Compound-Gaussian Clutter Model with Weibull-Distributed Textures and Parameter Estimation

Abstract

Compound-Gaussian models (CGMs) are widely used to characterize sea clutter. Various types of texture distributions have been developed so that the CGMs can cover sea clutter in different conditions. In this paper, the Weibull distributions are used to model textures of sea clutter, and the CGM with Weibull-distributed textures is used to derive the CGWB distributions, a new type of biparametric distribution. Like the classic K-distributions and Compound-Gaussian with lognormal texture (CGLN) distributions, the biparametric CGWB distributions without analytical expressions can be represented by the closed-form improper integral. Further, the properties of the CGWB distributions are investigated, and four moment-based estimators using sample moments, fractional-order sample moments, and generalized sample moments are given to estimate the parameters of the CGWB distributions. Their performance is compared by simulated clutter data. Moreover, measured sea clutter data are used to examine the suitability of the CGWB distributions. The results show that the CGWB distributions can provide the best goodness-of-the-fit for low-resolution sea clutter data as alternatives to the classic K-distributions.

1. Introduction

Subtle knowledge and cognition of characteristics of sea clutter form the foundation of performance evaluation and optimization and system design of maritime radars [1]. Typically, the type and parameters of the amplitude/intensity distribution of sea clutter determine the structure of the optimum/sub-optimum coherent detection and decision threshold to control false alarm rate [2]. The statistics of sea clutter in microwave-band radars are affected by too many factors, including sea states, radar parameters, and the viewing geometry of radar [3]. The diversity of sea clutter in statistics promotes the increasing development in modeling amplitude distributions. Early modeling of sea clutter is based on the statistical analysis of measured data. Rayleigh distributions, the biparametric lognormal distributions, and Weibull distributions are used to model sea clutter, ground clutter, and weather clutter [4,5,6], which basically meet the demands in non-coherent detection design in radars.

In view of the composite scattering mechanism of ocean surface [7], the compound-Gaussian model (CGM) represents sea clutter as the product of a fast-varying complex Gaussian speckle and a slowly varying texture. In a CGM, texture distribution determines the amplitude/power statistics and non-Gaussianity of sea clutter, and the temporal correlation of speckle sequence determines the Doppler spectral characteristics of sea clutter. The selection of texture distributions becomes the key to model amplitude distributions of sea clutter. Owing to the tractability in mathematics, the biparametric texture distributions have been developed in succession, from the classic K-distributions with gamma textures [1,8], the generalized Pareto (GP) intensity distributions with inverse gamma textures [9], the compound-Gaussian with inverse Gaussian textures (CGIG) distributions [10], the compound-Gaussian with lognormal texture (CGLN) distributions [11], to the recent compound-Gaussian with Nakagami texture (CGNG) distributions [12]. In sea clutter modeling, the wide use of the biparametric distributions benefits from the analytical optimum/near-optimum/suboptimum coherent detectors in these models [13,14,15,16,17]. In addition, the compound inverted exponentiated Rayleigh texture (CIER) distributions are also useful in sea clutter data [18]. Therefore, having more types of distributions is advantageous for modeling sea clutter under various conditions.

In addition to these biparametric distributions, two types of tri-parametric distributions are also available. The tri-parametric compound-Gaussian with generalized inverse Gaussian texture (CG-GIG) distributions include the K-distributions, GP distributions, and CGIG distributions as special examples [19]. The wide coverage avoids the troublesome type-selection of amplitude distributions in practical applications. Particularly, the analytical near-optimum coherent detectors in the CG-GIG distributions have been constructed [20]. Before the CG-GIG distributions, the generalized K (GK) distributions with the generalized gamma textures have been used to characterize sea clutter [21,22] for a long time. The modified GK distributions have recently been used to characterize airborne sea clutter data [23]. The absence of coherent analytical detectors and difficult parameter estimation makes the GK distributions overlooked in the long term.

The tri-parametric generalized gamma distributions include the biparametric gamma distributions, Weibull distributions [3], and Nakagami distributions as subclasses [24]. In other words, the tri-parametric GK distributions include the K-distributions and CGNG distributions [12] as the two biparametric subclasses. This paper focuses on the investigation of the sea clutter CGM with Weibull-distributed textures, a new biparametric subclass of the GK distributions. When the textures follow the Weibull distributions, the corresponding amplitude distributions are referred to as the CGWB distributions. Firstly, with the help of abundant results on the Weibull distributions, the properties of the CGWB distributions are analyzed in detail. Secondly, as a new type of biparametric amplitude distribution, the parameter estimators of the CGWB distributions using sample moments, fractional-order sample moments, and generalized sample moments are derived. Moreover, their performance comparison is performed by using simulated clutter data. Thirdly, experiments using measured sea clutter data are made to show the complementary of the CGWB distributions with the other types of biparametric distributions in sea clutter modeling. It is found that the CGWB distributions can provide better goodness-of-fit for low-resolution sea clutter data at small grazing angles than other types of biparametric distributions. The CGWB distributions enrich the family of biparametric amplitude distributions of sea clutter and provide an alternative other than the K-distributions in modeling low-resolution sea clutter data.

This paper is organized as follows. In Section 2, the biparametric CGWB distributions are presented, and the properties of the CGWB distributions are analyzed in detail. Section 3 gives several moment-based parameter estimators of the CGWB distributions and compares their performance by using simulated clutter data. In Section 4, low-resolution measured sea clutter data at small grazing angles are used to verify the validation of the CGWB distributions in the modeling of sea clutter. It is shown that the CGWB distributions can provide better goodness-of-the-fit to low-resolution sea clutter. Finally, we conclude our paper in Section 5.

2. Biparametric CGWB Distributions for Sea Clutter Modeling

The introduction of the tri-parametric distribution brings convenience to amplitude modeling but presents challenges in parameter estimation and analytical coherent detection methods [19]. Here, the biparametric CGWB distributions, as a biparametric subclass of the tri-parametric GK distributions [21], can provide an alternative to characterize sea clutter. The CGWB distributions are complementary to the other types of biparametric distributions.

Like the tri-parametric CG-GIG distribution, the GK distribution with the generalized Gamma distribution textures can be used in modeling sea clutter. The textures in the GK distributions follow the tri-parametric generalized Gamma distributions:

$$p\left(\tau ;\mu ,\upsilon ,\eta \right)={\displaystyle \frac{\eta {\upsilon }^{\upsilon }}{\mathrm{\Gamma }\left(\upsilon \right)\mu }}{\left({\displaystyle \frac{\tau }{\mu }}\right)}^{\upsilon \eta -1}\exp \left[-\upsilon {\left({\displaystyle \frac{\tau }{\mu }}\right)}^{\eta }\right],\tau >0;\mu ,\upsilon ,\eta \in \left(0,+\infty \right)$$

(1)

where μ stands for the scale parameter, and υ and η are two shape parameters that characterize the thickness of the tail of the probability density function (PDF). When τ→∞, its attenuation satisfies:

$$p\left(\tau ;\mu ,\upsilon ,\eta \right)\propto \exp \left[-\upsilon {\left({\displaystyle \frac{\tau }{\mu }}\right)}^{\eta }\right],\tau \to \infty ;\mu ,\upsilon ,\eta \in \left(0,+\infty \right)$$

(2)

which implies that the PDFs of the textures have power-law decay, giving a flexible decay mode and enhancing the variability of the tails of the PDFs.

