## Abstract

The Gaussian model adaptive processing in the time domain (GMAP-TD) method for ground clutter suppression and signal spectral moment estimation for weather radars is presented. The technique transforms the clutter component of a weather radar return signal to noise. Additionally, an interpolation procedure has been developed to recover the portion of weather echoes that overlap clutter. It is shown that GMAP-TD improves the performance over the GMAP algorithm that operates in the frequency domain using both signal simulations and experimental observations. Furthermore, GMAP-TD can be directly extended for use with a staggered pulse repetition time (PRT) waveform. A detailed evaluation of GMAP-TD performance and comparison against the GMAP are done using simulated radar data and observations from the Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) radar using uniform and staggered PRT waveform schemes.

## 1. Introduction

Ground clutter suppression is critically important for improving the radar data quality of any weather radar system. If not removed, the clutter may produce strongly biased estimates of the fundamental spectral moments such as mean power, mean Doppler velocity, and spectrum width. It is desirable to have a ground clutter filter providing good clutter suppression while minimizing the radar parameter estimation errors, namely, standard deviation and bias. In most cases, ground clutter signals have narrow spectrum width located at zero Doppler frequency. One important aspect of clutter handling is identifying if there is clutter in a region. This is handled in the clutter mitigation decision (CMD) step, which has found great use in practical implementation (Hubbert et al. 2009).

In the most simple transmission scheme where radar pulses are transmitted and received with uniform spacing, ground clutter can be filtered by using infinite impulse response (IIR) or finite impulse response (FIR) filters. These are high-pass filters having sharp narrow notches characterized by their type, rejection depth, and the notch width. In many applications, these filters are sufficient. However, when the signal and clutter overlap in the frequency domain as in the case of weather radar, the use of these filters removes clutter but also notches out a part of the signal and causes bias in the estimates. To mitigate this problem, advance filtering methods have been developed. Gaussian model adaptive processing (GMAP) (Siggia and Passarelli 2004) is a frequency domain method that not only filters out clutter points but also attempts to recover precipitation components that have been removed. However, the limitation of this or any spectral filtering method is the effect of spectral leakage caused by finite data length on the estimates of signal moments. In the GMAP algorithm, when clutter is strong, an aggressive time domain weighting function (Hamming or Blackman) is applied to reduce the spillover of power from the clutter into the weather spectrum via the clutter sidelobes (Siggia and Passarelli 2004). The use of such window functions reduces the energy of the received signal, and therefore reduces the information contained in the signal, causing a higher standard deviation of the signal parameters' estimates. Nguyen et al. (2008) proposed a parametric time domain method (PTDM) for clutter mitigation and precipitation spectral moment estimation. PTDM allows estimating clutter and precipitation echoes' spectral moments simultaneously. It is shown that this method leads to more accurate results than that of the clutter spectral filtering techniques. However, PTDM is computationally intensive, which currently prevents it from being implemented for real-time operation.

With the nonuniform pulsing techniques used for range–velocity ambiguity mitigation, such as staggered pulse repetition time (PRT) (Skolnik 2001), the clutter filtering problem becomes a challenge. Because of nonuniform sampling, standard clutter filters (IIR and FIR) cannot be applied directly to the staggered PRT sequences. Also, GMAP does not work directly with staggered PRT data, since it requires a Fourier transform that cannot be used with a nonuniform sampling sequence. Cho and Chornoboy (2005) have defined a finite impulse response time-varying filter that can be applied to staggered PRT sequences. Torres et al. (1998) proposed a family of regression filters that operates by applying the regression polynomials to the time series. In 2000, a spectral deconvolution algorithm (Sachidananda and Zrnić 2000) was introduced to solve the problem of clutter mitigation for staggered PRT sequences. However, those techniques suffer from large velocity errors in certain Doppler frequency bands. Moisseev et al. (2008) have extended PTDM (Nguyen et al. 2008) to the staggered PRT sampling case. It is shown that PTDM is able to mitigate ground clutter as well as accurately estimate signal spectrum moments even in the case of a very high clutter-to-signal ratio (CSR). The results are as good as the case of uniform sampling, since the covariance matrix model in PTDM adapts to the sampling scheme. Again, the drawback of PTDM is its computational load, which limits real-time implementation in the today's general purpose processors. Meymaris et al. (2009) discussed a technique that applies the GMAP algorithm to staggered PRT sequences. The principle of their technique is that two separate equispaced sequences that correspond to the odd and even samples of the staggered time series are filtered using GMAP. The resultant sequences are then combined together to create the staggered PRT sequence where parameters can be estimated in a standard fashion. This method is able to take the advantage of GMAP but loses the moment recovery accuracy because of the decreasing of Doppler interval.

