# Characterizing the Statistical Properties of SAR Clutter by Using an Empirical Distribution

Authors: Gui Gao; Gongtao Shi; Huanxin Zou; Shilin Zhou
Journal: International Journal of Antennas and Propagation
 (2013)

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## Abstract

The performances on the applications of synthetic aperture radar (SAR) data strongly depend on the statistical characteristics of the pixel amplitudes or intensities. In this paper, a new empirical model, called simplyℋ𝒢o, has been proposed to characterize the statistical properties of SAR clutter data over the wide range of homogeneous, heterogeneous, and extremely heterogeneous returns of terrain classes. A particular case of the ℋ𝒢odistribution is the well-known 𝒢o distributions. We also derived analytically the estimators of the presented ℋ𝒢o model by applying the “method of log cumulants” (MoLCs). The performance of the proposed model is verified by using some measured SAR images.


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## Body

## 1. Introduction

Nowadays, synthetic aperture radar (SAR) has become an advanced tool as compared to optical sensors for monitoring land or sea surfaces because of its advantages regardless of weather conditions [1, 2]. The interpretation of statistical properties of SAR clutter with various terrain classes is a crucial task for designing filtering [3], detection [4], segmentation [5], and classification [6–8] of algorithms to optimally exploit the contents of the processed SAR images. These analyses result in a growing interest in developing precise models for the statistics of the pixel amplitudes or intensities.It is known that SAR images are strongly affected by speckle noise due to its coherent imaging mechanism [1, 2, 9, 10]. This noise effect degrades the information of SAR images and limits the surveillance performances. To overcome this drawback, much work has been done to match amplitude or intensity statistics [3, 5, 6, 8, 11]. Among the existing statistical models, the parametric distributions have been intensively investigated because of their high accuracy and flexibility, which are usually obtained by two approaches.The first is to consider the physical mechanism of the backscattering from the land surfaces. The multiplicative model [1, 2] by combining the speckle noise and the terrain backscatter is commonly used in this class. Generally, multi-look SAR speckle noise intensity is assumed to obey a gamma distribution. Therefore, a central task for establishing this type of statistical models is to aim at the backscatter modeling of different terrain classes [1, 2, 11]. Two important distributions, consisting of 𝒦 as well as 𝒢0, which, in turn, in relation to the Gamma and inverse Gamma distribution for the intensity backscatter, have received a great deal of attention [2, 5–7, 9, 11, 12]. As compared to 𝒦, 𝒢0 agrees reasonably better with the heavy tail behavior coming from the extremely heterogeneous clutter like the cases of urban areas or other man-made structures [11].The second is to conduct the distributions from a purely mathematics view irrespective with physical property of radar clutter backscatter. Some known examples are the lognormal, weibull, and more recently the Fisher [6] (completely identical to the 𝒢0) distributions. The lognormal and weibull fit sometimes well with the SAR histogram of some heterogeneous and ocean regions [9]. However, they tend to occur a large deviation estimating the histograms with the heavy tail behavior.This paper is devoted to report an empirical model (denoted simply asℋ𝒢0) for characterizing the statistical properties of SAR clutter data to obtain the modeling ability of more heterogeneous clutter. The proposed model has the 𝒢0 distribution as a special case. Furthermore, using the second-kind statistics theory developed by Nicolas [13], which relies on the Mellin transform [14], that is, “method-of-log-cumulants” (MoLC), we derive the parameter estimators of the new distribution model.In the rest of this paper, the proposed distribution is first given in Section2. Section 3 derives the corresponding parameter estimators based on the MoLC. We provide the experimental results of the ℋ𝒢0 model using measured SAR data in Section 4, the comparisons with that 𝒢0 fits are also discussed in this section. The last section concludes this paper and give a perspective in the future work.