By varying the two shape parameters, the generalized gamma distribution encompasses a range of distributions with one or two parameters as subclasses [24], where the biparametric distributions include the gamma distribution (η = 1), the Weibull distribution (υ = 1) and the Nakagami distribution (η = 2) [12]. According to (1), the Weibull distribution can be obtained when υ = 1:

$$p\left(\tau ;\mu ,\eta \right)={\displaystyle \frac{\eta }{\mu }}{\left({\displaystyle \frac{\tau }{\mu }}\right)}^{\eta -1}\exp \left[-{\left({\displaystyle \frac{\tau }{\mu }}\right)}^{\eta }\right],\tau >0;\mu ,\eta \in \left(0,+\infty \right)$$

(3)

where μ and η are the scale and the shape parameters of the Weibull distribution, respectively. Considering that the scale parameter μ does not equal the mean of the texture, a new scale parameter is adopted as b = μΓ(1 + 1/η), where Γ(·) represents the gamma function. In the compound-Gaussian model, when the texture follows the Weibull distributions, the amplitude z of the complex clutter:

$$x=\sqrt{\tau }u,z=\left|{\tau }^{1/2}u\right|,\tau \sim p(\tau ;b,\eta ),u\sim ℂ\mathbb{N}(0,1)$$

(4)

follows the biparametric CGWB distributions:

$$\begin{array}{c}f\left(z;b,\eta \right)={\displaystyle \frac{2\eta {\left[\mathrm{\Gamma }\left(1+1/\eta \right)\right]}^{\eta }z}{b^{\eta }}}{\displaystyle {\int }_0^{+\infty }{\tau }^{\eta -2}}\exp \left\{-{\displaystyle \frac{z^2}{\tau }}-{\left[{\displaystyle \frac{\tau \mathrm{\Gamma }\left(1+1/\eta \right)}{b}}\right]}^{\eta }\right\}d\tau \\ ={\displaystyle \frac{2\mathrm{\Gamma }\left(1/\eta \right)}{\sqrt{b}}}{\displaystyle \frac{z}{\sqrt{b}}}{\displaystyle {\int }_0^{+\infty }{\tau }^{\eta -2}}\exp \left\{-\left[{\displaystyle \frac{\mathrm{\Gamma }\left(1+1/\eta \right)}{\tau }}{\left({\displaystyle \frac{z}{\sqrt{b}}}\right)}^2+{\tau }^{\eta }\right]\right\}d\tau ,z>0;b,\eta \in \left(0,+\infty \right)\end{array}$$

(5)

where z stands for the amplitude of the complex clutter and the texture τ and the complex Gaussian speckle u are mutually independent. It is easy to prove that the CGWB approaches the Rayleigh distribution, and the complex clutter approaches complex Gaussian clutter when the shape parameter η tends to infinity through a similar method [12]. Moreover, the shape parameter η reflects the non-Gaussianity of the clutter, and a smaller η corresponds to a stronger non-Gaussianity of the clutter and heavier tail of the amplitude distribution. The cumulative distribution function (CDF) of the amplitude of the CGWB-distributed clutter can be expressed after the integration over the amplitude:

$$\begin{array}{c}F\left(z;b,\eta \right)=1-{\displaystyle \frac{\eta {\left[\mathrm{\Gamma }\left(1+1/\eta \right)\right]}^{\eta }}{b^{\eta }}}{\displaystyle {\int }_0^{+\infty }{\tau }^{\eta -1}}\exp \left\{-{\displaystyle \frac{z^2}{\tau }}-{\left[{\displaystyle \frac{\tau \mathrm{\Gamma }\left(1+1/\eta \right)}{b}}\right]}^{\eta }\right\}d\tau \\ =1-\eta {\displaystyle {\int }_0^{+\infty }{\tau }^{\eta -1}}\exp \left\{-\left[{\displaystyle \frac{\mathrm{\Gamma }\left(1+1/\eta \right)}{\tau }}{\left({\displaystyle \frac{z}{\sqrt{b}}}\right)}^2+{\tau }^{\eta }\right]\right\}d\tau ,z>0;b,\eta \in \left(0,+\infty \right)\end{array}$$

(6)

It can be observed that both the PDF and CDF of the CGWB distribution are involved in improper integrals with parameter variables. Such a phenomenon is common in the biparametric subclasses of the GK distributions [21]. The K-distributions use the well-known modified Bessel functions of the second order, which have been thoroughly investigated. Therefore, the K-distributions are easy to handle in terms of computation and properties. However, the situation is quite different for the CGWB and CGNG distributions. Their computations and properties all involve improper integrals with parameter variables, which lack sufficient investigation in mathematics. As a result, it is a difficult task to implement fast computation, develop parameter estimation, and construct the corresponding near-optimal coherent detector of the CGWB distributions.

In what follows, we examine the difference between the CGWB distributions and Weibull distributions even though they are all biparametric distributions. As shown in Figure 1, the PDFs of the CGWB distributions with the scale parameter b = 1 and the shape parameter η = 0.2, 0.5, 1, 1.5, 2, and 5 in the linear scale and logarithmic scale. Wholly, with the increase of the shape parameter, the non-Gaussianity of the CGWB distribution becomes weaker and its tail becomes lighter. As the shape parameter tends to infinity, it approaches the Rayleigh distribution whose PDF is plotted in the black solid line for comparison. Furthermore, similar to the K-distributions and the CGNG distributions [12], the CGWB distributions exhibit a process that varies from monotonically decreasing functions to unimodal functions when the shape parameter alters from zero to infinity. Based on numerical computation, the CGWB distributions are monotonically decreasing functions when the shape parameter η is less than 0.528 and are unimodal functions when the shape parameter is greater than 0.528. As a comparison, Weibull distributions with the scale parameter b = 1 and the shape parameter η = 0.2, 0.5, 1, and 2 are also plotted in Figure 1 by colored dotted lines. It can be observed that the CGWB distributions and Weibull distributions are quite different. In the Weibull distributions, the shape parameter η takes values in the interval (0, 2]. Weibull distributions with η = 2 correspond to the Rayleigh distributions. When η > 2, the Weibull amplitude distributions have faster-decaying tails than the Rayleigh distributions, indicating that the clutter is sub-Gaussian [25]. It is known that non-Gaussian clutter hardly appears in radars. It can be seen from Figure 1b that the tails of the Weibull distributions fast vary when the shape parameter alters in the interval (0, 2]. Note that the Weibull distribution is a monotonically decreasing function when η ≤ 1, and an unimodal function when η > 1. This characteristic seems to be inherited by the CGWB distributions. In contrast, the CGWB distributions have a stronger ability to fit the tails of the amplitude distributions of data compared to the Weibull distributions.

