**On the Indicatrixes of Waves Scattering from the Random Fractal Anisotropic Surface**

Alexander A. Potapov

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68187

#### Abstract

Millimeter and centimeter wave scattering from the random fractal anisotropic surface has been theoretically investigated. Designing of such surfaces is based on the modifications of non-differentiable two-dimensional Weierstrass function. Wave scattering on a random surface is interesting for many sections of physics, mathematics, biology, and so on. Questions of a radar location and radio physics take the predominating position here. There are many real surfaces and volumes in the nature that can be carried to fractal objects. At the same time, the description of processes of waves scattering of fractal objects differs from classical approaches markedly. There are many monographs in the world on the topic of classical methods of wave scattering but the number of books devoted to waves scattering on fractal stochastic surfaces is not enough. These results of estimation of three-dimensional scattering functions are a priority in the world and are important in radar of low-contrast targets near the surface of the earth and the sea.

Keywords: fractal, fractal surfaces, Kirchhoff approach, radio waves scattering, Weierstrass function, radar, low-contrast targets

## 1. Introduction

There are a lot of scientific and engineering problems, which can be successfully solved only with deep understanding of wave-scattering characteristics for statistically rough surface (see, e.g., [1–3] and references). In this section, we consider the main issues of theory of fractal wave scattering on the statistically rough surface as applied to problems of image creation by radar methods (RMs). These issues are crucial for radio location of low-contrast targets on the background of earth and sea surface.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the general case, RM can be interpreted as a scattering specific effective squares (SESs), as a σ\* card (matrix) or as a signature (portrait) of object being sounded for the high angular resolution. SES card with fuzzy bounds corresponds to real RM for the wide-probing beam. RM resolution increase necessitates the use of complicated probing signals. Subject detail digital radar maps (DDRM or etalons) are often results of current image processing [4–8].

Currently, there are two general approaches of scattering on the statistically rough surface: method of small perturbation (SP) and Kirchhoff approach (tangent plane method (TPM)). These methods relate to two extreme cases of very small flat irregularities or smooth and large irregularities, respectively. Two-scale scattering model becomes a generalization of these methods. The model is a combination of small ripple (computations using SP) and large irregularities (computations using TPM). Review of these methods evolution is represented in Refs. [1–3].

Thus, before the present diffraction problems for the statistically rough surfaces took into account irregularities of only a single scale. Soon, it had been realized that multiscale surfaces lead to better fitting. As we have found out [6, 7] fractality accounting makes theoretical and experimental scattering patterns for earth cover in microwaves range closer. This fact is always interpreted (and has been interpreted now) as results of pure instrumental errors.

The aim of this work is to report systematically and consistently about theoretical solution of scattering problem for the random fractal anisotropic surface using Kirchhoff approach for the first time, to calculate scattering indicatrixes for radio microwaves, and to analyze the ensemble of indicatrixes obtained.

## 2. Formulation of the problem

Idea of fractal radio systems in the framework of fractal radio physics and radio electronics that was proposed and now is being consistently developed in the Institute of Radio Engineering and Electronics of the RAS (see, e.g., [5–48] and references) allows us to look at conventional radio physics methods in a new fashion. Currently, fractal radio physics and fractal radio location are the very active investigation areas, where significant applications have been obtained.

New problems that arise and being formulated are very important for every branch of science in the sense of its evolution. During the last 35 years, we succeeded in developing a number of important sections of fractal radio physics and fractal radio electronics that almost completes its main structure [6–8]. At once, these results reveal perspective of its modern applications and new relations between fractal physics and classical radio physics and electronics. It is necessary to note that for this course several monographs and more than 800 studies and 23 monographs were published (e.g., look at Refs. [5–48] and references).

Figure 1 shows us the main courses of works that are being carried out in the Institute of Radio Engineering and Electronics of the RAS and also information about the moment of its intensive growth beginning is demonstrated (for details, see Refs. [6, 7]). For such a "fractal" approach, it

Figure 1. Sketch of development of a new information technology.

In the general case, RM can be interpreted as a scattering specific effective squares (SESs), as a σ\* card (matrix) or as a signature (portrait) of object being sounded for the high angular resolution. SES card with fuzzy bounds corresponds to real RM for the wide-probing beam. RM resolution increase necessitates the use of complicated probing signals. Subject detail digital radar maps (DDRM or etalons) are often results of current image processing [4–8].

