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

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 simply ℋ𝒢^𝑜, 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 ℋ𝒢^𝑜distribution is the well-known 𝒢^𝑜 distributions. We also derived analytically the estimators of the presented ℋ𝒢^𝑜 model by applying the “method of log cumulants” (MoLCs). The performance of the proposed model is verified by using some measured SAR images.

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 as ℋ𝒢⁰) 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 Section 2. 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 𝑝_𝑍𝐴(𝑧)=−2𝜎𝜆^𝑣𝛼𝑛^𝑛Γ((𝛼−𝑛)/𝑣)𝑣Γ(𝑣𝛼)Γ(𝑛)×𝑧^2𝑛−1𝜆+𝑛𝑧^{−2𝑣(𝑛−𝛼)/𝑣},−𝛼,−𝑣,𝜆,𝜎,𝑛,𝑧>0, (1) 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 𝑝_𝑍𝐼(𝑧)=−𝜎𝜆^𝑣𝛼𝑛^𝑛Γ((𝛼−𝑛)/𝑣)𝑣Γ(𝑣𝛼)Γ(𝑛)×𝑧^𝑛−1𝜆+𝑛𝑧^{−𝑣(𝑛−𝛼)/𝑣},−𝛼,−𝑣,𝜆,𝜎,𝑛,𝑧>0. (2)

We refer to this distribution characterized by (1) or (2) as the ℋ𝒢⁰ distribution. Specifically, we call the ℋ𝒢^𝑜_𝐼 distribution and the ℋ𝒢^𝑜_𝐴 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 ℋ𝒢^𝑜_𝐴 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.

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Figure 1 

Plots of ℋ𝒢𝑜𝐴 versus different parameters.

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. Letting 𝑋_𝐼 and 𝑌_𝐼 indicates the backscattering RCS component and speckle noise one, respectively, 𝑍_𝐼 denotes the observed intensity of SAR data. Therefore, the relationship of this three variables is expressed by the multiplicative model as 𝑍_𝐼=𝑋_𝐼⋅𝑌_𝐼. (3)
If 𝑋_𝐼 obeys 𝑝_𝑋𝐼(𝑥)=𝜎𝜆^𝑣𝛼Γ()𝑥𝑣𝛼^𝛼−1exp−𝜆𝑥^𝑣,−𝛼,−𝑣,𝜆,𝜎,𝑥>0 (4) and the PDF of 𝑌_𝐼 is 𝑝_𝑌𝐼𝑦=𝑛^𝑛Γ(𝑛)𝑦^𝑛−1exp−𝑛𝑦^−𝑣,𝑦,𝑛,−𝑣>0. (5)
Then the distribution of the intensity return is characterized by the density shown in (2), that is, 𝑍_𝐼∼ℋ𝒢^𝑜_𝐼(𝛼,𝑣,𝜆,𝜎,𝑛).

Proof. Combining (4) and (5) via (3), the PDF of 𝑍_𝐼 can be easily derived as 𝑝_𝑍𝐼(𝑧)=^∞₀1𝑥⋅𝑝_𝑋𝐼(𝑥)⋅𝑝_𝑌𝐼𝑧𝑥=𝑑𝑥𝜎𝜆^𝑣𝛼𝑛^𝑛Γ()Γ(𝑛)𝑧𝑣𝛼^𝑛−1∞₀𝑥^{𝛼−𝑛−1}exp−𝜆−𝑛𝑧^−𝑣𝑥^𝑣𝑑𝑥. (6)
A variable change of 𝑡=1/𝑥 leads to 𝑥=1/𝑡 and 𝑑𝑥=−𝑡⁻²𝑑𝑡, (6) turns out be 𝑝_𝑍𝐼(𝑧)=𝜎𝜆^𝑣𝛼𝑛^𝑛Γ()Γ(𝑛)𝑧𝑣𝛼^𝑛−1∞₀𝑡^{−𝛼+𝑛−1}exp−𝜆−𝑛𝑧^−𝑣𝑡^−𝑣𝑑𝑡. (7)
Applying the integral formula ∫^∞₀𝑥^𝑎−1exp(−𝜇𝑥^𝑝)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 Appendix A, the 𝑚th order moments of the ℋ𝒢^𝑜_𝐼 are given by 𝐸𝑧^𝑚_𝐼=𝜎𝜆^{𝑣𝛼−(𝑚+𝛼)/𝑣}𝑛^{𝑛+(𝑚+𝑛)/𝑣}Γ(−(𝑚+𝑛)/𝑣)Γ((𝛼+𝑚)/𝑣)𝑣²Γ()Γ(𝑛).𝑣𝛼 (8)
The 𝑚th order moments of the corresponding amplitude random variable can easily be given by 𝐸(𝑧^𝑚_𝐴)=𝐸(𝑧_𝐼^𝑚/2).