Give x > 0, the xth moment of the Weibull texture distribution satisfies the following:

$$m_x\left(\tau ;b,\eta \right)={\displaystyle \frac{b^x\mathrm{\Gamma }\left(1+x/\eta \right)}{{\left[\mathrm{\Gamma }\left(1+1/\eta \right)\right]}^x}},x>0$$

(7)

In consideration of the independence of the texture and speckle components, the moment of the amplitude of the CGWB distribution can be easily derived, where the envelope of the complex Gaussian speckle is a Rayleigh distributed random variable with σ = 1, the original moments of the CGWB-distributed random variable satisfy [26]:

$$m_x\left(z;b,\eta \right)=m_{x/2}\left(\tau ;b,\eta \right)m_x\left(\left|u\right|\right)=b^{x/2}\mathrm{\Gamma }\left(1+{\displaystyle \frac{x}{2\eta }}\right)\mathrm{\Gamma }\left(1+{\displaystyle \frac{x}{2}}\right){\left[\mathrm{\Gamma }\left(1+{\displaystyle \frac{1}{\eta }}\right)\right]}^{-x/2},x>0$$

(8)

Further, the mean, variance, coefficient of variation (CV), skewness (skew), and kurtosis (kurt) of the amplitude z of the compound-Gaussian clutter with Weibull texture can be given analytically as follows:

$$\begin{array}{l}mean\left(z;b,\eta \right)={\displaystyle \frac{\sqrt{\pi b}\mathrm{\Gamma }\left(1/\left(2\eta \right)\right)}{4\sqrt{\eta \mathrm{\Gamma }\left(1/\eta \right)}}},\mathrm{var}\left(z;b,\eta \right)=b\left[1-{\displaystyle \frac{\pi }{16}}{\displaystyle \frac{{\mathrm{\Gamma }}^2\left(1/\left(2\eta \right)\right)}{\eta \mathrm{\Gamma }\left(1/\eta \right)}}\right],\\ \mathrm{CV}\left(z;\eta \right)={\displaystyle \frac{\sqrt{\mathrm{var}(z;b,\eta )}}{mean(z,b,\eta )}}=\sqrt{{\displaystyle \frac{16\eta \mathrm{\Gamma }\left(1/\eta \right)}{\pi {\mathrm{\Gamma }}^2\left(1/\left(2\eta \right)\right)}}-1},\\ skew\left(z;\eta \right)={\displaystyle \frac{3\sqrt{\pi }}{4}}{\displaystyle \frac{\left\{\frac{3}{2}{\eta }^{\frac{1}{2}}\mathrm{\Gamma }\left(\frac{3}{2\eta }\right)-{\eta }^{-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2\eta }\right)\mathrm{\Gamma }\left(\frac{1}{\eta }\right)+\frac{\pi }{24}{\eta }^{-\frac{3}{2}}{\mathrm{\Gamma }}^3\left(\frac{1}{2\eta }\right)\right\}}{{\left[\mathrm{\Gamma }\left(\frac{1}{\eta }\right)\left(1-\frac{\pi }{16}\frac{{\mathrm{\Gamma }}^2\left(1/2\eta \right)}{\eta \mathrm{\Gamma }\left(1/\eta \right)}\right)\right]}^{3/2}}}\\ kurt\left(z;\eta \right)={\displaystyle \frac{4\eta \mathrm{\Gamma }\left(\frac{2}{\eta }\right)-\frac{9\pi }{8}\mathrm{\Gamma }\left(\frac{1}{2\eta }\right)\mathrm{\Gamma }\left(\frac{3}{2\eta }\right)-\frac{3{\pi }^2}{256{\eta }^2}{\mathrm{\Gamma }}^4\left(\frac{1}{2\eta }\right)+\frac{3\pi }{8\eta }{\mathrm{\Gamma }}^2\left(\frac{1}{2\eta }\right)\mathrm{\Gamma }\left(\frac{1}{\eta }\right)}{{\mathrm{\Gamma }}^2\left(\frac{1}{\eta }\right){\left[1-\frac{\pi }{16}\frac{{\mathrm{\Gamma }}^2\left(1/2\eta \right)}{\eta \mathrm{\Gamma }\left(1/\eta \right)}\right]}^2}}\end{array}$$

(9)

The amplitude mean and the intensity mean/power of the clutter are determined by the scale and shape parameters. In contrast, the CV, skewness, and kurtosis, the quantities to determine the shape of the amplitude distribution, are fully determined by the shape parameter η. To examine the change of these numerical characteristics of the amplitude with the shape parameter, Figure 2 plots their curves when the scale parameter b is fixed at 1 and the shape parameter η ranges from 0.1 to 30. As can be observed, when the shape parameter becomes larger, the mean first fast increases and then becomes almost a constant, and the variance first fast decreases and then becomes almost a constant. The CV, skewness, and kurtosis describe the dispersion, asymmetry, and thickness of the tail of the amplitude PDFs, respectively. When the shape parameter increases, the CV, skewness, and kurtosis first fast decrease and then get almost constant. Additionally, the kurtosis of the CGWB distribution tends to be the same as that of the Rayleigh distribution, which is approximately 3.2451, given by (32 − 3π²)/(4 − π)². The fact that the five curves tend to be constants implies that the CGWB distribution fast approaches the Rayleigh distribution as the shape parameter becomes larger. Therefore, the CGWB clutter can be processed as complex Gaussian clutter as the shape parameter is slightly large. To determine this empirical value of the shape parameter of the CGWB distribution to be able to be processed as Gaussian clutter, the Kolmogorov-Smirnov distance (KSD) [27] and the Kullback-Leibler divergence (KLD) [28] between the CGWB distribution and the Rayleigh distribution are calculated and plotted in Figure 3. For comparison, the other four types of distributions are demonstrated in Figure 3. The four types exhibit a similar trend: the clutter can be regarded as Gaussian when the shape parameter is larger than 20. As depicted in Figure 3, for the four biparametric distributions, the corresponding KSDs and KLDs are provided when shape parameters equal 20, where the KSDs and KLDs are approximately equal to 0.01 and 0.001, respectively. Regarding the CGWB distribution, the shape parameter is larger than 5.86 when the KSD is less than 0.01, and larger than 6.11 when the KLD is smaller than 0.001. It can be observed that when the KSD and KLD of the CGWB distribution reach the same values as the other four types of distributions, the shape parameter is much smaller than 20. The diversity that makes the CGWB distributions strongly complementary to the other four types of distributions.

3. Moment-Based and [zlog(z)]-Based Parameter Estimation of CGWB Distributions

As a new family of biparametric distributions in the amplitude modeling of sea clutter, the parameter estimation of the CGWB distributions is important for the applications of the CGWB distributions in practice. In terms of the origin moments in (8), the joint of the original moments can derive different moment-based estimators. By using the second- and fourth-order moments, a moment-based estimator (MoM) is given as follows:

$$\begin{array}{l}{\Phi }_{24}(\widehat{\eta })\equiv {\displaystyle \frac{\widehat{\eta } \mathrm{\Gamma }\left(2/\widehat{\eta }\right)}{{\mathrm{\Gamma }}^2\left(1/\widehat{\eta }\right)}}={\displaystyle \frac{{\widehat{m}}_4\left(z\right)}{4{\widehat{m}}_2^2\left(z\right)}},\widehat{b}={\widehat{m}}_2\left(z\right),\\ {\widehat{m}}_x(z)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{z}_i^x},x>0\end{array}$$

