Statistical Modeling of Multi-channel SAR Images

Abstract

Currently, with the advancement of sophisticated SAR imaging modes, such as multitemporal interferometry and polarimetry, the return backscatter results are presented in two or more channels [1, 2]. The SAR interferogram, achieved by multiplying the first image by the complex conjugate of the second one [3, 4], has become an important tool for multiple-channel SAR image interpretation.

4.1 Introduction

Currently, with the advancement of sophisticated SAR imaging modes, such as multitemporal interferometry and polarimetry, the return backscatter results are presented in two or more channels [1, 2]. The SAR interferogram, achieved by multiplying the first image by the complex conjugate of the second one [3, 4], has become an important tool for multiple-channel SAR image interpretation. As a prominent example of interferogram applications, ground moving target indication (GMTI) has also received a great deal of attention and been intensively studied, e.g., [5,6,7,8,9].

Ground moving target identification (GMTI) using SAR has been a growing interest over the last couple of decades in many applications, such as military surveillance and reconnaissance of ground vehicles, and civilian ship monitoring of harbor [1,2,3]. The recent works [3, 4] reported in this field show that the multilook interferogram is an important tool for detecting moving targets. However, precise knowledge of the interferogram’s phase and magnitude statistics, i.e., the joint probability density function (PDF), is a major issue currently under study in the development of statistically based detector tests for distinguishing the moving targets from clutter [3,4,5].

Some investigations for statistical modeling of multilook SAR interferogram have been presented in the past, e.g., [3,4,5]. Lee et al. [5] firstly proposed the joint distribution of interferometric magnitude and phase with the condition of a constant radar cross section (RCS) background based on the complex Wishart distribution, presented by Goodman [6]. In the analysis of lot of literatures, it is shown [3,4,5] that the PDF is valid for modeling homogeneous areas, whereas it also tends to deviate strongly in most cases whose scenes contain heterogeneous or extremely heterogeneous regions.

Additionally, as the phase statistic is highly invariant against changes of the clutter type [3], the marginal PDFs of interferometric magnitude for heterogeneous and extremely heterogeneous regions are derived by Gierull [4]. Meanwhile, an original joint PDF of interferometric magnitude and phase for heterogeneous clutter is also given in afore-mentioned literature. Unfortunately, the joint PDFs of interferometric magnitude and phase for extremely heterogeneous regions are still a hard task by means of combining the marginal PDFs of magnitude and the ones of phase owing to that magnitude and phase are not statistically independent [3].

Meanwhile, since the theoretic foundation of the interferometric magnitude modeling is the complex Wishart distribution, some studies have been proposed based on this distribution in the past several years, such as [3, 4, 7,8,9,10,11]. Lee et al. [3] firstly proposed the interferometric magnitude distribution with the condition of a RCS background based on the complex Wishart distribution. Actually, the RCS of a homogeneous region (e.g., the agricultural areas) in either low-resolution or high-resolution SAR images can be expected to be a constant [12]. However, most scenes contain in-homogeneous regions with RCS fluctuations. Therefore, the magnitude’s probability density function (PDF) tends to deviate strongly in most cases. In practice, the product model [12, 13] has been widely and successfully used in the studies of the statistics of the single-channel image magnitude. Under this consideration, the interferometric magnitude can also be regarded as the product between an underlying RCS component with an uncorrelated multiplicative speckle noise component, which is similar with the polarimetric SAR images [14, 15].

In this chapter, our objective is to present a novel joint distribution of interferometric magnitude and phase for extremely heterogeneous clutter. We test the performance of the proposed distribution utilizing a representative dual-channel SAR image of urban area described as an extremely heterogeneous region.

Recently, Frery et al. [14,15,16] propose the reciprocal of the square root of a Gamma and the square root of a generalized inverse Gaussian to describe the signal components (RCS fluctuations) of SAR image magnitude for heterogeneous and extremely heterogeneous terrains. Motivated by this idea, based on famous complex Wishart distribution, this chapter utilizes the product model to deduce analytically the distribution models of interferogram’s magnitude under different environments, these distributions are simply referred to as the $\Gamma _{{{\mathcal{I}}n}}$ distribution, the ${\mathcal{K}}_{{{\mathcal{I}}n}}$ distribution and the $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distribution (corresponded to homogeneous, heterogeneous and extremely heterogeneous regions respectively). The presented distribution models are the successful generalization from the statistical model family of single-channel SAR images to the field of multi-channel SAR images. On this basis, using the second-kind statistics theory [17–19], i.e., MoLC, we derive the corresponding parameter estimators, as $\Gamma _{{{\mathcal{I}}n}}$_MoLC, ${\mathcal{K}}_{{{\mathcal{I}}n}}$_MoLC and $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$_MoLC. These estimators are capable of obtaining the iterative results accurately.

4.2 Normalized Interferogram

According to the central limit theorem, when the RCS of clutter background is constant, the in-phase and quadrature components of speckle are independent and identically distributed. Both of them obey the zero-mean complex Gaussian distribution [12], which meets the condition of the complex Wishart distribution. As the average of several independent samples, the $n$-look sample covariance matrix is given as [7]

$$ \hat{\varvec{R}} = \frac{1}{n}\sum\limits_{k = 1}^{n} {\varvec{Z}\left( k \right)\varvec{Z}\left( k \right)^{\text{H}} } = \frac{1}{n}\sum\limits_{k = 1}^{n} {\left[ {\begin{array}{*{20}c} {\left| {z_{1} \left( k \right)} \right|^{2} } & {z_{1} \left( k \right)z_{2} \left( k \right)^{*} } \\ {z_{1} \left( k \right)^{*} z_{2} \left( k \right)} & {\left| {z_{2} \left( k \right)} \right|^{2} } \\ \end{array} } \right]} $$

(4.1)

where $n$ represents number of looks, $\varvec{Z}\left( k \right) = \left[ {z_{1} \left( k \right),z_{2} \left( k \right)} \right]^{T}$ is the $k$th single-look image, the superscript * represents complex conjugate and ${\text{H}}$ means the complex conjugate transpose. The off-diagonal elements $\left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}} \right)\sum\nolimits_{{k = 1}}^{n} {z_{1} \left( k \right)z_{2} \left( k \right)^{*} }$ indicate the complex $n$-look interferogram.

