## 1. Introduction

Simultaneous transmission and reception of electromagnetic waves with horizontal and vertical polarizations have been recommended for a polarimetric prototype of the Weather Surveillance Radar-1988 Doppler (WSR-88D; Doviak et al. 2000). We will refer to the simultaneous transmission and reception of horizontally and vertically polarized waves as SHV. In this mode, six radar variables are measured in each radar resolution volume. These are reflectivity, Doppler velocity, spectrum width, differential reflectivity *Z*_{DR}, differential phase, and modulus of the copolar correlation coefficient *ρ*_{hv}. The first three values are the base radar products of the WSR-88D; the latter three are the polarimetric variables. Another polarimetric variable, the specific differential phase, is calculated from the differential phase.

The SHV mode has been implemented on the KOUN radar situated in Norman, Oklahoma, which was the preproduction model of the WSR-88D. On the WSR-88D, the reflectivity factor is calculated at signal-to-noise ratios (SNRs) larger than 2 dB; the same SNR threshold should be applied for the polarimetric variables. It is shown herein that in the SNR interval 2–15 dB, the *Z*_{DR} and *ρ*_{hv} estimates are susceptible to bias by noise. Low SNR is observed in distant precipitation, light rain, and snowfalls; therefore, it is desirable to eliminate noise biases in the network of weather radars.

Ryzhkov and Zrnić (1998) have pointed out that 0.1-dB accuracy of *Z*_{DR} is needed to distinguish between water droplets and snowflakes or crystals. Bringi and Chandrasekar (2001, section 8.3.3) have found that 0.2-dB *Z*_{DR} error and 0.8-dB reflectivity error lead to about 24% error in rain measurements. Illingworth (2004) and Ryzhkov et al. (2005) indicate that in order to measure light and moderate rain with accuracy of about 15%, *Z*_{DR} should be known within 0.1 dB. It appears that radar calibration can achieve such accuracy (Keenan et al. 1998; Hubbert et al. 2003; Zrnić et al. 2006). We will consider 0.1 dB to be the level under which biases of *Z*_{DR} should be kept.

Values of the copolar correlation coefficient vary from less than 0.5 for ground clutter to almost 1 for light rain. In clouds the usual interval of the coefficient is 0.8 to 1 with the lower limit observed in the bright band and hail (Balakrishnan and Zrnić 1990; Bringi and Chandrasekar 2001, section 7.5). Considering dependence of *ρ*_{hv} on drop size distribution, Illingworth and Caylor (1991) have come to the conclusion that the desired accuracy of measurements is better than 0.005. To achieve such accuracy, analyses of signal processing and antenna characteristics are needed (Liu et al. 1994; Doviak et al. 2000; Melnikov and Zrnić 2004; Wang et al. 2005). Herein we consider 0.01 as a more conservative accuracy of *ρ*_{hv} measurements.

Three sources of measurement errors are relevant to the SHV mode. First, this mode has intrinsic errors due to cross coupling of the orthogonally polarized signals, that is, due to depolarization by the scatterers and the antenna (Doviak et al. 2000; Moiseev et al. 2002; Wang et al. 2005). Second, difference of attenuations in the channels biases the estimates. Considerations of the propagation effects can be found in, for example, Bringi and Chandrasekar (2001), Doviak et al. (2000), and Torlaschi and Zawadzki (2003). Herein we focus on the third type of errors that occur at signal-to-noise ratios less than 15 dB. We demonstrate that the accuracy of *Z*_{DR} and *ρ*_{hv} measurements in the SNR interval of 2 to 15 dB depends on uncertainty of the noise levels used in the estimator. On the WSR-88D, noise is measured at high elevations prior to each volume scan and it is subsequently used to obtain SNR and spectral moment estimates. If actual noise changes during the scan the calculated radar moments become biased. Imperfections of radar devices, variation of thermal noise from the ground and precipitation, and wideband noise coming from electrically active clouds cause significant changes in the white noise power. We show that such uncertainties of system noise produce errors larger than the stipulated accuracies. Thus, it is desirable to devise estimators not biased by noise. Instead of using estimates of power we combine estimates of correlation function, which are not biased by noise similarly to the proposed estimation of spectrum widths from lag-1 and -2 autocorrelations (Srivastava et al. 1979).

In the next section we consider sources of noise uncertainties and influence of these on *Z*_{DR} and *ρ*_{hv} estimates. In section 3, we devise estimators free from noise biases and in section 4 we present the statistics of these estimators. In section 5 we discuss performance of the conventional and proposed estimators on radar data.

