields. The original "single pole"
PPI is given in Fig. 2.2a.

![](media/image8.png){width="6.5in" height="6.0in"}

Fig. 2.2 a) Reflectivity field, horizontal channel. Data which exceeds 2
dB in SNR is displayed. Elevation is 0.48 deg, the long PRT (3106.7 μs)
was used, and the clutter filter was not applied. Number of samples for
estimates is 17.

Doubling the noise power but keeping the previous threshold amounts to
lowering the threshold to -- 1dB with respect to the double noise power
as the test is 10log~10~(P-2N) \> 10log~10~(N)+2=10log~10~(2N)-1. Notice
the plethora of spurious speckles in the display. This is expected
because the PFA has increased from ≈1.2×10^-6^ to 3×10^-3^.

![](media/image9.png){width="6.5in" height="6.0in"}

Fig.2.2 b) Same as in 2.2a except the noise power was doubled so that
the equivalent SNR threshold is -1 dB.

The case for which the threshold is elevated to 2 dB above the noise
level is in Fig 2.2c. If we compare it to the original PPI we notice
that the significant portion of the features on the rim of the weather
returns has been lost.

For this data set the noise power *N~h~* = 1.21 *N~v~* (i.e., 0.82 dB)
and we find the threshold from Monte Carlo Simulations (for the desired
PFA = 10^-5^ and *M*=17 samples) is *T =* 10.83 *N~h~* (i.e., 10.34 dB).

![](media/image10.png){width="6.5in" height="6.0in"}

Fig.2.2 c) Same as in 2.2a except the noise power was increased by 3 dB,
and the threshold is set to 2 dB above the total noise power.

Application of the sum \[2.5b\] with this threshold produces the PPI in
Fig. 2.2d. We notice that all of the low reflectivity perimeter features
have been recovered, and quite a few of previously censored data are now
valid. This could be because the probability of false alarm (10^-5^) is
about seven times larger then if the Legacy (2.1) threshold is applied.
We plan to further investigate \[2.5b\] by varying the threshold and the
weights so that a closer match of the fields in Fig. 2.2d and Fig. 2.2a
would be achieved. This is very plausible and should be relatively
simple to do; just reducing the PFA for data in Fig. 2.2d (to the value
achieved in Fig. 2.2a) will inevitably shrink its detected area and
bring it closer to the area in Fig. 2.2a.

![](media/image11.png){width="6.5in" height="6.0in"}

Fig. 2.2d) Same as in 2.2a except the noise power was increased by 3 dB,
and the "best of the set" sum for detection has been applied.

Parameters that enter the choice of thresholds are SNR, number of
samples *M*, spectrum width, ratio of noise powers *N~h~/N~v~,* and
pulse repetition time *T~s~*. Discussion of the optimum choice of the
threshold and quantitative evaluation of the performance are the subject
of the PhD dissertation by Ivic (2008). He has computed the thresholds
for most Volume Coverage Patterns (VCPs). The section 2.3 of our report
is extracted from Ivic (2009) and should be consulted for the details of
implementation.

c\) *Velocity and spectrum width*

The velocity should be computed from

*v= λ* arg{*R~h~*(*T~s~*) + *R~v~*(*T~s~*)}/(4π*T~s~*). \[**2.7**\]

Similarly, the spectrum width should be computed from the average of the
two autocovariances. First the signal powers *P~sh~* and *P~sv~* are
computed; herein the *P~sh~* is listed and the same formula applies to
*P~sv~*.

![](media/image12.wmf). (2.8)

The formula (6.27) from Doviak and Zrnic (1993) can be used for
computing the spectrum width. Thus

![](media/image13.wmf), (2.9)

Where *P~s~* = *P~sh~* + *P~sv~* and *R*(*T*) = *R~h~*(*T*) +
*R~v~*(*T*).

Another good option for computing spectrum width in the Doppler mode is
from lag one autocovariances (Section 3, eq. 3.22).

Fig. 2.3 depicts the standard deviations of velocity estimates for four
situations. The figure is obtained from simulations by Melnikov and
Zrnic (2004) who also present a detailed study of the subject. It is
assumed that the noise powers in the two channels are the same. Note
that for a fixed signal power and a threshold of 3.5 dB the Legacy curve
indicates an SD of better than 0.9 m s^-1^. To obtain the SD in the dual
polarization mode we look at the value for the SNR = 3.5 -- 3 = 0.5 dB.
If nothing is done (H channel in the figure) the SD would be about 1.3 m
s^-1^. If the combined *R~h~, R~v~* are used to estimate the velocity as
in (2.7) the SD drops to about 1.2 m s^-1^ and 1.1 m s^-1^ at the values
of Z~DR~ equal to 2 and 0 dB respectively. The results are worse for
larger *Z~DR~*, but better for smaller ρ~hv~.

![](media/image14.wmf){width="5.995138888888889in"
height="4.709027777777778in"}

Fig. 2.3 Standard deviation of velocity estimates for the legacy system,
the dual polarization system with a 3 dB loss in power (H channel), and
the proposed solution (H&V channels) according to (2.7). Parameters in
the simulations are Z~DR~ of 2 and 0 dB, spectrum width of 5 m s^-1^,
and unambiguous velocity 25 m s^-1^ (PRF=1000 Hz and radar wavelength is
10 cm).

Clearly, it is possible to reduce the SD. So at the threshold of 0.5 dB
per channel in the dual polarization mode the error would be \~25%
larger than in the legacy (pre ORDA) system.

![](media/image15.png){width="5.998611111111111in" height="5.5375in"}

Fig. 2.3 a) Mean velocity field. SNR threshold is 3.5 dB. The short PRT
of 986.7 μs was used and the number of samples per velocity estimate is
52.

We have tested the effects of several threshold schemes on the velocity
field corresponding to data collected with the KOUN radar on March 19,
2006. In Fig. 2.3 a) velocity field obtained from a single
autocovariance (2.2, pulse pair) is displayed for SNRs \> 3.5 dB. This
is the standard for comparisons because it is currently the default
threshold on all WSR-88Ds.

Then, 3 dB of artificially generated noise was added to the time series
data (of each the horizontal and vertical channel) and the threshold for
displaying the fields was raised by 3 dB. This simulates the loss of 3
dB that would occur in each channel of the dual polarization radar. For
comparison we display in Fig. 2.3 b) the velocity field obtained by
applying (2.7) to this "noisier" data.

![](media/image16.png){width="5.998611111111111in" height="5.5375in"}

Figs. 2.3 b). Similar to 2.3 a) but for simulated dual polarization
case. Three dB of noise was added to time series data used to generate
Fig. 2.3a, and the SNR threshold was set to 3.5 dB with respect to the
total noise (existing and added). This is equivalent to lowering the
signal power by 3 dB and keeping the SNR at 3.5 dB; thus it represents
the loss of data that would occur in the dual polarization case.

To recuperate the lost data (in Fig. 2.3b) a lower SNR threshold of 0.5
dB could be applied. The result of such "simulation" is in Fig. 2.3 c).
One would expect the velocity field to be more cluttered by speckles
after lowering the threshold (as in the case of the surveillance scan).
This, however, does not happen; after lowering the threshold to 0.5 dB
(for simulating the dual-pol case) the image is still very similar to
the one in single-pol with the 3.5 dB threshold. The lack of cluttering
from speckles can be explained by examining the false alarm rates for
3.5 and 0.5 dB thresholds. These are very low, 2.3368×10^-26^ and
2.1429×10^-10^, respectively. Lowering the threshold to 0.5 dB (with *M*
= 52) significantly increases the rate of false alarms compared to the
rate if 3.5 dB threshold is applied. Nevertheless, the rate of false
detections still remains so low that no significant cluttering in the
PPI of the velocity fields is apparent.

![](media/image17.png){width="5.998611111111111in" height="5.5375in"}

Figs. 2.3 c). Same as in 2.3 b) but the SNR threshold is 0.5 dB.

Next, we show the field after application of the same type of
thresholding scheme as for the reflectivity. The velocity field thus
obtained is in Fig. 2.3 d). Obviously the area of "useful" data is
noticeably larger than in the cases whereby power thresholds were
applied. This is not surprising because the false alarm rate for the
"best" sum is 10^-5^ which is much higher than the false alarm rate for
any of the power thresholds. Consequently (see Appendix B, Fig. B.5) the
probability of detecting the signal is also higher compared to the
probability of detection in cases the power threshold are used. This
suggests that lowering the false alarm rate to less than 10^-5^ for the
"best" sum should be explored (as well as changing the weights); it
might be possible at lower false alarm rates to still detect much of the
data that would be lost if only the power threshold is used.

![](media/image18.png){width="5.998611111111111in" height="5.5375in"}

Fig. 2.3 d) Velocity field after application of the threshold scheme
given by (2.5b).

As stated, the weights in the detection criterion (eq. 2.5b) were set to
one, and quite satisfactory detection of weak signals was achieved.
Intuition suggests that unequal weights might outperform our simpler
scheme. Meanwhile the "best" threshold as in (2.5b) should be used for
displaying and processing the velocity data. The same holds for the
spectrum width. Test on the spectrum width and quantitative evaluation
of the effects of thresholding remain to be demonstrated. Other more
sophisticated possibilities based on the spectra should also be
explored.

In the "best" sum case the ratio of noise powers was *N~h~/N~v~* = 1.2
(0.8 dB). For reflectivity estimates *M=*17 (surveillance scan in the
split cut) and the threshold was *T*=4.7 *N~h.~.* For velocity estimates
(Doppler scan in the split cut) *M* = 52; the threshold in the optimum
sum was *T*=4.9 *N~h~*. If the signals are perfectly correlated all
three covariances would equal signal power; then substituting this in
(2.5b) reveals that detection occurs if SNR \> - 2.7 dB (for *T*=4.7
*N~h~*) and SNR \> -2.4 dB (for *T*=4.9 *N~h~*).

To get an idea at what SNRs the data will be displayed take the optimum
threshold as *T*=5 *N~h~* (rather than 4.9) and assume the following
extreme cases of weather signal parameters. Full coherency in time (very
small spectrum width) and across the channel (*ρ~hv~* =1), as in the
previous paragraph (see the Table below); this could be uniform wind
with no turbulence and small drops. Full coherency in time but no
coherency across the channels; uniform wind and chaff (has cross
correlation coefficient \~ 0.3). No coherency in time but full coherency
across channels; could be very turbulent flow containing small drops. No
coherency in time and no coherency across the channels.