According to the literature [8], random matrix $\varvec{B} = n\hat{\varvec{R}}$ obeys the complex Wishart distribution

$$ p_{\varvec{B}} \left( \varvec{B} \right) = \frac{{\left| \varvec{B} \right|^{n - 2} \exp \left[ { - tr\left( {\varvec{C}^{ - 1} \varvec{B}} \right)} \right]}}{{K\left( {n,2} \right)\left| \varvec{C} \right|^{n} }} $$

(4.2)

where $K\left( {n,2} \right) = \pi\Gamma \left( n \right)\Gamma \left( {n - 1} \right)$, the underlying covariance matrix $\varvec{C}$ is

$$ \varvec{C} = {\text{E}}\left[ {\varvec{ZZ}^{\text{H}} } \right] = \left[ {\begin{array}{*{20}c} {C_{11} } & {\sqrt {C_{11} C_{22} } \rho {\text{e}}^{j\theta } } \\ {\sqrt {C_{11} C_{22} } \rho {\text{e}}^{ - j\theta } } & {C_{22} } \\ \end{array} } \right] $$

(4.3)

where $\rho {\text{e}}^{j\theta }$ is the complex coefficient exported by the two channel. The magnitude of the complex correlation coefficient is represented by $\rho$, simply denotes as coherence magnitude in this chapter. Due to that the factors, such as thermal noise of receiver and speckle fluctuation noise, generate decorrelation, $\rho$ is normally 0.95–0.99 in ground scenes. Besides, $\theta$ only relates to the imaging geometry and scene altitude, therefore we often suppose $\theta$ to be zero on ground scene.

In (4.2), the 4 variables are processed to be normalized, then integrating the main diagonal elements to obtain the joint distribution of the normalized interferogram’s magnitude $\xi$ and the multi-look phase $\psi$:

$$ \begin{aligned} & p_{\xi ,\psi } \left( {\xi ,\psi } \right) = \frac{{2n^{n + 1} \xi^{n} }}{{\pi\Gamma \left( n \right)\left( {1 - \rho^{2} } \right)}}\exp \left( {\frac{{2n\rho \xi \,\cos \left( {\psi - \theta } \right)}}{{1 - \rho^{2} }}} \right)K_{n - 1} \left( {\frac{2n\xi }{{1 - \rho^{2} }}} \right) ,\\ & n,\xi ,\rho > 0\quad {\text{and}}\quad - \pi < \psi < \pi \\ \end{aligned} $$

(4.4)

where $K_{n - 1} \left( \cdot \right)$ is the second type modified Bessel function with order $\left( {n - 1} \right)$. $\Gamma \left( \cdot \right)$ indicates the Gamma function. Normalized magnitude $\xi$ of interferogram is defined as

$$ \xi = \frac{{\left| {\left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}} \right)\sum\nolimits_{k = 1}^{n} {z_{1} \left( k \right)z_{2} \left( k \right)^{*} } } \right|}}{{\sqrt {E\left( {\left| {z_{1} } \right|^{2} } \right)E\left( {\left| {z_{2} } \right|^{2} } \right)} }} = \frac{{\left| {\left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}} \right)\sum\nolimits_{k = 1}^{n} {z_{1} \left( k \right)z_{2} \left( k \right)^{*} } } \right|}}{{\sqrt {C_{11} C_{22} } }} $$

(4.5)

4.3 The Joint Distribution

4.3.1 The Known Joint Distribution for Heterogeneous Regions

Considering a $n$-look interferogram $I_{n}$, it is the average of $n$ single-look interferograms. Assuming the energy of two channels is identical, it’s well known that $I_{n}$ can be modeled by the multiplicative model as [4]

$$ I_{n} = A^{2} \xi {\text{e}}^{j\psi } $$

(4.6)

where $A$ represents the backscattering RCS magnitude of each channel.

As analyzed by Frery et al. [7], the random variable $A$ obeys a reciprocal of the square root of a Gamma distribution to characterize highly heterogeneous situation, i.e., $A\sim\Gamma ^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( { - \alpha ,\gamma } \right)$. Supposing $W = A^{2}$, the PDF of $W$ is given by

$$ p_{W} \left( w \right) = \frac{{\gamma^{ - \alpha } }}{{\Gamma \left( { - \alpha } \right)}}w^{\alpha - 1} \text{exp}\left( { - \frac{\gamma }{w}} \right),\quad - \alpha ,\gamma > 0 $$

(4.7)

where $\alpha$($- \alpha \in \left( {0,\infty } \right)$) is a shape parameter, which essentially reflects the degrees of homogeneity for processed areas. $\gamma$ is a scale parameter related to the mean energy of processed areas.

Therefore, the modified interferometric magnitude $\eta$ of heterogeneous clutter is given as $\eta = A^{2} \xi : = W\xi$ and the joint distribution of $\eta$ and $\psi$ can be expressed by

$$ p_{\eta ,\psi } \left( {\eta ,\psi } \right) = \int\limits_{0}^{\infty } {p_{W} \left( w \right)p\left( {\eta |w,\psi } \right)\text{d}w} $$

(4.8)

Applying (4.4), we get the right hand-side of the integral shown in (4.8) as

$$ \begin{aligned} p\left( {\eta |w,\psi } \right) & = \frac{1}{w}p\left( {\frac{\eta }{w},\psi } \right) \\ & = \frac{{2n^{n + 1} \eta_{{}}^{n} }}{{\pi\Gamma \left( n \right)\left( {1 - \rho^{2} } \right)}}\left( {\frac{1}{w}} \right)^{n + 1} \exp \left( {\frac{2n\eta \rho \,\cos \psi }{{1 - \rho^{2} }} \times \frac{1}{w}} \right)K_{n - 1} \left( {\frac{2n\eta }{{1 - \rho^{2} }} \times \frac{1}{w}} \right) \\ \end{aligned} $$

(4.9)

Combining (4.7) and (4.9) by (4.8), and utilizing the integral formula $\int\limits_{0}^{\infty } {x^{\mu - 1} {\text{e}}^{ - ax} K_{v} \left( {bx} \right){\text{d}}x} = \frac{{\sqrt \pi \left( {2b} \right)^{v} }}{{\left( {a + b} \right)^{\mu + v} }} \cdot \frac{{\Gamma \left( {\mu + v} \right)\Gamma \left( {\mu - v} \right)}}{{\Gamma \left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}}{}_{2}F_{1} \left( {\mu + v,v + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2};\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2};\frac{a - b}{a + b}} \right)$ [8], the joint distribution of magnitude and phase in the heterogeneous clutter is finally derived as [4]

$$ \begin{aligned} p_{\eta ,\psi } \left( {\eta ,\psi } \right) = & \frac{{\left( {2n} \right)^{2n} \gamma^{ - \alpha } }}{{2\sqrt \pi \left( {1 - \rho^{2} } \right)^{{ - \left( {n - \alpha } \right)}} }} \cdot \frac{{\Gamma \left( {2n - \alpha } \right)\Gamma \left( { - \alpha + 2} \right)}}{{\Gamma \left( n \right)\Gamma \left( { - \alpha } \right)\Gamma \left( {n - \alpha + {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}} \right)}} \cdot \eta^{2n - 1} \\ & \quad \cdot \left[ {f_{1} \left( {\eta ,\psi } \right)} \right]^{{ - \left( {2n - \alpha } \right)}} \cdot {}_{2}F_{1} \left( {2n - \alpha ,n - \frac{1}{2};n - \alpha + \frac{3}{2};f_{2} \left( {\eta ,\psi } \right)} \right) \\ \end{aligned} $$