3. The Parameter Estimators of the Proposed Distribution

3.1. The Log-Cumulants of ℋ𝒢⁰

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 𝑝_𝑋(𝑥), the second-kind first and second characteristic functions are, respectively, defined as [13] 𝜙_𝑋(𝑠)=ℳ𝑝_𝑋(𝑥)(𝑠)=^∞₀𝑥^𝑠−1𝑝_𝑋(𝑥)𝜑𝑑𝑥,_𝑋(𝑠)=log𝜙_𝑋(𝑠), (9) where ℳ is the Mellin transform operator. The 𝑟th order derivative of 𝜑_𝑋(𝑠) at 𝑠=1 is the log-cumulants of order 𝑟, that is, 𝑘_𝑟=𝑑^𝑟𝜑_𝑋(𝑠)𝑑𝑠^𝑟_𝑠=1. (10)

Hereafter, we take the intensity distribution ℋ𝒢^𝑜_𝐼 as an example to estimate its parameters 𝑣, 𝛼, and 𝜆. The processing of the amplitude one is similar. Owing to 𝜙_𝑋(𝑠)=𝐸(𝑋^𝑠−1), consequently, via (8) and (9), the second-kind second characteristic functions of the ℋ𝒢^𝑜_𝐼 distribution yields 𝜑_ℋ𝒢^𝑜_𝐼(𝑠)=log(𝜎)+log𝜆^{𝑣𝛼−(𝛼/𝑣)}+1−𝑠𝑣log(𝜆)+log𝑛^{𝑛+(𝑛/𝑣)}+𝑠−1𝑣(𝑛)Γ−log+log𝑠−1+𝑛𝑣+logΓ𝛼+𝑠−1𝑣−log𝑣²Γ(𝑣𝛼)Γ(𝑛) (11) which leads to that the log-cumulants of the ℋ𝒢^𝑜_𝐼 are expressed as 𝑘₁=log(𝑛/𝜆)−Ψ(−𝑛/𝑣)+Ψ(𝛼/𝑣)𝑣,𝑘_𝑟=Ψ(𝑟−1,−𝑛/𝑣)(−𝑣)^𝑟+Ψ(𝑟−1,𝛼/𝑣)𝑣^𝑟,𝑟≥2, (12) where Ψ(⋅) represents the digamma function, and Ψ(𝑟,⋅) is the 𝑟th order polygamma function.

Given a sample set {𝑧_𝑖}, 𝑖∈[1,𝑁], the log-cumulants can be estimated directly by 𝑘₁=1𝑁^𝑁_𝑖=1ln𝑧_𝑖,𝑘₁=1𝑁^𝑁_𝑖=1ln𝑧_𝑖−𝑘₁^𝑟,𝑟≥2. (13)

3.2. The Parameter Estimators of ℋ𝒢⁰

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 estimates 𝑣 and 𝛼 of the parameters 𝑣 as well as 𝛼 are obtained by solving the following equations resorting to the numerical computation:𝑘₂=Ψ(1,−𝑛/𝑣)+Ψ(1,𝛼/𝑣)𝑣²,𝑘₃=Ψ(2,𝛼/𝑣)−Ψ(2,−𝑛/𝑣)𝑣³. (14)

Second, according to the first equation of (12), the estimate 𝜆 of the parameter 𝜆 is 𝑛𝜆=exp𝑣𝑘₁(𝑣)(𝑣).+Ψ−𝑛/−Ψ𝛼/ (15)