  --------------------------------------------- -------------------------
  Case                                          Signal to noise ratio

  Fully coherent in time and across channels    SNR \> -2.2 dB (i.e., \>
                                                3/5)

  Fully coherent in time but incoherent across  SNR \> -1.2 dB ( \> 3/4)
  channels                                      

  Incoherent in time but fully coherent across  SNR \> 1 dB ( \> 1)
  channels                                      

  Incoherent in time and incoherent across      SNR \> 1.8 dB ( \> 3/2)
  channels                                      
  --------------------------------------------- -------------------------

**2.3 Computation of thresholds for various *M* and *N~v~*/*N~h~***

This section is a slightly modified version of a section in Ivic (2008).
It is self contained and explains how to compute the thresholds in real
time. Different threshold based on the ratio *N~v~*/*N~h~*, should be
computed for each *M* value used in the dwell time. Thus the threshold
computation must be done at the end of each volume scan because *N~v~*
and *N~h~* can change and are computed (for calibration of differential
reflectivity) at the end of volume scans.

First let us examine the sensitivity of a PFA to a change in
*N~v~*/*N~h~*. The analysis for the uniform sum (Fig. 2.4) indicates the
PFA to be very sensitive to change in the *N~v~*/*N~h~* ratio. Moreover,
the plots show that if we want to be within ±10% of the desired PFA, the
true ratio *N~v~*/*N~h~* value must not differ more than 0.5% from the
ratio for which the threshold is calculated. More important, though, is
how much the resulting POD varies as the actual ratio *N~v~*/*N~h~*
departs from the one for which the threshold was calculated. This is
shown in Fig. 2.5 and Table Error: Reference source not found which
suggest that the POD is moderately sensitive to variation in
*N~v~*/*N~h~*. For instance, if *M* is 17 the drop in the *N~v~*/*N~h~*
ratio of 2% results in only 0.5% loss in the detection rate. Moreover,
as *M* increases the loss in POD for the same change in noise power
ratio decreases, so if *M* = 52 the detection rate is merely 0.05%
smaller for the 0.02 drop in the *N~v~*/*N~h~*. To put this into
perspective and examine how this reflects on the detection of real
signals, the following experiment is performed. The actual measured
ratio of noise powers in V vs. H channel is 0.8269, so to assess the
consequence of having *N~v~*/*N~h~* different from the one for which the
threshold is found, , the amount of noise power added in the V channel
is as follows. Let Δ be the amount by which we want the *N~v~*/*N~h~* to
be skewed (i.e.,the desired difference between the *N~v~*/*N~h~* ratio
for which the threshold is given and the true one). Then the amount of
noise power that needs to be added in the V channel to achieve the
desired ratio is

![](media/image19.wmf) (2.10)

The results of choosing various values of Δ are given in Error:
Reference source not found and Error: Reference source not found, for
*M* = 17, and *M* = 52, respectively. These appear in agreement with the
simulation results in Error: Reference source not found.

The analysis from the previous paragraph implies that for each *M*, an
efficient method for threshold computation, based on the measured
*N~v~*/*N~h~* ratio, should be devised. In this regard, the plot in Fig.
2.6 shows the threshold as function of the *N~v~*/*N~h~* ratio for

*M* = 17

![](media/image20.wmf){width="6.377777777777778in"
height="5.097222222222222in"}

Fig. 2.4 Uniform sum PFA sensitivity to change in the ratio
*N~v~*/*N~h~* for threshold adjusted for PFA of 1.2×10^-6^, and the
unity ratio *N~v~*/*N~h~*.

and PFA = 1.2×10^-6^. The circle markers in the plot denote the
thresholds found for *N~v~*/*N~h~* ranging from 0.5 to 1 with the step
of 0.05. The dashed line is the least squares fit using the gamma
function

THR = *N~h~ x^B^e^A+Cx^* \[**2.11**\]

where *x* = *N~v~*/*N~h~*, to circle markers. As an assessment of the
interpolation quality, square red markers are placed on top of the
interpolation curve. These represent the estimated thresholds for
*N~v~*/*N~h~* ratio ranging from 0.6 to 0.98 with the step of 0.02.
Apparently, the gamma curve obtained as the least squares fit is an
excellent approximation and can be used to quickly calculate the
threshold for any ratio of noise powers in H and V channels.

![](media/image21.wmf){width="6.499305555555556in"
height="6.797916666666667in"}Fig. 2.5 Uniform sum POD sensitivity to
change in the ratio *N~v~*/*N~h~* for threshold set to PFA of
1.2×10^-6^, for the unity ratio *N~v~*/*N~h~*, and parameters *SNR~h~* =
0 dB, *σ~v~* = 2 m s^-1^, *Z~DR~* = 1 dB, *ρ~HV~* = 0.96.

Table 2.1 The estimated drop in uniform sum POD vs. the drop in the
ratio of *N~v~*/*N~h~* for threshold set to PFA of 1.2×10^-6^, for the
unity ratio *N~v~*/*N~h~*, and signal parameters *SNR~h~* = 0 dB, *σ~v~*
= 2 m s^-1^, *Z~DR~* = 1 dB, *ρ~HV~* = 0.96.

+---------+---------+---------+---------+---------+---------+---------+
| Δ       | -0.01   | -0.02   | -0.03   | -0.04   | -0.05   | -0.06   |
| *N~v~*  |         |         |         |         |         |         |
| /*N~h~* |         |         |         |         |         |         |
+---------+---------+---------+---------+---------+---------+---------+
| Δ POD   | -2.28   | -4.57   | -6.91   | -9.19   | -1.15   | -1.39   |
|         | ×10^-3^ | ×10^-3^ | ×10^-3^ | ×10^-3^ | ×10^-2^ | ×10^-2^ |
| (M=17)  |         |         |         |         |         |         |
+---------+---------+---------+---------+---------+---------+---------+
| Δ POD   | -1.44   | -2.86   | -4.5    | -5.84   | -7.65   | -9.11   |
|         | ×10^-3^ | ×10^-3^ | ×10^-3^ | ×10^-3^ | ×10^-3^ | ×10^-3^ |
| (M =    |         |         |         |         |         |         |
| 32)     |         |         |         |         |         |         |
+---------+---------+---------+---------+---------+---------+---------+
| Δ POD   | -2.45   | -4.71   | -6.9    | -9.28   | -1.16   | -1.41   |
|         | ×10^-4^ | ×10^-4^ | ×10^-4^ | ×10^-4^ | ×10^-3^ | ×10^-3^ |
| (M =    |         |         |         |         |         |         |
| 52)     |         |         |         |         |         |         |
+---------+---------+---------+---------+---------+---------+---------+

Table 2.2 The uniform sum detections vs. the difference between the
actual ratio of *N~v~*/*N~h~* and the one the threshold was adjusted
for, in case of *M* = 17, and PFA of 1.2×10^-6^.

  ------------- ---------- ---------- ---------- ---------- ---------- ----------
  Δ             0          -0.01      -0.02      -0.05      -0.1       -0.2

  *Ratio of     0.984366   0.983747   0.983924   0.983573   0.982610   0.980850
  total                                                                
  detections*                                                          

  *The Ratio of 0.822027   0.814692   0.814696   0.812835   0.800011   0.782148
  bounded                                                              
  detections*                                                          

  *Ratio of     0.021714   0.021760   0.021615   0.020041   0.018912   0.015892
  additional                                                           
  detections*                                                          
  ------------- ---------- ---------- ---------- ---------- ---------- ----------

Table 2.3 The uniform sum detections vs. the difference between the
actual ratio of *N~v~*/*N~h~* and the one the threshold was adjusted
for, in case of *M* = 52, and PFA of 1.2×10^-6^.

  ------------- ---------- ---------- ---------- ---------- ---------- ----------
  Δ             0          -0.02      -0.05      -0.1       -0.2       -0.4

  *Ratio of     1          1          1          1          1          1
  total                                                                
  detections*                                                          

  *The Ratio of 0.741758   0.741758   0.741758   0.741758   0.741758   0.741758
  bounded                                                              
  detections*                                                          

  *Ratio of     0.471704   0.466731   0.457801   0.451406   0.437871   0.404143
  additional                                                           
  detections*                                                          
  ------------- ---------- ---------- ---------- ---------- ---------- ----------

![](media/image22.wmf){width="6.498611111111111in"
height="4.79375in"}Fig. 2.6 Straight line approximation to function that
takes the *N~v~/N~h~* ratio as an input and outputs threshold for PFA =
1.2×10^-6^.

**Table 2.4** Parameters for threshold calculation as THR =
max(*N~h~,N~v~*)·\
(min(*N~h~*,*N~v~*)/max(*N~h~,N~v~*))^B^·exp(A+C·min(*N~h~*,*N~v~*)/max(*N~h~,N~v~*)).

  --------- --- --- -------- ----------- -------- ----------- -------- ----------- -------- ----------- -------- ----------- -------- ----------- -------- ------------ -------- ------------ -----
  *M*/PFA                    5×10^-7^             6×10^-7^             7×10^-7^             8×10^-7^             9×10^-7^             1×10^-6^             1.1×10^-6^            1.2×10^-6^   

  10            A            1.4207               1.4156               1.4127               1.4137               1.4071               1.4020               1.3988                1.3975       

                B            -0.1047              -0.1027              -0.1001              -0.0947              -0.0961              -0.0965              -0.0957               -0.0939      

                C            0.6079               0.6062               0.6030               0.5968               0.5990               0.5999               0.5994                0.5973       

  11            A            1.3904               1.3873               1.3762               1.3697               1.3709               1.3668               1.3653                1.3615       

                B            -0.0859              -0.0825              -0.0863              -0.0873              -0.0823              -0.0821              -0.0802               -0.0803      

                C            0.5857               0.5821               0.5872               0.5888               0.5832               0.5833               0.5811                0.5816       

  12            A            1.3534               1.3473               1.3443               1.3395               1.3410               1.3349               1.3338                1.3298       

                B            -0.0767              -0.0758              -0.0736              -0.0731              -0.0681              -0.0697              -0.0676               -0.0679      

                C            0.5774               0.5771               0.5745               0.5745               0.5687               0.5710               0.5684                0.5693       

  16            A            1.2242               1.2281               1.2252               1.2134               1.2204               1.2084               1.2048                1.2036       

                B            -0.0595              -0.0517              -0.0494              -0.0549              -0.0459              -0.0525              -0.0526               -0.0511      

                C            0.5650               0.5551               0.5530               0.5606               0.5496               0.5583               0.5587                0.5570       

  17            A            1.2201               1.2175               1.2099               1.1990               1.1965               1.1936               1.2064                1.2039       

