## Abstract

A radar antenna intercepts thermal radiation from various sources, including the ground, the sun, the sky, precipitation, and man-made radiators. In the radar receiver, this external radiation produces noise that constructively adds to the receiver internal noise and results in the overall system noise. Consequently, the system noise power is dependent on the antenna position and needs to be estimated accurately. Inaccurate noise power measurements may lead to a reduction of coverage if the noise power is overestimated or to radar data images cluttered by noise speckles if the noise power is underestimated. Moreover, when an erroneous noise power is used at low to moderate signal-to-noise ratios, estimators can produce biased meteorological variables. Therefore, to obtain the best quality of radar products, it is desirable to compute meteorological variables using the noise power measured at each antenna position. An effective technique that achieves this by estimating the noise power in real time from measured powers at each scan direction and in parallel with weather data collection has been proposed. Herein, the effects of such radial-based noise power estimation on spectral moment estimates are investigated.

## 1. Introduction

Accurate measurement of noise power is important for proper operation of censoring algorithms and for producing high-quality weather radar products from signals at low to moderate signal-to-noise ratios (SNR). Such SNRs are observed in distant precipitation, light rain, and snowfall. In those cases, an accurate noise power measurement is important, as it is integral in the estimation of returned signal power, used to compute multiple weather radar products, and thus potentially impacting the correct operation of automated algorithms and accurate forecasts derived from weather radar products. Currently, the majority of weather radar systems employ some type of calibration procedure to measure noise power levels while the radar is not scanning for weather. For instance, on the Weather Surveillance Radar-1988 Doppler (WSR-88D) network of radars [Next Generation Radar (NEXRAD)], the blue-sky noise power is measured before each volume scan,^{1} while the antenna points at a high-elevation angle, so that no ground clutter radiation is received by the antenna. The measured blue-sky noise power is adjusted for use at lower antenna elevations using predetermined correction factors to account for thermal radiation from the ground, which contributes to the system noise. Correction factors are determined via an offline maintenance calibration procedure at irregular time intervals. It consists of measuring the noise power at each antenna position while executing a full azimuthal scan at selected elevations with the transmitter turned off. At each elevation, the largest measured noise power is chosen, and the ratio of this power to the one at the highest elevation is taken as the noise correction factor for that elevation. This approach estimates the system noise power value particular to an elevation angle at time intervals that are dictated by the duration of volume scans (4–10 min, depending on the volume scan type) using correction factors that are determined even less often. Hence, such noise power measurement does not capture the temporal variations of noise power caused by changes in the system, as these can alter the system noise power by more than 1 dB within a time period of less than 3 min (Melnikov 2006). Noise power can also vary greatly at different antenna azimuth positions due to external noise sources (e.g., cell phone towers producing interference or ground clutter, such as mountains), particularly at low-elevation angles. Consequently, if noise power is elevated at several positions in azimuth, it causes the calibration procedure to produce a measurement that is based on the elevated noise power and is incorrect for the rest of the azimuth positions, where the noise power is smaller. Because of the lengthy execution time, the noise correction factor calculation routine is not run regularly, and it cannot capture potential temporal variations due to temperature and season. Also, thunderstorms emit radiation over a broad frequency range, which can also raise the noise floor (Fabry 2001; Melnikov and Zrnić 2004). Finally, anomalous propagation can cause the beam to intersect with the ground, resulting in the increased noise power. Clearly, such variations cannot be captured by the periodical noise power calibration procedure.

Inaccurate noise power measurements reduce the coverage of weather radar if the measured noise power is higher than the true value, or conversely produce speckling in the fields of meteorological variables if the noise power used for censoring is much lower than the actual one. As for the meteorological variables, if error in noise power measurement is significant and signal power is comparable to that of noise, then the resultant estimates can be noticeably biased. This is especially true for products that combine noise power measurements with other estimates in a combination that is nonlinear—for example, spectrum width as well as differential reflectivity and correlation coefficient in the case of dual-polarized radars. Thus, in order to generate high-quality weather radar products, noise power needs to be accurately measured at each scan direction. However, given the unpredictable nature of noise, it is evident that the variations in time and location can only be captured if noise power is measured in real time at every radial^{2} while the radar is scanning for weather. A technique that achieves this has been described in Ivić et al. (2013). Unlike the spectral-processing-based techniques proposed in the past (Hildebrand and Sekhon 1974; Urkowitz and Nespor 1992; Siggia and Passarelli 2004), this technique makes no attempt to compute the noise power at every range location (or bin) independently. Instead, it operates under the assumption that average noise power is constant along an entire radial and uses the shape of the radial power profile in range and estimates of the SNR to search for signal-free bins. The search algorithm is based on the assumption that the noise is additive white Gaussian noise (AWGN). Once noiselike bins are identified, the technique computes the average power of these range volumes to produce an estimated noise power. Because the technique does not extract the noise power from bins where signal is present, it is not subject to the limitations of the spectral-processing-based techniques in cases when the number of samples at each range volume is small or when weather signals are spread over most of the Nyquist cointerval. Also, because the end result is computed only from bins classified as signal free, the technique is less likely to incur bias in noise power estimates than the spectral-processing-based techniques given that the noise power is constant along the radial. At the same time, in cases where the majority of bins in a radial contain signals either because storms span the entire unambiguous range or because of the range folding, the radial-based estimator from Ivić et al. (2013) may be incapable of producing a reliable result because the number of signal-free bins is insufficient [e.g., if a short pulse repetition time (PRT) is used at a low to medium antenna elevations]. In such cases either the latest noise power measurement from the same (or the closest) antenna position or the calibration-supplied estimate can be used instead, if available in a system. Alternatively, many radars use a dual PRT scan where the PRT with the large unambiguous range (long PRT) is used for reflectivity computation as well as range unfolding of overlaid echoes, and the PRT with the large unambiguous velocity, but poor unambiguous range (short PRT), is used for Doppler data measurements (i.e., velocity and spectrum width) at the same elevation. In such situations noise power estimates from the long PRT scans can be used for computations based on the short PRT data. An additional benefit of the Ivić et al. (2013) technique is that it operates solely on power estimates obtained from the raw time series data and averaged along sample time at every bin. Hence, the technique does not utilize any signal properties in sample time (e.g., autocorrelation or spectral processing) but rather the shape of power profile in range to discriminate signals from noise. This makes it insensitive to system phase noise as well as any phase coding schemes used to resolve range overlaid signals (Laird 1981; Siggia 1983; Zrnić and Mahapatra 1985; Sachidananda and Zrnić 1999). It also makes it applicable to systems that do not provide Doppler measurements.