Currently, there are two general approaches of scattering on the statistically rough surface: method of small perturbation (SP) and Kirchhoff approach (tangent plane method (TPM)). These methods relate to two extreme cases of very small flat irregularities or smooth and large irregularities, respectively. Two-scale scattering model becomes a generalization of these methods. The model is a combination of small ripple (computations using SP) and large irregularities (computations using TPM). Review of these methods evolution is represented in

Thus, before the present diffraction problems for the statistically rough surfaces took into account irregularities of only a single scale. Soon, it had been realized that multiscale surfaces lead to better fitting. As we have found out [6, 7] fractality accounting makes theoretical and experimental scattering patterns for earth cover in microwaves range closer. This fact is always

The aim of this work is to report systematically and consistently about theoretical solution of scattering problem for the random fractal anisotropic surface using Kirchhoff approach for the first time, to calculate scattering indicatrixes for radio microwaves, and to analyze the ensem-

Idea of fractal radio systems in the framework of fractal radio physics and radio electronics that was proposed and now is being consistently developed in the Institute of Radio Engineering and Electronics of the RAS (see, e.g., [5–48] and references) allows us to look at conventional radio physics methods in a new fashion. Currently, fractal radio physics and fractal radio location are the very active investigation areas, where significant applications have been

New problems that arise and being formulated are very important for every branch of science in the sense of its evolution. During the last 35 years, we succeeded in developing a number of important sections of fractal radio physics and fractal radio electronics that almost completes its main structure [6–8]. At once, these results reveal perspective of its modern applications and new relations between fractal physics and classical radio physics and electronics. It is necessary to note that for this course several monographs and more than 800 studies and 23

Figure 1 shows us the main courses of works that are being carried out in the Institute of Radio Engineering and Electronics of the RAS and also information about the moment of its intensive growth beginning is demonstrated (for details, see Refs. [6, 7]). For such a "fractal" approach, it

monographs were published (e.g., look at Refs. [5–48] and references).

interpreted (and has been interpreted now) as results of pure instrumental errors.

Refs. [1–3].

obtained.

ble of indicatrixes obtained.

2. Formulation of the problem

188 Fractal Analysis - Applications in Physics, Engineering and Technology

is natural to focus on analysis and also on the processing of radio physical signals (fields) only in space of fractional measuring using hypothesis of scaling and distributions with "heavy tales" or stable distributions. Note that scale transformations using scaling effects are widespread in up-to-date physics when different relations between thermodynamical values in renormgroup theory of phase changes are setting up [49].

Fractals belong to sets, which have extremely branched and irregular structure. In December 2005 in the USA, Mandelbrot approved [34] fractal classification that was developed by the author and is presented in Figure 2, where fractal features are characterized so long as there is a fractal structure with fractal dimension D in the space with topological dimension. Physical mathematical problems of the fractals theory and fractional measuring are represented in monographs [6–8] in detail.

Figure 2. Classification and morphology of fractal sets and fractal set signatures.

In case of RM formation, the structure and parameters of wave field, which is generated by remote random surface at the field analysis area, depend on receiving point location and surface parameters. By taking into account these facts, we have to analyze the scattered field in a timespatial continuum [5]. Therefore in the late 1970s of the ХХ century, the author formulated the problem of creating a theoretical modeling the band of millimeter and centimeter waves (MMW and CMW, respectively) for radar time-spatial signal by taking into account radio channel "antenna's aperture–atmosphere–targets–chaotic covering without vegetation" and the problem of creating of new features classes for radar targets recognition or radar signatures [5].

## 3. "Diffraction by fractals" 6¼ "classical diffraction"

Effectiveness of radio physical investigations can be significantly improved by taking into account fractality of wave phenomena that are progressing at every stage of wave radiation, scattering, and propagation in different medium. In spite of pure scientific interest, there are practical applications to the radar and telecommunications problems solution and also to problems of mediums monitoring at different time-spatial scales.