We assume the observed amplitude PDF is 𝑝_𝑍𝐴(𝑧) from the actual data, which corresponds to the histogram of tested data. The theoretical amplitude PDF denotes as 𝑝_𝑍𝐴(𝑧). As the previous analysis, since 𝜎 indicates the whole proportion fluctuation of 𝑝_𝑍𝐴(𝑧), 𝑝_𝑍𝐴(𝑧)/𝜎 can be calculated by using 𝑣, 𝛼, and 𝜆. Given a sample amplitude set {𝑧_𝑖∣𝑖=1⋯𝑁}, let the symbol 𝑞_𝑍𝐴(𝑧) be equal to 𝑝_𝑍𝐴(𝑧)/𝜎, then the estimate 𝜎 of the parameter 𝜎 is simply given by 𝑝𝜎=𝐸^𝑍𝐴𝑧_𝑖𝑞_𝑍𝐴𝑧_𝑖,𝑧_𝑖∈ℤ, (16) where ℤ is a subset of {𝑧_𝑖∣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={𝑧_𝑖∣𝑝_𝑍𝐴(𝑧_𝑖)>𝑝_max/2,𝑝_max=max(𝑝_𝑍𝐴(𝑧_𝑖)),𝑖∈[1,𝑁]}.

4. Experimental Results

In this section, we aim at verifying the performance of the proposed ℋ𝒢⁰. 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.

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Figure 2 

The 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).

The estimated PDFs of the proposed ℋ𝒢⁰ 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] 𝑛=𝐸𝑁𝐿=𝑛_𝑎+𝑛_𝑟, (17) where 𝑛_𝑎 and 𝑛_𝑟 are the effective number of looks in the azimuth and range. For the previous “Sanchagang” and “Beijing” TerraSAR-X images used in this investigation, 𝑛_𝑎 and 𝑛_𝑟 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 Figure 2.

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Figure 3 

Plots of amplitude histograms and of the estimated PDFs: (a)–(f) are corresponded to the fitting results for the scenes shown in Figures 2(a)–2(f), respectively.

Furthermore, in order to quantitatively assess the performances that different distributions fit, we define an error ratio factor (ERF) as ∑𝛿=^𝑁𝑖=1𝑝_𝑐𝑧_𝑖−𝑞𝑧_𝑖²∑^𝑁_𝑖=1𝑝_𝑏𝑧_𝑖−𝑞𝑧_𝑖², (18) where {𝑧_𝑖∣𝑖=1⋯𝑁} is a sample set, 𝑝_𝑐(⋅) represents the compared theoretical PDF, 𝑝_𝑏(⋅) 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 𝑝_𝑐 and 𝑝_𝑏 to approximate 𝑞. The better the performances of 𝑝_𝑐 related to 𝑝_𝑏 fit, the smaller 𝛿 is, and vice verse. Specifically, 𝛿=1, if both 𝑝_𝑐 and 𝑝_𝑏 have the identical capability for fitting the measured data.

Let ℋ𝒢^𝑜_𝐴 represent the basic amplitude PDF, and let 𝒢^𝑜_𝐴 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 ℋ𝒢^𝑜_𝐴 better agrees with the amplitude histograms of all six terrains than the 𝒢^𝑜_𝐴, 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 ℋ𝒢^𝑜 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 the ℋ𝒢⁰ 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 model ℋ𝒢^𝑜_𝐴 shown in (1), its cumulative distribution function (CDF) is written as (see Appendix B) 𝐹_ℋ𝒢^𝑜_𝐴(𝑥)=−𝜎𝜆^{𝑣𝛼+(𝑛−𝛼)/𝑣}𝑛^𝑛−1Γ((𝛼−𝑛)/𝑣)()Γ(𝑛)𝑥𝑣Γ𝑣𝛼^2𝑛×₂𝐹₁𝛼−𝑛𝑣𝑛,−𝑣𝑛;1−𝑣𝑛;−𝜆𝑥^−2𝑣,−𝛼,−𝑣,𝜆,𝜎,𝑛,𝑥>0, (19) where ₂𝐹₁(⋅,⋅;⋅;⋅) is the Gauss hypergeometric function. For a given value of the false alarm probability, denoted by 𝑃_𝑓𝑎, the corresponding CFAR threshold 𝑇 for the ℋ𝒢^𝑜_𝐴 distribution can be obtained from 1−𝑃_𝑓𝑎=𝐹_ℋ𝒢^𝑜_𝐴(𝑇)=−𝜎𝜆^{𝑣𝛼+(𝑛−𝛼)/𝑣}𝑛^𝑛−1Γ((𝛼−𝑛)/𝑣)()Γ(𝑛)𝑇𝑣Γ𝑣𝛼^2𝑛×₂𝐹₁𝛼−𝑛𝑣𝑛,−𝑣𝑛;1−𝑣𝑛;−𝜆𝑇^−2𝑣. (20)

Considering 𝐹_ℋ𝒢^𝑜_𝐴(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 the ℋ𝒢⁰ distribution performs when extending it to other application fields.