In this paper the impacts, on spectral moment estimates, of inaccurate noise power measurements and the effects of radial-based noise power estimation, using the technique from Ivić et al. (2013), are investigated. The paper is structured as follows. Section 2 analyzes aspects of calibration and radial-based noise power estimation. The effects on radar coverage are investigated in section 3, and section 4 analyzes the sensitivity of spectral moment computation to inaccurate noise power measurements. Section 5 presents examples from the engineering evaluation of the technique. Finally, section 6 summarizes the main conclusions of the paper.

## 2. Calibration versus radial-based noise power estimation

As in the case of meteorological variables, the outcome of noise power measurement can be viewed as a random variable. This is because the estimate of noise power is computed as

where complex random variables *V*(*m*, *k*) are obtained by sampling receiver voltages. Bin positions in range are denoted by *k*, while *m* denotes the sample number, and the total number of samples, in sample time, per radial is *M*. The variable *K* signifies the total number of bins from which powers are averaged to produce the noise power estimate. When radial-based noise power estimation (RBNE) is used, the data contain both signals and noise, so is computed only from bins that are classified as signal free [e.g., by the algorithm described in Ivić et al. (2013)]. In case of the WSR-88D calibration procedure, data are collected with the transmitter turned off, so no signals are present. Thus, the estimate of noise power () at each radial is the sum of the true value (*N*) and instant error (). Therefore, at every radial it is characterized by the bias

and the standard deviation

where the angle brackets (〈 〉) denote mathematical expectation. Noise samples are independent both in sample time and in range, so the standard deviation of is

Hence, as the number of samples (*KM*), from which is extracted, increases, the standard deviation of noise power measurement decreases. Clearly, needs to be computed from a sufficient number of samples so that the uncertainty of estimates (due to the limited number of samples) does not affect the radar products adversely.

In the WSR-88D network, the calibration procedure computes noise power from a large number of samples (i.e., *KM* is more than 5000) when the radar is not scanning for weather. Consequently, the standard deviation of the measurement [i.e., as given in (3)] is negligibly small and can be neglected; thus, the error with respect to the true noise power value *N* can be observed only in terms of the bias Δ*N*. Because true noise power can vary significantly both in azimuth and elevation, but noise power produced by the calibration is constant at each elevation, Δ*N* may change appreciably with each scan position (Fig. 1). As a result, the influence of bias Δ*N* on the radar products may differ considerably with azimuth.

When RBNE is used, the sample size, from which noise power is computed, is limited by the number of samples per radial (*M*) and the number of bins in range. It is further decreased if noise power is computed only from bins that are classified as free of signal (Ivić et al. 2013; Dixon and Hubbert 2012). Consequently, care should be taken so that the bias and standard deviation of noise power estimates do not noticeably increase the overall bias and standard deviation of the meteorological variable estimates at low to moderate SNRs. In that regard, the bias and the standard deviation of RBNE must be minimized to realize the advantages of radial-based noise power computation. As shown in Ivić et al. (2013), the bias of noise power estimates produced by the RBNE technique is small enough so it can be neglected for all practical purposes, while the standard deviation is estimated to be 1.2% (0.052 dB) of the true noise power.

