## 1. Introduction

Range-oversampling processing can be used on pulsed Doppler weather radars to increase the signal-to-noise ratio (SNR) through digital matched filtering (Chiuppesi et al. 1980) or to reduce the variance of estimates by decorrelating oversampled data with a matrix transformation followed by incoherent range averaging (Torres and Zrnić 2003a,b). Early studies showed the feasibility of this type of processing using weather data collected with experimental setups and offline signal processing (Ivić et al. 2003; Torres and Ivić 2005). More recently, a real-time implementation was employed on the National Weather Radar Testbed phased-array radar (NWRT PAR) that resulted in faster scan times (by a factor of 2) with no significant loss in data quality (Curtis and Torres 2011). As range-oversampling processing is feasible on modern digital receivers and signal processing architectures, it becomes increasingly important to understand what calibration measurements are necessary and how accurate they need to be for optimal performance. This is the focus of this paper.

The decorrelation transformations used for range-oversampling processing (e.g., digital matched filtering and whitening) depend solely on the normalized range correlation matrix _{V} (Curtis and Torres 2011). This matrix captures the necessary range correlation information for computing appropriate matrix transformations; hence, it is important to measure it accurately. One of the advantages of range-oversampling processing is that the range correlation of oversampled signals can be measured a priori; that is, assuming uniform reflectivity in the radar resolution volume, the range correlation of oversampled signals depends only on radar parameters, namely, the modified pulse (Torres and Zrnić 2003a). The modified pulse is defined as the envelope of the transmitter pulse convolved with the baseband-equivalent receiver impulse response (i.e., the transmitter pulse shape after passing through the receiver filters); thus, it captures both components of the range correlation (Torres and Curtis 2012). In this paper, we show that accurately measuring the range correlation matrix (directly from the data or derived from the modified pulse) is critical for optimal performance of range-oversampling processing.

In principle, the correlation of samples in range can be measured directly from the actual radar data (Ivić et al. 2003), but this makes a systematic mismeasurement analysis less straightforward. Alternatively, the range correlation can be derived from the measured modified pulse. This method is adopted herein for convenience but without the loss of generality. The modified pulse can be measured in a couple of ways. One way is to simply capture the return from a strong point target (or from multiple ones to check for consistency). If the target is extremely narrow in range extent (e.g., a metal transmission tower), then the measured response will consist of the original transmitted pulse that is reflected off the target and is modified after passing through the receiver filters. This satisfies the definition of the modified pulse and is one way to accurately measure it. The second way to measure the modified pulse is to inject a delayed version of the transmitted pulse directly into the front end of the radar receiver. Note that an additional receiver channel is required if this measurement is to be carried out when the actual pulse is transmitted. However, this method eliminates the need to find a narrow point target, and the signal level can be set appropriately to accurately measure the modified pulse. Although the path through the antenna is not included, it should not have a significant effect on the measurement. An additional consideration when measuring the modified pulse is the receiver sampling rate. A more accurate measurement of the range correlation can be obtained if the pulse is sampled at a rate higher than that used for oversampling processing (normally a multiple of the range-oversampling rate). However, as will be demonstrated later, sampling at the oversampling rate should be sufficient to derive the range correlation matrix needed for range-oversampling processing.

Having established the intimate connection between range-oversampling processing and the range correlation matrix of oversampled signals, and having outlined procedures to measure the latter, it is essential to examine how inaccurate measurements of the range correlation can affect the performance of range-oversampling processing. As will be shown, one of the major effects of a mismeasured range correlation is reflectivity bias. This bias can be on the order of a few decibels (dB), which can lead, for example, to significantly inaccurate quantitative precipitation measurements. Another effect is an overall degradation in performance; that is, the expected increase in SNR or reduction in variance of estimates is not achieved if the range correlation is measured inaccurately. This affects all of the meteorological variables and can, for example, unexpectedly result in higher variance of estimates for whitened data compared to a digital matched filter, even at high signal-to-noise ratios.

This paper is structured as follows. Section 2 introduces a theoretical framework for studying the effects of mismeasurement of the range correlation matrix. In section 3, the theory is validated using simulations that show how different types of mismeasurements affect the reflectivity bias and degrade the performance of range-oversampling processing. Section 4 shows a case from the NWRT PAR that clearly illustrates the reflectivity biases observed when range-oversampling processing relies on an inaccurate measurement of the range correlation. Finally, the results are summarized along with suggestions for applying this work so that the benefits of range-oversampling techniques can be fully realized.