                B            -0.0402              -0.0381              -0.0395              -0.0439              -0.0425              -0.0419              -0.0300               -0.0293      

                C            0.5404               0.5369               0.5399               0.5468               0.5456               0.5450               0.5285                0.5285       

  21            A            1.1271               1.1217               1.1149               1.1103               1.1056               1.1040               1.1087                1.1026       

                B            -0.0351              -0.0353              -0.0359              -0.0364              -0.0369              -0.0356              -0.0292               -0.0319      

                C            0.5405               0.5406               0.5433               0.5439               0.5453               0.5437               0.5356                0.5396       

  25            A            1.0785               1.0733               1.0728               1.0703               1.0678               1.0644               1.0648                1.0622       

                B            -0.0141              -0.0142              -0.0109              -0.0097              -0.0088              -0.0089              -0.0064               -0.0062      

                C            0.5180               0.5181               0.5145               0.5133               0.5125               0.5129               0.5098                0.5100       

  28            A            1.0457               1.0408               1.0378               1.0352               1.0326               1.0313               1.0282                1.0280       

                B            -3.765e-3            -3.416e-3            -2.561e-3            -1.527e-3            -9.055e-4            5.054e-4             3.237e-4              2.192e-3     

                C            0.5064               0.5067               0.5055               0.5046               0.5041               0.5025               0.5031                0.5009       

  29            A            1.0227               1.0232               1.0189               1.0148               1.0099               1.0090               1.0068                1.0098       

                B            -9.285e-3            -5.248e-3            -5.108e-3            -5.286e-3            -6.399e-3            -4.801e-3            -4.322e-3             -2.64e-4     

                C            0.5171               0.5118               0.5122               0.5129               0.5147               0.5127               0.5125                0.5072       

  32            A            0.9939               0.9966               0.9869               0.9916               0.9907               0.9872               0.9847                0.9848       

                B            -1.213e-3            4.296e-3             4.25e-4              5.993e-3             8.099e-3             7.707e-3             7.02e-3               9.016e-3     

                C            0.5103               0.5028               0.5089               0.5005               0.4986               0.4994               0.4994                0.4970       

  37            A            0.9472               0.9462               0.9430               0.9422               0.9409               0.9374               0.9354                0.9350       

                B            4.742e-3             7.631e-3             8.143e-3             1.011e-2             1.143e-2             1.088e-2             1.135e-2              1.268e-2     

                C            0.5062               0.5030               0.5026               0.5001               0.4986               0.4996               0.4993                0.4976       

  41            A            0.9312               0.9431               0.9516               0.9339               0.9391               0.9365               0.9361                0.9352       

                B            0.0197               0.0317               0.0414               0.0297               0.0373               0.0371               0.0388                0.0398       

                C            0.4873               0.4710               0.4588               0.4737               0.4661               0.4662               0.4646                0.4634       

  43            A            0.9271               0.9382               0.9343               0.9375               0.9125               0.9247               0.9221                0.9155       

                B            0.0288               0.0399               0.0393               0.0441               0.0277               0.0395               0.0393                0.0359       

                C            0.4754               0.4604               0.4609               0.4542               0.4773               0.4625               0.4630                0.4677       

  46            A            0.9022               0.9014               0.8983               0.9041               0.9064               0.8942               0.8955                0.8969       

                B            0.0275               0.0301               0.0304               0.0372               0.0416               0.0340               0.0372                0.0394       

                C            0.4801               0.4768               0.4765               0.4679               0.4633               0.4734               0.4700                0.4665       

  52            A            0.8782               0.8715               0.8579               0.8564               0.8729               0.8735               0.8745                0.8572       

                B            0.0404               0.0377               0.0299               0.0310               0.0464               0.0480               0.0508                0.0394       

                C            0.4673               0.4699               0.4806               0.4793               0.4602               0.4573               0.4543                0.4704       

  56            A            0.8368               0.8422               0.8406               0.8431               0.8411               0.8359               0.8368                0.8342       

                B            0.0265               0.0330               0.0347               0.0385               0.0389               0.0369               0.0396                0.0388       

                C            0.4871               0.4778               0.4766               0.4713               0.4709               0.4741               0.4716                0.4722       

  60            A            0.8352               0.8348               0.8319               0.8338               0.8319               0.8207               0.8252                0.8252       

                B            0.0405               0.0430               0.0429               0.0469               0.0467               0.0405               0.0451                0.0467       

                C            0.4686               0.4659               0.4656               0.4610               0.4606               0.4700               0.4636                0.4618       

  63            A            0.8377               0.8412               0.8236               0.8336               0.8210               0.8276               0.8238                0.8246       

                B            0.0538               0.0588               0.0485               0.0576               0.0505               0.0567               0.0552                0.0576       

                C            0.4528               0.4457               0.4607               0.4476               0.4588               0.4497               0.4515                0.4492       

  64            A            0.8202               0.8211               0.8107               0.8160               0.8100               0.8085               0.8121                0.8052       

                B            0.0441               0.0474               0.0422               0.0478               0.0448               0.0456               0.0498                0.0461       

                C            0.4663               0.4619               0.4696               0.4616               0.4652               0.4648               0.4593                0.4646       

  70            A            0.8056               0.8211               0.7950               0.7969               0.8031               0.8024               0.7897                0.7985       

                B            0.0525               0.0673               0.0492               0.0532               0.0594               0.0602               0.0521                0.0606       

                C            0.4568               0.4381               0.4615               0.4575               0.4489               0.4475               0.4588                0.4487       

  87        A       0.7601               0.7510               0.7447               0.7379               0.7324               0.7542               0.7433                0.7718                

            B       0.0615               0.0586               0.0530               0.0519               0.0488               0.0668               0.0599                0.0827                

            C       0.4477               0.4544               0.4577               0.4630               0.4667               0.4423               0.4524                0.4216                

  88        A       0.7150               0.7596               0.7450               0.7438               0.7495               0.7417               0.7495                0.7407                

            B       0.0285               0.0663               0.0573               0.0592               0.0634               0.0593               0.0662                0.0610                

            C       0.4912               0.4429               0.4554               0.4551               0.4462               0.4525               0.4427                0.4505                
  --------- --- --- -------- ----------- -------- ----------- -------- ----------- -------- ----------- -------- ----------- -------- ----------- -------- ------------ -------- ------------ -----

In the batch mode there are cases whereby a small number of pulses are
used in the long PRT (i.e., surveillance) chunk. For instance, in VCP 11
a batch mode exists with 6 pulses for surveillance. If power based
censoring is used with the threshold set to 3.5 dB above the noise
power, the rate of false detections can be obtained from eq. (A.11) in
Appendix A, which gives the PFA of 1.1078×10^-4^. The coefficients for
LS gamma fit, for such case, are given in Table 2.5.

**Table 2.5** Parameters for threshold calculation as THR =
max(*N~h~,N~v~*)·\
(min(*N~h~*,*N~v~*)/max(*N~h~,N~v~*))^B^·exp(A+C·min(*N~h~*,*N~v~*)/max(*N~h~,N~v~*)).

  --------- ---- ---------- ---------- ---------- ---------- ---------- ---------- ------------ ------------
  *M*/PFA        5×10^-5^   6×10^-5^   7×10^-5^   8×10^-5^   9×10^-5^   1×10^-4^   1.1×10^-4^   1.2×10^-4^

  6         A    1.4783     1.4664     1.4758     1.4558     1.4507     1.4463     1.4424       1.4425

            B    -0.1114    -0.1114    -0.0975    -0.1054    -0.1027    -0.1011    -0.0991      -0.0956

            C    0.6228     0.6240     0.6050     0.6168     0.6151     0.6126     0.6105       0.6047
  --------- ---- ---------- ---------- ---------- ---------- ---------- ---------- ------------ ------------

Similarly, when 8 pulses are used, the resulting false alarm rate is
1.1713×10^-5^. The coefficients for LS gamma fit are given in Table 2.6

**Table 2.6** Parameters for threshold calculation as THR =
max(*N~h~,N~v~*)·\
(min(*N~h~*,*N~v~*)/max(*N~h~,N~v~*))^B^·exp(A+C·min(*N~h~*,*N~v~*)/max(*N~h~,N~v~*)).

  --------- --- ---------- ---------- ---------- ---------- ------------ ------------ ------------ ------------
  *M*/PFA       7×10^-6^   8×10^-6^   9×10^-6^   1×10^-5^   1.1×10^-5^   1.2×10^-4^   1.3×10^-4^   1.4×10^-4^

  8         A   1.4396     1.4429     1.4383     1.4358     1.4296       1.4170       1.4263       1.4067

            B   -0.0965    -0.0890    -0.0878    -0.0857    -0.0863      -0.0931      -0.0822      -0.0947

            C   0.6016     0.5918     0.5908     0.5881     0.5898       0.5983       0.5850       0.6012
  --------- --- ---------- ---------- ---------- ---------- ------------ ------------ ------------ ------------

**3. Computation of polarimetric variables for the simultaneous H, V
(SHV) scheme**

Computations of polarimetric variables described herein apply to a radar
that transmits and receives simultaneously horizontally and vertically
(SHV) polarized waves. Demonstration of such concept was made on the
KOUN radar and the same is proposed for the WSR-88D upgrade. The
computation of polarimeric variables can be done in the time domain or
frequency domain. The real time processing on the KOUN (implemented by
Sigmet) uses time domain computations. No clutter filter or schemes for
mitigating Doppler ambiguities were incorporated into the demonstration.
If spectral clutter filter is applied, a natural way to obtain the
polarimetric variables is from the complex spectral coefficients.
Similarly, if phase coding is applied, it becomes much more efficient to
use directly the complex spectral coefficients of the weak echo signal
than the time domain techniques for computing the polarimetric
variables. This is because the spectral replicas of the weak signal are
required for computing its spectral moments.

The purpose here is to present the equations for both time domain and
frequency domain computations. Moreover, in case of time domain, two
variants for computing the polarimetric variables are given. One is the
standard method (employed on the KOUN) the other involves use of
autocorrelations at lag 1.

The notation used follows the one in Section 2. It is repeated here for
convenience. Assume that the sequence is transmitted at a uniform PRT of
duration *T~s~*. *H~i~* is a complex signal (in-phase and quadrature
phase) of horizontally polarized echoes at a fixed range location (same
range gate) and, the first echo received is *H~0~*. The spacing between
*H~i~* samples is *T~s~* and the total number of H samples is *M* (that
is the index *i* goes from 0 to *M-*1). *V~i\ ~*is a complex signal of
vertically polarized echoes, these are transmitted at the same time as H
samples. Therefore spacing between *V~i~* s is also *T~s~* and total
number of *V~i~* samples is also *M*. So the sequence of samples is
(*H~0~ V~0~),* (*H~1~ V~1~*), (*H~2~ V~2~*), (*H~3~ V~3~*)...etc.