(10)

where ${\widehat{m}}_x(z)$ is the x-order sample moment of the amplitude data {z₁, z₂, …, z_N}. In this estimator, the scale parameter is estimated by the second-order sample moment. The shape parameter is estimated by solving the equation on $\widehat{\eta }$. It can be proved that the function Φ₂₄(·) in (10) is a monotonically decreasing function of $\widehat{\eta }$. The proof is given in Appendix A. Therefore, its inverse function exists, though it fails to be given analytically. The lookup table method can be used to estimate the shape parameter by the following:

$$\widehat{\eta }={\Phi }_{24}^{-1}\left({\displaystyle \frac{{\widehat{m}}_4\left(z\right)}{4{\widehat{m}}_2^2\left(z\right)}}\right)$$

(11)

Note that the usage of high-order sample moments involves large variances under spiky clutter, and the variances can be reduced by employing lower-order and fractional-order sample moments. For example, similar to [26], the moment-based estimator using the first-, second-, and third-order amplitude moments, namely, the method of low-order moments (MoLM) is given as follows:

$${\Phi }_{123}\left(\widehat{\eta }\right)\equiv {\displaystyle \frac{\widehat{\eta } \mathrm{\Gamma }\left(3/2\widehat{\eta }\right)}{\mathrm{\Gamma }\left(1/\widehat{\eta }\right)\mathrm{\Gamma }\left(1/2\widehat{\eta }\right)}}={\displaystyle \frac{2{\widehat{m}}_3\left(z\right)}{9{\widehat{m}}_1\left(z\right){\widehat{m}}_2\left(z\right)}},\widehat{b}={\widehat{m}}_2\left(z\right),$$

(12)

and the method of fractional-order moments (MoFM) estimators through the first and the 1/2-order sample moments are given by the following:

$${\Phi }_{1-1/2}\left(\widehat{\eta }\right)\equiv {\displaystyle \frac{\widehat{\eta }\mathrm{\Gamma }\left(1/2\widehat{\eta }\right)}{{\mathrm{\Gamma }}^2\left(1/4\widehat{\eta }\right)}}={\displaystyle \frac{{\mathrm{\Gamma }}^2\left(1/4\right){\widehat{m}}_1\left(z\right)}{64\sqrt{\pi }{\widehat{m}}_{1/2}^2\left(z\right)}},\widehat{b}={\widehat{m}}_2\left(z\right)$$

(13)

Similarly, it can be verified that the functions Φ₁₂₃(·) and Φ_1-1/2(·) for the MoLM and MoFM estimators are also implicit monotonically decreasing functions, and the lookup table method can be used to calculate their inverse functions and then for estimation of the shape parameter η.

In addition, the introduction of log-based moments into parameter estimation can improve performance [29]. By differentiating both sides of the original moments in (8) with respect to x, we obtain for x > 0:

$$\begin{array}{ll}E\left(z^x\ln \left(z\right)\right)& ={\displaystyle {\int }_0^{+\infty }{z}^x\ln \left(z\right)f\left(z;b,\eta \right)dz}\\ & ={\displaystyle \frac{d{m}_x\left(z\right)}{dx}}={\displaystyle \frac{d}{dx}}\left(b^{x/2}\mathrm{\Gamma }\left(1+{\displaystyle \frac{x}{2\eta }}\right)\mathrm{\Gamma }\left(1+{\displaystyle \frac{x}{2}}\right){\left[\mathrm{\Gamma }\left(1+{\displaystyle \frac{1}{\eta }}\right)\right]}^{x/2}\right)\\ & ={\displaystyle \frac{x^2}{8}}{b}^{{\displaystyle \frac{x}{2}}}{\eta }^{{\displaystyle \frac{x}{2}}-1}\mathrm{\Gamma }\left({\displaystyle \frac{x}{2\eta }}\right)\mathrm{\Gamma }\left({\displaystyle \frac{x}{2}}\right){\left[\mathrm{\Gamma }\left({\displaystyle \frac{1}{\eta }}\right)\right]}^{-x/2}\left\{\ln b+{\displaystyle \frac{4}{x}}+{\displaystyle \frac{1}{\eta }}\Psi \left({\displaystyle \frac{x}{2\eta }}\right)+\Psi \left({\displaystyle \frac{x}{2}}\right)+\ln \eta -\ln \left[\mathrm{\Gamma }\left({\displaystyle \frac{1}{\eta }}\right)\right]\right\}\end{array}$$

(14)

where Ψ(·) denotes the digamma function, and Ψ(1) = −γ, and γ is the Euler’s constant. Simultaneous system of the equations of m₁(z), m(zln(z)), and m(ln(z)) obtains the equation on the shape parameter:

$${\Phi }_{[z\log (z)]}\left(\eta \right)\equiv 2-\ln 2+{\displaystyle \frac{1}{2\eta }}\left[\Psi \left({\displaystyle \frac{1}{2\eta }}\right)-\Psi \left(1\right)\right]={\displaystyle \frac{m\left(z\ln \left(z\right)\right)}{m_1\left(z\right)}}-m\left(\ln \left(z\right)\right)$$

(15)

It can be proved that the function Φ_[zlog(z)](·) is monotonically decreasing, as shown in Appendix A. Therefore, its inverse function does exist and can be computed by the lookup table method. In this way, the [zlog(z)]-based estimators are given by using the four sample moments:

$$\begin{array}{l}\widehat{\eta }={\Phi }_{[z\log (z)]}^{-1}\left({\displaystyle \frac{\widehat{m}\left(z\ln \left(z\right)\right)}{{\widehat{m}}_1\left(z\right)}}-\widehat{m}\left(\ln \left(z\right)\right)\right),\widehat{b}={\widehat{m}}_2\left(z\right),\\ \widehat{m}\left(z^x\ln \left(z\right)\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{z}_i^x\ln z_i^x},x=0,1.\end{array}$$

(16)

Here, the scale parameter is estimated by the second-order sample moments of the data. In the above four estimators, the scale parameter is estimated in the same manner. The analytical expression for the estimation error of the scale parameter is as follows:

Property 1. 

In the four estimators above, the relative root mean squared error (RRMSE) of the scale parameter is independent of the scale parameter itself and is determined by the shape parameter and the sample size.

In terms of (10, 12, 13, 16), it is easy to prove that the estimate of the scale parameter is unbiased, and its RRMSE can be given by the following:

$$\mathrm{RRMSE}\left(b\right)={\displaystyle \frac{1}{b}}\sqrt{E\left[{\left(\widehat{b}-b\right)}^2\right]}=\sqrt{{\displaystyle \frac{1}{N}}\left[{\displaystyle \frac{4\eta \mathrm{\Gamma }\left(2/\eta \right)}{{\mathrm{\Gamma }}^2\left(1/\eta \right)}}-1\right]}$$

(17)

It is directly inversely proportional to the square root of the sample size and becomes larger as the shape parameter becomes smaller or the non-Gaussianity of the clutter increases. Owing to the analytical expression, the RRMSEs of the scale parameter are not exhibited in the figures for performance comparison.

Property 2. 

In the four estimators above, the RRMSE of the shape parameter is independent of the scale parameter and is determined by the shape parameter and sample size.