## 2. Uncertainty in the noise level

*Z*

_{DR}and

*ρ*

_{hv}are calculated as

*P̂*and

_{h}*P̂*are the estimates of the powers in the channels for horizontally (

_{υ}*h*) and vertically (

*υ*) polarized waves,

*N*and

_{h}*N*are the mean noise powers in these channels, values without the circumflex stand for true means, and

_{υ}*R̂*

_{hv}is the estimate of the copolar correlation function that is calculated from complex voltages

*e*

^{(h)}

_{m}and

*e*

^{(υ)}

_{m}in the H and V channels as

*M*is the number of samples used in the estimate,

*m*numerates the samples, and the asterisk denotes complex conjugate. The signal powers in the channels,

*Ŝ*and

_{h}*Ŝ*, are defined as

_{υ}*Ŝ*=

_{h}*P̂*−

_{h}*N*and

_{h}*Ŝ*=

_{υ}*P̂*−

_{υ}*N*. We will refer to (1) and (2) as conventional estimates.

_{υ}On the WSR-88Ds, noise is measured at high antenna elevation in absence of precipitation before each volume scan. The measured noise is then used in calculations of radar moments during the scan, that is, during 5 to 7 min depending on the type of scan. Imperfections of radar devices and noise from the clouds and ground cause deviations in noise. It is seen from (1) and (2) that if current noise deviates from measured *N _{h}* and

*N*, the estimates are biased. Here we consider the following sources of noise variations: (a) system noise drift due to imperfections of radar devices, (b) thermal noise of the ground and precipitation, and (c) noise of electrically active clouds.

_{υ}We begin with imperfections of radar devices. Figure 1a presents noise records in the KOUN’s H channel with the antenna in the park position (azimuth = 0°; elevation = 22°). Four hundred consecutive range locations along the radial were split into four equal parts, and the mean noise power was calculated for each part from 128 consecutive time samples so that four estimates of the mean noise power were obtained. This procedure was conducted during approximately 50 s and the result is presented in the figure in the form of four curves. It is seen that all curves are highly synchronous exhibiting time variations of the noise level. Time scale of these variations is of few seconds and variations are about 1 dB. Such variations are observed frequently but not all the time; most of the time they are within 0.5 dB of the mean value. Noise measurements during calibration cannot compensate for these rapid variations. Note that in the SHV mode, two receivers are employed and fluctuations in the independent channels may result in the net effect on (1) and (2) of larger than 1 dB

In Fig. 1b, time variations of system gain in the vertical channel of KOUN is shown. The power in the V channel was recorded injecting strong stable signal at the input to the low noise amplifier. It is seen that the gain experiences 0.8-dB deviation during 5 min. Gain deviations cause changes to the relative noise level at the output of the digital receiver. Because the radar utilizes the noise levels and gains measured before a volume scan wrong parameters are introduced in (1) during gain’s deviations and biased *Z*_{DR} and *ρ*_{hv} are obtained.

Thermal radiation coming to a radar antenna from precipitation adds to the receiver input noise power: *N* = *N*_{sys}+ *N _{p}*, where on the right side are the system noise and precipitation noise powers. On KOUN,

*N*

_{sys}= −113 dBm (Melnikov et al. 2003). The variable

*N*can be expressed as

_{p}*N*=

_{p}*kBT*(1 −

_{p}*l*

^{−1}), where

*k*is the Boltzmann constant,

*B*is the bandwidth (1 MHz for the WSR-88D),

*T*is the temperature of precipitation, and

_{p}*l*(≥1) is the loss factor in precipitation (Doviak and Zrnić 1993, Eq. 3.31). On KOUN, we have observed noise increase of 0.8 dB due to summer precipitation. At S band, Ryzhkov and Zrnić (1995) observed attenuation in excess of 8 dB, which would correspond to

*N*= −114 dBm and a noise increase of 2.5 dB (

_{p}*T*= 10°C = 283 K used for precipitation). At X band, Fabry (2001) observed 1-dB noise variations due to thermal noise of precipitation. Because attenuations of the H and V fields differ, contributions to thermal noise in the two channels are unequal. Thermal radiation from the ground also contributes to noise variations at lower antenna elevations.