## 2. Theoretical analysis

*L*times higher than with conventional sampling and performing incoherent averaging of transformed datasets. The goal of this processing may be to increase the SNR with a digital matched filter (Chiuppesi et al. 1980) or to reduce the variance of meteorological variable estimates (Torres and Zrnić 2003a). Mathematically, the transformation of range-oversampled signals can be expressed as

*L*oversampled signals for a given sample time

*mT*(

_{s}*m*= 0, 1, … ,

*M−1*, where

*M*is the number of pulses per dwell;

*T*is the pulse repetition time). Oversampled time series data are typically spaced by

_{s}*T*=

_{r}*τ*/

*L*in range time (

*τ*is the transmitter pulse width) and by

*T*in sample time. Here,

_{s}*L*-by-

*L*transformation matrix, and

**X**

_{m}is the vector of

*L*transformed samples with the same structure as

**V**

_{m}. In general,

*L*-by-

*L*complex-valued matrix normalized to satisfy the “power-preserving condition” given by Torres et al. (2004) as

*S*is the true signal power and

*E*is the statistical expectation operator. A special case of

*N*sampled at the same range-oversampling rate of

_{p}*L*-by-(

*N*+

_{p}*L*− 1) modified-pulse convolution matrix (referred to as the pulse matrix) given by

**p**. As shown next, any mismatches between the true and measured modified pulses will translate into improper normalizations of

**X**can be used to estimate the spectral moments of weather signals via estimates of the sample-time autocorrelation function at a few small lags (Doviak and Zrnić 1993); that is, from sample-time lag-

*k*autocorrelation estimates of the form

^{1}can be derived as

*υ*is the Nyquist velocity, and “arg” denotes the argument of a complex number. In the previous equations, subscript

_{a}*X*is used to indicate that autocorrelation estimates are derived from transformed data, and superscript

*S*is used to denote sample-time autocorrelation (as opposed to range-time autocorrelation, which will be denoted with a superscript

*R*). Because the effects of receiver noise are not the focus of this study, a high SNR is assumed for simplicity. Whereas the bias of (6) can be derived directly, the biases of (7) and (8) can only be approximated. Here, we follow the approach by Benham et al. (1972) and use first-order perturbations of the nonlinear functions of

*M*(Zrnić 1977). Thus, it is possible to find the biases of spectral moment estimators as functions of the expected values of autocorrelation estimates. Using the results from Benham et al. (1972) and Zrnić (1977),

*σ*

_{υn}is the spectrum width normalized by the Nyquist cointerval (2

*υ*); the quantities with no hat represent true values; and “Re” and “Im” stand for real and imaginary parts of a complex number, respectively. The expected values of sample-time autocorrelation estimates in the right-hand side of (9)–(11) can be determined from (5) as

_{a}*k*,

_{X}, note that the elements along the main diagonal of this matrix are all equal to

_{X}can be expressed in terms of the normalized range correlation matrix of raw samples as

_{X}and to the identity

_{X}=

_{X}, where

_{X}is defined as the normalized range correlation matrix of transformed samples with ones along its main diagonal. However, the power-preserving condition can be met only if

_{V}is accurately measured.

*mismeasured*range correlation matrix

*true*range correlation of transformed oversampled signals depends on

_{V}; thus, the correct power-preserving condition would be

_{V}). As a consequence of this mismatch,

*γ*contains the reflectivity calibration constant and all range-dependent terms (Doviak and Zrnić 1993), it can be shown that

_{V}and

The polarimetric variables (i.e., differential reflectivity *Z*_{DR}, differential phase Φ_{DP}, and magnitude of the copolar correlation coefficient *ρ*_{HV}) can be analyzed analogously. Perturbation formulas for the biases and variances of these estimators (e.g., Melnikov and Zrnić 2004) can be used to show that all polarimetric variables remain unbiased for any degree of mismatch. Without going into the mathematics, this can be intuitively explained by noticing that *Z*_{DR} and *ρ*_{HV}, like *σ*_{υ}, are based on ratios of auto- and cross-correlation estimates, and Φ_{DP}, similarly to *υ*, is based on the argument of a correlation estimate. Thus, any scaling errors arising from range correlation mismeasurements will cancel out in the ratios or become immaterial after taking the argument. Because of these similarities, it can also be shown that the variance ratios for the polarimetric variables are given by (24). In the next section, Eqs. (22)–(24) are validated with simulations and are used to quantify the effects of relying on an inaccurately measured range correlation.