Its proof is straightforward. Taking the [zlog(z)]-based estimator in (16) as an example, we prove that the RRMSE of the shape parameter is independent of the scale parameter. Without loss of generality, the amplitude z can be written as the following:

$$z=\sqrt{b}{z}_0,z_0~\mathrm{CGWB}\left(•;1,\eta \right)$$

(18)

where the normalized amplitude z₀ is independent of the scale parameter b. Further, the combination of the sample moments can be written into the following:

$$\begin{array}{ll}{\displaystyle \frac{\widehat{m}\left(z\ln \left(z\right)\right)}{{\widehat{m}}_1\left(z\right)}}-\widehat{m}\left(\ln \left(z\right)\right)& ={\displaystyle \frac{\widehat{m}\left(\sqrt{b}{z}_0\left(\ln \left(z_0\right)+1/2\ln b\right)\right)}{\sqrt{b}{\widehat{m}}_1\left(z_0\right)}}-\widehat{m}\left(\ln \left(z_0\right)+1/2\ln b\right)\\ & ={\displaystyle \frac{\widehat{m}\left(z_0\ln \left(z_0\right)\right)}{{\widehat{m}}_1\left(z_0\right)}}+{\displaystyle \frac{\ln b}{2}}-\widehat{m}\left(\ln \left(z_0\right)\right)-{\displaystyle \frac{\ln b}{2}}={\displaystyle \frac{\widehat{m}\left(z_0\ln \left(z_0\right)\right)}{{\widehat{m}}_1\left(z_0\right)}}-\widehat{m}\left(\ln \left(z_0\right)\right)\end{array}$$

(19)

Because the combination is independent of the scale parameter, the estimate of the shape parameter and its RRMSE are independent of the scale parameter. For the other three estimators, Property 2 can be proved in a similar manner.

The four estimators all use sample moments or generalized sample moments, which means that the estimators are all sensitive to outliers of large amplitude in data [30,31,32]. In addition to these moment-based estimators, the maximum likelihood (ML) estimator offers higher accuracy in parameter estimation. However, the complex expression of the CGWB distributions caused us to fail to obtain an analytical or iterative ML estimator. Alternative estimation, such as the numerical ML estimator, is of computational expense and initial-point dependence. To obtain robust parameter estimates of the CGWB distributions in the case with outliers, the curve-fitting estimation method for the truncated empirical cumulative distribution function (ECDF) of data is considered [30]. For a given 0 < β < 1, the fitting ECDF estimates are obtained by solving the following optimization:

$$\left(\widehat{\eta },\widehat{b}\right)=\underset{\eta ,b}{\arg \min }\left\{{\displaystyle \sum _{n=1}^{\left[N\beta \right]}{\left[F\left(z_{\left(n\right)};b,\eta \right)-\frac{n}{N}\right]}^2}\right\}$$

(20)

where z_(n) is the amplitude sample set sorted in ascending order as $z_{\left(1\right)}\le z_{\left(2\right)}\le \cdots \le z_{\left(N\right)}$, and β is the truncation ratio of the ECDF of the data. Nevertheless, the computational burden and dependency on initial points are two defects of the estimates in (20) [11]. In use, the estimates obtained from the moment-based estimator are often used as the initial points to relieve the initial-point dependency, and the exclusion of obvious outliers is used to reduce the influence of outliers.

In what follows, we compare the performance of the four moment-based estimators by using simulated CGWB-distributed data. Note that the RRMSE of the scale parameter is analytically given, and the RRMSE of the shape parameter is proved to be independent of the scale parameter. Thus, in all simulation experiments, it is justifiable to fix the scale parameter at 1, which helps to simplify the analysis and enables a more focused study of the effects of other factors on the estimation performance. The first experiment compares the performance of the four moment-based estimators when the sample size varies from 500 to 10⁴ and the shape parameters η = 0.5 (stronger non-Gaussianity) and η = 2 (weaker non-Gaussianity). In addition to the RRMSEs of the shape parameter, the KLD between the true PDF and the fitted PDF using estimated parameters is employed to evaluate the estimation performance. The KLD is calculated by [28] as follows:

$$\mathrm{KLD}\left(\widehat{b},\widehat{\eta }\right)={\displaystyle {\int }_0^{+\infty }{f}_z\left(z;b,\eta \right)\log \frac{f_z\left(z;b,\eta \right)}{f_z\left(z;\widehat{b},\widehat{\eta }\right)}dz}$$

(21)

As the KLD measures the difference between the two PDFs, particularly on tails, a KLD of zero indicates identical PDFs, and a larger KLD implies a greater difference. Consequently, the KLD serves as a more direct metric for assessing parameter estimation performance compared to the RRMSE. The RRMSE of the shape parameter and the average KLD of the fitted and true PDFs are calculated in 10⁴ independent trials. Figure 4 illustrates the RRMSEs and average KLD of the four moment-based estimators when the sample size varies from 500 to 10⁴. It can be observed that both the RRMSEs of the shape parameter and the average KLDs are approximately inversely related to the square root of the sample size N, although there are no analytical expressions for them. It is worth noting that the RRMSE and the average KLD decay as the sample size increases, which indicates that the four estimators exhibit better performance with a larger number of samples, both in terms of estimation accuracy and the goodness-of-the-fit of fitted PDFs. Unfortunately, many cases do not meet the demand for sufficient samples. Only a limited sample size is available in real radar environments. In this scenario, a large sample size, such as N = 10⁴, is adopted in the simulation because it is easy to generate enough large samples under the ideal conditions. As for the estimation performance of the shape parameter, two different shape parameters are given in the experiment. When the shape parameter η = 0.5, the MoM estimator behaves the worst and has the largest RRMSEs and average KLDs, while the [zlog(z)]-based and MoFM estimators achieve the best and locate the second place, respectively. Their performances are very close, and the MoLM estimator is in the middle. When the shape parameter η = 2, the situation is slightly different. The performances of the MoFM, MoLM, and [zlog(z)]-based estimators are comparable, and that of the MoM estimator is the worst.

In the second experiment, we examine the performance of the four estimators for the CGWB distribution when the shape parameter varies from 0.1 to 10 with an interval of 0.01, and the sample size is 10⁴. As shown in Figure 5, the RRMSEs of the shape parameter and the average KLDs for the four estimators are computed in the same manner as in the first experiment, where both the horizontal and vertical axes use a logarithmic scale. From Figure 5, the [zlog(z)]-based estimator attains the best performance, followed by the MoFM estimator, whose performance is quite close to that of the [zlog(z)]-based estimator. The other two estimators are much worse in performance, particularly the MoM estimator. In addition, it was found that the assessments of the RRMSE and average KLD are inconsistent. For each estimator, the RRMSE curves are all anti-unimodal functions of the shape parameter. Or rather, as the shape parameter increases, the RRMSE of the shape parameter first decreases, then achieves its minimum, subsequently increases, and finally decreases again. The RRMSE curves of the four estimators have different minimums. The average KLD curves of the four estimators have more complicated changes. Overall, when the shape parameter becomes larger, the average KLD gradually reduces and approximates the average KLD when the CGWB distribution degenerates to the Rayleigh distribution. In other words, the four estimators can attain better goodness-of-the-fit for CGWB-distributed clutter with weaker non-Gaussianity. The modeling of the probability distribution of data shows that the RRMSEs of the estimated parameters are indirect indexes to assess the performance, whereas the average KLD is a more effective direct index to assess the performance. It is worth noting that the four moment-based estimators are all sensitive to a small number of outliers of large amplitudes in data. When they are employed, the data must be clean without outliers, or outliers need to be removed in advance. For the other types of biparametric distributions, the outlier-robust estimators have been developed [30,31,32]. Therefore, it is an interesting issue to develop outlier-robust estimators for the CGWB distributions in the future. To avoid the influence of outliers, the parameter estimation of the CGWB distributions for measured sea clutter employs the numerical method in (20) to fit the truncated CDF of the data.

