## 1. Introduction

Proper measurement of noise power is of principal importance for the estimation and quality control of weather radar data at moderate-to-low signal-to-noise ratios (SNR). This, in turn, is critical for the correct operation of automated algorithms and accurate forecasts derived from such data. Inaccurate noise power measurements may lead to reduction of coverage in cases where the noise power is overestimated or to radar data images cluttered by speckles of noisy data if the noise power is underestimated. Consequently, accurate noise power measurement is essential for proper operation of quality control techniques in both single- and dual-polarized radars (Melnikov and Zrnić 2004, 2007; Ivić et al. 2009). Moreover, because estimators of reflectivity and spectrum width utilize noise power measurements, inaccurate noise power can introduce bias in these estimates at low-to-moderate SNRs. In the case of dual-polarization radar variables, such as differential reflectivity and correlation coefficient, biases are even more pronounced because these estimators use noise power measurements from both the horizontal and vertical channels, which renders them even more sensitive to mismeasured noise powers. In cases where the unambiguous velocities are in excess of 20 m s^{−1}, lagged estimators (Melnikov 2006; Melnikov and Zrnić 2007) can be employed to circumvent the use of noise powers in the computation of dual-polarization variables. At lower antenna elevations, however, long pulse repetition times (PRTs) are used to avoid range folding of the echoes and these result in unambiguous velocities well below 20 m s^{−1}, which prevents lagged estimators from producing quality estimates.

Typically, the noise power in weather radars can be measured in several ways. For instance, on the Weather Surveillance Radar-1988 Doppler (WSR-88D), the blue-sky noise power is routinely measured as part of the online system calibration performed after each volume scan,^{1} at a high antenna elevation angle. The blue-sky noise power is scaled 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 based on the maximum noise power at each elevation measured during an offline calibration routine. Thus, this approach measures a system noise power value particular to an elevation angle at time intervals that are dictated by the duration of volume scans (4 min or more). Furthermore, some systems do not have the capability to perform online calibrations, and the noise power must be measured offline, which significantly increases time between measurements. Clearly, the downside of these approaches is that they do 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 the time period of less than 3 min (Melnikov 2006). Moreover, they do not account for variations in the external noise sources, which contribute to the noise power variability in azimuth. For example, differences in landscape surrounding the radar can create strong azimuthal variations in noise power levels. Also, heavy precipitation emits radiation over a broad frequency that, when intercepted by the antenna, raises the noise power (Fabry 2001; Melnikov and Zrnić 2004). Finally, external interference sources can create high gradients in noise power levels at affected azimuths. These external interference sources fall into two basic categories. In the first case, interference affects data along the entire radial and its statistical properties are similar to those of white Gaussian noise; thus, noise power is elevated at all positions in range (in some cases more than 10 dB). In the second case, interference influences only certain volumes in range, thus showing as spike-type artifacts in the radial power profile (a.k.a. point clutter). Consequently, the benefit of noise power measurements at each radial^{2} becomes obvious, and the only way such measurements can be performed operationally is in conjunction with data collection. Thus, an efficient approach that estimates noise power from data is desirable.

In the past, several techniques have been proposed. Hildebrand and Sekhon (1974) described a method that subjects the Doppler spectrum coefficients to a series of tests whereby coefficients are recursively discarded until statistical conditions suggest only noise coefficients remain. Urkowitz and Nespor (1992) used the Kolmogorov–Smirnov test applied to the Doppler spectrum by successively discarding the coefficients until the criterion for the noise hypothesis is satisfied. Siggia and Passarelli (2004) used rank order statistics on Doppler spectrum estimates to determine the noise power at each radar resolution volume.^{3} The common thread in these approaches is discarding excess spectral coefficients until the remaining ones satisfy statistical conditions for noise. Inevitably, such an approach introduces biases even if no signal is present, particularly for radar volumes with weather signals that spread over most of the Nyquist cointerval, or if using a small number of samples. Unlike the aforementioned methods that inherently assume signal presence, we approach the system noise power estimation process by discarding those volumes where signal is detected and produce the estimation from the remaining ones for a single radial.