## 2. The Proposed Distribution

The proposed amplitude distribution is defined as(1)pZA(z)=-2σλvαnnΓ((α-n)/v)vΓ(vα)Γ(n)×z2n-1(λ+nz-2v)(n-α)/v,-α,-v,λ,σ,n,z>0,
where σ, α, and λ are the power, shape, scale parameters, respectively. v indicates the stretching parameter. Γ(·) represents the Gamma function. n is the number of looks. The corresponding intensity expression of (1) is further given by
(2)pZI(z)=-σλvαnnΓ((α-n)/v)vΓ(vα)Γ(n)×zn-1(λ+nz-v)(n-α)/v,-α,-v,λ,σ,n,z>0.We refer to this distribution characterized by (1) or (2) as the ℋ𝒢0 distribution. Specifically, we call the ℋ𝒢Io distribution and the ℋ𝒢Ao distribution, correspond to (2) and (1), respectively, to distinct the intensity statistic as well as the amplitude statistic. Figure 1 gives some examples of the ℋ𝒢Ao distribution with respect to the various parameters. From this figure, it can be seen clearly that the parameter α reflects the degree of homogeneity for the tested returns, which implies that the smaller value of |α| obtains, the more in-homogeneous (i.e., larger tails) they are. Meanwhile, v corresponds to a stretching of the SAR image amplitude or intensity, showing a strong effect for low values of the return. Moreover, as an independent parameter, σ indicates the whole fluctuation (contains magnifying or shrinking) of power of densities along the vertical axis. The parameter λ controls the peak value of the density.Plots ofℋ𝒢Ao versus different parameters.
(a)
(b)
(c)
(d)Although the probability density functions (PDFs) characterized by (1) or (2) are shown as the empirical models, the ℋ𝒢0 model also can be derived within the structure of the multiplicative model [1, 2, 11]. Herein, we account for the intensity distribution as an example, the following theorem is established.Theorem 1.
LettingXI and YI indicates the backscattering RCS component and speckle noise one, respectively, ZI denotes the observed intensity of SAR data. Therefore, the relationship of this three variables is expressed by the multiplicative model as
(3)ZI=XI·YI.
IfXI obeys
(4)pXI(x)=σλvαΓ(vα)xα-1exp(-λxv),-α,-v,λ,σ,x>0
and the PDF of YI is
(5)pYI(y)=nnΓ(n)yn-1exp(-ny-v),y,n,-v>0.
Then the distribution of the intensity return is characterized by the density shown in (2), that is, ZI~ℋ𝒢Io(α,v,λ,σ,n).Proof.
Combining (4) and (5) via (3), the PDF of ZI can be easily derived as
(6)pZI(z)=∫0∞1x·pXI(x)·pYI(zx)dx=σλvαnnΓ(vα)Γ(n)zn-1∫0∞xα-n-1exp{(-λ-nz-v)xv}dx.
A variable change oft=1/x leads to x=1/t and dx=-t-2dt, (6) turns out be
(7)pZI(z)=σλvαnnΓ(vα)Γ(n)zn-1∫0∞t-α+n-1exp{(-λ-nz-v)t-v}dt.
Applying the integral formula∫0∞xa-1exp(-μxp)dx=(1/p)μ-a/pΓ(a/p), Reμ>0, Rea>0, p>0 [15, 3.478, Equation 1], one can easily obtain the result that (7) is equal to (2).
Furthermore,ℋ𝒢0 has the charming property that the well-known 𝒢0, presented by Frery et al. [11] to model homogeneous, heterogeneous, and extremely heterogeneous terrains, is a special case of this proposed model when σ=1 and v=-1. Thus, the proposed model exhibits higher fitting ability as compared to 𝒢0.
As derived in AppendixA, the mth order moments of the ℋ𝒢Io are given by
(8)E(zIm)=σλvα-(m+α)/vnn+(m+n)/vΓ(-(m+n)/v)Γ((α+m)/v)v2Γ(vα)Γ(n).
Themth order moments of the corresponding amplitude random variable can easily be given by E(zAm)=E(zIm/2).


## 3. The Parameter Estimators of the Proposed Distribution

### 3.1. The Log-Cumulants ofℋ𝒢0

Nicolas [13] has proposed a parametric PDF estimation technique based on the MoLC for a function defined over ℝ+. Given a positive-valued random variable X with the PDF pX(x), the second-kind first and second characteristic functions are, respectively, defined as [13]
(9)ϕX(s)=ℳ[pX(x)](s)=∫0∞xs-1pX(x)dx,φX(s)=log(ϕX(s)),
where ℳ is the Mellin transform operator. The rth order derivative of φX(s) at s=1 is the log-cumulants of order r, that is,
(10)kr=drφX(s)dsr|s=1.Hereafter, we take the intensity distributionℋ𝒢Io as an example to estimate its parameters v, α, and λ. The processing of the amplitude one is similar. Owing to ϕX(s)=E(Xs-1), consequently, via (8) and (9), the second-kind second characteristic functions of the ℋ𝒢Io distribution yields
(11)φℋ𝒢Io(s)=log(σ)+log(λvα-(α/v))+1-svlog(λ)+log(nn+(n/v))+s-1vlog(n)+log(Γ(-s-1+nv))+log(Γ(α+s-1v))-log(v2Γ(vα)Γ(n))
which leads to that the log-cumulants of the ℋ𝒢Io are expressed as
(12)k1=log(n/λ)-Ψ(-n/v)+Ψ(α/v)v,kr=Ψ(r-1,-n/v)(-v)r+Ψ(r-1,α/v)vr,r≥2,
where Ψ(·) represents the digamma function, and Ψ(r,·) is the rth order polygamma function.Given a sample set{zi}, i∈[1,N], the log-cumulants can be estimated directly by
(13)k^1=1N∑i=1N[ln(zi)],k^1=1N∑i=1N[(ln(zi)-k^1)r],r≥2.