**3.1 Standard method from time series data**

The procedure written here is extracted form the paper by Zahrai and
Zrnic (1991) and the patent by Zrnic (1996). Estimators of the
polarimetric variables incorporated on the KOUN are listed next.

a\) *Estimation of reflectivity factor*

Two powers, one for horizontally the other for vertically polarized
signals, are computed as follows

![](media/image23.wmf) (3.1)

![](media/image24.wmf) (3.2)

From (3.1), the reflectivity factor for horizontal polarization *Z~h~*
is computed using system parameters and the radar equation. If the noise
power is known (measured) it should be subtracted from (3.1) and (3.2).
Otherwise it may be better to subtract the noise in the Radar Product
Generator (RPG), or leave it alone. If the signal-to-noise ratios
(linear scale) at two polarizations are measured, one can take the noise
out by multiplying

Unbiased *P~sh~* = *P~h~* \[*SNR~h~*/(1+*SNR~h~*)\], (3.3)

Unbiased *P~sv~* = *P~v~* \[*SNR~v~*/(1+*SNR~v~*)\]. (3.4)

Note: The best way is to estimate the noise power separately for each
radial; otherwise, use the calibration curve to obtain unbiased *P~h~*
and *P~v~*. If for some reason the unbiasing is not practical, use
equations (3.1) and (3.2); at SNR\>15 dB the noise bias should not be
significant.

b\) *Differential reflectivity Z~DR~*

*Z~DR~* = 10 log (Unbiased *P~h~ /* Unbiased *P~v~*) (dB) \[**3.5**\]

c\) *Differential phase* Φ~DP~

First compute autocorrelation *R~hv~* (between *H~i~* and *V~i~*
signals) as follows

![](media/image25.wmf). (3.6)

The total differential phase Φ~DP~ is computed from (3.6) as

Φ~DP~ = arg\[*R~hv~*(0)\] (deg) \[**3.7**\]

The differential phase computed from (3.7) could be ambiguous and should
be assigned to a contiguous 360 deg interval. But first the differential
phase of the system must be removed from (3.7) and 25 deg should be
added to the result. That way statistical fluctuation would not cause
discontinuities in differential phase of storms close to the radar; by
starting the phase 25 deg above zero allows for a large increase (close
to 360 deg) before aliasing occurs. Thus, the differential phase of the
system must be measured. This is done once by recording the histogram of
differential phase of ground clutter (Zrnic et al. 2006) and taking its
mean value. Alternately, it can be estimated from the value of Φ~DP~
measured in precipitation. Precipitation close to the radar causes small
differential phase shift (less than 1 deg) so that the bulk of the
difference is due to the system. To illustrate, let the system phase be
-40 deg. The measurement may indicate some value close to -40 deg (could
be a little more of less due to statistical uncertainty). Then 65 deg
should be added to all the differential phases, 40 deg to remove system
phase and 25 deg margin to eliminate aliasing at close ranges.

Note: Because in general the complex signals might not follow the
convention assumed here (that is they might be conjugates of what is
given herein) the differential phase could decrease in rain. If that
happens, change the sign or switch the I,Q lines, or ask Ann Landers.

d\) *Mean Doppler Velocity v*

The formulas presented here are a repeat from Section 2. These are
suitable in cases that do not involve the clutter filter or phase
coding. Two autocorrelations can be estimated to compute the Doppler
phase shifts.

![](media/image26.wmf) (3.8)

![](media/image27.wmf) (3.9)

Let

ψ = arg(*R~h~* + *R~v~*), (3.10)

then the Doppler velocity is computed as

*v* = - λψ/(4πT~s~), (m s^-1^) \[**3.11**\]

where λ is the wavelength (about 0.01 m, the exact value should be
computed from the known frequency, and *T~s~* should be in s). At SNRs
larger than 3.5 dB it may suffice to use only one autocorrelation and
that should be *R~h~* because it is usually stronger. Both should be
definitely used at SNRs between 0.5 and 3.5 dB (see Section 2 eq. 2.5).

e\) *Spectrum width σ~v~*

Use equation (3.8) and (3.1) or (3.3), depending on how you compute the
powers. The spectrum width is then

σ~v~ = λ \[ln (*P~h~*/\|*R~h~*(T~s~)\|)\]^1/2^ */*(2*πT~s~*√2) (m
s^-1^). \[**3.12**\]

If the argument under the logarithm is smaller than 1 set the spectrum
width to 0 or to missing value. (See equation 6.27 in Doviak and Zrnic
1993). There are other ways to compute the spectrum width, such as from
spectral processing, lag 2 and lag 1 autocorrelations (implemented in
the SZ-2 phase coding), and so on; you can choose any that you trust.

f\) *Correlation coefficient at zero lag ρ~hv~(0)*

To compute *ρ~hv~*(0) use (3.1), (3.2), or (3.3), (3.4) and (3.6). The
result is

![](media/image28.wmf). **\[3.13\]**

Equations numbered bold, bracketed with \[\], and offset to the right
represent the six variables that the primary signal processor should
provide. Computation of specific differential phase, various thresholds,
noise subtraction, and censoring should be done in the RPG.

**3.2 Method based on autocovariances**

Let *R~h~*(*T~s~*) and *R~v~*(*T~s~*) be the autocorrelations of the
horizontally and vertically polarized returns at lag *T~s~* (where
*T~s~* is the pulse repetition time). These autocorrelations are
computed from the time series data or from the spectra.

From time series *R~h~*(*T~s~*) is obtained as

![](media/image29.wmf) (3.14)

where *M* is the number of sample pairs, index *n* refers to time
(*nT~s~*), and *H* is the complex return at a specific range location.

From the spectrum *S~h~*(*k*)

![](media/image30.wmf) (3.15)

Identical formulas apply to the autocorrelations of the *V* signals.

We follow the notation in Doviak and Zrnic (1993) hence the
autocorrelation (3.14 or 3.15) has the form

*R~h~*(*T~s~*) = *P~sh~ ρ*. (3.16)

Similarly

*R~v~*(*T~s~*) = *P~sv~ ρ*, (3.17)

where *ρ* = │*ρ*│exp(*jω~d~T*), and *P~sh\ ~*and *P~sv~* are signal
powers (same as *S~h~* and *S~v~* in Doviak and Zrnic 1993).

Following Melnikov and Zrnic (2004) two cross correlation at lag *T~s~*
are expressed as:

![](media/image31.wmf), (3.18)

and

![](media/image32.wmf). (3.19)

where *ρ~hv~* is the copolar correlation coefficient.

From equations (3.16, 3.17, 3.18 and 3.19) the polarimetric variables
Z~dr~ (in linear units), and *ρ~hv~* should be computed as (see details
as well as errors in Melnikov and Zrnic 2004)

![](media/image33.wmf) \[**3.20**\]

and

![](media/image34.wmf). \[**3.21**\]

These estimates are not biased by noise, but have somewhat larger
standard errors.

Similarly, there is a spectrum width formula which is also not biased by
white noise (see Melnikov and Zrnic 2004). The formula reads

![](media/image35.wmf) , \[**3.22**\]

where ![](media/image36.wmf). (3.23)

The arithmetic average of the two autocorrelations in (3.21) and (3.22)
can be replaced with the geometric mean, that is

│*R~hv~*(*T~s~*) *R~vh~*(*T~s~*)│^1/2^. (3.24)

This was done and tested on simulated signals in the SNR range of 2 to
20 dB. Practically there were no differences between the errors obtained
with either of these two variants.

Formulas in this section produce estimates that are not biased by white
noise. Hence we recommend these for Doppler scans if the white noise is
not estimated at each radial. For the long PRT surveillance scans and at
larger spectrum widths these formulas could produce estimates with
larger statistical errors. In such cases adaptive estimators can be
applied, so that the estimate would be either from autocovariances or
standard depending on the values of the correlation coefficients.

Last a brief suggestion on how to use the covariance estimates in cases
of staggered PRT or batch mode. Take the staggered PRT. There are two
autocorrelation pairs, *R~h~*(*T~1~*), *R~v~*(*T~1~*) and

*R~h~*(*T~2~*), *R~v~*(*T~2~*). Each of the autocorrelations can be
estimated from the usual sum of lagged products. Then the combination of
ratios (for example to compute *Z~dr~*) would need to be used in either
an arithmetic mean or geometric mean like

![](media/image37.wmf) (3.25a)

or ![](media/image38.wmf). (3.25b)

Analysis of (3.25a) and (3.25b) needs to be made to determine under
which conditions one of these prevails. Adaptive estimators should be
also considered for this case.

**3.3 Spectral Computations**

Let the complex spectrum coefficients be *A~h~*(*k*) and *A~v~*(*k*),
where *k* is the spectral index. Because there are at most *M* spectral
coefficients, it is possible to obtain the polarimetric variables from
sums involving all of these coefficients. But often several spectral
coefficients are discarded because they are either heavily contaminated
by clutter, could be due to noise, or are not available (as in case of
weak trip spectra of phase coded signals). In such cases, the remaining
*L* spectral coefficients should be used with the following caveat. The
significant coefficients must come from the same subset of the spectrum
of both H and V signals. Preferably, there should be no additional
manipulations of the spectral coefficients (such as interpolation) other
than use of the same window on the H and V time series. The cross
correlation between the H and V spectra is computed as

![](media/image39.wmf), (3.26)

where *L* is the number of coefficients which are significant.
Similarly, the powers are computed from

![](media/image40.wmf) (3.27)

![](media/image41.wmf) (3.28)

The spectral noise powers should be subtracted from \|*A~v~*(*i*)\|^2^
and \|*A~v~*(*i*)\|^2^ to obtain unbiased values. Then, equations (3.5),
(3.7), and (3.13) are applicable for computing the spectral moments and
polarimetric variables.

**3.4 Estimation of the noise powers**

In the current WSR-88D, receiver noise power is estimated at the end of
every volume scan and applied to data from the subsequent scan. Such
procedure produces accurate results most of the time. Nonetheless, the
noise power is not constant. It increases if there is precipitation
along the beam because particles radiate noise. Further, at low
elevations, contribution from the earth increases the noise level and
there are incoherent radiation sourced that also appear as white noise.
In the polarimetric WSR-88D, noise powers in both receiving channels
must be estimated. If the estimates of noise powers are made at the end
of every volume scan, then it would be advantageous to use the unbiased
estimates of polarimetric variables (i.e., from autocovariances, section
3.3). Excepted are the cases where spectral computations (as with
clutter filters, or ambiguity mitigation techniques) are required.