In this paper, we present the GMAP in the time domain (GMAP-TD) method for clutter mitigation and spectral moment estimation for weather radars. The GMAP-TD algorithm operates similarly to the original GMAP but does so in the time domain. The proposed algorithm addresses two important things. First, it overcomes the disadvantages of the original GMAP filter as described above. Second, GMAP-TD can be applied to the staggered PRT technique used for range–velocity ambiguity mitigation. The computational requirement of GMAP-TD is reasonable, and it can be implemented for real-time application using current general purpose processors.

This paper is organized as follows. The GMAP-TD method, including filter design and interpolation technique, is described in section 2. In section 3, a technique to enable the application of GMAP-TD to staggered PRT data is introduced. Section 4 discusses the evaluation of the method based on signal simulation with various input parameters. Also in this section, a comparison between estimation errors by GMAP-TD and GMAP is given for the uniform sampling case. In section 5, Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) radar data in both uniform and staggered PRT transmission scheme are used to illustrate the performance of GMAP-TD and are compared against GMAP. The last section summarizes the important results of this paper.

## 2. GMAP-TD

### a. Signal model

For meteorological targets, the returned signal is the sum of the backscatter from individual hydrometeors in a radar pulse volume. Precipitation particles have widely different scattering amplitudes and move with different velocities relative to the radar. The distribution of the scatters' radial velocity can be approximated by a Gaussian distribution with a mean velocity and spectrum width. The ground clutter Doppler spectrum is also approximated by a Gaussian shape (Doviak and Zrnić 1993). Using those assumptions, one can write the Doppler spectrum of the radar signal of weather and clutter as follows:

where *υ* is Doppler velocity; , , , and are the Doppler spectrum of the received signal, precipitation, clutter, and noise, respectively; is the precipitation power, is the precipitation mean velocity, is the precipitation spectrum width, is the clutter power, and is the clutter width. Here we assume that the received radar signal is the summation of signals coming from ground clutter, precipitation, and white Gaussian noise that are independent. Given the Doppler spectral representation [Eq. (1)], the autocovariance functions for each component of the radar signal can be calculated using the inverse Fourier transform (Bringi and Chandrasekar 2001) as

where *R* denotes the autocovariance function, is the temporal variable, is wavelength, and is the noise power. The subscripts , , and denote precipitation, clutter, and noise, respectively.

Let be the complex time sample data; its autocovariance matrix can be written as

where , , and are autocovariance matrices of ground clutter, precipitation, and noise, respectively. In the case of uniform sampling with the pulse repetition interval of , the signal autocovariance matrix is expressed (Bringi and Chandrasekar 2001) as

For the staggered PRT transmission technique where the pulse repetition time alternates between two pulse spacings and , the signal autocovariance matrix is given by

Consider a time-varying filter of dimension ; the autocovariance matrix of the filter output can be written as

where is the expectation operator and superscript H denotes a complex conjugate transpose.

The first term in the sum (6) is the response of the filter to signal from ground clutter and noise. In the case of no weather echo in the received signal (i.e., ), we want the output signal to have the autocovariance matrix similar to that of a white noise process,

In other words, the filter attenuates the clutter signal to a level comparable to the noise level, and does not whiten the clutter signal. The result is that the noise level of the output signal is the same as that of the input signal. It is an important property of this filtering process and therefore, we do not have to reestimate the noise power and do not have to redesign the filter matrix for each step in the interpolation loop (see section 2c). The Eq. (7) is termed the *filter equation*.