In section 5, the bias and standard deviation of spectral moment estimates computed using noise power estimates produced by the WSR-88D calibration procedure (*N*_{c}) and the RBNE technique (*N*_{m}) from Ivić et al. (2013) are compared to those obtained using true noise powers. The analysis uses analytical derivations and simulations. To simulate noise power estimates (*N*_{m}), produced by the Ivić et al. (2013) technique, the probability density function (pdf) of such estimates was needed. In that regard, it was experimentally found that the best approximation to pdf of the outputs from the Ivić et al. (2013) technique is the three-parameter log-logistic distribution, defined as

with parameters *α* = 56.9, *β* = 0.334, and *γ* = 0.665 when the true noise power is unity. The approximation is based on the best fit to the pdf curve estimated from 159 490 realizations (Fig. 11 in Ivić et al. 2013). This approximation is used to simulate the estimates *N*_{m} as

where *U* is the uniformly distributed random variable and *N* is the true noise power value. Hence, (6) is used to produce realizations of random variable *N*_{m} with pdf as in (5).

Indicative examples using real data are given in Figs. 2–5. Figure 2a shows the reflectivity field obtained by the WSR-88D site located in Pendleton, Oregon (KPDT). The reflectivity field is computed using unfiltered power estimates from which noise power, supplied by the WSR-88D calibration, has been subtracted but no censoring has been applied, so that the large variations of noise power in azimuth are observable as varying densities of noise speckles. At the time of data collection, the radar operated at a PRT of 3.1 ms, with *M* = 15, and at an elevation of 0.87°. The calibration and the RBNE-supplied noise power estimates are presented in Fig. 2b. This particular radar site is surrounded by mountains on one side and plains on the other. The side where mountains are located is visible in Fig. 2b as the elevated RBNE noise power (*N*_{m}) and in Fig. 2a as the area with the increased density of noise speckles at azimuths between 50° and 180°. Notice that the largest RBNE noise power variation in azimuth, of roughly 2.5 dB*m*, is caused by an external interference source at approximately 120°. Also, the plot in Fig. 2b demonstrates the deficiency in the legacy calibration because it produces noise power that does not capture the azimuthal variations shown by the RBNE outputs. Figure 2c shows the ratios of *N*_{c} and *N*_{m} (dB) at each elevation, where the vertical bars stand for the ratio variations in azimuths. The top and the bottom of each bar are the maximum and the minimum ratio at each elevation, respectively. The horizontal lines connect the ratio median values at each elevation. Given that the *N*_{m} is on average more accurate than the *N*_{c}, the bar plot in Fig. 2c provides a good estimate of the noise power errors introduced by the calibration. As expected, the largest differences between the calibration and the RBNE noise powers are at lower elevations, whereas they decline as the elevation increases. Furthermore, the horizontal line, which connects bar medians, is very close to zero for elevations higher than 5°. This is because the NEXRAD calibration corrects the blue-sky noise power only for antenna elevations less than 5°. Another interesting piece of information about this particular site is that the correction factors computed using the described WSR-88D noise calibration procedure have been adjusted manually to mitigate the overall noise power bias caused by the large noise power variations in azimuth and the large peak caused by the external interference. Thus, the calibration noise power shown in Fig. 2b is somewhat lower than the one produced by the straightforward calibration procedure. In the following text, this example will be referred to as test case 1.

The next example is from the WSR-88D site located near Vance Air Force Base, Oklahoma (KVNX). Data were collected using a PRT of 3.1 ms, with *M* = 17, and at an elevation of 0.53°. The reflectivity field and the measured/calibration noise powers are given in Fig. 3. Figure 3b reveals the noise power to be fairly constant in azimuth except in the region from 240° to 250°, where a significant blockage from trees raised the noise power level. This caused the calibration procedure to set the noise power at a level significantly higher than levels at azimuth positions unaffected by the blockage. As in the previous example, Fig. 3c shows the median of the ratio *N*_{c}/*N*_{m} to be close to unity at elevations above 5°. This suggests that the median (in azimuth) of the RBNE noise power is roughly equal to the calibration noise power at these elevations. For the purpose of reference, this example is dubbed test case 2.

Another test case is presented in Fig. 4. This dataset was collected with the research WSR-88D radar located in Norman, OK (KOUN), at a PRT of 3.1 ms. The number of samples *M* is 17, and the elevation is 0.53°. This site is located in the plains area, and the noise power is fairly flat in azimuth when skies are clear. Thus, we speculate that the variations in *N*_{m}, shown in Fig. 4b, were caused by precipitation. Moreover, Fig. 4c shows the larger variations in differences between the legacy calibration and the RBNE noise powers to persist at higher elevations, which further corroborate the assumption of precipitation-induced noise power fluctuations. Hence, this is a good example that shows how the meteorological phenomena being observed can alter radar system noise power. Notice that in this case, the noise power variations cannot be visually observed in Fig. 4a, as areas with increased speckle density are due to the noise power variations being smaller than in the previous two cases. This is test case 3.