The technique proposed in this work is an extension of the research by Ivić and Torres (2010, 2011), where accuracy and efficiency are improved compared to the previous versions. The technique operates solely on estimated powers (obtained from the raw time series data); thus, it is even applicable to systems that do not provide Doppler measurements. It uses the shape of the power profile in range and estimates of the SNR to search for signal-free volumes. The rationale for such design is briefly discussed in section 2. A detailed algorithm description is given in section 3. In section 4, the technique is evaluated on simulated and real time series data. Results show that the proposed technique produces noise power estimates that are closely matched to the ones obtained from manually identified, signal-free radar volumes at far ranges from the radar, which provides an empirical validation of the technique. The bias and standard deviation of the technique are also examined.

## 2. Design considerations

As mentioned in the previous section, sources of noise in a system are internal and external. We base our approach on the assumption that the internal noise is additive white Gaussian (AWGN) and that most external sources emit wideband radiation that, when intercepted by the antenna, produce signals that have the properties of AWGN, thus constructively adding to the overall noise power in a system. Exceptions are man-made radiators that create either white-noise-like interferences (which affect all volumes in a radial) or signals that exhibit high gradients in range (i.e., usually referred to as pulsed interference or point clutter). The former can be observed in a system as an elevated noise power level at the affected azimuth positions. The latter is usually detected and removed by a specialized electromagnetic interference (or point clutter) filter; consequently, we do not aim to estimate these. The technique presented herein is designed to estimate the power of signals that have properties of AWGN and collectively produce a constant power level along the entire radial. Herein, these are referred to as noise and all others as signals. Also, we assume that the powers the technique operates on are unaffected by signal processing (e.g., if a notch filter is applied to remove ground clutter, then it will also be removing some of the noise); hence, the technique performs best on data not subjected to any prior processing that may violate the AWGN assumption.

^{−1}). This means that, at a large number of volumes in range, signal coherency does not meet the criterion necessary for accurate estimation of the first and second Doppler spectral moments. This criterion is given in Doviak and Zrnić (1993) as

*υ*

_{a}is the unambiguous velocity and

*σ*

_{υ}is the spectrum width. This is especially true at ranges farther away from the radar where the beam becomes larger and encompasses more scatterers, thus producing echoes with larger spectrum widths. Consequently, autocorrelation coefficient measurements are not good indicators of signal presence at ranges far away from radar when long PRT scans are used. At the same time, it is at those ranges that most volumes free of signal are located.

Another potentially useful measurement for finding volumes containing only noise is the Doppler phase (Dixon and Hubbert 2012). Namely, if signal is present, then Doppler phase should exhibit continuity in range as opposed to being uniformly distributed (between ±π) in the noise-only case. Clearly, this holds if the inequality (1) is satisfied. Because of the already stated reasons (narrow Nyquist cointerval and beam broadening in range), this is not typically the case at far ranges when long PRT scans are used. In addition, if the maximum unambiguous velocity is small (as is the case with the long PRT), then the continuity of Doppler phase is further diminished by aliasing. For these reasons, we chose not to use Doppler phase estimates in the algorithm.

## 3. Algorithm description

*M*) in the three cases is 15, 17 and 28, respectively. The test power profiles are shown in Fig. 1, where the power at each volume (or range gate) is obtained as

*V*(

*m*,

*k*),

*m*indexes the sample time, and

*k*denotes the range time (or range gate number). To get a preliminary assessment of the algorithm’s accuracy, a value dubbed as the “expert noise power” (

*N*

_{e}) is introduced. It is obtained by visually identifying a flat section in the power profile that is recognized by “expert” determination to be free of coherent signal. The average power at these range gates (shown in gray in Fig. 1) is taken to be

*N*

_{e}. In general, the algorithm steps are designed to mimic actions of an expert in identifying signal-free volumes in a power profile.