### b. Clutter transformation matrix

Estimates of the first three signal spectral moments—that is, the mean power, mean velocity, and spectrum width—depend solely on the autocovariance function of the filtered signal. It should also be noted that the autocovariance function only depends on the signal power spectral density via the inverse Fourier transform. In other words, since the phase response of the filter is not involved in the transformation of the power spectral density of the signal, it has no effect on the autocovariance function of the filter output and consequently, the phase response of the filter does not affect the signal spectral moment estimates. Therefore, in the design of the GMAP-TD filter, we can ignore the phase response of filter . It is relatively easy to prove that for any choice of matrix satisfying the filter Eq. (7), the power spectral density of the filter output is the same. For simplicity, in this work the clutter filter is chosen as

where (−) power denotes the matrix square root. The solution (8) always exists, since the autocovariance matrix is a Hermitian positive semidefinite matrix, as is the matrix . It is straightforward to verify that filter satisfies the filter Eq. (7).

The clutter filter shown in Eq. (8) is an adaptive filter. It is a function of the clutter power, clutter spectrum width, and noise power [Eqs. (2) and (8)] at a given range gate that is generally unknown. To design filter , these parameters must be estimated from the time series data. Methods to estimate clutter power () and signal noise power () for uniform sampling data are fairly straightforward and are described in the literature (e.g., Siggia and Passarelli 2004). The clutter spectrum width can be chosen from predetermined values based on the antenna beamwidth and scan rate (Doviak and Zrnić 1993). For instance, for CSU–CHILL S-band radar, the ground clutter width is approximate 0.2 m s^{−1} at a scan rate of 8° s^{−1}.

The frequency response of a time invariant filter can be defined by the Fourier transform of its impulse response. This definition cannot be applied to a general linear filter such as filter (8). However, a power frequency response at frequency *w* can be defined as the power of the output signal when a complex exponential signal is passed through the system. Figure 1 demonstrates the power frequency response of filter for the uniform pulse scheme. The depth of the notch at zero in Fig. 1 depends on the relative ratio between clutter power and noise power (in this case, 60 dB). When applying matrix to the time series data, the ground clutter spectrum will be transformed to a “flat” spectrum. The width of this transformed spectrum depends on the clutter spectrum width () and the clutter power (), and is much narrower than the signal Doppler range. The amplitude of this transformed spectrum depends on the ratio [Eqs. (2) and (8)], and if the clutter and noise power are estimated correctly, it equals the received signal noise power. In practice, the variance in clutter and noise power estimates may cause this level to vary within a few decibels. However, the width of the clutter-transformed spectrum is relatively small (a few tenths of a meter per second) and this error does not significantly affect the GMAP-TD performance.

### c. Signal interpolation procedure

In the second term of Eq. (6), one can see that the autocovariance of the precipitation signal is modified. This is because part of the precipitation echo overlapping the clutter is also transformed to noise. To mitigate this problem, an interpolation procedure is developed to recover the transformed part of the weather echo. The main goal of this procedure is to ensure that the signal autocovariance matrix has a Gaussian form (as assumed in section 2a). After applying the clutter filter , this property may not hold. By finding the difference between a Gaussian-fitted model before and after filtering and then compensating for the transformed part of the covariance function via a loop, it is possible to recover the lost portion of the signal. Figure 2 describes this concept. First, the covariance matrix is used as the initial input of the iteration, where is the sample autocorrelation matrix of the signal where clutter has been transformed to noise. It is similar to the spectral coefficients of weather echo after discarding clutter and noise coefficients in GMAP. Second, signal spectral moments [mean power (), mean velocity (), and spectrum width ()] are estimated from the filtered covariance matrix . It should be noted that the *k*th diagonal of the covariance matrix comes from samples of the autocovariance function at lag *k*. As the signal parameters are estimated, a Gaussian model for precipitation is constructed using Eqs. (2) and (4) [or (5) for the staggered PRT case]. Next, the difference of the model before and after applying filter is computed and used to update the filtered covariance matrix . This procedure is repeated until the power difference and/or the velocity difference between two consecutive iterations is/are less than certain predetermined thresholds. This can be considered as a cost function optimization. Tests conducted on radar data showed that thresholds of 0.1 dB and 0.5% of the Nyquist velocity provide good performance, and that the interpolation procedure will converge after a few steps. The final result will yield a signal after mitigating ground clutter.