The last example (i.e., test case 4) is also from KOUN. This dataset was collected at a PRT of 3.1 ms, with *M* = 15, and at an elevation of 0.53°. The reflectivity field in Fig. 5a and the measured noise power in Fig. 5b reveal the presence of strong interference sources at approximately 125°, 280°, and 340° as well as weaker interference sources at 235° and 260°. When interference is wideband and has constant power at all ranges in a radial, it behaves similarly to AWGN and can significantly raise the noise floor, as its power linearly adds to the background noise power. If calibration does not capture this contamination, then it is treated as signal by the data processing algorithms, making it visible in radar product displays. Consequently, all estimated products that use noise power measurements in the presence of such strong contamination are heavily biased and their use in further processing (e.g., rain-rate estimation) is undesirable. The radial-based noise power estimation technique, on the other hand, accurately measures the raised noise power level at azimuths where interference is similar to AWGN. This is visible in Fig. 5b for data at the lowest elevation but also in Fig. 5c, which shows the differences between the legacy calibration and the RBNE noise powers to be the largest at the first three elevations. This is an indication that the interference present at the lowest elevation persists at the next two elevations as well. To show the effect of RBNE on the censored radar products, the reflectivity fields produced using the calibration and the radial-based measured noise powers are presented next to each other in Fig. 6. Notice that the interference streak at about 260° is still visible in the field produced using the RBNE noise power. This is because the properties of the external interference differed significantly from AWGN at that location, so the radial-based noise power estimator failed to produce a result and the legacy-calibration-supplied noise power was used instead. Also, the streak at 300°, which is likely caused by a reflection of the weather echo, is not removed because it is classified as signal by the RBNE technique. Finally, the range on the ordinate is adjusted in Fig. 7, so that the fine variations in the RBNE noise power are visible. A visual inspection of the *N*_{m} reveals slightly elevated noise power between approximately 40° and 170°. Since the weather system to the east of the radar appears at these azimuths, we infer that the slightly elevated noise power is caused by the radiation from the storm.

## 3. Coverage effects

All weather radar systems implement some kind of censoring procedure to discard measurements at volumes that either contain only noise or do not contain sufficiently strong signal (i.e., from which useful information can be extracted). In general, censoring computes some function *f*(**V**) (where **V** = [*V*(0, *k*), … , *V*(*M* −1, *k*)]^{T} and *k* spans all range positions in a radial) at every bin in a radial and compares the outputs against a threshold to determine which bins contain significant signals. Generally, a threshold computation is based on noise power measurement and is always directly proportional to it. Hence, in terms of radar coverage, the effect of noise power measurement errors can be twofold. If true noise power is higher than the measured noise power, then the rate of false alarms can increase to the point where the radar display becomes cluttered with noise speckles. Cases like this are rather rare and are most commonly caused by interference, which can raise the noise floor by 10 dB or more; thus, only sections of radar display at azimuth positions where interference is present are affected (e.g., test case 4). If the estimated noise power value is higher than the true one, then the false alarm rate (FAR) drops but so does the signal detection rate. Hence, radar coverage decreases and signals that otherwise would be detected are discarded. In most systems the censoring rule is a simple threshold test of estimated SNR represented as

where is the noise power estimate (e.g., either *N*_{c} or *N*_{m}) and THR_{SNR} is the threshold (e.g., 2 dB) set to produce the desired FAR assuming an accurate estimate of the true noise power *N*. The power estimate at a position *k* in range is computed as

and represents the sum of signal and noise power. Taking into account the instantaneous errors in noise measurement (i.e., ), we get

where ΔTHR_{SNR} is the error in the actual threshold equal to

so the error in the noise power estimate is embedded into the censoring threshold. In the WSR-88D radars, the common censoring thresholds are 2 and 3.5 dB for the power and Doppler measurements, respectively. An illustration of how the threshold error changes with the ratio /*N* is shown in Fig. 8. A visual inspection of Fig. 8a reveals that the error in the ratio of only 0.6 dB (or 14%) results in an actual threshold that is 1 dB higher than the intended one of 2 dB. When the desired THR_{SNR} equals 3.5 dB, Fig. 8b suggests the same error occurs if is about 0.7 dB.

To show these effects on real data, the change in the number of detections and the coverage when using the SNR censoring with *N*_{c} versus *N*_{m} is shown in Table 1. In the first two cases, both metrics show an increase in both detections and coverage when RBNE is used instead of the calibration noise power. This is expected because the WSR-88D calibration value overestimated the noise power in both cases. Notice that, percentagewise, the difference is generally larger when coverage (km^{2}) is used as the metric. This is because most differences are in the detection of weak signals located on the fringes of the weather phenomena, and thus commonly located farther away from the radar. Consequently, because of the radar beam broadening, each such range volume covers more area than a range volume closer to the radar. In test case 3, *N*_{m} exceeded *N*_{c} at azimuths where weather was present, so this resulted in slightly less coverage when using RBNE. Nonetheless, we maintain that the use of the RBNE is still beneficial in this case because it preserves the desired signal quality in terms of the SNR and FAR. An interesting phenomenon occurs in test case 4, where the total number of detections and the coverage also decrease when using RBNE. This is caused by the numerous instances of interference in which the RBNE technique classifies them as noise and produces the noise power estimates that account for their presence; thus, interference signals are discarded by a censoring procedure. Conversely, the opposite occurs if using the calibration noise power, in which case the interferences are classified as signals, resulting in more detections than if RBNE is used.