Step 1: *Detect and remove volumes that exhibit sharp power discontinuities in range.*

*k*th volume is removed from further consideration if

*M*), is chosen from a table with entries based on

*M*. It is set so that the probability of falsely detecting noise as point clutter is 10

^{−4}. The mathematics used to calculate the clutter threshold is presented in appendix A. After removing volumes detected as point clutter, the resulting power profiles are presented in Fig. 2. Note that volumes are renumbered after discarding gates that meet the condition in (3). The effectiveness in removing point clutter is most visible in the first case (cf. Figs. 1a and 2a).

Step 2: *Detect flat sections in the range profile of power and estimate the mean power of each section. Take the smallest estimate as the intermediate noise power.*

*k*using a running window of length

*K:*

*k*th volume is considered a signal-free candidate. This approach leverages the fact that the expected value of (4) is small in regions where power fluctuations in range are minimal, which is one of the main properties of regions devoid of signal. When signal is present, this value generally increases. To demonstrate this, time series data containing only noise and both signal and noise with SNR linearly increasing from 0 to 5 dB were generated. The velocities and the spectrum widths of the signal were uniformly distributed between ±8.92 m s

^{−1}, and between 0 and 10 m s

^{−1}. Histograms showing probability density functions (pdfs) of Var

_{dB}(

*k*), for

*K*values of 16 and 32, are presented in Fig. 3. As expected, the mean of each pdf is visibly larger when signals are present as opposed to when they are not. Also, it is informative to note that the amount of overlap between the noise and the signal-plus-noise pdfs is inversely proportional to both the window length (

*K*) and the number of samples (

*M*, not shown here). In general, because this overlap always exists, some gates containing signal are classified as signal free and vice versa. Thus, to successfully implement this step, the threshold and the parameter

*K*need to be chosen properly. The rationale for choosing these is as follows. If the threshold is chosen too high, then too many gates containing signal will be classified as noise. On the other hand, if the threshold is chosen too low, then an excessive number of noise-only samples are discarded, and the resulting noise power estimate may be significantly underestimated. We choose to set this threshold to a value denoted as

*l*var_THR so that

*K*(the mathematics used to obtain values for the threshold is presented in appendix B). Clearly, at this stage gates that contain signal may still be classified as noise only. We remove these in subsequent steps.

When choosing *K*, two things need to be taken into consideration. First, if *K* is too large, then the shorter flat areas in the power profile can easily pass undetected, which can cause the algorithm to fail due to an insufficient amount of data or can result in degraded performance. On the other hand, if *K* is too small, then the overlap between the noise and the signal-plus-noise pdfs can be excessive, making this test a poor detector of regions with constant power. Clearly, a balanced value for *K* is needed. Given that volumes are spaced 250 m in range, this means that a region of noise (or the flat signal section) must be at least 8 km long to be detected when *K* is 32 versus 4 km if *K* is 16. On the other hand, visual inspection of Fig. 3 reveals that a *K* value of 32 ensures less overlap in pdfs, resulting in smaller probabilities of classifying volumes with signal as noise only (i.e., 0.25 for *K* = 32 vs 0.5 for *K* = 16). We assert that 8 km is a small enough span to detect short flat sections while providing sufficient separation between signal and noise; hence, we choose *K* to be 32.

Because variance Var_{dB}(*k*) is estimated from powers in logarithmic units, it depends only on the power gradients and fluctuations within the range interval used to compute it, but not on the mean power level in the range interval; thus, no prior knowledge of noise power is required to execute this step. Consequently, portions of the power profile where strong signals are present can sometimes pass this test if power variations/gradients are small (e.g., stratiform precipitation). Such sections are usually discarded in the subsequent step. Once all volumes are classified, contiguous ones labeled as signal free are grouped into flat sections, and the mean power is computed for each group. Out of these estimates the smallest one is taken to be the intermediate noise power estimate (*N*_{int}). The results of applying step 2 to the test data are shown in Fig. 4. Sections of contiguous volumes classified as flat are shown in different shades of gray. The group with the minimal mean power (used to compute *N*_{int}) is enclosed by a rectangle. Notice in Fig. 4b that regions that clearly contain strong signals have been classified as flat. These, however, are discarded in the following step.