_{p}Lightning emits radiation in a broad frequency band, which if intercepted by the antenna causes excess noise as seen in Fig. 2 beyond 68 km. The time interval between the two records is 263 ms and the number of samples in the estimate is 256. The gradual increase of the noise level with increasing distance is a result of the range-squared weighting applied in reflectivity calculations. One can see a jump of about 10 dB in the noise level. Such large noise jumps are less frequent than smaller ones. This type of noise, because of its broad band, can be considered as white in the radar receiver.

Figure 3 depicts biases in *Z*_{DR} for noise increase in one and both channels computed from (1). Allowed bias of 0.1 dB is marked with the horizontal dashed lines. In Fig. 3a, the bias due to a 1.5-dB noise increase in the H channel is shown. It is seen that at SNR lower than 12 dB the bias is larger than the desired error. The decrease of noise in the H channel or the increase of noise in the V channel result in negative noise biases. If noise variations in the H and V channels are opposite the noise biases will be larger than in Fig. 3a. Figure 3b shows the biases for the 1.5-dB noise increase in both channels. This case corresponds to noise increase due to additional thermal noise from the ground and precipitation. For simplicity, noise increases in the channels are considered equal. The biases in *ρ*_{hv} computed from (2) are in Fig. 4, where it can be seen that at SNR lower than 15 dB the biases exceed 0.01. We conclude that in the SNR interval 2–15-dB noise variations can bias *Z*_{DR} and *ρ*_{hv} estimates beyond the acceptable errors.

The considered noise biases do not depend on the number of samples in the estimates and thus are asymptotic. Such biases cannot be reduced with additional spatial or temporal averaging and therefore should be kept as low as possible. One way to eliminate these biases is by estimating noise power along each radial during actual dwell time. Noise power could be computed in the time domain by averaging powers from ranges void of signal and/or in the spectral domain by identifying the white portion of the Doppler spectrum. Either case involves considerable processing and might not be doable for each radial. Another way to combat this problem is by devising estimators free from the asymptotic noise biases. In the next section, we propose such estimators.

## 3. Estimators free from the noise bias

*T*as

*h*and

*υ*denote the parameters that are calculated using the pulse trains in the H and V channels,

*T*is the pulse repetition interval [

*T*= 1/pulse repetition frequency (PRF)],

*υ*is the unambiguous velocity (

_{a}*υ*=

_{a}*λ*/4

*T*, where

*λ*is the wavelength),

*ρ*(

_{h}*T*),

*ρ*(

_{υ}*T*) are the temporal correlation coefficients, and

*j*is the imaginary one. The moduli of correlation functions (4) do not depend on the Doppler velocities:

*ρ*(

_{h}*T*) =

*ρ*(

_{υ}*T*) =

*ρ*(

*T*), we obtain

*Z*

_{DR}from (5) as

Estimators (6) and (10) were obtained under the assumption of equal temporal correlation coefficients or spectrum widths in the polarimetric channels. The widths *σ _{υ}*

_{(h)}and

*σ*

_{υ}_{(υ)}can differ due to different fall speeds of hydrometeors with different sizes and oblateness and also because the hydrometeors are not perfect tracers of the wind. The first effect becomes pronounced at elevation angles where both the difference in oblateness and fall speed are sensed by the radar. Atlas et al. (1973) and Martner and Battan (1976) found that in rain, the fall speed contribution to the spectrum width varies from 0.5 to 1.5 m s

^{−1}. At elevation angles less than 20° (the maximal elevation angle in operational observations with the WSR-88D), fall speed contribution is less than 1.5 sin(20°) = 0.5 m s

^{−1}. In the presence of hail the spectrum width can reach 7 m s

^{−1}(Martner and Battan 1976). But in hail,

*Z*

_{DR}is close to zero (Balakrishnan and Zrnić 1990; Bringi and Chandrasekar 2001, section 7.5) and no difference between

*σ*

_{υ}_{(h)}and

*σ*

_{υ}_{(υ)}should be expected.

The effects of imperfect tracing of winds by scatterers have been studied by Stackpole (1961) and Bohne (1982). This might lead to different widths *σ _{υ}*

_{(h)}and

*σ*

_{υ}_{(h)}, but the magnitude of the effect has not been estimated.

To verify the assumption of equal temporal correlations in the polarimetric channels, we have measured the spectrum widths in both channels using time series data collected in thunderstorms. To eliminate noise effects, data with SNR ≥ 20 dB was selected. The number of samples in the estimates was 128. Spread of measured widths estimates was 1 to 10 m s^{−1}. In Fig. 6, distributions of the difference of the spectrum width estimates are presented for elevations lower than 20°. The data were collected in May through August 2004. It is seen from the figure that there is no apparent bias in the difference of the widths; the distributions have the median very close to zero. We did not notice any statistically significant biases in the data. Thus we conclude that our data support the assumption of equal spectral widths in the channels.