If noise powers are estimated in each radial then the standard method
would work well.

**4. Polarimetric measurements and mitigation of range velocity
ambiguities**

General overview of computations to obtain the polarimetric variables
and mitigate ambiguities in range and velocity are presented herein. For
the most part the proposed computations have been tested on time series
data and/or via simulations. The discussion indicates preferred
computations which are explained in sufficient detail to enable
incorporation into the signal processing system of the ORDA. It should
be understood that the capability of the signal processor and time
constraints of the developers might require a phased approach. Such
would initially incorporate less complicated, faster to run, test, and
code procedures. These would then be improved through an evolutionary
process.

**4.1 Processing flow in the ORDA**

In Fig. 4.1 plotted is a block diagram of pertinent subsystems
envisioned for the time of polarimetric upgrade. Two channels, one for
horizontal the other for vertical polarization are indicated. The
current design contains one channel consisting of the first two blocks
in the H path. Addition of the vertical polarization requires a second
channel. There might be few variants in the actual implementation but
these would not change the essence depicted in Fig. 4.1. For example the
operations on the I & Qs could be done sequentially, or a physical
separation into parallel streams for processing might be made. Decision
on the optimum configuration should be made at a time closer to
implementation when the tradeoff between cost and performance will be
easier to quantify.

Fig 4.1. Data flow and major computation blocks in the dual polarization
radar with simultaneous transmission/reception o horizontally and
vertically polarized signals.

Fig. 4.2 (Previous page). More detailed flow of information and
computations. The left side represents the flow for a single (H) channel
and in the middle are operations and data flow (dashed lines with
arrows) for dual polarization. The flow for the V channels is on the
right side, and it is essentially identical to the H channel. The
control above the red curve signifies that some computations in the V
channel depend on the computations in the H channel. The two blocks
"Computation of spectral moments" are actually connected as some of the
moments (i.e., mean velocity and spectrum width) require combination of
signal autocovariances at the two polarizations.

The Digital receiver block performs down conversion of IF signals to
base band and digital filtering (matched, if needed). The Sigmet design
allows output sample spacing, length of the filter (in range time), and
pulse length to be independent. Therefore the length of the filter can
be larger than the output sample spacing. This is important for
conventional processing of oversampled (in range) signals. The sequence
of operations on the I,Q samples is as follows (see the blocks on the
left of Fig. 4.2).

*Procedure*

1- Samples are corrected for phase discontinuity in range. This is now
part of the standard procedure implemented on the RVP8.

2 - Oversampling -- initially 2.1 (see next paragraph) might be used. At
a later time 2.2 or a variant thereof could supersede it.

2.1 - Oversampling with subsequent averaging.

The samples could be spaced 50 m apart so that there would be 5 samples
within one pulse (250 m). The digital filter will be matched to 50 m (in
the digital receiver block). In the processing block there will be two
steps, one consists of identical processing (as without oversampling) on
each I,Q (oversampled), the other will recombine the intermediate
results to a resolution of 250 m.

2.1.1 *First step:* will depend on the choice technique for R/V
mitigation.

*2.1.1.1 Phase Coding -- split cut:* The procedure is exactly as
documented in the NSSL report (Torres et al. 2004). It includes
determining location of overlaid echoes, decoding for the desired
recovery, and ground clutter filtering with the GMAP filter.

*2.1.1.2* *Staggered PRT:* The procedure (without clutter filter) has
been tested in real time and documented in the NSSL's report (Torres et
al. 2003). The proposed clutter filter will use GMAP (or variant) to
identify clutter components. Removal of these at or close to zero can be
done with the GMAP or a similar procedure. Replacement of clutter
spectral components with correct weather components at other four
locations (where clutter had been superposed due to staggered PRT) will
be done according to the procedure developed at NSSL (Sachidananda 1999,
Torres et al. 2005). This procedure has been tested on data collected
with the RRDA and it works very well.

2.1.2 *Second step:*

Intermediate results such as autocovariances and spectra at 5
consecutive range locations will be combined (averaged) into one
autocovariance (spectrum) from which spectral moments at 250 m range
interval will be computed. Prior to averaging the intermediate results,
data quality control could be applied to determine if some (or all) of
the intermediate results should be censored. A simple consensus can be
applied so that only estimates which agree (say three or more out of
five) are combined. Other refinements should be examined. Suppose four
out of five estimates are heavily contaminated by ground clutter (or
AP), then it would make sense to retain the uncontaminated estimate.
Clearly there will be several improvements which will evolve once
oversampling is implemented.

2.2 Oversampling and whitening or pseudo whitening

The principal feature of whitening or pseudo whitening is that blocks of
5 consecutive oversampled signals will be transformed into blocks of 5
independent (I,Q) signals (almost independent in case of pseudo
whitening). Each of these five is processed to obtain intermediate
results which are then combined into spectral moment estimates at 250 m
interval. Phase coding will be included in the processing similar to the
procedure in 2.1.

*Issues* -- [Ground clutter filtering in the case of oversampled signals
needs to be examined.]{.underline}

Thus far, the described procedure is equally applicable to single and
dual polarization radar. But, in dual polarization mode there are
additional computations and some constraints that need attention. These
are explained next.

**4.2 Polarimetric variables**

In the simultaneous transmission scheme the polarimetric variables can
be estimated from sums of operations on returns from single pulses
(Section 3.1). That is, the estimation is similar as done for
reflectivity in that no cross product of returns from successive pulses
is required. Hence, if it were not for velocity measurements,
polarimetric variables could be obtained with no range ambiguities
whatsoever. That is how the data were obtained during the JPOLE
experiment. Uniform PRTs have been the norm during JPOLE because at the
time Sigmet processor was configured to accept only uniform pulse
trains. Because Doppler measurements are also required, there are some
specifics pertaining to estimation of polarimetric variables that need
to be addressed if using a scheme to mitigate the effects of range
velocity ambiguities.

Polarimetric variables can also be estimated from cross products of
returns on successive pulses (Melnikov and Zrnic 2004, see also Section
3.2) or spectral processing (Section 3.3) can be used to enhance the
quality of estimates. These techniques have been tested on times series
data.

Two choice techniques to mitigate R/V ambiguities to be soon implemented
on the WSR-88D are systematic phase coding and staggered PRT. NSSL
recommends a volume coverage pattern with phase coding at the lowest
elevations (one or two) and staggered PRT at higher elevations. Thus,
herein we discuss ways to compute polarimetric variables and also
mitigate the effects of range and velocity ambiguities. First, we
briefly outline what procedure to employ in standard VCPs that have
surveillance and Doppler scans at low elevations, batch at intermediate
elevations, and Doppler at higher elevations.

**4.2.1 Standard VCP**

Polarimetric variables should be computed in both surveillance and
Doppler scans. Then they should be sent to the RPG for further
preprocessing. In the RPG, we recommend combining the polarimetric
variables (and/or various correlation functions) from the two scans in
the following manner.

If there are no overlaid echoes or the strongest trip power is larger by
20 dB (depending on the variable this number could be 15 to 25 dB) from
the sum of weaker powers take the average of each variable (correlation
functions if appropriate) from the two scans. If there are overlaid
echoes take the variables from the long PRT. Then preprocess the new
composite PPI to obtain K~DP~ and smooth data for classification and
quantitative precipitation estimation.

The main thing here is to produce two streams of polarimetric data (or
correlations) and, in the future, combine the two into one in the RPG.

**4.2.2 Phase coding**

Ground clutter requirements and possibility to perform spectral analysis
suggest phase coding at the lowest one or two elevations. Further,
requirement for un-obscured measurement of reflectivity to \~ 460 km
favors separate scans for Z and v; thus long and short PRTs as in the
current VCP are recommended. In this case, the polarimetric variables
can and should be computed from both PRTs. The main thing here is to
produce two streams of polarimetric data and combine the two into one in
the RPG.

a\) *Surveillance scan*

Estimates from the long PRT are free of overlaid echoes but have larger
statistical errors. Standard formulas for computing these estimates are
listed in Section 3.1. And this is how the computations were made on the
Sigmet processor which we used in the JPOLE experiment. Improvements in
estimates might be achieved through spectral processing, and this needs
to be explored. Note that the spectrum of the long PRT data would occupy
a relatively large part of the Nyquist interval therefore filtering of
noise would be less effective compared to what is possible with Doppler
PRTs. For the same, reason the autocovariance estimators of the
polarimetric variables (Section 3.2) might not be as effective as in the
case of Doppler scans. This should be quantified.

b\) *Doppler scan*

Estimates from short PRTs in regions where there is no overlay of echoes
can be used to supplement the estimates from long PRTs. These estimates
could be the same as currently done, or might involve other more
advanced techniques which will be examined in the near future.
Improvements in sensitivity, noise rejection, and some reduction in
statistical errors can be achieved by spectral processing of the
signals.

*We recommend one lag estimators for the polarimetric variables, because
these are free of noise bias* (Melnikov and Zrnic 2004). *The functional
description is in Section* 3.2*.*

Regions of overlaid echo should be recognized- as currently done in the
R/V mitigation scheme (SZ-2 or the legacy system).

Processing to obtain the polarimetric variables requires some additional
steps. Ground clutter filter should be as recommended in the NSSL report
(Torres et al. 2004) and subsequent modifications (Torres et al. 2005).
It is the Sigmet's GMAP filter with minor enhancements (such as
preservation of phases of the amplitude spectrum) needed for phase
coding. The sequence of operation remains as suggested in the report.

Section 4.5 contains the description of computations of the polarimeric
variables from the short PRT. The essence is to compute the complex
cross correlation from the complex spectrum. That is the echoes are
cohered for the strong trip. The complex coefficients (two or more
spectral replicas) for the weak trip furthest away from the strong are
identified. These coefficients are injected into formulas (3.25, 3.26.
3.27) for computing intermediate results and then the differential
reflectivity, the correlation coefficient, and the differential phase
are calculated (as in Section 3.1). Censoring criteria should be similar
(possibly more stringent) to the ones proposed for velocity and spectrum
width estimates.