Step 3: *Discard all volumes at range locations where the power estimate is larger than a threshold based on the intermediate power (N*_{int}*).*

*N*

_{int}and the value chosen from a table with entries based on

*M*. Table values are computed using the following formula (Ivić et al. 2009):

*N*is the noise power, Γ

_{inc}is the incomplete gamma function, and PFA is the probability of false alarm. Given the value of SNR_THR, (6) computes the probability that the power estimate in (2) exceeds that value when samples

*V*(

*m*,

*k*) are independent (e.g., white noise). Hence, for a known noise power

*N*, (6) produces the probability at which noise volumes are falsely mistaken for signal when power-based signal censoring is applied. Given

*M*and the desired PFA, we can compute the SNR_THR using (6) for the unit value of

*N*. For an arbitrary mean power

*N*, the value can simply be multiplied by

*N*to produce the threshold value that yields the desired PFA. In this particular case,

*N*

_{int}is used in place of

*N*, and the SNR_THR for each value of

*M*is computed to produce a PFA of 10

^{−3}. This step discards most of the volumes containing strong signals. The resulting profile after the third step can be seen in Fig. 5. After this step, the ratios of the mean power to the expert noise powers for the three test cases are −0.21, 0.039, and 0.243 dB, respectively.

Step 4: *Apply a “range persistence” filter that detects and labels 10 or more consecutive volumes in range with powers larger than the median.*

The fourth step applies a range persistence filter that detects and labels 10 or more consecutive volumes in range with powers larger than the median. The rationale for choosing 10 consecutive volumes in range is as follows. The probability that any power sample is larger than the median is 0.5; hence, the probability that 10 randomly chosen independent samples are larger than the median is 0.5^{10} = 9.76 × 10^{−4}. Consequently, this filter detects volumes with larger powers (evidence of signal-like returns) that exhibit some continuity in range while retaining those in predominantly noise areas. Range locations that are labeled by the range persistence filter are highlighted in Fig. 5. Notice that no locations are marked in Fig. 5b.

Step 5: *Discard volumes labeled by the range persistence filter and compute the mean power from the remaining data.*

Step 6: *Discard all volumes at range locations where the power estimate is larger than a threshold based on the mean noise power from step 5.*

Next, we discard volumes labeled by the range persistence filter and compute the mean power from the remaining data. Step 5 is followed by another power-based censoring as in step 3. The only difference is that, instead of *N*_{int}, the mean power from step 5 is used to compute the censoring threshold. Thus, in step 6 we discard all volumes at range locations where the power estimate is larger than a threshold based on the mean noise power from step 5. The ratios of the mean power to the expert noise power after this step are −0.24, 0.038, and 0.16 dB for the three cases, respectively.

Step 7: *Remove remaining weak signals by applying running sums on the volumes in range and discard data until the remaining powers meet the criterion for noise.*

In the final step, we remove remaining weak signals by applying running sums on the volumes in range and discard data until the remaining powers meet the criterion for noise. A running sum is performed over the array of remaining powers to make weak signals detectable. The assumption is that the remaining volumes with signals are contiguous in range. Thus, by applying a running sum with *W*, regions where signals are still present “emerge” above those that contain only noise. These regions can be detected by comparing each running sum to a threshold. If it exceeds the threshold, then all volumes used to compute the sum are discarded. In this implementation, we choose to base the threshold on the mean power. However, if the mean power obtained in step 6 is still well above the true noise power, then a single application of the running sum does not guarantee removal of all signals. Consequently, we choose to make this step iterative, which compels us to devise a statistical criterion to determine when the remaining data contain only noise. If such a criterion is not met, then all volumes used in the calculation of running sums larger than the threshold are discarded and the new mean power is computed. The new running sums are computed over the remaining volumes and the statistical criterion is checked again. The process ends when the remaining volumes meet the requirement for noise or until a maximum number of iterations (e.g., 10) is reached.