### 3.2. The Parameter Estimators ofℋ𝒢0

We notice that (12) is independent of the parameter σ and the rth log-cumulants shown in (12) are irrespective with the parameter λ on the condition that r≥2. In addition, we regard the parameter n as a known constant, which can be replaced by the equivalent number of looks (ENLs) [4, 11] or obtained from some prior knowledge about processed SAR images, hence, allowing us to divide the parameter estimates to three stages.First, the estimatesv^ and α^ of the parameters v as well as α are obtained by solving the following equations resorting to the numerical computation:(14)k^2=Ψ(1,-n/v^)+Ψ(1,α^/v^)v^2,k^3=Ψ(2,α^/v^)-Ψ(2,-n/v^)v^3.Second, according to the first equation of (12), the estimate λ^ of the parameter λ is
(15)λ^=nexp(v^k^1+Ψ(-n/v^)-Ψ(α^/v^)).We assume the observed amplitude PDF isp^ZA(z) from the actual data, which corresponds to the histogram of tested data. The theoretical amplitude PDF denotes as pZA(z). As the previous analysis, since σ indicates the whole proportion fluctuation of pZA(z), pZA(z)/σ can be calculated by using v^, α^, and λ^. Given a sample amplitude set {zi∣i=1⋯N}, let the symbol qZA(z) be equal to pZA(z)/σ, then the estimate σ^ of the parameter σ is simply given by
(16)σ^=E(p^ZA(zi)qZA(zi)),zi∈ℤ,
where ℤ is a subset of {zi∣i=1⋯N}. Here, in order to facilitate the stable fitting, we choose empirically all points of histogram within 3 dB bandwidth to calculate σ, such that ℤ={zi∣p^ZA(zi)>p^max/2,p^max=max(p^ZA(zi)),i∈[1,N]}.


## 3.1. The Log-Cumulants ofℋ𝒢0

Nicolas [13] has proposed a parametric PDF estimation technique based on the MoLC for a function defined over ℝ+. Given a positive-valued random variable X with the PDF pX(x), the second-kind first and second characteristic functions are, respectively, defined as [13]
(9)ϕX(s)=ℳ[pX(x)](s)=∫0∞xs-1pX(x)dx,φX(s)=log(ϕX(s)),
where ℳ is the Mellin transform operator. The rth order derivative of φX(s) at s=1 is the log-cumulants of order r, that is,
(10)kr=drφX(s)dsr|s=1.Hereafter, we take the intensity distributionℋ𝒢Io as an example to estimate its parameters v, α, and λ. The processing of the amplitude one is similar. Owing to ϕX(s)=E(Xs-1), consequently, via (8) and (9), the second-kind second characteristic functions of the ℋ𝒢Io distribution yields
(11)φℋ𝒢Io(s)=log(σ)+log(λvα-(α/v))+1-svlog(λ)+log(nn+(n/v))+s-1vlog(n)+log(Γ(-s-1+nv))+log(Γ(α+s-1v))-log(v2Γ(vα)Γ(n))
which leads to that the log-cumulants of the ℋ𝒢Io are expressed as
(12)k1=log(n/λ)-Ψ(-n/v)+Ψ(α/v)v,kr=Ψ(r-1,-n/v)(-v)r+Ψ(r-1,α/v)vr,r≥2,
where Ψ(·) represents the digamma function, and Ψ(r,·) is the rth order polygamma function.Given a sample set{zi}, i∈[1,N], the log-cumulants can be estimated directly by
(13)k^1=1N∑i=1N[ln(zi)],k^1=1N∑i=1N[(ln(zi)-k^1)r],r≥2.