## 3. GMAP-TD filter for staggered PRT

The staggered PRT scheme (Skolnik 2001) is used to resolve the range–velocity ambiguity problem and is described in Fig. 3. In this scheme, the pulse spacing alters between two pulse intervals and , and the maximum unambiguous velocity is defined by () (Moisseev et al. 2008). Generally, and are chosen as multiples of a certain unit time, . It is shown that for the pulse-pair estimator, the optimal PRT ratio *T*_{1}/*T*_{2} is (Zrnić and Mahapatra 1985). Theoretically, GMAP-TD can be directly applied to staggered PRT data. However, there are some practical considerations that must be addressed to make GMAP-TD work well in this scheme. In this section, the application of GMAP-TD to a pulse ratio (staggered PRT ) is studied.

The first consideration is clutter and dynamic noise power estimates for staggered PRT. In the uniform sampling case, GMAP-TD uses the method described in Siggia and Passarelli (2004). This method estimates clutter power and dynamic noise power from signal spectrum points. However, in the case of staggered PRT, a signal spectrum is not available or at least cannot be obtained directly from the time series data. Here, the staggered PRT time series is zero interpolated to form a uniform sequence with a sampling rate of . The spectrum of this zero-interpolated data includes some replicas of clutter and weather echoes at certain frequencies (Bringi and Chandrasekar 2001). However, the clutter echo is still located at zero frequency, and the method used by Siggia and Passarelli (2004) can be applied to this data to obtain clutter and noise power. It should be noted that GMAP-TD uses the signal power spectrum to calculate only clutter and dynamic noise powers.

The second consideration is estimation accuracy. Clutter filters that have been designed for staggered PRT sequences (Cho and Chornoboy 2005) suffer from unwanted notches in certain Doppler frequency bands. These unwanted notches introduce errors in signal estimates that limit the application of these algorithms in practice. The top panel in Fig. 4 shows the power response of the GMAP-TD filter for the staggered PRT waveform. It can be seen that there are unwanted notch centers at velocities and (Fig. 4b). When the weather echo is present in the unwanted notches regions, part of the signal will be affected and estimation errors will occur. Because part of the signal is removed by the unwanted notch, its power will be underestimated. In section 2c, we have explained how the GMAP-TD interpolation procedure (Fig. 2) recovers the portion of the signal removed by the main notch of the filter. Similar discussion can be applied to the unwanted notches of the filter. Hence, this bias error can be mitigated by the interpolation loop in GMAP-TD. In addition to the power bias, GMAP-TD velocity estimates show an increase in standard deviation that is similar to the analysis in Sachidananda and Zrnić (2000). As we observe in the case of staggered PRT , at those specific cases, estimated velocities can be off by or , which is equal to the intervals between unwanted notches. In general, it has the form of , where for staggered PRT . This error occurs because the signal and its spectral replicas fall into unwanted notches with different depths and the estimator may pick up a replica signal, instead of the correct one. To mitigate this problem, we modify filter to , such that filter will have equal notch depths at the frequencies and . The modified filter is shown as follows:

where and are frequency-shifted versions of . That is, the frequency response of has the main notch at and the main notch of is at :

Matrices and are formed based on Eq. (5) and their elements are defined as

The power frequency response of the resulting filter is shown in Fig. 4c. It should be noted that the depth of the notches in the modified filter are greater than that of the filter at zero frequency because placing additional notches has an impact on the notch at zero frequency. The modified filter is only used to remove the potential offset in the estimated velocity, while the rest of the algorithm uses the original filter [Eq. (8)]. The performance of this technique is shown in the next section using radar simulation. Beside these modifications, GMAP-TD processing for a staggered PRT sequence is similar to the case of uniform data and it would provide a similar performance when applied to the uniform sample case.