*M*, the length of

*W*is chosen by rounding the ratio 500/

*M*to the nearest integer. In case the number of remaining volumes (i.e., volumes not discarded by previous steps as contaminated by signals) times

*M*is less than a predetermined minimum number of samples (800 in our implementation), the algorithm exits with the “no estimate” fail code. Moreover, this criterion is applied after each iteration in the final step. In most cases, however, estimates are produced from a significantly larger number of samples than the minimum, resulting in a rather small global estimate variance (Fig. 11). The statistical criterion for noise is met if

*L*is the number of volumes in an iteration,

*N*is the mean power, and Γ

_{inc}is the incomplete gamma function. Each running sum is computed as

After the first iteration, running sums are shown for all three test cases in Fig. 6. Running sums that are larger than the threshold are highlighted along with all points to the left and right that are larger than the mean. These are all considered to be “contaminated” with signal if the criterion in (7) is not met. Consequently, all volumes used to compute these running sums are discarded. The rationale for devising such a criterion is explained in detail in appendix C. In all three test cases, the noise criterion was met after volumes marked in the first iteration had been discarded; thus, the algorithm terminated after the first iteration. For the first test case (*M* = 15), the final noise power estimate came out 0.3 dB below the expert noise power. This can be explained by the fact that this test case was inundated by the point clutter (Fig. 1a), which impaired reliable visual determination of the flat section from which the expert noise power was extracted. In the other two cases, when *M* was 17 and 28, noise powers came out 0.014 and 0.0043 dB below the expert noise powers, respectively. This is very close to the expert noise powers, so the differences may be attributed to statistical variations. Finally, volumes determined by the algorithm to contain only noise are shown in Fig. 7.

## 4. Accuracy assessment

In this section, we will subject the proposed noise estimation technique to several tests in order to assess its accuracy. Because the algorithm works by sequentially discarding data, it inevitably introduces a negative bias if the input is pure white noise; thus, we apply it to simulated white noise samples to determine whether this bias is significant. The results in Fig. 8 show that the algorithm introduces a negligible negative bias compared to the plain power estimate [computed using (2)].

Next, we evaluate the performance on real time series data by comparing the noise power estimates to the expert noise powers computed from the radar volumes at far ranges from the radar where the absence of signal is visually determined; hence, we also refer to this noise as far range. Note that this approach of computing the expert noise is sensitive to contamination by weak signals or pulsed interference not visible in the reflectivity image. Nonetheless, a reasonable agreement between the two can serve as a confidence measure of the validity of the technique in various data cases. The first test case is from the KPDT radar site in Pendleton, Oregon. Data were collected with a PRT of 3.1 ms, *M* = 15, and at an elevation of 0.87 °. Figure 9a shows the reflectivity field obtained using the noise power value produced by the Next Generation Weather Radar (NEXRAD) calibration procedure (one value for all azimuths). No censoring has been applied, so noise variations with azimuth are apparent. Visual inspection of the reflectivity field in Fig. 9a reveals no significant returns beyond 350 km; thus, the expert noise power for each radial direction is produced by computing the mean power from all volumes at ranges beyond 350 km. Volumes classified as containing signal are presented in Fig. 9b. Measured versus expert (i.e., far range) noise powers are plotted in Figs. 9c and 9d for H and V channels. Notice that they overlap to such a degree that it is difficult to visually differentiate between them. It is interesting to note that noise power variations in azimuth are in excess of 1 dB. This is due to the mountainous terrain on one side of the radar site and plains on the other. In addition, a prominent peak in the noise power is visible in the horizontal channel at about 110°. It is most likely caused by an unknown interference source. Hence, this presents a compelling example where radial-based noise power estimation is important for proper product estimation at low-to-moderate SNRs.