4. Suitability of CGWB Distributions for Measured Sea Clutter

In addition to the biparametric CGWB distributions, there are at least five types of biparametric distributions, including the K-distributions [1], the GP distributions [9], the CGIG distributions [10], the CGLN distributions [11], and the recent CGNG distributions [12], and two types of tri-parametric distributions, namely the CG-GIG distributions [15] and the GK distributions [21]. These types of distributions have all been developed within the compound-Gaussian model to characterize the amplitude/intensity distributions of sea clutter data under various conditions. One of the main reasons for developing new types of amplitude distributions is that no single type of amplitude distribution can be suitable for sea clutter data under various conditions because the statistics of sea clutter are affected by a multitude of factors. As a new type of biparametric distribution, what we are concerned about is that the CGWB distributions are suitable for sea clutter data under which conditions.

For sea clutter data X that are assumed to follow the compound-Gaussian model, for a given type of biparametric distributions, the goodness-of-the-fit of the type of distributions to the data can be measured by the KSD [27] of the empirical CDF of the data and fitted CDF and the KLD [28] of the empirical PDF of the data and the fitted PDF:

$$\begin{array}{l}\mathrm{KSD}\left(\mathrm{X};type\right)=\underset{z\in \left(0,+\infty \right)}{\max }\left\{\left|F_z\left(z;\mathrm{X}\right)-F_z\left(z;\widehat{b},\widehat{\eta };type\right)\right|\right\}\\ \mathrm{KLD}\left(\mathrm{X};type\right)={\displaystyle {\int }_0^{+\infty }{f}_z\left(z;\mathrm{X}\right)\left[\log f_z\left(z;\mathrm{X}\right)-\log f_z\left(z;\widehat{b},\widehat{\eta };type\right)\right]dz}\end{array}$$

(22)

where $f_z\left(z;\mathrm{X}\right)$ and $F_z\left(z;\mathrm{X}\right)$ are the empirical PDF and CDF of the data and $\widehat{b}$ and $\widehat{\eta }$ are estimated scale and shape parameters from the data. It is worth noting that the KSD highlights the fitness of the empirical PDF of the data on the body, whereas the KLD highlights the fitness of the empirical PDF of the data on the tail [12,33]. For measured sea clutter data, to reduce the influence of the parameter estimation methods on the selection of the type of distribution, outliers in data are manually removed by simple detection methods, and then the iterative/numerical maximum likelihood estimators or the curve-fitting estimation method (20) are used to estimate the parameters from the data. Like in [12,33], the selection of the best type of data X uses the geometric mean of the KSD and KLD:

$$\text{Best-type}\left(\mathrm{X}\right)=\underset{type\in \left\{\mathrm{K},\mathrm{GP},\mathrm{CGIG},\mathrm{CGLN},\mathrm{CGNG},\mathrm{CGWB}\right\}}{\arg \min }\left\{\sqrt{\mathrm{KSD}\left(\mathrm{X};type\right)\cdot \mathrm{KLD}\left(\mathrm{X};type\right)}\right\}$$

(23)

where the best type is selected from the six types of biparametric distributions, including the K-distributions, the GP distributions, the CGIG distributions, the CGLN distributions, the CGNG distributions, and the CGWB distributions. It is known that the six types of distributions all degenerate into Rayleigh distributions as the shape parameter tends to infinity. Therefore, each type is suitable for sea clutter data that are close to complex Gaussian clutter, and the selection of the best type hardly brings any benefit. The maximal selection benefit (MSB) [33] of the data X is defined by the following:

$$\mathrm{MSB}\left(\mathrm{X}\right)={\displaystyle \frac{\underset{type\in \left\{\mathrm{K},\mathrm{GP},\mathrm{CGIG},\mathrm{CGLN},\mathrm{CGNG},\mathrm{CGWB}\right\}}{\max }\left\{\sqrt{\mathrm{KSD}\left(\mathrm{X};type\right)\times \mathrm{KLD}\left(\mathrm{X};type\right)}\right\}}{\underset{type\in \left\{\mathrm{K},\mathrm{GP},\mathrm{CGIG},\mathrm{CGLN},\mathrm{CGNG},\mathrm{CGWB}\right\}}{\min }\left\{\sqrt{KSD\left(\mathrm{X};type\right)\times KLD\left(\mathrm{X};type\right)}\right\}}}$$

(24)

Equation (24) is the ratio of the synthetic measure of the worst type to that of the best type. A larger MSB(X) indicates a greater benefit from selecting the type. When the MSB is close to one, it means that every type is suitable for the data X, which often occurs when the clutter is approximated to complex Gaussian clutter.

In the first experiment of measured sea clutter data, the C-band shored-based radar data are used. The radar, which is installed on the top of the mountain with a height of about 1000 m, operates in electronic scanning mode with a range resolution of 60 m and a grazing angle of 2°. Figure 6 illustrates the amplitude map (dB) of the radar returns from an oceanic region with radial distance from 28.0 km to 52.0 km and the azimuth angle from 100° to 176° in the northeast sky coordinate system. The size of the data is 5 × 401 × 154, where 401 is the number of range cells, 154 is the number of beam positions, and the azimuth angle interval of two adjacent beam positions is 0.5 degrees, and at each beam position, the radar transmits five pulses of a bandwidth of 2.5 MHz. The statistics of sea clutter alter with grazing angle, sea state, and the viewing geometry of the radar; therefore, the scene in Figure 6 is segmented into 30 clutter map cells (CMCs), which are labeled by white lines, and the numbers from one to thirty marked in Figure 6 respectively represent these 30 CMCs. Each CMC consists of the radar returns at 200 contiguous range cells and ten consecutive beam positions. An amplitude distribution is used to model sea clutter in each CMC. To reduce the impact of outliers in the data, the spatial resolution cells containing outliers of high amplitude and power are manually removed from the data by simple detection methods. There is no outlier-robust estimator for the CGWB distribution; therefore, the fitting ECDF method using truncated data is used to estimate the parameters, and the initial point is from the [zlog(z)]-based estimator. The removed spatial resolution cells are labeled with dark squares in Figure 6b. For the 30 CMCs of the data, their best types are determined by (23) of the six types. The results of the best types are listed in

Table 1. Numbers of the CMCs of each type to be the best and the mean MSBs of each type on its best CMCs of the low-resolution sea clutter data.

Models	K	GP	CGIG	CGLN	CGNG	CGWB
Number	5	6	0	2	7	10
Mean MSB	2.0752	4.5051	-	4.4329	2.4776	2.2861

The bold font denotes the largest number of optimal models and the largest mean MSB.