Recently, interest to investigate wave scattering by rough surfaces that have non-Gaussian statistics has also grown. They often argue that correlation spatial coefficient of dispersive surface rðΔx ¼ x<sup>2</sup> � x1,Δy ¼ y<sup>2</sup> � y1Þ cannot be exponential due to non-differentiability of respective random process. Sometimes in this case they use regularizing function about a zero

point. Fundamental physical foundation of non-differentiable functions application for wavescattering analysis was developed only after taking into account fractal theory, fractionalmeasuring theory, operators of integro-differentiation, and scaling relations in radio physical problems [6, 7, 20].

It is significant to note that Gaussian model is parabolic near the angle of incidence θ ≈ 0, while the exponential model is linear near the same point. Below, we consider in detail the approach to scattering of MMW and CMW by fractal random surface [5–7, 20, 39–44, 47].

At the present time, many works of foreign authors are related with wave interaction with fractal structures (see, e.g., respective chapters in monographs [6, 7]). Fractal surface implies the presence of irregularities of all scales with respect to scattered wavelength. Therefore, fractal wave front being non-differentiable does not have normal. In that way, conceptions of "ray trajectory" and "ray optics effects" are excluded. However, chords, which connect values of typical irregularity heights at the certain horizontal distances, still have finite root-meansquare slope. For this case, "topoteza" of fractal random surface is introduced; it is equal to the length of surface slope closeness to the unity [6, 7, 20].

Subject to all features, there are scattering models in the west of author works: (1) model of fractal heights and (2) model of fractal irregularities slopes. Thus, model No. 2 is once differentiable and has a slope that is changing continuously from point to point. This model leads to ray optics or to effects that are described using the conception of "ray." Such a kind of scattering was investigated together with radio waves propagation in the ionosphere [6, 7].

Electromagnetic waves scattering by fractal surfaces was investigated in detail in Refs. [50–58]. In Ref. [50], it was shown that diffraction by fractal surfaces fundamentally differs from diffraction by conventional random surfaces and some of classical statistical parameters like correlation length and root-mean-square deviation go to infinity. It is due to self-similarity of fractal surface. In Ref. [52], band-limited Weierstrass function was used. Less restrictions were imposed than the ones in Ref. [50]. The proposed function possesses both self-similarity property and still finite number of derivatives over a certain range under consideration. This relaxation of conditions of Weierstrass function allows performing analytical and numerical calculations.

Though there are many works on the creation and analysis of chaotic surfaces with the fractal structure [6, 7, 55–58], only few of them consider two-dimensional (2D) fractal surfaces. Corrugated surfaces that possess fractal properties only for one dimension (1D) were characterized in some works [52, 53, 59, 60]. In Refs. [39–44, 47, 61–63], modified Weierstrass function was used for designing 2D fractal chaotic surface. This function was derived from bandlimited Weierstrass function. General solution for scattered field was obtained using Kirchhoff theory [1–3, 5–7, 61–65]. On this basis, we will carry out further calculations.

## 4. Fractal model of 2D chaotic surface

In case of RM formation, the structure and parameters of wave field, which is generated by remote random surface at the field analysis area, depend on receiving point location and surface parameters. By taking into account these facts, we have to analyze the scattered field in a timespatial continuum [5]. Therefore in the late 1970s of the ХХ century, the author formulated the problem of creating a theoretical modeling the band of millimeter and centimeter waves (MMW and CMW, respectively) for radar time-spatial signal by taking into account radio channel "antenna's aperture–atmosphere–targets–chaotic covering without vegetation" and the problem

Effectiveness of radio physical investigations can be significantly improved by taking into account fractality of wave phenomena that are progressing at every stage of wave radiation, scattering, and propagation in different medium. In spite of pure scientific interest, there are practical applications to the radar and telecommunications problems solution and also to

Recently, interest to investigate wave scattering by rough surfaces that have non-Gaussian statistics has also grown. They often argue that correlation spatial coefficient of dispersive surface rðΔx ¼ x<sup>2</sup> � x1,Δy ¼ y<sup>2</sup> � y1Þ cannot be exponential due to non-differentiability of respective random process. Sometimes in this case they use regularizing function about a zero

of creating of new features classes for radar targets recognition or radar signatures [5].