The next example is from the KVNX radar site near Vance Air Force Base, Oklahoma. Data were collected with a PRT of 3.1 ms with *M* = 17 at an elevation of 0.53°. As in the previous example, the measured and the expert noise powers in Figs. 10c and 10d overlap for the most part (i.e., as in the previous case, it is difficult to visually distinguish between the two). Unlike the previous example, there is a noticeable difference of about 0.5 dB in the overall noise power levels between the two channels. In the H channel, the only noticeable difference between the measured and the far range noise is the slight deficit (~0.25 dB) in the measured noise located at ~310°. In this particular case, the histogram of powers at range locations determined by the algorithm to contain only noise significantly departs from the expected Gaussian curve (compared to other azimuth locations). This is likely caused by the presence of weak interference at this azimuth. The spike in the V channel differences at 350° is due to artifacts in the data beyond 350 km. The noise power estimator recognizes those as signals and removes them as opposed to the simple averaging that produces the expert noise powers. The same outcome causes several other moderate spikes in the expert noise powers shown in Fig. 10d.

The next test consists of subtracting the noise power estimates at each radial and setting powers to zero at all volumes classified as noise only. Then, simulated white noise (with known power) is added to the data (the details of this operation are given in appendix D). Thus, this test can be used to assess the technique’s accuracy, precision, and its ability to distinguish between gates with noise and gates with weak signal and noise. In total, data from 20 volume coverage patterns (VCP) (U.S. Department of Commerce 2006) were processed in this manner to obtain the statistical properties of the estimator. This produced 159 490 realizations from which the histogram in Fig. 11 is obtained. Note that the algorithm failed to produce an estimate 40 times (0.025%). These were the cases in which radials were so inundated with signal that the algorithm could not locate a sufficient number of signal-free volumes to produce a reliable noise power estimate. The number of cases in which the algorithm failed would have been significantly higher had we chosen to process all radials regardless of the PRT used to collect the data. Instead, whenever an azimuth position was scanned using two PRTs, at the same elevation, the data collected by the longer one was used to compute the noise power. Generally, cases when the noise power estimator fails to produce a result, at an antenna position, can be handled by using a calibration noise power or the latest noise power measurement from the same or the closest antenna position. The histogram shows the noise power estimator to be very accurate, as the tiny negative bias of 0.004 dB can be neglected, and the standard deviation is only 0.052 dB (or 1.2% of the true noise power). The statistics show that 86% of the noise power estimates fall within the standard deviation bounds (i.e., ±0.052 dB).

## 5. Discussion

In this work, we propose a novel technique that produces more accurate radial-specific and range-independent additive white Gaussian noise power measurements. These radial-specific system noise power measurements impact the weather radar coverage, as well as the accuracy of the radar products. Furthermore, the case for accurate noise power measurement is strengthened by the fact that at lower antenna elevations, external noise contributors add the most to the system noise power budget and can produce significant noise power variations in azimuth. The customary approach to noise power measurement is to compute the blue-sky noise power while the radar is not transmitting (e.g., in between volume scans), and to adjust this value to account for external noise contributors at lower antenna elevations. Unfortunately, such an approach does not account for the noise power variations in azimuth. Alternatively, a noise power map could be created in weather-free situations and used when weather is present. This would account for noise power azimuth variations due to stationary external noise contributors (e.g., ground clutter), but it neglects momentary noise power increases from heavy precipitation. In addition, neither of the listed approaches has the potential to entirely capture temporal variations of noise power caused by changes in the system gain. Improved accuracy in measuring the noise power translates in improved performance of nonsignificant-data thresholding. This is especially true in mountainous regions, where the noise power is typically overestimated. It also benefits the estimators of radar products that are based on estimates of signal power (i.e., spectrum width, differential reflectivity, and correlation coefficient). Although alternative estimators can be used that are based on correlations at lag 1 and therefore immune to inaccurate noise power estimates, they require sufficient correlation in sample time in order to produce quality estimates, making them incompatible with the long PRTs used at the lowest elevation angles. Hence, it is imperative to have the noise power measured at each antenna position while scanning for weather in order to produce accurate radar products at low-to-moderate SNRs.

The proposed approach estimates the noise power from raw power estimates computed using samples collected during weather scans; hence, it does not interfere with the existing scanning strategies. It is robust and feasible for real-time implementation on operational weather radars. The technique uses a set of criteria to detect radar volumes that do not contain significant signals and calculates the system noise power from samples at these locations. Because it does not use the Doppler spectrum, this method is not limited by the spectral extent of weather signals or the small number of samples, which is usually the case when scanning using long PRTs. The technique consists of seven steps, where each is designed to gradually remove volumes that contain signals. It begins by first discarding volumes containing strong signals and continues with the removal of weak ones. Each step is computationally simple, so it can be easily and efficiently programmed into the radar signal processor.