## 4. Performance of GMAP-TD system using signal simulation

Extensive analysis of the performance of GMAP-TD has been carried out with radar signal simulation. The simulation procedure follows the work of Chandrasekar et al. (1986) with various input parameters. To evaluate GMAP-TD, we implemented the GMAP algorithm (Siggia and Passarelli 2004) to compare side by side against GMAP-TD for the case of uniform sampling. The implementation of GMAP is extensively tested, and it is demonstrated that it can provide similar results to that reported in Ice et al. (2004). In the present work, where the clutter suppression performance is emphasized, attention is devoted to the cases of moderate CSR () and very high CSR (). In GMAP-TD and GMAP, dynamic noise power estimation is used. The input parameters for simulation are given in Table 1.

### a. Uniform sampling case

For a uniform pulsing scheme with pulse repetition time , the maximum unambiguous velocity is (Bringi and Chandrasekar 2001). In this case, GMAP-TD performance will be directly compared to GMAP. In the first scenario, where the clutter is moderate (), both GMAP-TD and GMAP perform well. It is noted that in GMAP, a Blackman window is used (Siggia and Passarelli 2004). Figure 5 summarizes the performance of GMAP-TD and presents a comparison against GMAP. In Fig. 5, GMAP-TD results are shown in solid lines, while dashed lines represent GMAP. The first row in Fig. 5 shows the errors for power estimates. For large velocity (), GMAP-TD and GMAP perform equally in term of power biases. Both perform well as power biases are within 0.5 dB, even for this high CSR. However, when the signal locates close to clutter (), the difference in power bias is more obvious between the two methods. At small velocity, where clutter strongly overlaps precipitation echo in the GMAP method, the clutter spectrum was broadened because of the impact of the Blackman window used in GMAP; therefore, more signal will be removed compared to the GMAP-TD method. This explains why GMAP-TD has lower power biases compared to that of GMAP for small velocities. For velocity and spectrum width estimates, GMAP-TD biases are less than 0.5 m s^{−1} at all Doppler ranges for the case of CSR = 40 dB.

The left column in Fig. 5 shows a comparison in standard deviations of the three spectral moment estimates. Clearly, GMAP-TD shows an improvement over the GMAP in the standard deviation of the estimates in all parameters analyzed here. The higher standard deviation in the spectral-based method can be explained by the effect of the data window applied to the time series—in this case, the Blackman window (Siggia and Passarelli 2004). On the other hand, GMAP-TD uses time domain processing and avoids the use of any data window. The standard deviations in power and velocity estimates of GMAP-TD are about 0.5 dB and 0.5 m s^{−1} less than that of GMAP, respectively.

For the second scenario, both the GMAP-TD and GMAP methods are tested with simulated data with very strong clutter contamination and fairly weak precipitation echo strength ( and ). The results are summarized in Fig. 6. In GMAP, a Blackman window was applied to the data. Because the peak sidelobe level of a Blackman window is −57 dB (Oppenheim and Schafer 2009), the clutter sidelobe is only 2 dB below the clutter-to-signal level; thus, the spectral leakage from clutter is significantly strong when compared to the weather echo. Applying a Gaussian curve to fit three central components of clutter (step 3 in Siggia and Passarelli 2004) will not completely remove these leakage points because they are outside the Gaussian curve. These clutter spectral leakage signals will add to the signal spectrum. This explains the overestimation in power estimate and the underestimation in velocity estimate by GMAP (Fig. 6). The GMAP performance is compromised in such extreme scenarios. In contrast, GMAP-TD does not use a Blackman window; therefore, it does not suffer from its effect. The limitation of spectral filtering techniques caused by the effect of spectral leakage has been addressed with GMAP-TD.