(4.10)

where $_{2} F_{1}$ is the Gauss hypergeometric function and

$$ \left\{ \begin{aligned} f_{1} \left( {\eta ,\psi } \right) = \left( {1 - \rho^{2} } \right)\gamma + 2n\eta \left( {1 - \rho \,\cos \psi } \right) \hfill \\ f_{2} \left( {\eta ,\psi } \right) = \frac{{\left( {1 - \rho^{2} } \right)\gamma - 2n\eta \left( {1 + \rho \,\cos \psi } \right)}}{{\left( {1 - \rho^{2} } \right)\gamma + 2n\eta \left( {1 - \rho \,\cos \psi } \right)}} \hfill \\ \end{aligned} \right. $$

(4.11)

4.4 The Proposed Distribution for Interferogram’s Magnitude of Homogenous Clutter

4.4.1 The $\Gamma _{{{\mathcal{I}}n}}$ Distribution for Homogeneous Clutter

To obtain probability distribution of interferogram’s magnitude under homogeneous clutter background, the phase variable in (4.4) needs to be integrated. Via the following equation [21]:

$$ \int\limits_{0}^{\pi } {\exp \left( {z\,\cos x} \right){\text{d}}x} = \pi I_{0} \left( z \right) $$

(4.12)

The marginal PDF of normalized interferogram’s magnitude can be obtained as [3]

$$ p_{\xi } \left( \xi \right) = \frac{{4n^{n + 1} \xi^{n} }}{{\Gamma \left( n \right)\left( {1 - \rho^{2} } \right)}}I_{0} \left( {\frac{2n\rho \xi }{{1 - \rho^{2} }}} \right)K_{n - 1} \left( {\frac{2n\xi }{{1 - \rho^{2} }}} \right),\quad n,\xi ,\rho > 0 $$

(4.13)

where $I_{0} \left( \cdot \right)$ is the first type modified Bessel function of order zero. (4.13) is originally derived by Lee et al. [3].

The (4.13) has the following properties:

• The value of the first type modified Bessel function of order zero will tend to infinite rapidly [21] and $I_{0} \left( x \right) \to 1$ on the condition that $x \to 0^{ + }$. See Fig. 4.1;

  Fig. 4.1

  

  Plot of the first type modified Bessel function of order zero with $n = 4$ and $\rho = 0.9$

• Contrarily, the value of the second type modified Bessel function will tend to zero rapidly [21], see Fig. 4.2. Whereas $K_{n - 1} \left( x \right) \approx \left\{ {\begin{array}{*{20}l} { - \ln x,} \hfill & {n = 1} \hfill \\ {\frac{{\Gamma \left( {n - 1} \right)}}{2}\left( {\frac{2}{x}} \right)^{n - 1} ,} \hfill & {n > 1} \hfill \\ \end{array} } \right.$ when $x \to 0^{ + }$;

  Fig. 4.2

  

  Plot of the second type modified Bessel Function with $n = 4$ and $\rho = 0.9$

The two modified Bessel functions shown in (4.13) are generally evaluated by numerical methods. Owing to the complicated expressions of two types of modified Bessel function, the practical applications such as moving target detection by this distribution shown in (4.13) are limited, i.e., computation time, the analytical expression of detection threshold etc. It is a hard task to obtain the CFAR detection threshold by making an integral for (4.13).

Herein, our objective is to simplify the (4.13). Applying the asymptotic expansion formulas [21] of the first type and the second type modified Bessel functions, i.e.,

$$ \begin{aligned} & I_{v} \left( x \right) \cong \frac{\exp \left( x \right)}{{\sqrt {2\pi x} }}\sum\limits_{k = 0}^{K} {\left[ {\left( { - 1} \right)^{k} \frac{{\Gamma \left( {v + k + \frac{1}{2}} \right)}}{{\Gamma \left( {k + 1} \right)\Gamma \left( {v + \frac{1}{2} - k} \right)}}\frac{1}{{\left( {2x} \right)^{k} }}} \right]} \\ & K_{v} \left( x \right) \cong \frac{{\sqrt \pi \,\exp \left( { - x} \right)}}{{\sqrt {2x} }}\sum\limits_{k = 0}^{K} {\left[ {\frac{{\Gamma \left( {v + k - \frac{1}{2}} \right)}}{{\Gamma \left( {k + 1} \right)\Gamma \left( {v + \frac{1}{2} - k} \right)}}\frac{1}{{\left( {2x} \right)^{k} }}} \right]} \\ \end{aligned} $$

(4.14)

Then $I_{0} \left( {c\rho } \right)K_{n - 1} \left( c \right)$ $\left( {c = \frac{2n\xi }{{1 - \rho^{2} }}} \right)$ in (4.13) can be denoted as

$$ \begin{aligned} I_{0} \left( {c\rho } \right)K_{n - 1} \left( c \right)& \cong \frac{{\exp \left( { - \left( {1 - \rho } \right)c} \right)}}{2\sqrt \rho c}\left( {\sum\limits_{k = 0}^{K} {\left( { - 1} \right)^{k} \frac{{\Gamma \left( {k + \frac{1}{2}} \right)}}{{\left( {2\rho c} \right)^{k}\Gamma \left( {k + 1} \right)\Gamma \left( {\frac{1}{2} - k} \right)}}} } \right) \\ & \quad \cdot \left( {\sum\limits_{k = 0}^{K} {\frac{{\Gamma \left( {n + k - \frac{1}{2}} \right)}}{{\left( {2c} \right)^{k}\Gamma \left( {k + 1} \right)\Gamma \left( {n - \frac{1}{2} - k} \right)}}} } \right) \\ & = \frac{{\exp \left( { - \left( {1 - \rho } \right)c} \right)}}{2\sqrt \rho } \\ & \quad \sum\limits_{k = 0}^{K} {\sum\limits_{m = 0}^{K} {\frac{{\left( { - 1} \right)^{k}\Gamma \left( {k + \frac{1}{2}} \right)\Gamma ^{ - 1} \left( {k + 1} \right)\Gamma \left( {n + m - \frac{1}{2}} \right)}}{{2^{k + m} \rho^{k} c^{k + m + 1}\Gamma \left( {\frac{1}{2} - k} \right)\Gamma \left( {m + 1} \right)\Gamma \left( {n - \frac{1}{2} - m} \right)}}} } \\ \end{aligned} $$

(4.15)