3. "Diffraction by fractals" 6¼ "classical diffraction"

Figure 2. Classification and morphology of fractal sets and fractal set signatures.

190 Fractal Analysis - Applications in Physics, Engineering and Technology

problems of mediums monitoring at different time-spatial scales.

Modified 2D band-limited Weierstrass function has the view [6, 7, 20, 39–44, 47, 61–63]

$$W(\mathbf{x}, \mathbf{y}) = c\_w \sum\_{n=0}^{N-1} q^{(D-3)n} \sum\_{m=1}^{M} \sin \left\{ Kq^n \left[ \mathbf{x} \cdot \cos \left( \frac{2\pi m}{M} \right) + \mathbf{y} \cdot \sin \left( \frac{2\pi m}{M} \right) \right] + \phi\_{nm} \right\}, \tag{1}$$

where cw is the constant that provides unit normalization; q > 1- is the fundamental spatial frequency; D is the fractal dimension (2<D<3); K is the fundamental wave number; N and M are the number of tones; φnm- is an arbitrary phase that has a uniform distribution over the interval [�π, π].

Eq. (1) is a combination of random structure and determined period. Function W(x, y) is anisotropic in two directions if M and N are not very large. It has derivatives, and at the same time, it is self-similar. Respective surface is multiscale and roughness can vary depending on the scale being considered. Since the natural surfaces are neither purely random nor periodical and are often anisotropic [5, 40], the function that was proposed above is a good candidate for characterizing natural surfaces.

## 5. Relationships between statistical parameters of roughness measurements and fractal surface parameters

Such parameters as correlation length Γ, mean-root-square deviation σ, and spatial autocorrelation coefficient r(τ) are conventionally used for numerical characterization of rough surface. In this section of our work, these statistical parameters are introduced for the estimation of fractal dimension D influence and other fractal parameters influence on the surface roughness. Similar relationships are presented in Refs. [6, 7, 20] for 1D fractal surfaces. Derivations of σ and r(τ) for 2D fractal surfaces are cumbersome and tedious [61], and so we present here only some final results.

#### 5.1. Mean square deviation

The mean-root-square deviation σ is determined as

$$
\sigma = \left( \langle \mathcal{W}^2(\overrightarrow{r}) \rangle\_{\sf s} \right)^{1/2} \tag{2}
$$

where Wðr !Þ ¼ <sup>W</sup> <sup>ð</sup>x, yÞ; r! <sup>¼</sup> <sup>x</sup> <sup>I</sup> ! þ yJ ! . Angle bracket implies ensemble averaging.

From Eqs. (1) and (2), we have

$$\sigma = \mathfrak{c}\_w \left[ \frac{M \left( 1 - q^{2(D-3)N} \right)}{2 \left( 1 - q^{2(D-3)} \right)} \right]^{\frac{1}{2}}.\tag{3}$$

If σ = 1, then Eq. (3) is as follows:

$$\mathbf{c}\_w = \left[ \frac{2\left(1 - q^{2(D-3)}\right)}{M\left(1 - q^{2(D-3)N}\right)} \right]^{\frac{1}{2}}.\tag{4}$$

Thus, Eqs. (1) and (4) are as follows:

Wðx, yÞ ¼ cw

characterizing natural surfaces.

interval [�π, π].

some final results.

where Wðr

5.1. Mean square deviation

From Eqs. (1) and (2), we have

If σ = 1, then Eq. (3) is as follows:

N X�1 n¼0

192 Fractal Analysis - Applications in Physics, Engineering and Technology

qðD�3Þ<sup>n</sup>

X M

sin Kqn <sup>x</sup> � cos

where cw is the constant that provides unit normalization; q > 1- is the fundamental spatial frequency; D is the fractal dimension (2<D<3); K is the fundamental wave number; N and M are the number of tones; φnm- is an arbitrary phase that has a uniform distribution over the

Eq. (1) is a combination of random structure and determined period. Function W(x, y) is anisotropic in two directions if M and N are not very large. It has derivatives, and at the same time, it is self-similar. Respective surface is multiscale and roughness can vary depending on the scale being considered. Since the natural surfaces are neither purely random nor periodical and are often anisotropic [5, 40], the function that was proposed above is a good candidate for

Such parameters as correlation length Γ, mean-root-square deviation σ, and spatial autocorrelation coefficient r(τ) are conventionally used for numerical characterization of rough surface. In this section of our work, these statistical parameters are introduced for the estimation of fractal dimension D influence and other fractal parameters influence on the surface roughness. Similar relationships are presented in Refs. [6, 7, 20] for 1D fractal surfaces. Derivations of σ and r(τ) for 2D fractal surfaces are cumbersome and tedious [61], and so we present here only

2πm M � �

� � � �

� �

. Angle bracket implies ensemble averaging.