## 4. Effects on spectral moment estimates

Excessive errors in determining system noise power can contribute to errors of signal power estimates at moderate to low SNR, and because both signal and noise powers are used in the computation of reflectivity and spectrum width, the quality of these estimates is susceptible to inaccuracies in noise power measurements. In general, the accuracy of any estimator that is a function of the vector of samples **V** and the noise power measurement () is susceptible to an error in noise power measurement expressed in terms of the bias and the variance of . To see how the overall bias of a generic estimator is affected, we use perturbation analysis, defined as

where

Hence, the expected value of the bias is approximated by

When the calibration noise power is used, we can assume that ≈ 0, so the bias induced by the noise power error depends only on Δ*N*. If the radial-based noise power estimate is used, then (13) suggests that the overall estimate bias is affected by both the bias and, to a lesser degree, by the variance of a radial-based noise power estimator (the variance of has less impact than the bias Δ*N* because it appears in the second order of expansion).

To see how the standard deviation of an estimator is affected, we use the Taylor expansion of and the expected value of (11) squared. We get

In (14), SD[*g*(**V**, *N*)] is the standard deviation, assuming the estimator uses the true noise power and the rest of the expression accounts for the standard deviation change due to the bias Δ*N* and the variance of a noise power estimate.

### a. Reflectivity

In the WSR-88D, the reflectivity factor *Z*_{e} is estimated as

where *C* is the constant calculated from the radar equation (Doviak and Zrnić 1993) and *Z*_{e} is usually expressed in decibel units (dB*Z*) as

Using perturbation analysis we have

where *Z*_{e} (dB*Z*) is the true reflectivity factor and

In (18) *S* is the true signal power and *M*_{I} is the equivalent number of independent samples (Doviak and Zrnić 1993),

where *ρ*(*l*) is the autocorrelation coefficient at lag *l*. The previous analysis shows that there are two contributors to the bias of the reflectivity factor estimate (dB*Z*). The first contributor is the bias that depends on the SNR and the spectrum width (i.e., *M*_{I}), and the second contributor is the bias and the variance of the noise power estimate. An example of the bias is given in Fig. 9. The curves were computed using both simulations (shown using lines) and perturbation analysis [shown using symbols and computed from (17) and (18)]. Weather samples were simulated using the approach from Zrnić (1975), and the estimates of *N*_{m} were generated using (6). Results for a hypothetical noise power estimator where both Δ*N* and SD(*N*_{m}) are zero are shown in black. Notice that the term −5/[ln(10)*M*_{I}] in (17) causes a constant negative bias of about −0.4 dB (or −8.8% of the true value) seen at the signal power levels, where the influence of noise can be neglected. Note that this bias is an artifact of expressing the reflectivity factor in reflectivity decibels (dB*Z*). Hence, it does not influence the rainfall-rate estimation, which uses *Z*_{e} expressed in mm^{−6} m^{−3} (Doviak and Zrnić 1993). The case when Δ*N* = 1 dB and SD(*N*_{c}) = 0 dB is indicative of the situation where the inaccurate calibration-supplied noise power measurement is used for reflectivity factor computation. This shows a visible difference compared to the case when Δ*N* and SD(*N*_{m}) are zero, hence demonstrating that the significant error in *N*_{c} can cause additional bias in reflectivity factor at low SNRs. The case when Δ*N* = 0 dB and SD(*N*_{m}) = 0.052 dB is indicative of the instance when the Ivić et al. (2013) estimator is used. The results for these settings overlap entirely with those shown in black [i.e., if Δ*N* = 0 and SD(*N*_{m}) = 0]. Hence, this indicates that the standard deviation of the Ivić et al. (2013) estimator has no adverse influence on the overall bias of reflectivity factor estimates.

Next, we evaluate how the standard deviation of reflectivity factor estimates behaves in the presence of errors in noise power estimates. We use the formula for the standard deviation of reflectivity factor given in Doviak and Zrnić (1993) and modify it to include the effects of noise power measurement errors. Using perturbation analysis we have

Expression (20) demonstrates that the large variance of noise power estimates can have adverse effects on the overall standard deviation of reflectivity factor measurements. An assessment of these effects is shown in Fig. 10. The standard deviation for a hypothetical estimator where noise power is known exactly is shown in black. As in the case of bias, the results when the RBNE technique is used [i.e., Δ*N* = 0 dB and SD(*N*_{m}) = 0.052 dB] overlap entirely with those shown in black. This implies that the standard deviation of the Ivić et al. (2013) estimator has no adverse influence on the overall standard deviation of reflectivity factor estimates. The overall standard deviation for a hypothetical case where the standard deviation of noise estimates is 1.76 dB (or 50% of the true noise power) is shown with gray circles. It demonstrates that the variance of noise power estimates can add to a visible increase in the overall standard deviation if not kept within reasonable limits. The differences between the full black curve and the gray circles in Fig. 10 start to become significant at an SNR of about 10 dB. For instance, at an SNR of 10 dB, the standard deviation of a *Z*_{e} (dB*Z*) estimate increases by 5.7% when SD(*N*_{m}) rises from 0 to 1.76 dB. Using standard values of *C* from the WSR-88D to convert power estimates into reflectivity factors shows that the signals with an SNR of 10 dB translate into reflectivity factor values in excess of 20 dB*Z* 300 km away from the radar. Thus, even a fairly weak signal (susceptible to noise power measurement errors) located far away from the radar can represent a return with significant reflectivity. In that regard, NEXRAD specifications state that “for a true spectrum width of 4 m s^{−1} the standard deviation in the estimate of the reflectivity will be less than or equal to 1 dB*Z* at SNR ≥ 10 dB (including only the error due to meteorological signal fluctuations)” (Baron Services Inc. 2008). Thus, if the standard deviation of a signal with 20-dB*Z* reflectivity factor increases by 5.7%, then it amounts to a standard deviation increase of 1.14 dB*Z*, which clearly exceeds the desired accuracy of 1 dB*Z*. This indicates that the standard deviation in noise power estimates in excess of 1.7 dB can appreciably diminish the quality of the reflectivity factor measurements far away from the radar.