## 3.2. The Parameter Estimators ofℋ𝒢0

We notice that (12) is independent of the parameter σ and the rth log-cumulants shown in (12) are irrespective with the parameter λ on the condition that r≥2. In addition, we regard the parameter n as a known constant, which can be replaced by the equivalent number of looks (ENLs) [4, 11] or obtained from some prior knowledge about processed SAR images, hence, allowing us to divide the parameter estimates to three stages.First, the estimatesv^ and α^ of the parameters v as well as α are obtained by solving the following equations resorting to the numerical computation:(14)k^2=Ψ(1,-n/v^)+Ψ(1,α^/v^)v^2,k^3=Ψ(2,α^/v^)-Ψ(2,-n/v^)v^3.Second, according to the first equation of (12), the estimate λ^ of the parameter λ is
(15)λ^=nexp(v^k^1+Ψ(-n/v^)-Ψ(α^/v^)).We assume the observed amplitude PDF isp^ZA(z) from the actual data, which corresponds to the histogram of tested data. The theoretical amplitude PDF denotes as pZA(z). As the previous analysis, since σ indicates the whole proportion fluctuation of pZA(z), pZA(z)/σ can be calculated by using v^, α^, and λ^. Given a sample amplitude set {zi∣i=1⋯N}, let the symbol qZA(z) be equal to pZA(z)/σ, then the estimate σ^ of the parameter σ is simply given by
(16)σ^=E(p^ZA(zi)qZA(zi)),zi∈ℤ,
where ℤ is a subset of {zi∣i=1⋯N}. Here, in order to facilitate the stable fitting, we choose empirically all points of histogram within 3 dB bandwidth to calculate σ, such that ℤ={zi∣p^ZA(zi)>p^max/2,p^max=max(p^ZA(zi)),i∈[1,N]}.


## 4. Experimental Results

In this section, we aim at verifying the performance of the proposedℋ𝒢0. In order to assess how the ℋ𝒢0  performs, several space-borne TerraSAR-X geocoded scenes with various land-over typologies as examples are reported. Figures 2(a)–2(d) show four selected typical patches from a large “Sanchagang” TerraSAR-X image with low resolution, which are the portions of water body, drying riverbed, mountain, and a town returns, respectively. The four types of scenes are related to the homogeneous and heterogeneous terrains. For simplicity, we denote them as “water-body”, “drying riverbed”, “mountain”, and “town.” Additionally, two urban areas shown in Figures 2(e)-2(f), extracted from a large “Beijing” TerraSAR-X scene with high resolution, are further carried out to demonstrate the effectiveness of the proposed model on the extremely heterogeneous terrains. Likewise, the two urban images are denoted by “urban1” and “urban2.” The main parameters of TerraSAR-X systems for “Sanchagang” and “Beijing” data are listed in Table 1.Table 1
The main parameters of  TerraSAR-X space-borne systems for the data in this study.
Testing site
Acquisition mode
Polarization
Resolution(az × rg)
Date
Size
Azimuth looks
Range looks
Beijing, China
SpotLight
HH
1.92 m × 2.26 m
2008-01-31T10:03 UTC
21333 px × 20000 px
1.19
1.02
Sanchagang, China
StripMap
VV
6.11 m × 5.98 m
2008-01-01T22:20 UTC
11909 px × 22363 px
1.02
2.67The Tested TerraSAR-X SAR image. (a) “Water body” patch with a size of 332 px × 322 px, (b) “drying riverbed” image (size: 1597 × 1734 pixels), (c) “mountain” image (size: 1996 × 2034 pixels), (d) “town” image (size: 525 × 560 pixels), (e) “urban1” image (size: 1198 × 1342 pixels), and (e) “urban2” image (size: 705 × 728 pixels).
(a)
(b)
(c)
(d)
(e)
(f)The estimated PDFs of the proposedℋ𝒢0 model for the histograms of six selected areas indicated in Figure 2 are shown in Figure 3. As the corresponding comparisons, the fitting results with the intensively used 𝒢0 distribution are also provided, where parameter estimates of this distribution are derived by Tison et al. [6] based on the MoLC. Herein, the estimate n in ℋ𝒢0 is first replaced by the ENL, that is, [4]
(17)n^=ENL=na+nr,
where na and nr are the effective number of looks in the azimuth and range. For the previous “Sanchagang” and “Beijing” TerraSAR-X images used in this investigation, na and nr are listed in the last two columns of Table 1 according to reading their metadata files. Next, the estimates of all other parameters in ℋ𝒢0 are accomplished by the estimators derived in Section 3, which are shown at Table 2.Table 2
Parameter estimations of noted areas in Figure2.
Area
ℋ
𝒢
A
°
(
α
,
v
,
λ
,
σ
,
n
)
“Water body”
(−6.6029, −1.1001, 6.3048, 1.0709, 3.683)
“Drying riverbed”
(−6.0382, −1.0555, 5.6813, 1.1897, 3.683)
“Mountain”
(−2.9788, −0.5779, 2.4694, 0.1426, 3.683)
“Town”
(−5.062, −0.7271, 4.5872, 0.4861, 3.683)
“Urban1”
(−1.1774, −0.4895, 0.6905, 11.1945, 2.22)
“Urban2”
(−0.8641, −0.7065, 0.3227, 2.3517, 2.22)Plots of amplitude histograms and of the estimated PDFs: (a)–(f) are corresponded to the fitting results for the scenes shown in Figures2(a)–2(f), respectively.
(a)
(b)
(c)
(d)
(e)
(f)Furthermore, in order to quantitatively assess the performances that different distributions fit, we define an error ratio factor (ERF) as
(18)δ=∑i=1N∥pc(zi)-q(zi)∥2∑i=1N∥pb(zi)-q(zi)∥2,
where {zi∣i=1⋯N} is a sample set, pc(·) represents the compared theoretical PDF, pb(·) is the basic theoretical PDF, and q(·) indicates the actual PDF from the observed data. The symbol ∥·∥ denotes 2-norm. It is obvious that the numerator and denominator in (18) separately imply the total fitting errors with the pc and pb to approximate q. The better the performances of pc related to pb fit, the smaller δ is, and vice verse. Specifically, δ=1, if both pc and pb have the identical capability for fitting the measured data.Letℋ𝒢Ao represent the basic amplitude PDF, and let 𝒢Ao be the compared amplitude one. The δ values of the previous six areas in this study are given in Figure 3 (see the yellow textbox). It can be clearly seen that the proposed ℋ𝒢Ao better agrees with the amplitude histograms of all six terrains than the 𝒢Ao, as expected, because the δ values are larger than 1 for all six areas. The same conclusion can be confirmed from a visual point of view and implies the higher fitting precision, using ℋ𝒢o than using 𝒢0, over homogeneous, heterogeneous, and extremely heterogeneous regions.