�

3 5

1 2

: ð3Þ

�

<sup>þ</sup> <sup>y</sup> � sin <sup>2</sup>π<sup>m</sup>

M

þ φnm

, ð1Þ

ð2Þ

m¼1

5. Relationships between statistical parameters of roughness

σ ¼ � 〈W<sup>2</sup> ðr !Þ〉<sup>s</sup> �<sup>1</sup>=<sup>2</sup>

σ ¼ с<sup>w</sup>

M �

2 4

2 �

<sup>1</sup> � <sup>q</sup><sup>2</sup>ðD�3Þ<sup>N</sup>

1 � q<sup>2</sup>ðD�3<sup>Þ</sup>

! þ yJ !

measurements and fractal surface parameters

The mean-root-square deviation σ is determined as

!Þ ¼ <sup>W</sup> <sup>ð</sup>x, yÞ; r! <sup>¼</sup> <sup>x</sup> <sup>I</sup>

$$\mathcal{W}\_{\mathbf{n}}(\mathbf{x},\mathbf{y}) = \left[\frac{2(1-q^{2(D-3)})}{M(1-q^{2(D-3)N})}\right]^{1/2} \sum\_{n=0}^{N-1} q^{(D-3)n} \sum\_{m=1}^{M} \sin\left\{Kq^{n} \left[\mathbf{x} \cdot \cos\left(\frac{2\pi m}{M}\right) + \mathbf{y} \cdot \sin\left(\frac{2\pi m}{M}\right)\right] + \phi\_{nm}\right\}.\tag{5}$$

Eq. (5) is normalized with σ = 1. A normalized function will be used in the following sections for the analysis and modeling of wave field scattered by fractal surfaces. Surface becomes more isotropic with the increase of N and M. It is important to notice that Wu(x, y) characterizes mathematical fractals only if N ! ∞ и M ! ∞.

#### 5.2. Coefficient of spatial autocorrelation and of correlation length

Now, let us turn to the consideration of spatial autocorrelation coefficient r(τ) and correlation length Г. By definition

$$\rho(\mathbf{r}) = \frac{\langle \mathcal{W}\_{\rm \rm tr}(\overrightarrow{r} + \overrightarrow{\pi}) \mathcal{W}\_{\rm \rm tr}(\overrightarrow{r}) \rangle\_{s}}{\sigma^{2}} \tag{6}$$

$$\text{where } \mathbf{r} = (\Delta \mathbf{x}^2 + \Delta y^2)^{\frac{1}{2}}. \tag{7}$$

From Eqs. (5) and (6), we have

$$\rho(\mathbf{r}) = \left[ \frac{\left(1 - q^{2(D-3)}\right)}{M\left(1 - q^{2(D-3)N}\right)} \right] \sum\_{n=0}^{N-1} q^{2(D-3)n} \sum\_{m=1}^{M} \cos\left[\mathbf{K} \boldsymbol{\eta}^{n} \boldsymbol{\tau} \cdot \cos\left(\boldsymbol{\Theta} - \frac{2\boldsymbol{\pi} \cdot \boldsymbol{m}}{M}\right)\right],\tag{8}$$

$$\text{where } \sin \theta = \frac{\Delta y}{\pi}, \text{ } \cos \theta = \frac{\Delta x}{\pi}. \tag{9}$$

The average spatial autocorrelation coefficient

$$
\tilde{\rho}(\pi) = \langle \rho(\pi) \rangle\_s = \left[ \frac{\left( 1 - q^{2(D-3)} \right)}{\left( 1 - q^{2(D-3)N} \right)} \right] \sum\_{n=0}^{N-1} q^{2(D-3)n} J\_0(Kq^n \pi), \tag{10}
$$

where <sup>J</sup>0ðKq<sup>n</sup>τ<sup>Þ</sup> is the zero-order Bessel function of the first kind.