Further, the PDF of $\xi$ in (4.13) is approximated by

$$ \begin{aligned} p_{\xi } \left( \xi \right)& \cong \frac{{n^{n} }}{{\Gamma \left( n \right)}}\xi^{n - 1} \exp \left( { - \frac{2n}{1 + \rho }\xi } \right) \\ & \quad \cdot \sum\limits_{k = 0}^{K} {\sum\limits_{m = 0}^{K} {\left[ {\frac{{\left( { - 1} \right)^{k}\Gamma \left( {k + \frac{1}{2}} \right)\Gamma ^{ - 1} \left( {k + 1} \right)\Gamma \left( {n + m - \frac{1}{2}} \right)}}{{\Gamma \left( {\frac{1}{2} - k} \right)\Gamma \left( {m + 1} \right)\Gamma \left( {n - \frac{1}{2} - m} \right)\rho^{{k + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }} \cdot \left( {\frac{{1 - \rho^{2} }}{4n\xi }} \right)^{k + m} } \right]} } \\ \end{aligned} $$

(4.16)

As some examples shown in Fig. 4.3, we found that the dual summing part of (4.16) has a very small fluctuation in the range of $\xi \in {\mathbb{R}}^{ + }$, which can be approximated¹ by a constant value $m_{0}$. Utilizing $\int_{0}^{\infty } {p\left( \xi \right){\text{d}}\xi } = 1$, $m_{0} = {{2^{n} } \mathord{\left/ {\vphantom {{2^{n} } {\left( {1 + \rho } \right)^{n} }}} \right. \kern-0pt} {\left( {1 + \rho } \right)^{n} }}$. We have the PDF of the $\xi$ as

Fig. 4.3

The values of the dual summing part in (4.16) under different number of looks with $K = 50$ and $\rho = 0.95$

$$ p_{\xi } \left( \xi \right) = \frac{{\beta n^{n} }}{{\Gamma \left( n \right)}}\left( {\beta \xi } \right)^{n - 1} \exp \left( { - n\beta \xi } \right),\quad \beta ,n,\xi > 0 $$

(4.17)

where $\beta = {2 \mathord{\left/ {\vphantom {2 {\left( {1 + \rho } \right)}}} \right. \kern-0pt} {\left( {1 + \rho } \right)}}$.

Making a Definition of $\tau = \sqrt {C_{11} C_{22} } \xi : = \sigma \xi$, hence $\tau$ represents the interferogram’s magnitude. The PDF of $\tau$ is given analytically by transforming (4.17) as $\Gamma _{{{\mathcal{I}}n}} \left( {n,\beta ,\sigma } \right)$, i.e.,

$$ p_{\tau } \left( \tau \right) = \frac{1}{{\Gamma \left( n \right)}}\left( {\frac{n\beta }{\sigma }} \right)^{n} \tau^{n - 1} \exp \left( { - \frac{n\beta }{\sigma }\tau } \right),\quad \beta ,n,\sigma ,\tau > 0 $$

(4.18)

The (4.18) is the distribution of interferogram’s magnitude of homogeneous clutter, hereafter simply as the $\Gamma _{{{\mathcal{I}}n}}$ distribution.

Once $\rho = 1$ (i.e., two channel outputs are completely correlated), $\Gamma _{{{\mathcal{I}}n}}$ is degenerated as

$$ p_{\tau } \left( \tau \right) = \frac{1}{{\Gamma \left( n \right)}}\left( {\frac{n}{\sigma }} \right)^{n} \tau^{n - 1} \exp \left( { - \frac{n}{\sigma }\tau } \right),\quad n,\sigma ,\tau > 0 $$

(4.19)

The (4.19) is exactly the classical Gamma distribution $\Gamma \left( {n,{n \mathord{\left/ {\vphantom {n \sigma }} \right. \kern-0pt} \sigma }} \right)$ [12, 16].

4.4.2 Parameter Estimators of $\Gamma _{{{\mathcal{I}}n}}$

• (1) Coherence magnitude estimation

Normally, the estimation of coherence magnitude $\rho$ is derived from the following equation [10]

$$ \hat{\rho } = {{\left| {\sum\limits_{{k = 1}}^{N} {z_{1} \left( k \right)z_{2} \left( k \right)^{*} } } \right|} \mathord{\left/ {\vphantom {{\left| {\sum\limits_{{k = 1}}^{N} {z_{1} \left( k \right)z_{2} \left( k \right)^{*} } } \right|} {\sqrt {\sum\limits_{{k = 1}}^{N} {\left| {z_{1} \left( k \right)} \right|^{2} } \sum\limits_{{k = 1}}^{N} {\left| {z_{2} \left( k \right)} \right|^{2} } } }}} \right. \kern-\nulldelimiterspace} {\sqrt {\sum\limits_{{k = 1}}^{N} {\left| {z_{1} \left( k \right)} \right|^{2} } \sum\limits_{{k = 1}}^{N} {\left| {z_{2} \left( k \right)} \right|^{2} } } }} $$

(4.20)

where $N$ is sample number, the PDF of estimation $\hat{\rho }$ is given as

$$ f\left( {\hat{\rho }} \right) = 2\left( {N - 1} \right)\left( {1 - \rho^{2} } \right)^{N} \hat{\rho }\left( {1 - \hat{\rho }^{2} } \right)^{N - 2} {}_{2}F_{1} \left( {N,N;1;\rho^{2} \hat{\rho }^{2} } \right) $$

(4.21)

where ${}_{p}F_{q}$ is the generalized hypergeometric function. In [10], the first moment of $\hat{\rho }$ is

$$ {\text{E}}\left( {\hat{\rho }} \right) = \frac{{\Gamma \left( N \right)\Gamma \left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}} \right)}}{{\Gamma \left( {N + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}}\left( {1 - \rho^{2} } \right)^{N} \cdot {}_{3}F_{2} \left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2},N,N;N + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2},1;\rho^{2} } \right) $$

(4.22)

This classical estimation shown in (4.20) is biased [10]. Therefore, starting from the definition of normalized interferogram ${\mathcal{I}}_{n} = {{\left( {{1 \mathord{\left/ {\vphantom {\varvec{1} n}} \right. \kern-0pt} n}} \right)\sum\nolimits_{k = 1}^{n} {z_{1} \left( k \right)z_{2} \left( k \right)^{*} } } \mathord{\left/ {\vphantom {{\left( {{\varvec{1} \mathord{\left/ {\vphantom {\varvec{1} n}} \right. \kern-0pt} n}} \right)\sum\nolimits_{k = 1}^{n} {z_{1} \left( k \right)z_{2} \left( k \right)^{*} } } {\sqrt {\text{E}\left( {\left| {z_{1} } \right|^{2} } \right)\text{E}\left( {\left| {z_{2} } \right|^{2} } \right)} }}} \right. \kern-0pt} {\sqrt {\text{E}\left( {\left| {z_{1} } \right|^{2} } \right)\text{E}\left( {\left| {z_{2} } \right|^{2} } \right)} }}$, this chapter applies (4.4) to derive 1th order moment of ${\mathcal{I}}_{n}$ (see Appendix 4.1). The coherence magnitude estimation can be described exactly as

$$ \hat{\rho } = \left| {\text{E}\left( {{\mathcal{I}}_{\text{n}} } \right)} \right| $$

(4.23)

This estimation is unbiased as $\hat{\rho } = \text{E}\left( {\hat{\rho }} \right) = \left| {\text{E}\left( {{\mathcal{I}}_{\text{n}} } \right)} \right|$. After finishing the estimation of $\rho$, $\beta$ is a constant in the $\Gamma _{{{\mathcal{I}}n}}$ distribution.