## 3. Simulations

The theoretical results derived in the previous section are validated with simulations in this section. Range-oversampled weatherlike signals were simulated using the procedure described by Torres and Zrnić (2003a) and the modified pulse of the NWRT PAR (Zrnić et al. 2007) located in Norman, Oklahoma, where the range-oversampling factor was *L* = 4 (as in the NWRT PAR). Simulated range-oversampled signals were processed using the approach described in the previous section to obtain spectral moment estimates for 10 000 realizations of simulated signals with a high SNR of 30 dB and a spectrum width of 2 m s^{−1}, which is typical of many weather phenomena (Fang et al. 2004). Without loss of generality, the Doppler velocity was set to zero because it has no effect on the biases and variances under analysis [cf. (15)–(20)]. A long pulse repetition time (PRT) was used for signal power (reflectivity) estimates, where the number of samples per dwell was *M* = 15 and the Nyquist velocity was *υ _{a}* = 7.5 m s

^{−1}. A short PRT was used for mean Doppler velocity and spectrum width estimates, where

*M*= 40 and

*υ*= 23.7 m s

_{a}^{−1}. Range-oversampling processing was carried out using (i) a whitening transformation (whitening-transformation-based estimates are denoted by WTB), (ii) pseudowhitening transformations (PTBs) with sharpening parameter

*α*= 0.6 and 0.8 (Torres et al. 2004), and (iii) a digital-matched filter transformation (MFB). As explained next, transformations were derived using mismeasured modified pulses with varying degrees of mismatch in either magnitude or phase.

A model for the modified-pulse shape was chosen so that amplitude and phase mismatches could be systematically varied. The amplitude of the modified pulse was assumed to be symmetric and was obtained by resampling the piecewise-cubic Hermite-polynomial interpolation with control points on the normalized time axis defined at (0,1), (*w*/2 − *r*_{0},0.9), (*w*/2,0.5), (*w*/2 + *r*_{1},0.1), and (1,0). This type of interpolation is particularly suitable for our purpose because of its shape-preserving properties; that is, it results in no overshoots and less oscillation if the data are not smooth (Fritsch and Carlson 1980). Figure 1 shows a depiction of the amplitude model, where *w* can be recognized as the 3-dB normalized pulse width and *r*_{0} + *r*_{1} as its normalized fall time. The phase of the modified pulse was derived from the phase of the NWRT PAR modified pulse (*φ*_{NWRT}) by adding constant and linearly increasing phase terms; that is, arg[*p*(*t*)] = *φ*_{NWRT} + *φ*_{0} + *φ*_{1}*t*.

Figure 2 shows the NWRT PAR modified pulse (dotted line) measured using the return from a tower and its best match using our model (solid black line). Also shown in this figure are the amplitudes and phases of the maximally mismatched pulses (solid gray lines) used in the simulation analyses. The model parameters that best match the NWRT PAR pulse are *w* = 0.79, *r*_{0} = 0.19, *r*_{1} = 0.2, *φ*_{0} = 0, and *φ*_{1} = 0. These are the nominal model parameters used for the simulations; that is, mismatches are obtained by deviating from the nominal parameters, where deviations are denoted by appending Δ to the parameter being changed (e.g., Δ*w* for *w*). For the maximally mismatched pulses, the normalized widths are *w* = 0.5 and 1.1 (i.e., a Δ*w* of about ±0.3), respectively, and the slopes of the linear phase shifts are *φ*_{1} = ±90° per sample (i.e., for *N _{p}* = 8, this results in a maximum shift of ±720° that occurs on the last sample).

Figure 3 shows theoretical (thick lines) and simulation (thin lines) results for MFB, PTB, and WTB spectral moment estimates as a function of the normalized width mismatch (Δ*w*). The left plots show the biases of reflectivity [cf. (22)] (top) and mean Doppler velocity [cf. (16)] (bottom). The right plots show the ratio of standard deviations obtained from a mismeasured modified pulse to those obtained from the true modified pulse for signal power and Doppler velocity estimates [cf. (23) and (24)]. A series of 50 modified pulses with normalized widths ranging from 0.5 to 1.1 are used in this analysis. The phases of the modified pulses are set to zero so that the effects of an amplitude mismatch can be independently quantified. Note that Δ*w* = 0 indicates no mismeasurement; that is, the measured pulse is exactly the true pulse and, as expected, this corresponds to zero reflectivity bias and unit standard deviation ratios. In general, as the degree of mismatch increases, reflectivity biases increase (in absolute value) and range-oversampling processing becomes less effective; that is, the standard deviations of WTB and PTB estimates increase with respect to the expected levels when there are no mismeasurements. The reflectivity bias for MFB estimates increases when the modified-pulse width is measured to be narrower than it actually is. WTB estimates exhibit the opposite behavior, and PTB estimates fall in between, depending on the degree of pseudowhitening (i.e., 0 < α < 1, ranging from MFB-like to WTB-like performance). For the range of width mismatches used in this simulation, the maximum absolute reflectivity bias is about 1.5 dB, which occurs for WTB estimates and a measured modified pulse narrower than the true one. The ratios of standard deviations for signal power estimates follow the same trends as the reflectivity biases. It is important to observe that although standard deviation ratios less than one would imply that transformations derived from a mismeasured range correlation matrix are more effective at reducing the variance of estimates, this is not the case for signal power estimates (top-right panel) since these are severely biased (top-left panel). Conversely, the ratios of standard deviations for velocity (and spectrum width) estimates (bottom-right panel) are more meaningful since these estimates are unbiased (bottom-left panel) irrespective of any mismeasurements. It is also worth noting that MFB velocity estimates are almost unaffected by the modified-pulse width mismatch, PTB estimates are minimally affected, and WTB estimates are affected with very modest departures from the ideal behavior (a maximum of about 10% degradation in the standard deviation of velocity estimates).