## 4. Statistical biases and standard deviations of the 1-lag estimates

In section 2 we consider the asymptotical bias of differential reflectivity, that is, the one that does not depend on the number of samples in the estimate. The statistical nature of scattered radar signal introduces biases, which depend on the number of samples in the estimates. In this section, we consider the statistical biases and standard deviations (SDs) in the 1-lag estimators and compare those with corresponding quantities for the conventional algorithms.

Three approaches can be used to estimate the statistical biases and SD. 1) Probability distributions of estimates can be used to obtain the first (bias) and second (SD) moments. The distributions of *Z*_{DR} and *ρ*_{hv} estimates are known for independent samples (Lee et al. 1994). Weather signal samples are usually highly correlated so that those distributions cannot be applied directly. 2) Many useful results on signal statistics have been obtained with the perturbation analysis (see sections 6 in Doviak and Zrnić 1993 and Bringi and Chandrasekar 2001). The method produces analytical forms for biases and SDs and works well for a sufficiently large number of samples, which is usual in weather radar observations. 3) Signal simulations can serve as a tool for obtaining statistical characteristics of estimates. To simulate statistically correlated sequences in the two polarization channels the method of Jenkins and Watts (1968, section 8.4.1) or Chandrasekar et al. (1986) can be applied. We used the first one to generate signals with given *Z*_{DR} and *ρ*_{hv} and to add independent white noises to each signal. In this section, we present results obtained with the perturbation analysis and simulations.

Our simulations show that the 1-lag estimates are free from asymptotic noise bias and have a small bias dependent on the number of samples *M*. We have confirmed this at SNR as low as −5 dB and this is one of the main results of the simulations. An example of simulation results of the bias in cross-correlation estimates is in Fig. 7. It is evident that the simulation confirms the theoretical results. Expressions for statistical biases are presented in the appendix, and these show that for SNR > 2 dB the bias of *Z*_{DR1} is below 0.1 dB and of *ρ*_{hv1} it is lower than 0.01. At lower SNR the *M*-dependent bias should be accounted for.

*Ẑ*

_{DR1}and

*Ẑ*

_{DR}using equations from the appendix. For equal noise levels in the channels, (A7) and (A10) are written as follows:

where *σ*_{υn} = *σ _{υ}*/

*υ*is the normalized spectrum width and

_{a}*Z*

_{dr}is differential reflectivity in linear units. In Fig. 8, the SDs are compared. The symbols represent the simulation results. Comparing the curves with the symbols we conclude that the perturbation analysis gives sufficiently accurate results for SNR as low as 2 dB. Equations (11) and (12) slightly underestimate the SDs at spectral width narrower than 1.5 m s

^{−1}. At such narrow spectra, the number of independent samples becomes small and the perturbation analysis underestimates the SDs. It is seen from the figure that the SDs are almost identical. Due to

*ρ*(

*T*) in the denominator of (12) SD in the 1-lag estimator increases with spectrum width at widths wider than 6 m s

^{−1}. At wide spectra the multiplicative increase in the SD is about 1.3. In various types of storms Fang and Doviak (2002) have found that more than 80% of spectrum widths are smaller than 6 m s

^{−1}.

*ρ̂*

_{hv1}and

*ρ̂*

_{hv}requires (A15) and (A17) from the appendix. For equal noise levels in the channels these equations can be written as follows:

The SDs are presented in Fig. 9. Comparing the curves with the symbols we conclude that the perturbation analysis slightly underestimates the SDs at narrow spectra. Due to *ρ*(*T*) in the denominator of (14) the SD in the 1-lag estimator increases with increasing spectrum width at widths wider than 6 m s^{−1}.

*n*-lag estimator, differential reflectivity can be obtained by analogy with (6):

*R̂*

_{h(υ)}(

*nT*) is the n-lag correlation function in the horizontal (vertical) channel. Results for the SDs remain similar to (12) and (14), but

*ρ*(

*T*) should be replaced with

*ρ*(

*nT*). SDs for the n-lag estimators are larger than either 1-lag or conventional estimators due to

*ρ*(

*nT*) in the denominators of (12) and (14). Combinations of the mean values of the autocorrelation magnitudes or of (15) can also be constructed. Our simulations indicate that the simple 1-lag estimators produce acceptable results.