Correlation length Г is defined as the first root of r(τ) = 1/e when τ increases from zero. From relationship (8)

$$\mathbb{E}\left[\frac{(1-q^{2(D-3)})}{M(1-q^{2(D-3)N})}\right] \sum\_{n=0}^{N-1} q^{2(D-3)n} \sum\_{m=1}^{M} \cos\left[Kq^n \Gamma \cdot \cos\left(\Theta - \frac{2\pi \cdot m}{M}\right)\right] = 1/\varepsilon. \tag{11}$$

Similarly from Eq. (10), the average correlation length is defined Γ�:

$$\frac{(1-q^{2(D-3)})}{M(1-q^{2(D-3)N})} \sum\_{n=0}^{N-1} q^{2(D-3)n} J\_0(Kq^n \tilde{\Gamma}) = 1/e. \tag{12}$$

From Eqs. (10)–(12), one can find relationships between average correlation length Γ�, fractal dimension D, and also q. There are dependences Γ� on q and D shown in Figures 3 and 4, respectively. It is shown that with an increased value of D, Γ� decreases more rapidly for the same variation of q. It is shown in Figure 4 that the value of Γ�reduces steadily with the increase of D value. However, Γ� does not change when q = 1.01.

Consequently, the mean correlation length Γ� is sensitive to fractal dimension D with the exception of cases when q is close to unity. These results imply that the value of fractal surface irregularities is mainly determined by fractal parameter D.

Figure 3. Average correlation length Γ~ as function of q.

Figure 4. Average correlation length Γ~ as function of D.

## 6. Memoir about the basic foundation of wave-scattering theory by fractal surfaces

As mentioned above, Kirchhoff approach has been already used for the analysis of wave scattering by fractal surfaces [6, 7, 20, 39–44, 47, 50–63]. This theory will be used in our work for numerical analysis of a field scattered by fractal chaotic surfaces. Conventional conditions of Kirchhoff approach are the following: irregularities are large scale, irregularities are smooth and flat. In the following calculations, we assume that observation is carried out from Fraunhofer zone, incident wave is plane and monochromatic, there are no points with infinite gradient on the surface, Fresnel coefficient V<sup>0</sup> is constant for this surface, and surface scales are much greater than incident wavelength.

#### 6.1. Scattered field

<sup>ð</sup><sup>1</sup> � <sup>q</sup><sup>2</sup>ðD�3<sup>Þ</sup>

Mð1 � q<sup>2</sup>ðD�3Þ<sup>N</sup>Þ � � <sup>N</sup>

Þ

194 Fractal Analysis - Applications in Physics, Engineering and Technology

of D value. However, Γ� does not change when q = 1.01.

irregularities is mainly determined by fractal parameter D.

Figure 3. Average correlation length Γ~ as function of q.

X�1 n¼0

Similarly from Eq. (10), the average correlation length is defined Γ�:

<sup>ð</sup><sup>1</sup> � <sup>q</sup><sup>2</sup>ðD�3<sup>Þ</sup>

Mð1 � q<sup>2</sup>ðD�3Þ<sup>N</sup>Þ

q<sup>2</sup>ðD�3Þ<sup>n</sup>

Þ

X M

cos Kq<sup>n</sup><sup>Γ</sup> � cos <sup>θ</sup> � <sup>2</sup><sup>π</sup> � <sup>m</sup>

� � � �

M

<sup>q</sup><sup>2</sup>ðD�3ÞnJ0ðKqnΓ~Þ ¼ <sup>1</sup>=e: <sup>ð</sup>12<sup>Þ</sup>

¼ 1=e: ð11Þ

m¼1

N X�1 n¼0

From Eqs. (10)–(12), one can find relationships between average correlation length Γ�, fractal dimension D, and also q. There are dependences Γ� on q and D shown in Figures 3 and 4, respectively. It is shown that with an increased value of D, Γ� decreases more rapidly for the same variation of q. It is shown in Figure 4 that the value of Γ�reduces steadily with the increase