• (2) $\Gamma _{{{\mathcal{I}}n}}$_MoLC

The second-kind first characteristic function and the second-kind second characteristic function of the $\Gamma _{{{\mathcal{I}}n}}$ distribution are:

$$ \left\{ {\begin{array}{*{20}l} {\phi_{{\Gamma _{{{\mathcal{I}}n}} }} \left( s \right) = \left( {\frac{\sigma }{\beta n}} \right)^{s - 1} \frac{{\Gamma \left( {n + s - 1} \right)}}{{\Gamma \left( n \right)}}} \hfill \\ {\zeta_{{\Gamma _{{{\mathcal{I}}n}} }} \left( s \right) = \left( {s - 1} \right)\ln \left( {\frac{\sigma }{\beta n}} \right) + \ln\Gamma \left( {n + s - 1} \right) - \ln\Gamma \left( n \right)} \hfill \\ \end{array} } \right. $$

(4.24)

The $\Gamma _{{{\mathcal{I}}n}}$ distribution’s log-cumulants are given by

$$ \left\{ {\begin{array}{*{20}l} {\tilde{c}_{1} = - \ln \left( {{{\beta n} \mathord{\left/ {\vphantom {{\beta n} \sigma }} \right. \kern-0pt} \sigma }} \right) +\Psi \left( n \right)} \hfill \\ {\tilde{c}_{k} =\Psi \left( {k - 1,n} \right),\quad k \ge 2} \hfill \\ \end{array} } \right. $$

(4.25)

where $\Psi \left( \cdot \right)$ represents the digamma function (i.e., the logarithmic derivative of the Gamma function), and $\Psi \left( {k, \cdot } \right)$ is the $k$ th order polygamma function (i.e., the $k$ th order derivative of the digamma function).

Given a sample set $\left\{ {x_{i} } \right\},i \in \left[ {1,N} \right]$, The estimation expressions of parameters $n$ and $\sigma$ in the $\Gamma _{{{\mathcal{I}}n}}$ distribution can be derived as

$$ \left\{ {\begin{array}{*{20}l} {\Psi \left( {\hat{n}} \right) - \ln \left( {\frac{{\beta \hat{n}}}{{\hat{\sigma }}}} \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\ln \left( {x_{i} } \right)} \right]} } \hfill \\ {\Psi \left( {1,\hat{n}} \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\left( {\ln \left( {x_{i} } \right) - \hat{\tilde{c}}_{1} } \right)^{2} } \right]} } \hfill \\ \end{array} } \right. $$

(4.26)

This equation shows the method of log-cumulants of the $\Gamma _{{{\mathcal{I}}n}}$ distribution, called $\Gamma _{{{\mathcal{I}}n}}$_MoLC.

4.5 Statistics of Multilook SAR Interferogram for In-homogeneous Clutter Based on $\Gamma _{{{\mathcal{I}}n}}$

4.5.1 Extremely Heterogeneous Clutter

For heterogeneous terrain like forest, cultivated farmland, RCS of them have fluctuations. Meanwhile, for extremely heterogeneous clutter like the urban areas, the histograms show the heavy trail [22,23,24,25]. The performance of the $\Gamma _{{{\mathcal{I}}n}}$ distribution decreases and it generates large deviations in fitting these magnitude data. The product model has been testified that it is valid for statistical modeling of single-channel SAR magnitude. Its expression is as follows [12, 16],

$$ Y_{i} = A_{i} X_{i} ,\;\;i = 1,2\quad {\text{and}}\quad X_{i} \sim\mathcal{N}^{\mathbb{C}} \left( {0,1} \right) $$

(4.27)

where $A_{i}$ represents the backscattering RCS magnitude component. $\mathcal{N}^{\mathbb{C}} \left( {0,1} \right)$ denotes the complex normal distribution with expectation 0 and variance 1. $X_{i}$ indicates speckle noise component, $i$ is the independent receiving channel.

Assuming the energy of two channel is equal to each other, i.e., $A_{i} \equiv A$, the interferogram’s magnitude could be expressed as

$$ \varXi = A^{2} \xi = W\xi $$

(4.28)

Frery et al. recently proposed a reciprocal of the square root of a Gamma distribution for the modulating random variable $A$ of heterogeneous clutter [14,15,16], i.e., $A\sim\Gamma ^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( { - \alpha ,\gamma } \right)$, It can be shown that the PDF of $W$ is

$$ p_{W} \left( w \right) = \frac{{\gamma^{ - \alpha } }}{{\Gamma \left( { - \alpha } \right)}}w^{\alpha - 1} \text{exp}\left( { - \frac{\gamma }{w}} \right),\quad - \alpha ,\gamma > 0 $$

(4.29)

where $\alpha$($- \alpha \in \left( {0,\infty } \right)$) is a shape parameter, which essentially reflects the degrees of homogeneity for processed areas. $\gamma$ is a scale parameter related to the mean energy of processed areas.

Herein, within the structure of product model, combining the $\Gamma _{{{\mathcal{I}}n}}$ distribution and (4.29), and using the equation $\int_{0}^{\infty } {x^{m - 1} {\text{e}}^{{ - \left( {a + 1} \right)x}} {\text{d}}x} = \frac{{\Gamma \left( m \right)}}{{\left( {a + 1} \right)^{m} }} ,\;m > 0,a > - 1$ [21], the PDF of $\varXi$ can be derived as $\mathcal{G}_{{{\mathcal{I}}n}}^{0} \left( {n,\alpha ,\beta ,\gamma } \right)$:

$$ p_{\varXi } \left( \varXi \right) = \frac{{\beta n^{n} \gamma^{ - \alpha }\Gamma \left( {n - \alpha } \right)}}{{\Gamma \left( n \right)\Gamma \left( { - \alpha } \right)}} \cdot \frac{{\left( {\beta \varXi } \right)^{n - 1} }}{{\left( {\gamma + n\beta \varXi } \right)^{n - \alpha } }},\quad \beta , - \alpha ,\gamma ,n,\varXi > 0 $$

(4.30)

Here (4.30) is called as $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distribution. This model enables the modeling of areas with varying degrees of homogeneity owing to the importing of the reciprocal of a square root of Gamma distribution.