Figure 4 is the same as Fig. 3, but the results are plotted as a function of the phase–slope mismatch (Δ*φ*_{1}). A series of 50 modified pulses with linear phase shifts ranging from −90° to 90° per sample are used in this analysis. In this case, the amplitudes of the modified pulses are always set to the amplitude of the NWRT PAR modified pulse (Fig. 2), and the phases are perturbed using those of the NWRT PAR modified pulse as a reference; that is, Δ*φ*_{1} = 0 indicates no mismeasurement (in amplitude or phase), which again corresponds to zero bias and unit standard deviation ratios. As before, reflectivity estimates become more biased and range-oversampling processing becomes less effective as the degree of mismatch increases. However, whereas the reflectivity bias for MFB estimates becomes more negative (referred to as the “cold” radar effect), those of WTB estimates become more positive (“hot” radar effect). For the range of phase mismatches in this simulation, Doppler velocity estimates are unbiased and the maximum reflectivity bias is about ±5 dB, and positive for WTB estimates and negative for MFB estimates. As expected, the ratio of standard deviations for signal power estimates follows the same trends as the reflectivity bias and, again, standard deviation ratios less than one are not an indication of higher effectiveness at reducing the variance of estimates because of the negative bias of signal power estimates. The ratios of standard deviations for velocity (and spectrum width) estimates exhibit a different behavior. It can be observed that MFB velocity estimates are, like before, almost unaffected by modified-pulse width mismatches. However, WTB estimates are maximally affected with up to 60% degradation from the ideal behavior, and, not surprisingly, the effects on PTB velocity estimates depend on the degree of pseudowhitening.

Similar analyses were conducted for the polarimetric variables, but they are not shown here because the bias and standard deviation ratio plots look nearly identical to those corresponding to mean Doppler velocity. We also conducted simulations varying the other modified-pulse model parameters. Fall-time mismatches (nonzero Δ*r*_{0} and Δ*r*_{1}) lead to performance degradations similar to those observed with pulse width mismatches. However, phase measurement errors in the form of constant biases (nonzero Δ*φ*_{0}) do not affect the performance of any of the range-oversampling techniques. It is important to note that, in all cases, the agreement between theoretical and simulation results is remarkable. In the next section, real data are used to illustrate the impact of these results.

## 4. Effects on data

The effects of mismeasurement of the modified pulse can be shown using collected time series data. For this comparison, data from 11 February 2009 were acquired using a long PRT *T _{s}* = 3104 ms (

*υ*= 7.5 m s

_{a}^{−1}) and

*M*= 15 samples per dwell for reflectivity, and a short PRT

*T*= 984 ms (

_{s}*υ*= 23.7 m s

_{a}^{−1}) and

*M*= 40 samples per dwell for mean Doppler velocity and spectrum width. Long- and short-PRT datasets were processed with both an accurately measured modified pulse and a “mismeasured” modified pulse. To obtain the mismeasured modified pulse, the phase of the NWRT PAR modified pulse was altered as described in section 3 with Δ

*φ*

_{1}= 90° per sample. Figures 5 and 6 show plan position indicator (PPI) displays of reflectivity and Doppler velocity fields at ~0159 UTC. In each of these figures, the three left panels show data processed with the accurately measured modified pulse, and the three right panels show data processed with the “mismeasured” modified pulse. The top two panels show the results utilizing a digital-matched filter, the middle two panels show the results for pseudowhitening with

*α*= 0.6, and the bottom two panels show the results for whitening. This allows us to see the differing effects of mismeasurement on the three types of processing.