Consequently, the mean correlation length Γ� is sensitive to fractal dimension D with the exception of cases when q is close to unity. These results imply that the value of fractal surface

> Scattering geometry is presented in Figure 5. Then, scattered field ψрðr !Þ that interacts with surface square S of 2Lx � 2Ly when �Lx ≤ x ≤Lx and �Ly ≤ y ≤Ly are equal to [1–3, 5–7, 20, 61]:

$$\psi\_{\mathbf{p}}(\overrightarrow{r}) = -\frac{i\mathbf{k}\cdot\mathbf{e}\exp(ikr)}{4\pi\cdot r} 2F(\Theta\_1, \Theta\_2, \Theta\_3) \int\_S \exp[ik\phi(\mathbf{x}\_0, y\_0)]d\mathbf{x}\_0 dy\_0 + \psi\_\mathbf{x}.\tag{13}$$

In Eq. (13), we used the following notations:

Figure 5. Scattering geometry: θ1, incident angle; θ2– scattering angle; and θ<sup>3</sup> azimuth angle.

$$F(\Theta\_1, \Theta\_2, \Theta\_3) = \frac{1}{2} \left( \frac{Aa}{\mathbb{C}} + \frac{Bb}{\mathbb{C}} + c \right),\tag{14}$$

$$\Phi(\mathbf{x}\_{0\prime}\,\boldsymbol{y}\_{0}) = A\mathbf{x}\_{0} + B\boldsymbol{y}\_{0} + Ch(\mathbf{x}\_{0\prime}\,\boldsymbol{y}\_{0})\,\tag{15}$$

$$h(\mathbf{x}\_0, y) = \sigma \cdot \mathcal{W}\_\mathbf{n}(\mathbf{x}\_0, y\_0),\tag{16}$$

$$a = V\_0(\sin\Theta\_1 - \sin\Theta\_2\cos\Theta\_3),\tag{17}$$

$$b = V\_0(\sin\Theta\_2 \cdot \sin\Theta\_3),\tag{18}$$

$$c = V\_0(\cos\Theta\_1 + \cos\Theta\_2),\tag{19}$$

$$A = \sin\Theta\_1 - \sin\Theta\_2 \cos\Theta\_3 \,\tag{20}$$

$$B = -\sin\Theta\_2 \sin\Theta\_3 \tag{21}$$

$$\mathcal{C} = - (\cos \Theta\_1 + \cos \Theta\_2),\tag{22}$$

$$\begin{split} \psi\_{\kappa} &= -\frac{ik \cdot \exp(ikr)}{4\pi \cdot r} \left\{ \frac{ia}{kc} \int \left[ \exp(ik\phi(\mathbf{X}, y\_0)) - \exp(ik\phi(-\mathbf{X}, y\_0)) \right] \cdot dy\_0 + \\ &+ \frac{ib}{kc} \int \left[ \exp(ik\phi(\mathbf{x}\_0, Y)) - \exp(ik\phi(\mathbf{x}\_0 - \mathbf{Y})) \right] \cdot d\mathbf{x}\_0 \right\}. \end{split} \tag{23}$$

Component ψ<sup>к</sup> relates to edge effect. From Eqs. (15) and (16), we have

$$\exp[\mathrm{ik}\Phi(\mathbf{x}\_{0\prime}y\_0)] = \exp\{\mathrm{ik}[\mathbf{A}\mathbf{x}\_0 + \mathbf{B}y\_0 + \mathsf{C}\sigma \cdot \mathcal{W}\_\mathbf{u}(\mathbf{x}\_{0\prime}y\_0)]\}.\tag{24}$$