Especially, when the two channels are completely correlated, $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ is degenerating to the well-known intensity distribution ${\mathcal{G}}_{I}^{\text{0}} \left( {\alpha ,\gamma ,n} \right)$ [16, 26, 27]:

$$ p_{\varXi } \left( \varXi \right) = \frac{{n^{n} \gamma^{ - \alpha }\Gamma \left( {n - \alpha } \right)}}{{\Gamma \left( n \right)\Gamma \left( { - \alpha } \right)}} \cdot \frac{{\varXi^{n - 1} }}{{\left( {\gamma + n\varXi } \right)^{n - \alpha } }},\quad - \alpha ,\gamma ,n,\varXi > 0 $$

(4.31)

4.5.2 Heterogeneous Clutter

For heterogeneous clutter in a single-channel SAR image, the ${\mathcal{K}}$ distribution is obtained when assuming a multiplicative noise model for SAR amplitude by expressing the RCS component as a square root of Gamma distribution [14,15,16, 12], i.e., $A\sim\Gamma ^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {\alpha ,\lambda } \right)$. We here rewrite (4.28) as $\ell = A^{2} \xi = \upsilon \xi$, the corresponding PDF of $\upsilon$ is expressed as

$$ p_{\upsilon } \left( \upsilon \right) = \frac{{\lambda^{\alpha } }}{{\Gamma \left( \alpha \right)}}\upsilon^{\alpha - 1} \exp \left( { - \lambda \upsilon } \right),\quad \alpha ,\lambda ,\upsilon > 0 $$

(4.32)

Thus, the deriving process is similar as that of $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distribution. We utilize the product model and combine the $\Gamma _{{{\mathcal{I}}n}}$ distribution and (4.32), thus resulting in the following distribution (named for convenience as ${\mathcal{K}}_{{{\mathcal{I}}n}}$) via $\int_{0}^{\infty } {x^{\alpha - 1} \exp \left( { - \frac{\gamma }{x} - \lambda x} \right)dx} = 2\left( {\frac{\gamma }{\lambda }} \right)^{{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}} K_{\alpha } \left( {2\sqrt {\gamma \lambda } } \right)$ for the magnitude of interferogram:

$$ p_{\ell } \left( \ell \right) = \frac{2\lambda \beta n}{{\Gamma \left( n \right)\Gamma \left( \alpha \right)}} \cdot \left( {\lambda \beta n\ell } \right)^{{{{\left( {\alpha + n} \right)} \mathord{\left/ {\vphantom {{\left( {\alpha + n} \right)} 2}} \right. \kern-0pt} 2} - 1}} \cdot K_{\alpha - n} \left( {2\sqrt {\lambda \beta n\ell } } \right),\quad \beta ,\alpha ,\lambda ,n,\ell > 0 $$

(4.33)

This distribution can fit a wide range of experimental data well because the square root of Gamma distribution is a well known model for RCS component in homogeneous and heterogeneous clutter. Especially, when the two channels are completely correlated, ${\mathcal{K}}_{{{\mathcal{I}}n}}$ is degenerating to the well-known ${\mathcal{K}}$ intensity distribution ${\mathcal{K}}_{I}^{{}} \left( {\alpha ,\lambda ,n} \right)$ [16, 12]:

$$ p_{\ell } \left( \ell \right) = \frac{2\lambda n}{{\Gamma \left( n \right)\Gamma \left( \alpha \right)}} \cdot \left( {\lambda n\ell } \right)^{{{{\left( {\alpha + n} \right)} \mathord{\left/ {\vphantom {{\left( {\alpha + n} \right)} 2}} \right. \kern-0pt} 2} - 1}} \cdot K_{\alpha - n} \left( {2\sqrt {\lambda n\ell } } \right),\quad \alpha ,\lambda ,n,\ell > 0 $$

(4.34)

4.5.3 Parameter Estimators of In-homogeneous Clutter Statistics

• (1) ${\mathcal{K}}_{{{\mathcal{I}}n}}$_MoLC

The second-kind first characteristic function and the second-kind second characteristic function of the ${\mathcal{K}}_{{{\mathcal{I}}n}}$ distribution are

$$ \left\{ \begin{aligned} \phi_{{{\mathcal{K}}_{{{\mathcal{I}}n}} }} \left( s \right) & = \left( {\lambda \beta n} \right)^{{ - \left( {s - 1} \right)}} \frac{{\Gamma \left( {\alpha + s - 1} \right)\Gamma \left( {n + s - 1} \right)}}{{\Gamma \left( \alpha \right)\Gamma \left( n \right)}} \\ \zeta_{{{\mathcal{K}}_{{{\mathcal{I}}n}} }} \left( s \right) & = - \left( {s - 1} \right)\ln \left( {\lambda \beta n} \right) + \ln\Gamma \left( {\alpha + s - 1} \right) \\ & \quad + \ln\Gamma \left( {n + s - 1} \right) - \ln\Gamma \left( \alpha \right) - \ln\Gamma \left( n \right) \\ \end{aligned} \right. $$

(4.35)

The estimation of parameters in the ${\mathcal{K}}_{{{\mathcal{I}}n}}$ distribution can be derived as

$$ \left\{ {\begin{array}{*{20}l} {\Psi \left( {\hat{n}} \right) +\Psi \left( {\hat{\alpha }} \right) - \ln \left( {\hat{\lambda }\beta \hat{n}} \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\ln \left( {x_{i} } \right)} \right]} } \hfill \\ {\Psi \left( {1,\hat{n}} \right) +\Psi \left( {1,\hat{\alpha }} \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\left( {\ln \left( {x_{i} } \right) - \hat{\tilde{c}}_{1} } \right)^{2} } \right]} } \hfill \\ {\Psi \left( {2,\hat{n}} \right) +\Psi \left( {2,\hat{\alpha }} \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\left( {\ln \left( {x_{i} } \right) - \hat{\tilde{c}}_{1} } \right)^{3} } \right]} } \hfill \\ \end{array} } \right. $$

(4.36)

This estimator is called as ${\mathcal{K}}_{{{\mathcal{I}}n}}$_MoLC.