The technique relies on having a sufficient number of volumes free of signal from which noise power estimates can be computed. This may not always be the case, as storms can span the entire unambiguous range or range folding can cause all range volumes in a radial to contain signals when short PRTs are used at low-to-medium antenna elevations, thus causing the algorithm to fail. Nonetheless, this should not be an issue because, in such situations, a noise power measurement at a given antenna position is usually available from the long PRT scan. In general, if the algorithm fails, then either the latest noise power measurement from the same (or the closest) antenna position or the calibration noise power can be used instead.

The performance of the technique was assessed on real data by comparison to the mean power of data volumes located far away from the radar (i.e., where no visible signal was present in the test cases). This test showed remarkable agreement between the two power measurements. Furthermore, bias and standard deviation were estimated using combinations of real signals and simulated noise inputs, which yielded known noise powers on a large dataset sufficient for a reliable statistical assessment. This showed that the technique produces noise power estimates with low variance and a small bias that can be neglected for practical purposes. The first means that, on average, the noise power estimator accurately reflects variations in noise power levels as the antenna points to different directions. The second ensures that the use of the measured noise power in radar product estimation does not noticeably increase their variance.

The technique has been applied to numerous test cases to validate its behavior in a variety of situations that can occur in operations. These tests have proven the technique’s ability to accurately produce noise power estimates. The largest noise power variations were observed in mountainous regions (up to 2.5 dB) and from external interference. To a lesser extent, radiation emitted by storms also contributed to noise power increase. In the case of aggressive external interference affecting the entire radial, the technique measured elevated noise power. As this could be perceived as unusual behavior, such cases can be automatically detected by the abrupt increase in radial-by-radial noise levels and reported to an operator. This technique 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. It is expected that the technique will significantly improve the quality of products on the network of WSR-88D radars.

## Acknowledgments

The authors thank Alan Free of the Radar Operations Center and David Warde of the National Severe Storms Laboratory for providing the radar data used to validate the technique. We also thank Valery Melnikov and David Warde 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.

## APPENDIX A

### Threshold Calculation for the High Gradient Echo Detection

*N*is the mean power. Moreover, because the first two terms in (A1) have equal value, we can write

*Mx*/

*N*=

*z*, we get

*My*/

*N*=

*t,*we arrive at

_{inc}(

*x*,

*α*) is the incomplete gamma function, defined as

^{−4}in case of white noise and given the number of samples

*M*, we need to determine the value of the threshold PCT(

*M*). From the right-hand side in (A5), it is obvious that the value of PCT(

*M*) cannot be found directly. Therefore, Newton’s iterative method (Ypma 1995) is applied. In this method, an initial guess reasonably close to the root of a function is selected. Then, the function is approximated by its tangent line (which can be computed using the tools of calculus), and the

*x*intercept of this tangent line is computed easily using elementary algebra. This

*x*intercept will typically be a better approximation to the function’s root than the original guess, and the method can be iterated until the result is found with desired accuracy. An update formula is given by

*p*.

_{n}## APPENDIX B

### Threshold Calculation for the Flat Section Detection

_{dB}(

*k*) when all samples are white noise. Because Var

_{dB}(

*k*) is a function of numerous random variables, the analytical derivation of the pdf using variable transformation becomes intractable. We approach this problem by attempting to match the pdf shapes in Fig. 3 to a known distribution. Visual inspection reveals that these shapes are similar to that of the gamma distribution,

_{dB}(

*k*) estimator, we can produce the pdf model to be used in the search. We start by deriving the first two moments as

*N*is the mean noise power. Then,

Next, we check how well the gamma distribution models the pdf of Var_{dB}(*k*) by comparing the third and the fourth moments of Var_{dB}(*k*) obtained using Monte Carlo simulations (Zrnić 1975) and those computed using the gamma distribution. The comparison shows that the gamma distribution matches the actual distribution up to the fourth moment. This means that the skewness and the kurtosis of the gamma approximation matches that of the Var_{dB}(*k*) distribution, making it an excellent choice for an efficient computation of the threshold that satisfies (5).