In summary, for uniform sampling, GMAP-TD biases were shown to be as good or better than that of GMAP. In addition, by avoiding the use of data window functions, GMAP-TD provides lower standard deviations in all signal parameter estimates. It performs well even in cases of very strong clutter contamination (e.g., CSR is as high as 55 dB) where GMAP does not perform favorably.

### b. Application to staggered PRT 2/3 sampling case

The other major advantage of GMAP-TD over GMAP is that it can be directly extended to nonuniform sampling schemes such as the staggered PRT . For staggered PRT techniques, velocity estimate is the most challenging process (Sachidananda and Zrnić 2000). For that reason, in this case simulation was carried out with a signal spectrum width of 4 m s^{−1}. The input parameters are and , and the staggered PRT sequence has a length of 64 samples. Figure 7 shows a scatterplot for velocity estimates using GMAP-TD. One hundred realizations comprise the simulations. It is shown that the estimated velocities are very close to the true values and there is almost no outlier. At the Doppler regions where the replicas of ground clutter occur (Sachidananda and Zrnić 2000), there is no increase in the estimation bias. More detailed analysis of GMAP-TD performance in this scenario is given in Fig. 8. In addition to the GMAP-TD results, pulse pair (PP) estimates for simulated staggered PRT data with the same input parameters but without clutter are also shown. These are considered to be the baselines and will be used to gauge GMAP-TD performance in this analysis. As shown in Fig. 8, in most cases, GMAP-TD power and velocity biases are very close to pulse-pair performance in the nonclutter environment. Moreover, the standard deviations of velocity and spectrum width estimates are better with GMAP-TD (Figs. 8e and 8f). This can be explained by the use of the interpolation loop in GMAP-TD (Fig. 2). Basically, the loop updates the Gaussian model of the signal at each step and GMAP-TD outputs are obtained when the loop converges (after several iterations). The convergence helps reducing the variation in the estimated values. On the other hand, the PP method computes signal velocity and spectrum width directly from lags 1 and 2 of the signal autocovariance function.

## 5. Implementation and demonstration using CSU–CHILL radar data

The performance of the GMAP-TD filter is also verified using CSU–CHILL radar measurements using two sampling schemes collected on 20 December 2006. The first dataset considered a staggered PRT pulsing scheme taken at 2358:19 UTC with and . A minute later, a uniform sampling dataset was recorded at 2359:20 UTC with . The small time difference between the two observations ensures a reasonable comparison of the results for the two schemes. With the data setup, the uniform sampling data has an unambiguous velocity of ±27.5 m s^{−1}, while the staggered PRT would attain double the range, that is, from −25 to 55 m s^{−1}. The data were collected during a snowstorm event, where the signal-to-noise ratio was often less than 20 dB and spectrum widths larger than 4 m s^{−1} were observed in many regions. In addition, at 1.0° of elevation, observations came from the Rocky Mountain region west of the radar that showed a large amount of strong clutter. To gauge the GMAP-TD performance, GMAP was applied to the uniformly sampled dataset. Results are shown in Fig. 9. One can observe that GMAP-TD removed ground clutter fairly well in both the uniform sampled and staggered PRT datasets, especially at ranges less than 20 km and at the Rocky Mountain regions. Differentiating between original and filtered power plots shows a clutter suppression ratio up to 50 dB for staggered PRT data from the CHILL radar. Reflectivity estimates from GMAP-TD and GMAP are comparable (first column in Fig. 9). At a range of 60 km and an azimuth of 270° where strong clutter due to the mountains is present, GMAP-TD shows less variation in reflectivity estimates than does GMAP. At the ranges from 80 to 100 km and the azimuth angles from 240° to 270°, reflectivity estimates from the staggered PRT data by GMAP-TD show some clutter residues. This can be attributed to the power saturation of the CHILL radar when working in staggered PRT mode. The uniform pulsing data show velocity folding at the northeastern quadrant (inside the highlighted ellipse). This means the precipitation mean velocity is over the maximum unambiguous velocity of the uniform sampling scheme (27.5 m s^{−1}). The color scale was selected from −32 to 32 m s^{−1} for both uniform and staggered sampling data for comparison while still showing the velocity fields in full detail. The middle plan position indicator (PPI) plot in the last row (Fig. 9) shows that the velocity folding problem was solved by applying GMAP-TD to the staggered PRT observations. In this case, GMAP-TD provides correct velocity estimates without any folded velocity points.