• (2) $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$_MoLC

The second-kind first characteristic function and the second-kind second characteristic function of the $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distribution, i.e.,

$$ \left\{ \begin{aligned} \phi_{{\mathcal{G}_{{{\mathcal{I}}n}}^{0} }} \left( s \right) & = \left( {\frac{\gamma }{\beta n}} \right)^{s - 1} \cdot \frac{{\Gamma \left( {n + s - 1} \right)\Gamma \left( { - \alpha - \left( {s - 1} \right)} \right)}}{{\Gamma \left( n \right)\Gamma \left( { - \alpha } \right)}} \\ \zeta_{{\mathcal{G}_{{{\mathcal{I}}n}}^{0} }} \left( s \right) & = \left( {s - 1} \right)\ln \left( {\frac{\gamma }{\beta n}} \right) + \ln\Gamma \left( {n + s - 1} \right) + \ln\Gamma \left( { - \alpha - \left( {s - 1} \right)} \right) \\ & \quad - \ln\Gamma \left( n \right) - \ln\Gamma \left( { - \alpha } \right) \\ \end{aligned} \right. $$

(4.37)

Calculating the derivative of $\zeta_{{\mathcal{G}_{{{\mathcal{I}}n}}^{0} }} \left( s \right)$ at $s = 1$, the log-cumulants of the $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distribution is described by

$$ \left\{ {\begin{array}{*{20}l} {\tilde{c}_{1} = \ln \left( {\frac{\gamma }{\beta n}} \right) +\Psi \left( n \right) -\Psi \left( { - \alpha } \right)} \hfill \\ {\tilde{c}_{k} =\Psi \left( {k - 1,n} \right) + \left( { - 1} \right)^{k}\Psi \left( {k - 1, - \alpha } \right),\quad k \ge 2} \hfill \\ \end{array} } \right. $$

(4.38)

The expressions of estimating parameters $\alpha$, $\gamma$, $n$ in the $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distribution by (4.38) and (4.26) are given as

$$ \left\{ {\begin{array}{*{20}l} {\ln \left( {{{\hat{\gamma }} \mathord{\left/ {\vphantom {{\hat{\gamma }} {\left( {\beta \hat{n}} \right)}}} \right. \kern-0pt} {\left( {\beta \hat{n}} \right)}}} \right) +\Psi \left( {\hat{n}} \right) -\Psi \left( { - \hat{\alpha }} \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\ln \left( {x_{i} } \right)} \right]} } \hfill \\ {\Psi \left( {1,\hat{n}} \right) +\Psi \left( {1, - \hat{\alpha }} \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\left( {\ln \left( {x_{i} } \right) - \hat{\tilde{c}}_{1} } \right)^{2} } \right]} } \hfill \\ {\Psi \left( {2,\hat{n}} \right) -\Psi \left( {2, - \hat{\alpha }} \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\left( {\ln \left( {x_{i} } \right) - \hat{\tilde{c}}_{1} } \right)^{3} } \right]} } \hfill \\ \end{array} } \right. $$

(4.39)

Similarly, the above equation is called as $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$_MoLC.

4.5.4 Relationship Between Distributions

Figure 4.4 shows the relationship of presented distributions mentioned-above. It is clear that these distributions have the following properties.

Fig. 4.4

The relationship of distributions

• (1) The interferogram’s magnitude PDFs are “downward compatible”. It can be proved that the $\mathcal{G}_{{{\mathcal{I}}n}}^{0} \left( {\beta ,\alpha ,\gamma ,n} \right)$ converges in distribution to the $\Gamma _{{{\mathcal{I}}n}} \left( {\beta ,n,\sigma } \right)$ when $- \alpha ,\gamma \to \infty$ and ${{ - \alpha } \mathord{\left/ {\vphantom {{ - \alpha } \gamma }} \right. \kern-0pt} \gamma } \to {1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-0pt} \sigma }$. The ${\mathcal{K}}_{{{\mathcal{I}}n}} \left( {\beta ,\alpha ,\lambda ,n} \right)$ converges in distribution to the $\Gamma _{{{\mathcal{I}}n}} \left( {\beta ,n,\sigma } \right)$ when $\alpha ,\lambda \to \infty$ and ${\alpha \mathord{\left/ {\vphantom {\alpha \lambda }} \right. \kern-0pt} \lambda } \to \sigma$. Moreover, $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ encompasses the modeling abilities of ${\mathcal{K}}_{{{\mathcal{I}}n}}$ whilst extending them to enable the modeling of extremely heterogeneous data, because of the empirical evidence [16] describing the relationship of the square root of Gamma distribution and the reciprocal of a square root of Gamma distribution. The properties stated in Fig. 4.4 show that either homogeneous, heterogeneous, or extremely heterogeneous interferogram’s magnitude statistics can be treated as the outcome of the $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distribution.

• (2) The interferogram’s magnitude PDFs are “channel compatible”. Herein, the “channel compatible” means that the interferogram’s magnitude PDF of multi-channel SAR images is compatible with intensity PDF of single channel SAR images. When the two channel outputs are completely correlated, the $\Gamma _{{{\mathcal{I}}n}}$, the ${\mathcal{K}}_{{{\mathcal{I}}n}}$, and the $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distributions are respectively degenerated to the $\Gamma$, the ${\mathcal{K}}_{I}^{{}}$, and the ${\mathcal{G}}_{I}^{\text{0}}$ distributions, see (4.19), (4.34), and (4.31). These three known distributions are extensively used in modeling single channel SAR intensity images [16, 20].

Additionally, from Fig. 4.4 and Eqs. (4.18), (4.19), (4.30), (4.31), (4.33) and (4.34), due to $\beta$ can be regarded as a constant, we have the following results:

$$ \begin{aligned}\Gamma _{{{\mathcal{I}}n}} \left( {n,\beta ,\sigma } \right) & =\Gamma \left( {n,{\sigma \mathord{\left/ {\vphantom {\sigma \beta }} \right. \kern-0pt} \beta }} \right) \\ {\mathcal{K}}_{{{\mathcal{I}}n}} \left( {n,\alpha ,\beta ,\lambda } \right) & = {\mathcal{K}}_{I} \left( {n,\alpha ,\lambda \beta } \right) \\ \mathcal{G}_{{{\mathcal{I}}n}}^{0} \left( {n,\alpha ,\beta ,\gamma } \right) & = {\mathcal{G}}_{I}^{\text{0}} \left( {n,\alpha ,{\gamma \mathord{\left/ {\vphantom {\gamma \beta }} \right. \kern-0pt} \beta }} \right) \\ \end{aligned} $$

(4.40)

This property indicates the presented distribution models successfully generalize the powerful statistical model family of single-channel SAR images to the field of multi-channel SAR images.

4.5.5 Experimental Analysis

In this section, our major objective is to investigate how the proposed distribution models perform well on the really measured InSAR data. The tested InSAR data used in this study were collected by NASA/JPL’s airborne platform (known as AirSAR) operated in C band and standard dual-baseline ATI mode, with the spatial resolution 3.331 m × 3.898 m (range × azimuth). The name of this flight-line (site name) is usamacinta245-1, and the imaging time of ground scene is March 4, 2004. The original ground scene is very large and abundant terrain classes are included. The detailed imaging parameters corresponding to this scene are listed in Table 4.1.