. The CGWB distribution is the best type of 10 CMCs out of 30 CMCs, and its percentage is 33%. The CGNG distributions, GP distributions, and K-distributions are the best types for 7, 6, and 5 CMCs, respectively, and their percentages are 23.3%, 20%, and 16.7%, respectively. Only on two CMCs are the CGLN distributions the best, and their percentages are 6.7%. The CGIG distributions are not the best on all the CMCs. From the result, the CGWB distributions are more suitable for modeling low-resolution sea clutter data at small grazing angles. Further, for each type, the mean MSB is calculated on its best CMCs by (24). As can be seen in Table 1, the mean MSB of the GP distributions is the largest, followed by the CGLN distribution, while the mean MSBs of the other three types are comparable. Moreover, a larger mean MSB, which is apparently larger than 1, implies that the selection of the type is necessary for the measured data.

Therefore, we conclude that the CGWB distributions are suitable to model the low-resolution C-band sea clutter data. Below, the sixth and eighth CMCs in Figure 6 are used to illustrate the effect to fit the empirical PDF of the data by the six types of distributions. As shown in Figure 7a,b, the CGWB distributions are highly in accord with the empirical PDFs of the data, while the K-distributions and the CGNG distributions can also fit the data well. The other three types of distributions deviate significantly from the empirical PDFs of the data. The results confirm that the K-distributions, CGNG distributions, and CGWB distributions are suitable for low-resolution sea clutter, while the GP distributions, CGIG distributions, and CGLN distributions are more suitable for moderate/high-resolution sea clutter [33].

Table 2. Estimated parameters on the sixth and eighth CMCs under the six types of distributions and quantitative assessment of their goodness-of-the-fit.

	Models	Shape Parameter	Scale Parameter	KSD	KLD	(KSD × KLD)0.5
Sixth CMC	K	1.5168	8.7405 × 104	0.0184	0.0926	0.0413
	GP	1.3888	1.5988 × 105	0.0430	0.1567	0.0821
	CGIG	0.7571	1.0030 × 105	0.0311	0.1386	0.0656
	CGLN	1.1154	9.9411 × 104	0.0286	0.1275	0.0604
	CGNG	0.5508	8.5196 × 104	0.0178	0.0770	0.0371
	CGWB	1.3323	8.5235 × 104	0.0154	0.0853	0.0363
Eighth CMC	K	1.7166	3.6063 × 104	0.0155	0.0774	0.0346
	GP	1.5554	5.5670 × 104	0.0327	0.1239	0.0636
	CGIG	0.9139	4.0383 × 104	0.0227	0.1047	0.0488
	CGLN	1.2899	4.0050 × 104	0.0207	0.0960	0.0466
	CGNG	0.5838	3.6350 × 104	0.0153	0.0740	0.0336
	CGWB	1.4401	3.5151 × 104	0.0137	0.0743	0.0320

The bold font denotes the smallest KSD, KLD, and the geometric mean of the KSD and KLD.

lists the estimated parameters under the six types of distributions. In terms of the shape parameter, the low-resolution sea clutter exhibits remarkable non-Gaussian characteristics, with the shape parameters of the fitted K-distributions, CGNG distributions, and CGWB distributions having relatively small values. Moreover, from the KSD and KLD and their geometric average, the CGWB distributions, K-distributions, and CGNG distributions indeed attain a much better fitting effect compared to the other three distributions. Particularly, the scale parameters in the six types of distributions represent the power of sea clutter. From the fourth column of Table 2, it can be seen that the estimated clutter powers under the K-distributions, CGNG distributions, and CGWB distributions are comparable. However, for the other three types, the estimated clutter powers severely deviate from the true clutter power. Therefore, the selection of the type is crucial, and the mismatch of type may lead to the deviation of the basic parameters of sea clutter from the true ones. As shown in Figure 6, the data in the sixth and eighth CMCs are clean sea clutter data. Therefore, we assess the performance of the four estimators based on moments and log-based moments of the CGWB distributions using measured sea clutter data, and the results of the fitting ECDF method are given as well. The PDFs fitted using the four estimators and the method to fit the truncated ECDF of the data, as well as the empirical PDFs of the data, are plotted in Figure 7c,d. As can be observed, all five fitted PDFs are highly consistent with the empirical PDF on the eighth CMC, while on the sixth CMC, the situation is different. The corresponding estimates and assessment measures are listed in

Table 3. Performance comparison of the five estimators of the CGWB distributions on the sixth and eighth CMCs.

	Estimators	Shape Parameter	Scale Parameter	KSD	KLD	(KSD × KLD)0.5
Sixth CMC	Fitting ECDF	1.3323	8.5235 × 104	0.0154	0.0853	0.0363
	MoM	1.6009	7.9933 × 104	0.0282	0.1034	0.0540
	MoLM	1.5342	7.9933 × 104	0.0245	0.0994	0.0493
	MoFM	1.3543	7.9933 × 104	0.0228	0.0947	0.0465
	[zlog(z)]	1.3510	7.9933 × 104	0.0232	0.0948	0.0469
Eighth CMC	Fitting ECDF	1.4401	3.5151 × 104	0.0137	0.0743	0.0320
	MoM	1.4174	3.4829 × 104	0.0122	0.0744	0.0301
	MoLM	1.4362	3.4829 × 104	0.0131	0.0746	0.0313
	MoFM	1.3842	3.4829 × 104	0.0153	0.0743	0.0337
	[zlog(z)]	1.3842	3.4829 × 104	0.0153	0.0743	0.0337

The bold font denotes the smallest KSD, KLD, and the geometric mean of the KSD and KLD.

. The fitted PDFs by using the fitting ECDF, MoLM, MoFM, and [zlog(z)]-based estimators can accord with the empirical PDF of the data, but the fitted PDF of the MoM estimator deviates from the empirical PDF. Even for clean sea clutter data and the best type of distributions, the MoM estimator probably results in inaccurate estimates because of higher-order sample moments. Additionally, from the eighth CMC, it can be deduced that the fitting ECDF method also provides inaccurate estimates owing to the truncation of some normal clutter samples.

In what follows, we examine the performance of the estimator of the CGWB distributions in the case with outliers. On the 23rd CMC, the most suitable type is the CGWB distribution, as shown in

Table 4. Estimated parameters on the 23rd CMC under the six types of distributions and quantitative assessment of their goodness-of-the-fit.

Types	Shape Parameter	Scale Parameter	KSD	KLD	(KSD × KLD)0.5
K	1.5066	3.1167 × 105	0.0197	0.1093	0.0464
GP	1.2961	6.7835 × 105	0.0492	0.1595	0.0886
CGIG	0.6814	3.6984 × 105	0.0378	0.1397	0.0727
CGLN	1.0616	3.6138 × 105	0.0338	0.1288	0.0660
CGNG	0.6541	2.7355 × 105	0.0236	0.1146	0.0520
CGWB	1.3221	3.0459 × 105	0.0165	0.1083	0.0423

The bold font denotes the smallest KSD, KLD, and the geometric mean of the KSD and KLD.

, and the CMC contains a small number of outliers with larger amplitudes. In the CMC, when outliers are not removed, the fitted PDFs by using the six estimators and the empirical PDF of the data are plotted in Figure 8, and the fitting results are listed in

Table 5. Performance comparison of the five estimators of the CGWB distributions on the 23rd CMCs.