*Issue*: Censoring criteria and thresholds need to be determined.

c\) *Ground clutter filter and polarimetric variables*

The ground clutter filter should be a spectral filter in which the
coefficients containing clutter have been removed (Fig. 4.3). There
should be no interpolation over the zero velocity gap to reconstruct the
weather signal power. This applies to all the polarimetric variables.
The exception could be in computing Z~h~, v, and σ~v~ for which the GMAP
could be used as is. The data in the **H** *channel* should determine
the parameters of the clutter filter; these must be replicated exactly
(range gate to range gate) in the **V** *channel* as depicted in Fig.
4.3. That way, each of the remaining spectral coefficients (after
clutter filtering) in the H channel will have a matching coefficient in
the V channel. If v and σ~v~ are computed as in (2.7) and (2.9) then
GMAP should be applied to both H and V channels before combining the
autocovariances. Elements of the GMAP (as in Fig. 4.3) should be used to
identify which coefficients (notch width W) must be removed prior to
computation of the polarimetric variables.

Fig. 4.3 A possible clutter filtering procedure. Alternatives are to
interchange H and V in this figure or to use adaptively as master the
channel in which clutter has more power.

After removal of the clutter spectral components the remaining complex
spectra of the H and V signals should be used to compute the Z~DR~ and
complex ρ~hv~. A spectral SNR threshold should be applied to reduce
noise contamination.

*Issue*: [Clutter canceling followed by computation of polarimetric
variables should be demonstrated on time series data.]{.underline}

*Issue:* Ground clutter power in the V channel is more often higher than
its power in the H channel. Therefore an adaptive (gate to gate) choice
of which channel to assume master control might be a better option than
exclusive control with the H channel.

**4.2.3 Staggered PRT**

The 2/3 stagger ratio scheme is illustrated in Fig. 4.4. This scheme is
recommended for elevations where echoes can not extend beyond a delay of
*T~2~*. In case that the echo extends up to 2*T~1~* censoring can be
made so that such data are removed.

Fig. 4.4 Overlaid echoes in the staggered PRT scheme. It is assumed that
there the scatterers are present up to the time delay *T~2~*.

Examination of powers within the short PRT interval (regions I) and in
region I of the long PRT interval can reveal overlay from the short PRT
into the long PRT interval. Comparison of powers in region II of *T~1~*
and region II of *T~2~* establishes if the echo extends beyond the
*T~2~* interval. The choice *T~1~* and *T~2~* should be such that this
does not happen. Therefore and for simplicity assume that the echo does
not extend beyond the *T~2~* interval (this limit is shown in the figure
4.4). Then the powers, differential reflectivity, and cross correlation
can be computed in the standard manner for intervals I and II within
*T~1~*, and II and III within *T~2~*. The estimates from the interval II
of each PRT should be combined to reduce statistical errors.
Computations of Z~DR~ and ρ~hv~ should also be made in the region I of
*T~2~* at range locations where there are no overlaid echoes. If there
is no overlay it is advantageous to sum the powers and to sum the
intermediate autocorrelations from regions I of *T~1~* and *T~2~.* Then
appropriate ratios can be taken (to obtain Z~DR~ and ρ~hv~). Other
techniques to compute the polarimetric variables should be explored.

**4.4. Filtering ground clutter**

NWS has accepted GMAP for clutter filtering on uniform PRT sequences.
This filter or a modification will also be used on phased coding
sequences. For example application of the filter should be different for
estimation of reflectivity from application for estimation of
differential reflectivity. Normally, before computing Z~h~, the notch
identified by GMAP is filled with a portion of a Gaussian shaped
spectrum which approximates the spectrum of the weather signal. The same
notch filter determined for the H signal must be used for the V signals
(this is indicated with the control line in Fig. 4.2 or withd W in Fig.
4.3). Further, to avoid possible bias in Z~DR~ (which would be caused by
artificial addition of the interpolated powers) there should be no
interpolation of the spectral coefficients across these notches.
Identical reasoning applies to the computation of the cross-correlation
coefficient ρ~hv~.

Another improvement dealing with clutter that the GMAP can make is in
the mode whereby clutter is filtered in an operator-defined sector.
Whereas now, at the short PRTs, the filter is applied to all data in the
sector it would be advantageous to selectively apply the GMAP. This can
be done by using the GMAP on the long PRT to determine which locations
have clutter power (that is the powers above some threshold that the
GMAP has removed). Then these locations could serve as instantaneous
clutter map for application of GMAP at the short PRT. This would be
important for regions where clutter is intermittent such as in cases of
AP.

Polarimetric variables that we have used routinely had been obtained
without clutter filtering. We have also examined polarimetric data after
the time series have been filtered with the GMAP and other spectral
clutter filters.

The polarimetric variables obtained before clutter filtering could be
used as a proxy for an instantaneous clutter map. That might require two
data streams within the RDA, one not filtered and the other filtered.
Suggestions on how to generate the instantaneous clutter map using
polarimetric spectral analysis are in conference paper by Zrnic and
Melnikov (2007).

Finally, to reiterate, filtering data for estimating Z~h~ should differ
from filtering data for estimating Z~DR~, ρ~hv~, and Φ~DP~. Full GMAP is
suitable for Z~h~. But for computing the other polarimetric variables
only a part of GMAP should be used. That is, only the notch provided
within GMAP should be applied to the time series data and there should
be no reconstruction of powers within the notch.

*Issue:* [The effect on polarimetric variables of this proposed
scheme]{.underline}. How the polarimetric variables are affected with
this proposed scheme and/or any other clutter filtering scheme need to
be explored.

Clutter filtering of staggered PRT sequences (Sachidananda 1999) should
be modified for application on dual polarization data. Aside from having
to filter two sequences (H and V), we must preserve the phases of
reconstructed spectra which are needed for computation of differential
phase. The phase reconstruction is described in a recent paper and it
works very well (Sachidananda and Zrnic 2006). As previously argued, in
filtering of the sequence pair, one sequence (H polarization) determines
the exact filtering procedure which is then applied in an identical
manner to both sequences.

*Issue*: [Threshold for each polarimetric variable, including spectral
moments. These need to be quantified.]{.underline}

**4.5 Phase coding and computation of polarimetric variables**

Computation and censoring of polarimetric variables should be similar to
what is currently done on the velocity and spectrum width. The censoring
thresholds might have to be different for each of the polarimetric
variables.

In case of the strong trip echo overlaid with a weak echo the lag one
estimators (Section 3.2) should be used to avoid bias.

For the weaker trip, the essence is to compute the variables from the
complex spectrum. That is, the echoes from the H *channel* are cohered
for the strong trip. The complex coefficients (two spectral replicas)
for the weak trip farthest away from the strong are identified. The weak
trip spectral coefficients in the V channel must occupy the same band as
in the H channel (that is indicated by the control line in Fig. 4.2).
That is, the notch width position for the V channel *must be the same*
as for the H channel. That way, the two spectral replicas in the H and V
channels would overlap completely. These coefficients are injected into
formulas for computing the differential reflectivity, the correlation
coefficient, and the differential phase (see Section 3.3 and 3.1).
Censoring criteria should be similar (possibly more stringent?) to the
ones proposed for velocity and spectrum width estimates.

*Issue:* [Censoring thresholds for the polarimetric variables in the
SZ-2 algorithm]{.underline}.

The functional description of processing the signals to obtain the
polarimetric variables and use the systematic phase code (SZ) follows.
For simplicity and to highlight the essence only two echoes are
considered.

a\) *Assumptions and notation*

Assume that two echoes are overlaid and have powers P1 and P2 measured
with the long PRT.

We start with horizontally polarized signals h(*i*) at a range gate,
where *i* is time index. We do not list the windowing to shorten this
discourse, but a suitable window (as in the implementation of the SZ-2
on the ORDA) should be applied.

Notation: h1, h2 horizontally polarized signals cohered for trip 1 and
trip 2 (that is the non cohered trip signal has the modulation code);

H1, H2 complex spectra of h1, h2;

v1, v2 vertically polarized signals, trip 1 and 2;

V1, V2 complex spectra of v1, v2;

R1 and R2 are autocorrelations of signals (at lag 1) cohered for trip 1
and 2 respectively, but without separation of trip powers;

P1h, P1v, P2h, P2v powers are for trip 1, 2 but the powers have been
separated in the spectral domain; subscripts h and v refer to
polarization;

Similarly ρ1~hv~, ρ2~hv~ refer to complex correlations;

Φ1~DP~ = arg (ρ1~hv~), Φ2~DP~ = arg (ρ2~hv~).

b\) *Algorithm*

1 Cohere trip 1 to get h1 and estimate autocorrelation at lag 1, R1(1).
Cohere trip 2, to get h2 and estimate R2(1).

2 IF \|R1(1)\| and \|R2(1)\| are within X dB from each other.

3 Fourier Transform h1 and h2 to obtain H1 and H2

4 Fourier Transform v1 and v2 to obtain V1 and V2

5 IF \|R1(1)\| \> \|R2(1)\|

6 Compute the mean velocity from R1(1) and thus the spectral index k
where the spectrum H1(i) is centered.

7 Center the notch processing filter on k and retain at least two
spectral replicas, total 2d coefficients farthest from k (on the
circle). That is, take the index j from jmin = k+32-d to jmax = k+32+d.
Note, an odd or even number of coefficients could be used, and there
could be three spectral replicas!

8 Compute the powers by summing over j from jmin to jmax.

$P2_{h} = \sum_{j\text{min}}^{j\text{max}}{H1(j)H1^{}(j)}$ and
$P2_{v} = \sum_{j\text{min}}^{j\text{max}}{V1(j)V1^{}}(j)\text{.}$

9 Compute Z2~DR~ as Z2~DR~ = 10log(P2~h~/P2~v~)

10 Compute the complex correlation ρ~hv~ from

![](media/image42.wmf)

11 take the magnitude, and the argument equals Φ2~DP~

12 Estimate the second trip velocity (and spectral index) from filtered
data by recohering the spectral replicas from jmin to jmax.

Take the H2 (FFT(h2) obtained in 3) and repeat the steps 7 to 11 (that
is interchange the numerals 1 and 2) to get Z1DR, ρ1~hv~, and Φ1~DP~.
Skip step 12!

At this point the steps 7 to 11 have been done twice, first on the
cohered spectra H1 and V1 to obtain the pol variables of trip 2!, then
on H2 and V2 to obtain the variables of trip 1.

ELSE

Do the same computation as indicated by 6 to 12 except start with h2 and
v2

First pass from 6 to 11 yields Z1~DR~, ρ1~hv~, and Φ1~DP~ then in step
12 the first trip velocity is obtained from filtered data. A second pass
7 to 11 will give Z2~DR~, ρ2~hv~, and Φ2~DP~.

ELSE

Next are computations if one signal is considerably (X dB to be
determined) stronger than the other. This is the case where the weaker
signal might not be recoverable? Note -- next we start arbitrarily with
line No 20.