The digital-matched filtered data with the accurately measured modified pulse is used as the baseline for comparisons (top-left panel). Since the pseudowhitening and whitening transformations should be power preserving, the reflectivity levels shown in the middle and bottom right panels of Fig. 5 should match but should also look smoother because of the variance reduction obtained from these transformations. We do see that the data become progressively smoother when moving from digital-matched filtering to pseudowhitening to whitening, and the reflectivity levels also seem to match reasonably well. To quantify the effects of both modified pulses and all processing transformations, reflectivity differences between the datasets were utilized. The datasets were compared to the baseline digital-matched filtered dataset with an accurately measured modified pulse, and an SNR threshold of 20 dB was used to ensure that the comparisons were made using strong signals. Because of some large outliers in the reflectivity differences, the median was used to measure the central tendency. In the whitening case with an accurately measured pulse (bottom-left panel), the median difference in reflectivity values is 0.05 dB, which shows that the reflectivity levels are basically the same. For pseudowhitening, the median reflectivity difference is 0 dB.

For the “mismeasured” modified pulse, the differences in reflectivity are striking. The digital-matched filtered data (top-right panel) looks significantly “colder” than those from the accurately measured case. The median difference in reflectivity is −5.36 dB, which is close to the theoretically predicted value of −5.20 dB (see Fig. 4). As expected, the whitened data (bottom-right panel) look “hotter” than the accurately measured case. The median reflectivity difference is 4.43 dB, which also matches well compared to the theoretically predicted bias of 4.54 dB. The pseudowhitened data (middle-right panel) exhibit a smaller bias of 0.24 dB, nearly matching the theoretical value of 0.27 dB. A secondary effect of these biases can be seen in the different amounts of censoring between the left and right panels of Figs. 5 and 6; this censoring is based on the MFB signal power for all transformations, and thus all the “mismeasured” fields are affected in the same way. Although it is more difficult to see the effects of increased standard deviation, the whitened data using the mismeasured pulse do look less smooth than the whitened data using the accurately measured pulse. Even though this is an extreme example, substantial reflectivity biases can occur for smaller measurement errors, a fact that reinforces the case for accurate measurement of the range correlation. For instance, after changing a trigger amplifier on the NWRT PAR and before updating the range correlation measurement to reflect this change, we observed reflectivity biases of −3.16 and 3.46 dB when using a digital-matched filter and a whitening transformation, respectively.

As predicted by the theory, the biases in the mean Doppler velocity data are close to zero. As with the reflectivity, the velocity data become progressively smoother when moving from digital-matched filtering to pseudowhitening to whitening. Although fields of spectrum width are not included, the results are similar to those in Fig. 6. In a dual-polarimetric radar, we would also expect to see no biases in differential reflectivity, differential phase, and magnitude of the copolar correlation coefficient.

## 5. Conclusions

Range-oversampling techniques rely on accurate knowledge of the range correlation to derive time series data transformations that can improve the signal-to-noise ratio (e.g., digital matched filter) or reduce the variance of meteorological data estimates (e.g., whitening). The range correlation can be measured directly from the data or derived from measurements of the modified pulse. In this paper, we explored the effects of having a mismeasured range correlation on the performance of range-oversampling techniques.

A theoretical framework was developed to quantify the effects of mismeasurement in terms of the statistical performance of range-oversampling techniques based on a digital matched filter, whitening, and pseudowhitening transformations. Whereas reflectivity estimates can be severely biased depending on the degree of mismeasurement, all other meteorological variables were shown to remain unbiased under any mismeasurement conditions. However, range-correlation mismeasurements cause the standard deviations of all estimates to increase, except for reflectivity, which may have lower standard deviations because of the negative biases. Simulations based on a parametric model of the modified pulse were used to validate the theoretical results. Examples of both amplitude and phase mismatches were studied and shown to agree with the theoretical predictions. A real-data case was used to illustrate the performance degradation that can occur when there is a severe mismeasurement of the modified pulse; this also agreed well with simulations and theoretical results.

As range-oversampling techniques become more widely used, it is important to understand how a mismeasurement of the range correlation can affect their performance. The framework provided in this paper can be used to translate radar data quality requirements into concrete specifications for the accuracy of range correlation measurements on modern operational weather radars, and thus ensure that the benefits of range-oversampling techniques are fully realized.

## Acknowledgments

The authors thank Dusan Zrnić, Igor Ivić, and three anonymous reviewers for providing comments to improve the manuscript. Funding was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce.

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^{1}

Herein, we focus on the classical lag 0–1 spectrum-width estimator; however, the results also apply to other spectrum-width estimators that are based on the ratio of the autocorrelation at two different lags.