To show how these effects reflect on *Z*_{e} estimates, the histograms of the differences in *Z*_{e}, for data with SNRs larger than 2 dB when *N*_{c} and *N*_{m} are used, are presented in Fig. 11. These show that in all four test cases, the majority of differences are within ±0.1 dB*Z*. In the first two cases, the differences are predominantly negative, which is caused by the WSR-88D calibration noise power overestimation. The differences range between −0.6 and 0.1 dB*Z* but most are situated in the −0.1 and 0 dB*Z* interval. In the last two cases, the differences between *N*_{c} and *N*_{m} are both positive and negative, which is reflected in the similar behavior of the corresponding histograms shown in Figs. 11c,d, where most differences span the ±0.1-dB*Z* interval. The NEXRAD specifications state that the precision of reflectivity measurements should be no less than ±0.5 dB*Z*. Thus, given that the majority of differences are well below this value, it is conjectured herein that no large differences in were present in the analyzed test cases. Nonetheless, if differences presented by histograms in Fig. 11 are compounded with other sources of *Z*_{e} (dB*Z*) estimation errors, they might cause the overall precision to be worse than the desired ±0.5 dB*Z*. For instance, if the inherent constant bias of −0.4 dB*Z* [caused by the term −5/[ln(10)*M*_{I}] in (17) at *σ*_{υ} of 2 m s^{−1}] is added to errors caused the WSR-88D noise calibration, then it might cause an appreciable percentage of *Z*_{e}(dB*Z*) estimates to have precision less than ±0.5 dB*Z*.

### b. Spectrum width

To analyze the influence of the bias and the variance of the noise power estimates on the spectrum width measurements, we use the estimator as given in Doviak and Zrnić (1993):

where

Expression (21) implies that, when , an error in a noise power estimate can introduce additional bias and alter the standard deviation of a spectrum width estimate, and in the event that , the corresponding spectrum width estimate is marked as invalid in the WSR-88D signal processor. Consequently, significant errors in noise power measurements influence the number of valid *σ*_{υ} estimates, so that if Δ*N* > 0, then the number of valid estimates decreases and vice versa. These effects are quantized using simulations in Fig. 12. As in the previous analysis, weather samples were simulated using the approach from Zrnić (1975) and the estimates of *N*_{m} were generated using (6). Additional bias, introduced by the bias in noise power measurements, is computed as the difference between and , where the first denotes spectrum width estimates computed using the noise power estimates and the second using the true noise power. Results presented in Fig. 12a imply that the use of the RBNE technique introduces no additional bias in *σ*_{υ} estimates, whereas significant bias results when Δ*N* is significantly different from zero. Figure 12b shows the ratio of to . Results show this ratio to be one when the RBNE is used, which demonstrates that the Ivić et al. (2013) estimator does not alter the standard deviation of *σ*_{υ} estimates. On the other hand, the ratio is smaller than one when the SNR falls below about 18 dB when Δ*N* is different from zero. This can be explained by the fact that only valid estimates are used in the computation of the standard deviation; so, as Δ*N* increases, the number of valid estimates decreases and only those with the smaller fluctuations are used in the computation, thus artificially decreasing the estimated standard deviation. This is corroborated in Fig. 12c, which shows the ratio of valid estimates when *N*_{m} and *N*_{c} are used versus the case when the true noise power (*N*) is used to compute *σ*_{υ} estimates. When the RBNE technique is used, no difference in the number of valid estimates is observed, but when Δ*N* is not equal to zero, the corresponding ratios in Fig. 12c significantly decrease.

To validate the effects on real data [when estimator (21) is used], we note that in the NEXRAD operational setting, the spectrum width at lower elevations is computed from Doppler scan data usually executed after a surveillance scan. Surveillance scans use a long PRT, which yields long unambiguous range but modest unambiguous velocity due to phase folding. Doppler scans use a short PRT, which yields larger unambiguous velocity but produces echoes that contain sum returns from several trips (i.e., overlaid echoes). Data from the two consecutive scans are combined, where data from the surveillance scan are used for range unfolding. Hence, this is a procedure, used in weather radars, that produces velocity and spectrum width measurements at long ranges from Doppler scans. Signal power measurements are obtained from the long PRT in such a manner that estimated noise power is subtracted from the total computed power. Signal power is set to zero, where subtraction yields a negative value. So, obtained measurements are used to determine ranges at which echoes from several trips are overlaid and whether valid Doppler information can be extracted from estimates at such range locations (i.e., if echoes at such range locations can be resolved). A range location that contains several overlaid echoes is deemed resolved only if the power, obtained from the surveillance scan, of one of the overlaid echoes is larger than the sum of the rest by a user-specified value (usually 5 dB), and the echo is not censored. In such cases, the spectrum width estimate, obtained from the Doppler scan, is assigned to the range location having the echo with the largest power. Range locations having weaker powers are then flagged, as obscured and Doppler data are suppressed at those locations (usually shown as “purple haze” in spectrum width and velocity fields). In Table 2a parameters for the surveillance and the Doppler scans are listed for all four test cases. Also, the change in the number of valid estimates is listed in Tables 2b,c.