20 IF \|R1(1)\| \> \|R2(1)\|

21 Compute the mean velocity from R1(1) and thus the spectral index k
where the spectrum H1(i) is centered.

22 Retain about 2d spectral coefficients of H1 centered on k. An
adaptive scheme can be used to find d, similar to GMAP, or Hildebrands
Sekhon procedure or else. Also, one could retain an even or odd number
of coefficients.

23 Compute the powers by summing j symmetrically with respect to k

$P1_{h} = \sum_{j = k - d}^{j = k + d}{H1(j)H1^{}(j)}$ and
$P1_{v} = \sum_{j = k - d}^{j = k + d}{V1(j)V1^{}}(j)\text{.}$

24 Compute Z1~DR~ as Z1~DR~ = 10log(P1~h~/P1~v~)

25 Compute the complex correlation ρ~hv~ from

$\rho 1_{\text{hv}} = \frac{\sum_{k - 8}^{k + 8}{H1^{}(j)V1(j)}}{P1_{h}^{\frac{1}{2}}P1_{v}^{\frac{1}{2}}}$

26 take the magnitude, and the argument equals Φ1~DP~

ELSE

Repeat steps 21 to 25 but with the H2 to obtain Z2~DR~, ρ2~hv~, and
Φ2~DP~.

END

**5. Recommendations for initial implementation of polarimetric
capability**

This section contains the recommendation for processing in the RDA that
should be implemented at the time of deployment. Its essence is to
retain all the existing capabilities of the single polarization system,
apart from a 3 dB loss of power per channel. Thus, potent capabilities
offered by polarimetry are not utilized at this time, nor are other
advances recommended. These are left for evolutionary enhancements over
the forthcoming years. Current state of development on the ORDA
guaranties that the capability at the time of dual polarization upgrade
would include super resolution, and phase coding at the lowest two
elevation scans. There is a slim chance that staggered PRT might be
available in some VCPs. Sections herein contain the specific
recommendation in terms of reference to either this report or other
documents.

**5.1 Calibration**

Calibration of the reflectivity factor for horizontal polarization Z~h~
should be the same as on the current ORDA. Differential reflectivity
should be calibrated according the report No. 2 by Zrnic et al. (2007).
Determination of the system differential phase at time of deployment
should be from the histogram of ground clutter (Zrnic et al. 2006) and
unwrapping of phase should follow discussion on page 19, first paragraph
after Eq. \[3.7\]

**5.2 Determination of noise level and threshold for the variables**

The two noise levels N~h~ and N~v~ should be measured at end of volume
scans as currently done for N~h~. Adjustment for various elevations
should be as is for N~h~ and this procedure should be repeated for
adjustment of N~v~.

To recover the 3 dB sensitivity loss, the ORDA should combine H & V
channel autocorrelations to compute velocity and spectrum width as in
\[2.7\] and (2.9). A novel thresholding scheme for displaying Z and the
other spectral moments will use the sum of powers and correlations as in
\[2.5b\] with the weights (α, β, and γ) set to 1. The threshold in
\[2.5b\] could be set to *T*=5 *N~h~* in case that *N~h~* \> *N~v~* and
to *T=*5 *N~v~* otherwise, but a better option might be 5(*N~h~ +
N~v~*)/2.

[]{.mark}

**5.3 Computations of polarimetric variables**

At low elevations where surveillance scan (long PRT) precedes the
Doppler (short PRT) scan, the polarimetric variables should be computed
at both scans. Censoring thresholds for the polarimetric variables
should be the same as for reflectivity estimates.

*Issue*: The threshold settings and combining the polarimetric variables
from the two scans need to be further analyzed.

The variables should be computed from standard formulas: (3.1 and 3.2)
for Z~DR~ with adjustment for noise as explained in the paragraph
following (3.2). Differential phase is computed from \[3.7\] and
adjusted to span an interval from 0 to 360 deg as explained in the
paragraph following \[3.7\]. Cross correlation coefficient is computed
directly as in \[3.13\], after the noise powers have been removed from
the denominator terms in that equation.

Ground clutter filter must be identical for the H and V channels.
Therefore minor modification of the GMAP filter needs to be made so that
it can identify the coefficients to be removed; this has already been
done on the version of the GMAP for SZ-2 scheme. The coefficients
identified for removal in the H channel must be also removed in the V
channel. And the polarimetric variables should then be computed from the
remaining complex spectral coefficients in the two channels. See section
4.2.2 c) and equations (3.25, 3.26, and 3.27).

Computation of polarimetric variables in case of SZ-2 phase coded data
should parallel the computation of Doppler velocity. That is, the
polarimetric variables will follow the same paths (logic and analogous
computations) as the mean velocities. In case of no overlaid echo,
computation is from standard formulas. With overlaid echoes, for the
strong trip use direct computations (possibly in the spectrum domain and
take one half of the spectral coefficients around the peak of the strong
trip spectrum). For the weak trip, use spectral computation on the two
replicas as explained in sections 4.2.2 and 4.5.

For Batch and higher angle CD (continuous Doppler) scans, polarimetric
variables should be calculated for 1 deg azimuth intervals, the same as
the legacy moments. To reduce the errors in estimates, all pulses from
longer and shorter PRT should be used. That is: calculate P~h1~ from the
long batch PRT and use these to identify range overlay that can occur in
the short batch PRT. The powers should be accumulated across the two
PRTs so that the total P~h~ = P~h1~(from long PRT) + P~h2~ (from short
PRT). The Z~v~, and R~hv~ should be computed from the continuous
accumulation of values over the two batch PRTs unless range overlaid
echoes are detected in the long PRT. In that case there are two
possibilities. If the echoes are present but contamination is acceptable
use all samples (batch of long and short PRT) for computing the
variables. If the overlaid powers preclude accurate estimation from the
batch of short PRTs only the batch from the long PRT should be used for
computing the polarimetric variables. These should then be passed to the
RPG. Decision will be made in the RPG whether and how to use these data.

**Appendix A**

**Probability of false alarm for the Legacy Threshold scheme of the Z
field**

The threshold (2.1) on the signals in the surveillance mode produces
very low probability of false alarm (PFA) so that false detections do
not clutter the PPI images. Also, because meteorological radars deal
with distributed scatterers, missing a few out of 1000 might be an
acceptable trade-off for achieving low PFA. Consequently, calculation of
PFA for a given threshold requires integral evaluation within the tail
of probability density functions (*pdf*). Thus, Monte Carlo methods for
evaluating these become very time consuming making it advantageous to
obtain analytical expressions for the *pdf*. Herein we present an
analytic expression from which one can evaluate the *pdf.*

The probability density of power samples from a single return is
exponentially distributed. Let's denote this pdf as *f* =
exp(-*y/θ*)*/θ*, where *θ* is the mean power.

It is a well known that the sum

![](media/image43.wmf), (A.1)

of *M* independent exponentially distributed random variables (in our
case powers) *X~m~* with:

![](media/image44.wmf), (A.2)

has gamma distribution. For completeness the proof follows.

*Proof*: The moment generating function for each *X~m~* is:

![](media/image45.wmf). (A.3)

Hence,

![](media/image46.wmf). (A.4)

This is the same as the moment generating function for gamma
distribution with parameters:

![](media/image47.wmf). (A.5)

The formula used for power estimate is:

![](media/image48.wmf), (A.6)

with the mean and variance *P* and *P^2^*/*M*, respectively. Equating
these values with the mean and the variance for (A.5), (i.e., *Mθ* and
*Mθ^2^*) we get:

![](media/image49.wmf), (A.7)

which after replacing *P* with the noise power *N* gives us the
following PFA for a certain threshold *T*:

![](media/image50.wmf), (A.8)

where *P*(*x*,*α*) is the incomplete gamma function defined as:

![](media/image51.wmf). (A.9)

We can express *T* as a function of *N* as:

![](media/image52.wmf), (A.10)

where *T~dB~* is the threshold expressed in dB relative to the noise
power *N*. Note that in the WSR-88D the known noise power is always
subtracted from the estimate. Hence, the effective threshold for (A.6)
is expressed as in (A.10). Now we can calculate the PFA for the power
threshold as:

![](media/image53.wmf). (A.11)

If we put the power threshold to be 2 dB above the noise level this
gives us low PFA of only 1.1749×10^-6^. If the threshold is lowered to
-1 dB above noise the PFA increases to 3×10^-6^.

**Appendix B**

**Combinations of correlation functions for censoring radar data**

Ten candidates to threshold the data are compared herein. These are:

1.  ![](media/image54.wmf)

2.  ![](media/image55.wmf)

3.  ![](media/image56.wmf)

4.  ![](media/image57.wmf)

5.  ![](media/image58.wmf)

6.  ![](media/image59.wmf)

7.  ![](media/image60.wmf)

8.  ![](media/image61.wmf)

9.  ![](media/image62.wmf) OR ![](media/image63.wmf) OR
    ![](media/image64.wmf)

10. ![](media/image65.wmf) OR ![](media/image66.wmf) OR
    ![](media/image67.wmf)

All thresholds were calculated with respect to the desired Probability
of False Alarm (PFA). Given that a PPI has about 360\*1000 = 360000
points, if PFA is taken to be 10^-5^ that would yield only 3.6 false
detections per sweep. Consequently, this PFA value was taken as
reference for comparisons using real data.

From the available radar data, noise powers were calculated for both the
horizontal and the vertical channel. These values are (in the internal
processor power units):

-   N~h~ = 3.4174 10^-6^ for the horizontal channel

-   N~v~ = 2.8259 10^-6^ for the vertical channel.

To choose the threshold for each approach and obtain a desired PFA
(e.g., 10^-5^) we simulated time series data of the H and V components.
Then we computed the various correlations, combined these into sums, and
compared to thresholds. A total of 10^8^ trials were used to calculate
the PFAs. For the first 8 approaches the results are shown in Fig. B.1
for the case when *M* = 17. In Fig. B.2 these approaches are compared in
terms of Probability Of Detection (POD). From these plots it becomes
obvious that the approach

![](media/image68.wmf) (B.1)

yields the best results; it is referred to as the *best of the set* or
simply "*best"*.

![](media/image69.wmf){width="7.013194444444444in"
height="7.245833333333334in"}

**Figure B.1**. PFAs for different approaches.

To evaluate the OR combinations (i.e. the last two approaches)
thresholds were changes through a set of values. For each threshold
combination PFA was found. Then, POD was calculated for each
combination. The results together with the outcome for the best sum are
in Fig. B.3. It is apparent that the best sum performs significantly
better.