Table 4.1 The imaging parameters of the ground scene on usamacinta245-1 site

For convenience of displaying, a local image of normalized interferometric magnitude from this scene is shown in Fig. 4.5 in dB format. The horizontal and vertical axes are the directions of azimuth and range, respectively. Four patches indicated by boxes in Fig. 4.5, numbered A–D, are the main areas of this investigation. The chosen patches consist of very different terrain, form homogeneous grassland area (Patch A) to heterogeneous area (Patch B) dominated by vegetation of trees, even extremely heterogeneous urban areas (Patches C and D), and hence, provide a wide variety of scattering properties.

Fig. 4.5

The normalized interferometric magnitude image collected by the AirSAR on tested site

Figures 4.6 show the fitting results of the proposed distributions for the four areas indicated in Fig. 4.5. As seen from Fig. 4.6, in order to reveal the details more clearly, the square root of the normalized interferometric magnitude is adopted to suppress the dynamic range. The parameter estimations of each distribution are all accomplished by the proposed estimators based on the MoLC, which are shown at Table 4.2.

Fig. 4.6

Plots of interferometric magnitude histogram for the selected areas and of the estimated $\Gamma _{{{\mathcal{I}}n}}$, ${\mathcal{K}}_{{{\mathcal{I}}n}}$ and $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ PDFs: (a–d) are corresponded to Patches A–D, respectively

Table 4.2 Parameter estimations of noted clutter areas in Fig. 4.5

In order to quantitatively assess the fitting result, we adopt KL (Kullback-Leibler) distance [28] as a similarity measurement. This measurement is the means of the global comparison of PDFs [28].

When the actual density equals to theoretical density, the KL distance $D_{KL}$ is zero. Otherwise, $D_{KL}$ is a positive value. The KL distance measurement reflects the overall similarity of the actual and theoretical densities. The smaller the value of KL distance measurement obtains, the higher similarity they have, which shows that fitting accuracy is better.

The KL values of the fitting results shown in Fig. 4.6 are compared in Table 4.3. It is evident that the following conclusions can be drawn, i.e.,

Table 4.3 Statistical Characteristics of Different Patches in Fig. 4.5

• (1) For all four typical areas, the performance of $\Gamma _{{{\mathcal{I}}n}}$ is the relatively worst. However, the $\Gamma _{{{\mathcal{I}}n}}$ distribution tends to agree well with the areas with higher degree of homogeneity, which implies that $\Gamma _{{{\mathcal{I}}n}}$ is a proper statistical model for the homogeneous areas. Also, all the three densities tend to be alike for fitting areas with higher degree of homogeneity.

• (2) With the increasing in-homogeneity degree of the data, the higher precision of fitting is obtained by using ${\mathcal{K}}_{{{\mathcal{I}}n}}$ than by using $\Gamma _{{{\mathcal{I}}n}}$. However, both ${\mathcal{K}}_{{{\mathcal{I}}n}}$ and $\Gamma _{{{\mathcal{I}}n}}$ give very poor statistical description of extremely heterogeneous urban areas, whilst $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ is a good model, whose KL value have a very small fluctuation in all.

• (3) All the four areas with varying degrees of homogeneity are fitted well by $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$. In another words, the $\mathcal{G}_{{{\mathcal{I}}n}}^{0}$ distribution shows greater capability of modeling different clutter data with a broad variety of scattering properties.

Appendix 4.1

The Estimation of Coherence Magnitude

The 1th order moment estimation of normalized interferogram ${\mathcal{I}}_{n}$ is

$$ \begin{aligned} {\text{E}}\left( {{\mathcal{I}}_{n} } \right) & = \int\limits_{0}^{\infty } {\int\limits_{ - \pi }^{\pi } {\xi {\text{e}}^{j\psi } \cdot p_{\xi ,\psi } \left( {\xi ,\psi } \right){\text{d}}\psi {\text{d}}\xi } } \\ & = \int\limits_{0}^{\infty } {\frac{{2n^{n + 1} \xi^{n + 1} }}{{\pi\Gamma \left( n \right)\left( {1 - \rho^{2} } \right)}}K_{n - 1} \left( {\frac{2n\xi }{{1 - \rho^{2} }}} \right)\left( {\int\limits_{ - \pi }^{\pi } {{\text{e}}^{j\psi } \exp \left( {\frac{2n\rho \xi \,\cos \psi }{{1 - \rho^{2} }}} \right){\text{d}}\psi } } \right){\text{d}}\xi } \\ \end{aligned} $$

(4.41)

Supposing $V = \int_{ - \pi }^{\pi } {{\text{e}}^{j\psi } \exp \left( {\frac{2n\rho \xi \,\cos \psi }{{1 - \rho^{2} }}} \right){\text{d}}\psi }$, according to integral rules of odd and even functions, the definite integral of odd function is zero in a symmetry section, so it can be simplified as [21]

$$ V = j2\pi J_{1} \left( { - j\frac{2n\rho \xi }{{1 - \rho^{2} }}} \right) $$

(4.42)

where $J_{1} \left( \cdot \right)$ denotes the first type Bessel function of order 1. Plugging (4.42) to (4.41), then

$$ {\text{E}}\left( {{\mathcal{I}}_{n} } \right) = \frac{{j4n^{n + 1} }}{{\Gamma \left( n \right)\left( {1 - \rho^{2} } \right)}}\int\limits_{0}^{\infty } {\xi^{n + 1} K_{n - 1} \left( {\frac{2n\xi }{{1 - \rho^{2} }}} \right)J_{1} \left( { - j\frac{2n\rho \xi }{{1 - \rho^{2} }}} \right){\text{d}}\xi } $$

(4.43)

From [21], (4.43) can be integrated as

$$ {\text{E}}\left( {{\mathcal{I}}_{n} } \right) = \rho \left( {1 - \rho^{2} } \right)^{n + 1} {}_{2}F_{1} \left( {n + 1,2;2;\rho^{2} } \right) $$

(4.44)

where $_{2} F_{1}$ is the Gauss hypergeometric function. Using $_{2} F_{1} \left( {n + 1,m;m;z} \right) = \left( {1 - z} \right)^{{ - \left( {n + 1} \right)}}$, the 1st order moment estimation of ${\mathcal{I}}_{n}$ is finally given as

$$ E\left( {{\mathcal{I}}_{n} } \right) = \rho $$

(4.45)

To make $\rho$ be a positive real number, the estimation of $\rho$ is expressed as

$$ \rho = \left| {E\left( {{\mathcal{I}}_{n} } \right)} \right| $$

(4.46)