The results in Tables 2b,c show that the most notable increase in the number of valid *σ*_{υ} estimates is obtained in the first two test cases, when the RBNE technique is used instead of the legacy-calibration-supplied noise power. This is expected, since the largest difference between *N*_{c} and *N*_{m} is found in these test cases. Fields of *σ*_{υ} for the two cases are given in Fig. 13, and the histograms of differences in *σ*_{υ} estimates for signals with SNRs between 3.5 and 20 dB are given in Fig. 14. In the test case 1 fields, the difference in the number of valid estimates is apparent by a simple visual comparison, as shown by the reduced amount of invalid estimates in Fig. 13b versus Fig. 13a. In the test case 2 fields, the differences are most apparent between 180° and 230° about 200 km from the radar. The histograms quantify differences between valid estimates computed using *N*_{c} and *N*_{m}. They show most differences to be negative, which is an expected consequence of noise power overestimation by the legacy calibration. This demonstrates that the use of the RBNE technique results in significantly more accurate estimates of spectrum width when using the estimator (21).

We note that other estimators of spectrum width use autocorrelation estimates at lags other than zero and are therefore impervious to noise power mismeasurements. However, such estimators require that sufficient information about signal properties is present at lags higher than lag 1. Consequently, they underperform if the normalized spectrum width (Doviak and Zrnić 1993) is large. A good example is the improved hybrid spectrum width estimator proposed by Meymaris et al. (2009) and implemented on the WSR-88D network. It combines spectrum width estimates obtained from (21) with those computed by the estimator using autocorrelation estimates at lags 0, 1, and 2 to determine the amount of signal spread. If it determines that a spectrum width is large, it outputs the result computed by the estimator presented in (21); otherwise, it supplies the result from estimators that use autocorrelation estimates at lags one and higher based on the signal spread classification. This makes it less sensitive to errors in noise power measurements. Nonetheless, that fact that it still uses the estimator (21) for spectrum width classification and to produce results in certain regimes implies that the radial-based noise power estimator will have a positive effect on the performance of this hybrid spectrum width estimator. For this reason [and the fact that many weather radars still use (21) for *σ*_{υ} estimation] the estimator (21) has been chosen herein to analyze the RBNE impact on spectrum width estimation.

## 5. NEXRAD engineering evaluation examples

The radial-based noise estimation technique from Ivić et al. (2013) has been endorsed by the NEXRAD Technical Advisory Committee for an engineering evaluation and has been accepted for operational implementation by the NEXRAD Radar Operations Center (ROC). The technique has been implemented in the WRS-88D signal processor and has been approved after a thorough evaluation by the ROC engineering team (Ice et al. 2013). In the course of the evaluation, numerous data cases were examined, and the radar products obtained using the RBNE technique were compared with those computed using the noise power estimates produced by the legacy calibration. The most indicative results are presented in Figs. 15, 16. Data for the case shown in Fig. 15 are from the site located in Kauai, Hawaii (PHKI), and were collected with a PRT of 3.1 ms, with *M* = 15 at an elevation of 0.5°. The second case shown in Fig. 16 is from the site located in Duluth, Minnesota (KDLH), and was collected with a PRT of 3.1 ms, with *M* = 64 at an elevation of 0.5°. Visual comparison between the reflectivity fields shown in Figs. 16b,c and the zoomed-in fields in Figs. 16d,e reveals an increase in the radar coverage. This is expected, as *N*_{c} is higher than *N*_{m} in both cases. The increase in the coverage is more visible in the second case, mostly because the observed weather system consists of predominantly weaker echoes as opposed to the first case. Investigation by the ROC evaluation team found no potential negative impacts to the system from using the RBNE technique and corroborated the positive visible impacts, such as the increase in radar coverage (especially in case of mountainous sites) and the removal of certain types of interference (e.g., from cell towers). In case of the latter, it was ascertained that the legacy snow algorithm falsely accumulates precipitation from artifacts such as cell tower interference when *N*_{c} is used, while the inclusion of the RBNE technique in the NEXRAD system has the potential to remove those false accumulations. This is demonstrated by the example given in Fig. 17, where the streaks of false accumulations, visible when the legacy calibration is used, disappeared when the RBNE technique supplied the noise power measurements.