Appendices

A. The Derivation of 𝑚th Order Moments of the ℋ𝒢^𝑜_𝐼 Distribution

Via (2), the 𝑚th order moments of the ℋ𝒢^𝑜_𝐼 are expressed as 𝐸𝑧^𝑚_𝐼=^∞₀𝑧^𝑚𝑝_𝑍𝐼(𝑧)𝑑𝑧=−𝜎𝜆^𝑣𝛼𝑛^𝑛Γ((𝛼−𝑛)/𝑣)𝑣Γ(𝑣𝛼)Γ(𝑛)×𝜆^{(𝑛−𝛼)/𝑣∞}₀𝑧^{𝑚+𝑛−1}𝑛1+𝜆𝑧^{−𝑣(𝑛−𝛼)/𝑣}𝑑𝑧. (A.1)

A variable replacement 𝑥=𝑧^−𝑣 leads to 𝑧=𝑥^−1/𝑣 and 𝑑𝑧=−(1/𝑣)𝑥^{−(1/𝑣)−1}𝑑𝑥; thus (A.1) can be rewritten as 𝐸𝑧^𝑚_𝐼=−𝜎𝜆^𝑣𝛼𝑛^𝑛Γ((𝛼−𝑛)/𝑣)𝑣²Γ(𝑣𝛼)Γ(𝑛)×𝜆^{(𝑛−𝛼)/𝑣∞}₀𝑥^{((𝑚+𝑛)/𝑣)−1}𝑛1+𝜆𝑥^{(𝑛−𝛼)/𝑣}𝑑𝑥. (A.2)

According to the integral formula ∫^∞₀𝑥^𝜇−1𝑑𝑥/(1+𝛽𝑥)^𝑣=𝛽^−𝜇Γ(𝜇)Γ(𝑣−𝜇)/Γ(𝑣), |arg𝛽|<𝜋, Re𝑣>Re𝜇>0 [15, 3.194, Equation 3], one can obtain (8).

B. The Derivation of the Cumulative Distribution Function of ℋ𝒢^𝑜_𝐴

Based on the definition of the CDF, the CDF of ℋ𝒢^𝑜_𝐴 is 𝐹_ℋ𝒢^𝑜_𝐴(𝑥)=^𝑥_−∞𝑝_𝑍𝐴(𝑧)𝑑𝑧=^𝑥₀𝑝_𝑍𝐴(𝑧)𝑑𝑧=−2𝜎𝜆^𝑣𝛼𝑛^𝑛Γ((𝛼−𝑛)/𝑣)𝑣Γ(𝑣𝛼)Γ(𝑛)×𝜆^{(𝑛−𝛼)/𝑣𝑥}₀𝑧^2𝑛−1𝑛1+𝜆𝑧^{−2𝑣(𝑛−𝛼)/𝑣}𝑑𝑧. (B.1)

Similarly, utilizing a symbol change 𝑡=𝑧^−2𝑣, (B.1) turns out to be 𝐹_ℋ𝒢^𝑜_𝐴(𝑥)=𝜎𝜆^𝑣𝛼𝑛^𝑛Γ((𝛼−𝑛)/𝑣)𝑣²Γ(𝑣𝛼)Γ(𝑛)×𝜆^{(𝑛−𝛼)/𝑣𝑥−2𝑣}₀𝑡^{(−𝑛/𝑣)−1}𝑛1+𝜆𝑡^{(𝑛−𝛼)/𝑣}𝑑𝑡. (B.2) Likewise, applying the integral formula ∫^𝑢₀𝑥^𝜇−1𝑑𝑥/(1+𝛽𝑥)^𝑣=(𝑢^𝜇/𝜇)₂𝐹₁(𝑣,𝜇;1+𝜇;−𝛽𝑢), |arg(1+𝛽𝑢)|<𝜋, Re𝜇>0 [15, 3.194, Equation 1], we arrive at (19) by simplifying (B.2).