	Estimators	Shape Parameter	Scale Parameter	KSD	KLD	(KSD × KLD)0.5
data without outliers	Fitting ECDF	1.3221	3.0459 × 105	0.0165	0.1083	0.0423
	MoM	1.6557	2.8285 × 105	0.0278	0.1235	0.0586
	MoLM	1.5992	2.8285 × 105	0.0229	0.1193	0.0523
	MoFM	1.4763	2.8285 × 105	0.0184	0.1130	0.0456
	[zlog(z)]	1.4969	2.8285 × 105	0.0178	0.1137	0.0450
data with outliers	Fitting ECDF	1.4284	3.1740 × 105	0.0115	0.1422	0.0405
	MoM	0.3555	4.5538 × 105	0.4008	0.7819	0.5598
	MoLM	0.4408	4.5538 × 105	0.3000	0.6064	0.4265
	MoFM	1.1258	4.5538 × 105	0.0814	0.1948	0.1260
	[zlog(z)]	1.1127	4.5538 × 105	0.0808	0.1947	0.1254

The bold font denotes the smallest KSD, KLD, and the geometric mean of the KSD and KLD.

. It can be seen that the fitted PDFs using the MoM, MoFM, MoLM, and [zlog(z)]-based estimators all severely deviate from the empirical PDFs of the data because they are sensitive to outliers. In contrast, the fitted PDF of the fitting ECDF estimator of the data can still accord with the empirical PDFs of the data because of its robustness to outliers. However, the numerical method to fit the truncated ECDF of the data is time-consuming and initial-point-dependent. Therefore, it is necessary to develop the outlier-robust estimators of the CGWB distributions in the future, just as has been done for other types of biparametric distributions [12,30,31,32].

From the analysis above, we find that the CGWB distributions are suited to model low-resolution sea clutter data like the K-distributions. To further examine this point, the second experiment uses measured sea clutter data from the open CSIR radar database [34] collected in high-range resolution and high sea states and at small grazing angles. The carrier frequency of the radar is 9 GHz, the pulse repetition frequency (PRF) is 5 kHz, and the range resolution is 15 m, and the data were collected at VV polarization. The simple range resolution conversion [35] is used to transfer the data into low-resolution data by summation of complex radar returns at eight contiguous range cells. In this way, the conversed data have a range resolution of 8 × 15 = 120 m. For the dataset TFC15_014, the power maps before and after the range resolution conversion are plotted in Figure 9, where fine textures at high resolution are smoothed out [36].

After the range resolution conversion, the original 96 range cells are synthesized into 12 range cells or fewer because the range cells that contain outliers with large amplitudes are manually removed, just as in the first experiment. Each range-resolution lowered data are modeled by a compound-Gaussian model.

Table 6. Numbers of the data of each type to be the best and the mean MSBs of each type on its best data.

Types	K	GP	CGIG	CGLN	CGNG	CGWB
Number	11	9	7	4	5	15
Mean MSB	4.6828	2.2588	2.4479	1.7627	3.6109	4.9927

The bold font denotes the smallest KSD, KLD, and the geometric mean of the KSD and KLD.

lists the fitting results for the six types of biparametric distributions for the range resolution lowered datasets [35] of the 51 datasets from the CSIR radar database. On 26 datasets out of the 51 datasets, the CGWB and K-distributions are the best. The CGWB, K, GP, CGIG, CGNG, and CGLN distributions are the best on 29%, 22%, 18%, 14%, 10%, and 8% datasets, respectively. Moreover, the mean MSBs of the CGWB and K-distributions are rather large. This shows that the CGWB and K-distributions are suitable for low-resolution sea clutter at high sea states and small grazing angles. Note that, even at low-range resolution, sea clutter at high sea states exhibits strong non-Gaussianity. Figure 10 illustrates the empirical PDF of the low-resolution data from the TFC15_014, the different types of fitted PDFs, and the fitted CGWB distribution using different parameter estimators.

As depicted in Figure 10a, the CGWB distributions, K-distributions, and CGNG distributions provide better goodness-of-the-fit than the other three distributions.

Table 7. Estimated parameters of the low-resolution data from the TFC15_014 by using the six types of distributions and quantitative assessment of the goodness-of-the-fit.

Types	Shape Parameter	Scale Parameter	KSD	KLD	(KSD × KLD)0.5
K	2.0321	3.3789	0.0057	0.0323	0.0136
GP	1.7876	4.5947	0.0258	0.0796	0.0453
CGIG	1.1422	3.7063	0.0170	0.0628	0.0327
CGLN	1.5447	3.6756	0.0149	0.0543	0.0284
CGNG	0.7355	3.2674	0.0034	0.0317	0.0104
CGWB	1.6043	3.2911	0.0026	0.0300	0.0089

The bold font denotes the smallest KSD, KLD, and the geometric mean of the KSD and KLD.

lists the estimated parameters and the assessment measures. The shape parameters of the CGWB, K-, and CGNG distributions are relatively small, meaning that the low-resolution sea clutter at high sea states keeps strong non-Gaussianity and heavier tails. Particularly, when the types are suitable for the data, the estimated clutter powers under the CGWB, K-, and CGNG distributions are comparable. For the other three types, particularly for GP distributions, the estimated clutter powers are upward biased because of the mismatch of shapes. Figure 10b illustrates the fitted CGWB distributions using different parameter estimators. Owing to the absence of outliers and sufficient samples, the five estimators yield the CGWB distributions that accord with the empirical PDF of the data. However, there exist some small differences between the estimators. To illustrate this difference,

Table 8. Performance comparison of the five estimators of the CGWB distributions on the low-resolution data.

Estimators	Shape Parameter	Scale Parameter	KSD	KLD	(KSD × KLD)0.5
Fitting ECDF	1.6043	3.2911	0.0026	0.0300	0.0089
MoM	1.4293	3.3340	0.0145	0.0468	0.0260
MoLM	1.4883	3.3340	0.0082	0.0381	0.0177
MoFM	1.5670	3.3340	0.0052	0.0315	0.0128
[zlog(z)]	1.5897	3.3340	0.0057	0.0312	0.0133

The bold font denotes the smallest KSD, KLD, and the geometric mean of the KSD and KLD.

lists the estimated parameters by the different estimators on the data in Figure 9. The method to fit the truncated ECDF gives the best estimates. The MoFM and [zlog(z)]-based estimators give better estimates, while the MoM and MoLM estimators perform poorly. Therefore, for clean clutter data, the MoFM and [zlog(z)]-based estimators are recommended. For data with outliers, outlier-robust estimators of low computational cost need to be developed in the future.

5. Conclusions

In this paper, a CGM with Weibull-distributed textures, named the CGWB distributions, is proposed to meet the needs of various tails in sea clutter modeling. Moreover, several estimators are given to estimate the parameters of the CGWB distributions, and their performance comparison is made to show that the MoFM and [zlog(z)]-based estimators exhibit better performance for clean clutter data. The experiments using measured sea clutter data are made to examine the suitability of the CGWB distributions. The results show that, like the K-distributions and CGNG distributions, the CGWB distributions are suitable for low-resolution sea clutter data at small grazing angles. In conclusion, the CGM family of biparametric distributions is enriched to realize more accurate modeling of various sea clutter data.