Next, we examine how the optimal sum behaves in terms of the POD for the
preset PFA of 10^-5^. Using the values obtained through simulations
(shown in Fig. B.1) exact threshold was calculated by straight-line
interpolation between two values closest to 10^-5^. In total, 50000
trials were used to calculate each POD. Using the Chebyshev's
inequality, we get the confidence level of 99.982%, for *ε* = 10^-2^,
and *p* = 0.1. When normal approximation is used, the confidence level
obtained is 100% for all practical purposes. In Figs. B.4 and B.5, PODs
are examined for a various SNR and Z~DR~, respectively.

![](media/image70.wmf){width="5.256944444444445in" height="4.41875in"}

![](media/image71.wmf){width="5.227083333333334in"
height="4.3590277777777775in"}

**Figure B.2**. POD comparison.

![](media/image72.wmf){width="5.59375in" height="4.481944444444444in"}

![](media/image73.wmf){width="5.5625in" height="4.307638888888889in"}

**Figure B.3**. POD comparison for OR combination.

![](media/image74.wmf){width="5.346527777777778in"
height="4.259722222222222in"}

**Figure B.4** POD versus SNR. For comparison, the optimal sum curve is
plotted against the case if there were no 3 dB loss, and the 3 dB loss
with the threshold 2 dB above noise.

![](media/image75.wmf){width="5.361805555555556in"
height="4.063888888888889in"}

**Figure B.5** POD versus ZDR.

**Appendix C**

**Computation of the noise level**

Radar antenna intercepts thermal radiation from various sources
including ground, precipitation, sun, sky, or man-made radiators.
Further, abrupt change in receiver gain can cause wrong estimation of
the SNR. Consequently, and depending on where the antenna is pointing,
the receiver noise power will routinely vary by few dBs. In extreme
cases of man-made interference (e.g., from other radars, etc.), the
noise can increase by tens of dB. Therefore, to obtain best quality of
polarimetric radar data it is desirable to estimate receiver noise for
each radial and account for it in the computation of the radar
variables.

Besides removal of noise from reflectivity, the noise should also be
removed from the cross correlation coefficient and the differential
reflectivity estimates. Noise can be estimated from powers in the
several range gates at the end of PRTs. Here, we examine the number of
samples needed to obtain accurate noise power which when subtracted from
the total power will produce acceptable accuracy of *ρ~hv~* and *Z~DR~*.
Because noise power samples are uncorrelated, the standard deviation of
the estimates is *N~1~/√M,* where *N~1~* is the mean power for a single
return.

*Testing if noise is white*

A possible procedure to estimate the noise level could be as follows.
Start from the last range gate in the surveillance scan (long PRT) and
move forward by about 20 range gates; the suggested numbers are in this
appendix.

Test the cross correlation function obtained from 17 samples (standard
number per radial in the long PRT) to see if it is above a threshold.
Let *N* be the noise power estimate from the previous radial or a
default from calibration at the end of volume scan. The value at the end
of volume scan is needed for cases where the noise can not be determined
in a single radial.

Therefore,

if │*R~hv~*│\> γ*N* \-\-\-\-\-\-\-\-\-- discard that point. (C.1)

If (C.1) does not hold, the noise is most likely white and that data
point could be accepted as due to noise. This remains to be tested. It
is known that returns from chaff, or ground (under anomalous propagation
conditions) have low cross correlation coefficient, and it is possible
that some storms at such large range would exhibit small values of the
cross correlation function.

Thus, there might be a convincing argument to test for whiteness. In
that case the Kolmogorov-Smirnov test on the spectral coefficient can be
performed. How to precisely apply this test on the Doppler spectra of
weather signals is describes in detail by Urkowitz and Nespor (1992).
This algorithm is also part of the GMAP ground clutter filter
implemented on the ORDA. Thus, the code exists, and could be applied
with minor modifications (elimination of the DC removal and clutter
identifier) to determine the color of the signal. NSSL will provide (if
needed) to the NWS the functional description of how the white noise
power should be obtained.

*Requirement for the correlation coefficient*

The number of samples needed for a specified accuracy of the correlation
coefficient is discussed next. Because we are concerned only with the
accuracy of noise power which must be subtracted, we assume that the
signal plus noise powers *P~h~* and *P~v~* and cross correlation
function *R~hv~* in the formula have no error. Further, we assume that
the error in noise power equals the standard deviation and can be
positive or negative. Consequently, the signal powers can be biased high
or low. Thus, for short, we condense the notation, express the signal
powers as *S~h~* = *P~h~ ±* *N~1h~/√M,* *S~v~* = *P~v~ ±* *N~1v~/√M* ,
and present the following equation for the biased![](media/image76.wmf)

![](media/image77.wmf) (C.2)

Consider the case *Z~dr~* \> 1 (in linear units) which prevails in weak
signal regions. Further, for our purpose, set *R~hv~* = (*P~h~*
*P~v~*)^1/2^ (because for precipitation the *ρ~hv~* is high) and
*N~1h~*= *N~1v~*= *N~1~.* The formula (C.2) reduces to

![](media/image78.wmf) (C.3)

This biased ![](media/image79.wmf) should not deviate from the true
value by more than a specified tolerance *δ* and the worst case is if
the symbol ± is replaced with the minus sign (-). Assume that the true
value of *ρ~hv~* is 1; one could use other values and reduce
proportionately *R~hv~* but this has little bearing on the minimum *M*
needed for estimating the noise power. Therefore, we impose the
condition

> ![](media/image80.wmf). (C.4)

It follows that if

> ![](media/image81.wmf), (C.5)

the inequality (C.4) will be satisfied. Thus, from (C.5), we find the
value of *M*

*M≥*(*Z~dr~/(δ snr)*)^2^, (C.6)

where *snr* = *S~h~/N~1~* in linear units and both the signal and noise
powers are per pulse (single sample of I,Q).

Before discussing the choice of *M* we turn to the required noise
accuracy for *Z~dr~* estimates. This is because the more stringent
condition on the accuracy of the two determines the needed *M*.

*Requirement for the differential reflectivity*

Similar to the correlation coefficient the equation for the biased
differential reflectivity *Z~drb~* in linear units is

![](media/image82.wmf) (C.7)

and the number of samples is found from the inequality

1-ε ≤ *Z~drb~* ≤1+ε . (C.8)

Consideration of all the possible combinations of the plus and minus
signs produces the following inequality for the number of samples *M.*

![](media/image83.wmf) (C.9)

*Discussion*

Clearly *δ* and ε have very significant influence on *M.* The desirable
*δ* is in the range 0.01 to 0.02 and *ε* is in the range of 0.02 to 0.05
(corresponding to 0.1 to 0.2 dB). Therefore, it follows that if (C.6) is
satisfied (C.9) will be true.

Let's next take parameters relevant to the WSR-88D. Polarimetric
variables should provide superior measurements of precipitation to about
100 -- 150 km; say 150 km. In the SHV mode the reflectivity factor
corresponding to the receiver noise power at 150 km is 5 dBZ. We can
further stipulate that snow with *Z*=15 dBZ should be quantified and/or
discriminated with the polarimetric radar. This is an illustrative and
reasonable assumption. Thus the accuracy should be maintained down to
the SNR of at least 10 dB. Value of 3 dB (2 in linear units!) is
reasonable for the differential reflectivity in snow. Then, (C.6) with
*δ* = 0.01, *snr =* 10, and *Z~dr~ =* 2, produces *M* ≥400. With 17
returns per dwell time (radial) it follows that one would need 24 range
gates to achieve the desirable accuracy. This is realistic and very
practical.

If one relaxes *δ* to 0.02, *M* becomes 100. Suppose that one keeps *M*
= 400; it follows that the SNR at which the 0.02 bias is maintained
drops to 7 dB (i.e., linear 5). Therefore, at 150 km range snow with *Z*
= 2 dBZ should be measured very well.

**References:**

Doviak, R.J. and D. S. Zrnic, 1993: *Doppler Radar and Weather
Observations*. Academic Press. Second Edition, San Diego Cal., 1993, p.
562.

Doviak, R. J. and D. S. Zrnic, 1998: WSR-88D Radar for Research and
Enhancement of Operations: Polarimetric Upgrades to Improve Rainfall
Measurements, NOAA/NSSL Report, 110 pp.

I. Ivic, 2008: Decision thresholds for spectral moments and polarimetric
variables. PhD dissertation, University of Oklahoma, Department of
Electrical and Computer Eng. p 152.

Melnikov, V. M., and D.S. Zrnic, 2004: Simultaneous transmission mode
for the polarimetric WSR-88D -- statistical biases and standard
deviations of polarimetric variables. NOAA/NSSL Report, 84 pp.

Sachidananda, M., 1999 (part 3)**:** Signal Design and Processing
Techniques for WSR-88D Ambiguity Resolution, NOAA/NSSL Report, Part 3,
81 pp.

Sachidananda, M., and D.S. Zrnic, 2006: Ground clutter filtering
dual-polarized staggered PRT sequences. *J. Atmos. Oceanic Technol.,*
**23,** 1114-1130.

Torres S., M. Sachidananda, and D.S. Zrnic, 2005 (part 9): Signal Design
and Processing Techniques for WSR-88D Ambiguity Resolution: Phase coding
and staggered PRT. NOAA/NSSL Report, 112 pp.

Torres S., M. Sachidananda, and D.S. Zrnic, 2004 (part 8): Signal Design
and Processing Techniques for WSR-88D Ambiguity Resolution: Phase coding
and staggered PRT, implementation, and clutter filtering. NOAA/NSSL
Report, 157 pp.

Urkowitz, H. and J.D. Nespor, 1992: Obtaining Spectral Moments by
Discrete Fourier Transform with Noise Removal in Radar Meteorology.
Proceedings of the IEEE GRS Conference, Huston, 12-14.

Zrnic, D., V. Melnikov, and A. Ryzhkov, 2006: Correlation coefficients
between vertically and horizontally polarized returns from ground
clutter. *J. Atmos. Oceanic Technol.,* **23**, 381-394.

Zrnic, D., S. V. M. Melnikov, J. K. Carter, and I. Ivic, 2007:
[Calibrating Differential Reflectivity on the WSR-88D, Part
II](http://publications.nssl.noaa.gov/wsr88d_reports/ZDR_Rpt.pdf),
NOAA/NSSL Report, 16 pp.

Zrnic, D., S. V. M. Melnikov, 2007: Ground clutter recognition using
polarimetric spectral parameters. 33^rd^ International Conference on
Radar Meteorology, CD. AMS, Cairns, Au.