## 6. Summary

Errors in noise power measurement are present regardless of whether the system noise power is obtained using a calibration procedure executed in between weather scans or estimated on a radial basis in parallel with weather data collection. The conceptual advantage of the latter is that such an approach captures noise power variations in azimuth, elevation, and time. A possible caveat lies in the fact that radial-based noise power estimation produces estimates from sample sizes that can be significantly smaller than those available to a calibration procedure, especially if the majority of range volumes contain signals. Consequently, if noise power is estimated on a radial basis, then care needs to be taken so that the bias and the standard deviation of such estimates are small enough so that they exert no adverse effect on the estimates of spectral moments. In general, it is desirable that the impact of radial-based noise power estimation be positive compared to legacy calibration procedures.

In this work, the effects of noise power measurement errors and the impact of radial-based noise power estimation on spectral moment estimates were analyzed. The analysis was carried out using the legacy calibration procedure (implemented on the network of WSR-88D radars) and the radial-based noise power estimation technique proposed by Ivić et al. (2013). It was demonstrated that the radial-based noise power estimates are more precise than those produced by the legacy system calibration procedure, which are performed while the radar is not scanning for weather. Two factors contributed to this: the first is the conceptual advantage of radial-based noise power estimation, and the second is that the particular technique, utilized in the analysis, produces noise power estimates with low standard deviation and a small bias that can be neglected for practical purposes. The most apparent positive effects are at lower antenna elevations, where system noise power can differ significantly from that of a blue-sky noise power measurement. The largest variations of system noise power are observed at radar sites located in mountainous regions, where differences in terrain surrounding the radar can create strong azimuthal variations in noise power levels (up to 2.5 dB). In such cases, the WSR-88D legacy system calibration is incapable of supplying noise power measurements that take into account strong azimuthal variations and thus commonly yields results that are significantly higher than the true system noise power. Application of the radial-based noise power estimation in such cases results in visibly improved radar coverage and an increased number of valid spectrum width estimates. To a lesser extent, a radiation emitted by storms also contributed to changes in noise power and the radial-based noise power estimation technique was able to adequately measure those variations. In the test cases where the radiation emitted by storms was the dominant contributor to the noise power variations, the legacy-calibration-supplied noise power was closer to that estimated at each radial. Hence, improvements yielded by the radial-based noise power estimation were less pronounced than in the instances where the ground clutter contributed to noise power intensification. Strong variations in noise power were also observed in the presence of aggressive external interference. In cases of AWGN-like wideband interference affecting all range locations in a radial equally, the radial-based noise power estimation technique measured elevated noise power, which resulted in the removal of interference streaks from meteorological variable fields, thus effectively rendering such interference invisible to radar operators as opposed to situations when the legacy-calibration-supplied noise power was used in computations. As this could be perceived as unusual behavior, such cases can be automatically detected by an abrupt increase in radial-by-radial noise power levels and reported to an operator. At the same time, the accuracy of radar products affected by interference is questionable if the incorrect noise power produced by calibration is used in such cases. Accordingly, the more accurate radial-based noise power estimates significantly enhanced measurements at azimuth locations where strong wideband interference was present.

The effects of erroneous noise power measurements on the reflectivity factor were examined using theoretical analysis, simulations, and real data. Theoretical analysis, supported by simulations, showed that strong biases in noise power estimates can contribute to the overall bias of reflectivity factor measurements. It also indicated that large standard deviations (e.g., in excess of 1.7 dB) in noise power estimates can visibly increase the variability in the estimates of reflectivity factor. Consequently, the variance of a radial-based noise power estimation procedure must be small enough so as not to have negative impacts on data quality. In that regard, it has been shown that the variance of the radial-based noise estimation technique, used in the analysis, imposes no detectable increase on the reflectivity factor standard deviation. Examination of differences between the reflectivity factor estimates computed from real time series using the calibration-supplied noise power and the radial-based noise power estimator, via histograms, showed that the majority were within ±0.1 dB*Z*. Such differences were considerably smaller than the NEXRAD specifications, which require the precision of reflectivity measurements to be within ±0.5 dB*Z*.

In the case of spectrum width, simulations showed that the errors in noise power measurement impact not only the bias and standard deviation but also the number of valid estimates because noise power overestimation results in an increased number of invalid spectrum width estimates. Conversely, if the noise power is underestimated, then the number of valid estimates increases, but the resultant spectrum width estimates are biased high and data appear less coherent. In both cases, data quality is diminished. Two instances of noise power overestimation by the WSR-88D system calibration were examined and the results, derived from the real data, showed a visible increase in the number of valid estimates resulting from the use of the radial-based noise power estimation technique.

The radial-based noise power estimation technique from Ivić et al. (2013) has been implemented on the operational network of WSR-88D radars and had been thoroughly evaluated by the ROC engineering team as part of the implementation procedure. The evaluation corroborated the positive effects of radial-based noise power estimation and found no potential for adverse impacts on the WSR-88D network.

## Acknowledgments

The authors thank Christopher Curtis and Bradley Isom for providing comments that improved 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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## Footnotes

^{1}

A volume scan consists of the radar making 360° scans of the atmosphere while stepping through multiple elevation angles.

^{2}

A radial is a set of data originating from *M* consecutive transmissions, which are used to produce a ray of meteorological variables.