*L*is the number of trials and

*M*) using a gamma distribution approximation and arrange those into a table so they can be easily retrieved during real-time processing.

## APPENDIX C

### Running Sum Window Length and Threshold Derivation

*l*is computed as

*l*takes values from 0 up to

*L*−

*W*− 1,

*L*is the length of the dataset, and

*W*is the length of the running sum window. From the last term in (C1), we can see that the pdf of a running sum if all volumes used in the sum contain only white noise with power

*N*is (Ivić et al. 2009)

*W*and the value of the threshold, which in turn determine the expected value of the ratio. To do this, we generate the running sum pdf for both noise and signal plus noise using uniformly distributed SNRs and spectrum widths in the range of −3 to 2 dB, and 0–10 m s

^{−1}, respectively. This is demonstrated in the case of unit noise for

*M*of 15, and lengths of

*W*of 17 and 33. The bias is the prime concern in designing this step; thus, we want to choose the threshold so that no significant negative bias is introduced when operating on pure white noise data. We can easily achieve this by placing the threshold in the tail of the noise pdf function. The problem arises if the pdf of signal plus noise overlaps too much with that of the pure noise, so that a considerable portion of it is on the left side of the threshold; this causes a positive bias in the noise power estimate. Clearly, we need a balanced approach, in the sense that we want to lower the threshold as much as possible but not cause a negative bias in the estimates when data are pure noise. It was empirically found that no significant bias is introduced if the threshold is set, so that the expected value of the ratio in (C4) is few in a thousand. Next, we need to choose the length of

*W*. When

*W*is 17, we set the THR to yield a probability of 3.6 × 10

^{−3}that the running sum with pure noise exceeds the threshold. At the same time, 8 out of 100 signal-contaminated running sum values fall below the threshold, on average, and are classified as noise. On the other hand, when

*W*is 33, this happens only 2 times out of 1000 if we set the THR, so that the probability of the pure noise causing a running sum to exceed the threshold is 4.5 × 10

^{−3}. Clearly, setting

*W*to 33 results in much better detection of signal-contaminated running sum points. Setting the THR value to 37, we write the expected value of the ratio as

*M*is 15 and

*W*is 33, the total number of power estimates producing each running sum point is 495. Hence, to obtain

*W*, and the THR value, for any number of samples that produce comparable results, we round this to 500 and compute the window length for any

*M*by rounding 500/

*M*to the nearest integer. Since 37/33 ≈ 1.12, we get

^{−3}to 7.5 × 10

^{−3}for

*M*that spans from 3 to 200. Another approach is to set the expected value of the ratio to a fixed value (e.g., 4 × 10

^{−3}) and find the corresponding threshold value for each

*M*. For simplicity, however, we choose to implement the former approach.

## APPENDIX D

### Description of the Procedure for Creating a Power Profile with Known Noise Power

*k*to obtain the estimated signal power

*L*. Then the censoring vector

**V**, of the same length as the power data, is created where each position is set as

*V*(

*k*) being one with only noise present (i.e., no signal) is 10

^{−4}. Then, vector

**V**is updated as

^{−8}probability of falsely detecting noise as signal. The minimum power,

*S*

_{min}, is found next out of the volumes that correspond to ones in vector

**V**. A simulated noise power vector with mean power 3 times larger than

*S*

_{min}is created as

*A*(

*m*,

*k*) and

*B*(

*m*,

*k*) are Gaussian-distributed random variables with zero mean and unit variance, and

*j*is the square root of −1. A new power profile is created as

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^{1}

A volume scan consists of the radar making multiple 360° scans of the atmosphere, where elevation angles are gradually increasing.

^{2}

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

^{3}

In the interest of briefness, radar resolution volume is referred to as volume in the remainder of the text.