## 5. Conclusion and Perspective

We have developed an empirical model,ℋ𝒢0, to exploit the knowledge of statistical characteristics of SAR amplitude or intensity images over the wide terrain classes with homogeneous, heterogeneous, and extremely heterogeneous backscattering properties. The parameter estimators of this model based on the MoLC are also provided. Consequently, we report the performances of different land-over typologies with ℋ𝒢0 distribution fits. The experimental results show that the ℋ𝒢0 distribution is a more advanced model compared with the known 𝒢0 distribution to characterize the multilook processed SAR data.As we know, a preliminary statistical analysis of SAR clutter data is important for designing signal processing algorithms, such as speckle filtering, target detection, building extraction, image segmentation, and classification. In future, it is worth expecting to use theℋ𝒢0 distribution in these fields. Herein, we firstly attempt to give an analytical derivation for constructing a constant false alarm rate (CFAR) detector to promote the upcoming studies of target detection in SAR images.Given the propose amplitude modelℋ𝒢Ao shown in (1), its cumulative distribution function (CDF) is written as (see Appendix B)
(19)Fℋ𝒢Ao(x)=-σλvα+(n-α)/vnn-1Γ((α-n)/v)vΓ(vα)Γ(n)x2n×F21(α-nv,-nv;1-nv;-nλx-2v),-α,-v,λ,σ,n,x>0,
where F1(·,·;·;·)2 is the Gauss hypergeometric function. For a given value of the false alarm probability, denoted by Pfa, the corresponding CFAR threshold T for the ℋ𝒢Ao distribution can be obtained from
(20)1-Pfa=Fℋ𝒢Ao(T)=-σλvα+(n-α)/vnn-1Γ((α-n)/v)vΓ(vα)Γ(n)T2n×F21(α-nv,-nv;1-nv;-nλT-2v).ConsideringFℋ𝒢Ao(T) is strictly monotonously increasing, the threshold T can be accurately calculated with the help of the numerical solution or a simple bisection method [16].Our future work will focus on demonstrating the performances of the proposed CFAR detector using some measured SAR data and investigating how theℋ𝒢0 distribution performs when extending it to other application fields.



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