The third column in Fig. 9 shows the estimated signal spectrum width from uniform and staggered PRT modes. One can see that the results from uniform sampling data using GMAP and GMAP-TD and from staggered PRT data using GMAP-TD are comparable. In this dataset, the signal spectrum width varied from 1 to 7 m s^{−1}. Good results from GMAP-TD for both uniform and staggered PRT data validate the performance of the GMAP-TD method.

Although the estimated fields from uniform PRT observations, obtained using GMAP-TD and GMAP appear to be similar, there are differences that can be shown by a quantitative comparison. In Fig. 10, reflectivity and velocity profiles at the azimuth angles of 120° and 280° from Figs. 9b and 9c are plotted. Figure 10 shows that GMAP-TD performs better than GMAP. The main improvement we expect is the smaller variance in the estimates of reflectivity and velocity. One way to see this is by observing the fluctuation of the profiles. At the azimuth angle of 120° (Figs. 10b and 10d) where no ground clutter is present, the GMAP-TD and GMAP results are almost the same. At the azimuth angle of 280° (Figs. 10a and 10c), for the ranges from 40 to 60 km where strong ground clutter due to the mountains is present, GMAP-TD estimates (gray lines) show less variation than GMAP estimates (black lines). It is consistent with the inference from simulation (Fig. 5).

## 6. Summary

In this paper, a new method for ground clutter filtering, GMAP-TD, was presented. Similar to the GMAP algorithm, GMAP-TD assumes that the radar signal consists of ground clutter and weather echoes whose spectral shapes are approximately Gaussian. GMAP-TD adopts many advanced techniques used in GMAP such as adaptive clutter power estimation and interpolation procedure. By using adaptive clutter power estimate, GMAP-TD does very little or nothing to the signal in cases of no clutter. This eliminates the requirement of applying a clutter map, which often requires regular updates. When the signal and clutter overlap, the interpolation loop enables GMAP-TD to recover the part of the signal that was filtered out.

The primary difference between the two methods is that the GMAP algorithm uses the signal power spectrum, while GMAP-TD works with a signal covariance matrix in the time domain. It has several advantages over the GMAP method. First, because there is no need for any data window in GMAP-TD, lower standard deviations in signal parameters estimates are observed. Second, since GMAP-TD does not use the signal power spectrum, it is not affected by spectral leakage due to ground clutter. GMAP-TD was shown to perform well for the cases where signal and clutter strongly overlap. This property allows GMAP-TD to work well in scenarios of very high clutter contamination () and weak signals where GMAP does not. And last but most important, GMAP-TD can be directly applied to staggered PRT data with some very simple modifications. GMAP-TD performance for the staggered PRT case was shown to be as good as in the case of uniform sampling.

Based on the experience with CSU–CHILL implementation, GMAP-TD computational complexity is similar to that of GMAP. Normally, it converges after a few iteration loops. Our first attempt at implementing GMAP-TD for weather radars shows that, we process a ray of 300 range gates within a dwell time of 40 ms. The processing was split into many parallel threads and run in a server with two Intel Xeon Quad Core E5530 (2.4 GHz) processors and 16 GB of RAM.

In summary, GMAP-TD is shown to be a promising method for ground clutter filtering for weather radar systems. It can be adapted for uniform sampling as well as staggered PRT observations. GMAP-TD is able to retrieve a weak signal in severe clutter contamination.

## Acknowledgments

This work is supported by the National Science Foundation (Grant ATM-0735110).