## 5. Performance on radar data

To assess the 1-lag estimators, time series data (the in-phase and quadrature components or I and Q signal samples) from both H and V channels have been recorded. Vertical cross sections (RHI) have been chosen for comparing the 1-lag and conventional estimators because in RHIs 1) noise can be measured accurately at elevations higher than 3° where the influence of the ground is weak, and 2) there are a sufficient number of range samples, beyond radar echoes, free from weather signals so that actual noise can precisely be measured in those regions. Noise powers *N _{h}* and

*N*were obtained at heights above 15 km.

_{υ}The top two panels in Fig. 10 show RHIs of *Ẑ*_{DR1} and *ρ̂*_{hv1} from widespread light precipitation observed on 7 October 2005. The melting layer at heights of about 4.5 km is clearly seen in both fields. Corresponding fields of the conventional estimates (not shown) look identical. To demonstrate closeness of the conventional and 1-lag estimates, histograms of the differences of the estimates have been generated using the data. Four distributions of *Ẑ*_{DR} − *Ẑ*_{DR1} are presented in Fig. 11 for SNR_{h}_{(υ)} larger than 2 dB and in the interval 2 to 5 dB (Fig. 11a) and for SNR_{h}_{(υ)} larger than 5 and l0 dB (Fig. 11b). Figure 11c presents two distributions of *ρ̂*_{hv} − *ρ̂*_{hv1}. Clearly there are no noticeable biases in the 1-lag estimators because the noise powers have been precisely measured as stated earlier. The widths of the distributions decrease with increasing SNR. Note that the difference *Ẑ*_{DR} − *Ẑ*_{DR1} does not depend on absolute *Z*_{DR} calibration and thus can be measured with accuracy higher than 0.1 dB.

Another distribution of *Ẑ*_{DR} − *Ẑ*_{DR1} for SNR interval 2 to 5 dB is shown in Fig. 12a with the thick line. The data were collected in light rain on December 2004. The noise levels in the channels were recorded 21 min before the data were collected (KOUN is a research WSR-88D and at the time did not have a 5-min update of noise power measurement). The thick line was obtained with the system noise levels and gains measured 21 min earlier. Negative bias of about −0.4 dB can be easily seen from the figure. Such bias for light rain is substantial. Data analysis at higher elevations in regions free from precipitations showed that the apparent noise level dropped by about 0.6 dB in the V channel. We attribute this drop to the change in gain; after correcting (digitally) the gain by 0.6 dB we obtained the symmetric distribution (with respect to zero) represented with the thin line. We stress again that the difference *Ẑ*_{DR} − *Ẑ*_{DR1} is measured with accuracy higher than 0.1 dB, and therefore small biases in the difference can be recognized. We conclude that the histogram of the difference along a radial can be employed to recognize the noise bias.

The data on 7 October 2005 were used to simulate noise deviations in the channels. The solid line in Fig. 12b represents the distribution of *ρ̂*_{hv} − *ρ̂*_{hv1} for SNR* _{υ}* in the interval 2 to 6 dB and the 0.5-dB noise drop in the H channel. An apparent bias to the left from zero is seen in the median of the distribution. The dashed line in Fig. 12b shows a distribution for a 0.5-dB noise increase in both channels. A substantial shift of the median to the right from zero manifests the increase. Figure 12b demonstrates a possibility to monitor small noise deviations utilizing simultaneous measurements of

*ρ̂*

_{hv}and

*ρ̂*

_{hv1}. From the analysis of

*Ẑ*

_{DR}−

*Ẑ*

_{DR1}and

*ρ̂*

_{hv}−

*ρ̂*

_{hv1}distributions, deviations in

*N*and

_{h}*N*could be obtained. Those two deviations could be inferred from the two distributions. Comprehensive analysis of these noise measurements using the conventional and 1-lag

_{υ}*Z*

_{DR}and

*ρ*

_{hv}is deferred to the future.

The bottom two panels in Fig. 10 demonstrate one more feature of the 1-lag algorithm—the estimates are not biased by interference signals. The left bottom panel in Fig. 10 displays, in the RHI format, the field of *ρ̂*_{hv} estimates obtained with the conventional estimator; data above the SNR threshold of 1 dB are displayed. The melting layer was seen in the reflectivity field (not shown) and it is very well defined with the drop in the cross-correlation coefficient. The cloud produced light rain. At the top of the cloud, one can see radially aligned strips of reduced *ρ*_{hv}. Analysis of noise powers (beyond echo range) demonstrates that the radially aligned structures are the same in the two channels, which strengthens our suspicion that external interference causes these fluctuations. Enhanced noise strips (not shown) coincide with well-pronounced strips of reduced values of the correlation coefficient (left bottom panel), that is, interference signals cause the drop of correlation coefficient. This interference is not so visible in the field of *Z*_{DR}. The right bottom panel in Fig. 10 presents the 1-lag estimates of the coefficient. It is seen that the influence of interference is eliminated. To achieve the same elimination with conventional processing requires estimation of noise power in each radial. Although we strongly advocate for radial-by-radial estimation of noise powers we have no robust algorithm to do this routinely.

## 6. Conclusions

On the current WSR-88D the SNR threshold for display and use of reflectivity factor is 2 dB. The same threshold should apply to the polarimetric variables. At the SNR interval 2 to 15 dB, the conventional *Z*_{DR} and *ρ*_{hv} estimates can be biased by variations of white noise powers. It is cumbersome to estimate the noise levels in the channels with accuracy better than 1.5 dB due to change of system noise powers, variable thermal noise from outside sources, and drift of the system gain. Noise variations of 1.5 dB can bias the conventional *Z*_{DR} and *ρ*_{hv} estimations beyond 0.1 and 0.01 dB, respectively. The 0.1 dB accuracy in *Z*_{DR} is desirable to distinguish between rain and snow and to measure light rain and snowfall type. To eliminate noise-dependent bias, 1-lag estimators have been devised using the correlation functions, which have no noise biases. Simulation results and radar data indicate that the 1-lag estimators eliminate the asymptotic noise biases, that is, the ones that are independent from the number of samples.

The 1-lag estimators are based on the assumption of equal spectral widths in the channels. Presented radar data support this assumption, which is also expected to hold for most precipitation observed at elevation angles lower than 20°.

Statistical biases, that is, the ones that depend on the number of samples in the estimates, and standard deviations in the conventional and 1-lag estimators were obtained with the perturbation analysis. The results agree well with the signal simulations. At spectral widths less than 6 m s^{−1} (and an unambiguous velocity interval of 50 m s^{−1}) and SNRs > 2 dB the standard deviations of the 1-lag estimates and conventional estimates are practically identical. At widths wider than 6 m s^{−1} the standard deviations of the 1-lag estimators are slightly larger than the ones from the conventional algorithm.

Comparison of the conventional and 1-lag *Z*_{DR} and *ρ*_{hv} estimates along a radial can reveal noise power variations in the two polarimetric channels and thus monitor these variations.

Radar observations demonstrate that the 1-lag estimators are less sensitive to interference signals than the conventional algorithms.

## Acknowledgments

One of the authors (VM) would like to thank Dr. E. Torlaschi for the discussions on statistics of the correlation coefficient. Part of the funding for this study was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreement NA17RJ1227, U.S. Department of Commerce, and from the U.S. National Weather Service, the Federal Aviation Administration (FAA), and the Air Force Weather Agency through the NEXRAD Product Improvement Program. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of NOAA or the U.S. Department of Commerce. We are grateful to A. Zahrai who led the engineering team that performed modifications on the research WDR-88D radar to allow collection of dual polarization data. Mike Schmidt and Richard Wahkinney maintained the radar in impeccable condition.

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## APPENDIX

### Statistical Biases and Standard Deviations in the 1-Lag Estimators

*Z*

_{DR1}the variation equation applied to (6) is

*R̂*

_{h(υ)}(

*T*) =

*δR̂*

_{h(υ)}(

*T*)/

*R*

_{h(υ)}(

*T*), ℜ denotes the real part of a complex number, and the second-order terms have been retained. The bias is the expectation of (A1) hence,

*M*

_{I}_{1}is

*M*

_{I}_{1}, the following approximation holds within a 10% error (Melnikov and Zrnić 2004):

^{−1}and

*υ*= 25 m s

_{a}^{−1}.

*M*is the equivalent number of independent samples (Doviak and Zrnić 1993, section 6.3.1). For

_{I}*M*, the following approximation holds within a 10% error:

_{I}*ρ̂*

_{hv1}estimate. The perturbation equation applied to (10) is

*ρ*

_{hv}, we get

_{c}= 0 (common noise in the channels) but incorporates a correction in the noise term (E. Torlaschi 2005, personal communication). For equal noise level in the channels, substitutions of (A6) and (A11) into (A8), (A9), (A15), and (A17) produce (11) through (14).