In Eq. (24), the third exponent is expressed as

$$\begin{split} \exp[i\hbar \mathbb{C}\sigma \cdot W\_{u}(\mathbf{x}\_{0}, y\_{0})] &= \exp\left\{i\hbar \mathbb{C}\sigma \left[\frac{2\left(1 - q^{2(D-3)}\right)}{M\left(1 - q^{2(D-3)}\right)}\right]^{1/2} \sum\_{n=0}^{N-1} q^{(D-3)n} \sum\_{m=1}^{M} \times \\ &\times \cos\left[Kq^{n} \cdot \left(\mathbf{x}\_{0} \cdot \cos\left(\frac{2\pi \cdot m}{M}\right) + y\_{0} \sin\left(\frac{2\pi \cdot m}{M}\right) + \phi\_{nm} - \frac{\pi}{2}\right)\right] \right\} = \\ &= \prod\_{n=1}^{N-1} \prod\_{m=1}^{M} \exp\left\{i\hbar \mathbb{C}\sigma \left[\frac{2\left(1 - q^{2(D-3)}\right)}{M\left(1 - q^{2(D-3)N\_{0}}\right)}\right]^{1/2} q^{(D-3)n} \times \\ &\times \cos\left[\mathbf{x}\_{0} \cdot \cos\left(\frac{2\pi \cdot m}{M}\right) + y\_{0} \sin\left(\frac{2\pi \cdot m}{M}\right) + \phi\_{nm} - \frac{\pi}{2}\right] \right\}. \end{split} (25)$$

$$\exp(\text{izcosq}) = \sum\_{\mu=-\infty}^{+\infty} \text{i} \mathfrak{l}\_{\mu}(\text{z}) \exp(\text{iuq}),\tag{26}$$

$$c\_f = k \text{C} \sigma \left[ \frac{2 \left( 1 - q^{2(D-3)} \right)}{M \left( 1 - q^{2(D-3)N} \right)} \right]^{1/2} \text{.} \tag{27}$$

$$\begin{split} & \exp[\mathrm{ikC}\sigma \cdot W\_{\mathrm{u}}(\mathbf{x}\_{0}, y\_{0})] = \prod\_{n=0}^{N-1} \prod\_{m=1}^{M} \sum\_{n=-\infty}^{+\infty} I\_{u\_{nm}} \left( c\_{f} q^{(D-3)n} \right) \times \\ & \times \exp\left\{ iu \left[ Kq^{\mathrm{u}} \cdot \left( \mathbf{x}\_{0} \cdot \cos\left(\frac{2\pi \cdot m}{M}\right) + y\_{0} \sin\left(\frac{2\pi \cdot m}{M}\right) + \phi\_{nm} - \frac{\pi}{2} \right) \right] \right\}. \end{split} \tag{28}$$

$$\begin{split} & \left[ \exp[ik\mathbb{C}\sigma \cdot W\_{\mathbf{u}}(\mathbf{x}\_{0}, y\_{0})] = \sum\_{u\_{0}, \mathbf{o} = -\infty}^{+\infty} \dots \sum\_{u\_{0, \mathbf{o}} = -\infty}^{+\infty} \dots \sum\_{u\_{0, \mathbf{o}} = -\infty}^{+\infty} \dots \sum\_{u\_{0, \mathbf{o}} = -\infty}^{+\infty} \dots \sum\_{u\_{0, \mathbf{o}} = -\infty}^{+\infty} \times \\ & \times \left[ \prod\_{n=0}^{N-1} \prod\_{m=1}^{M} f\_{u\_{m}} \left( c\_{q} q^{(D-3)n} \right) \right] \cdot \exp \left\{ iK \left[ \sum\_{n=0}^{N-1} q^{(D-3)n} \sum\_{m=1}^{M} u\_{m} \cos \left( \frac{2\pi \cdot m}{M} \right) \right] \mathbf{x}\_{0} \right\} \times \\ & \times \exp \left\{ iK \left[ \sum\_{n=0}^{N-1} q^{(D-3)n} \sum\_{m=1}^{M} u\_{m m} \sin \left( \frac{2\pi \cdot m}{M} \right) \right] \mathbf{y}\_{0} \right\} \times \\ & \times \exp \left( i \sum\_{n=0}^{N-1} \sum\_{m=1}^{M} u\_{m n} \phi\_{mn} \right) . \end{split} (29)$$

ψрðr !޼� iLxLyk � expðikr<sup>Þ</sup> πr 2Fðθ1, θ2,θ3Þ � <sup>X</sup> þ∞ u1,0¼�∞ … X þ∞ u1,N�1¼�∞ … X þ∞ u2, <sup>0</sup>¼�∞ … X þ∞ u2,N�1¼�∞ … X þ∞ uM,N�1¼�∞ � � N Y�1 n¼0 Y M m¼1 Jumn <sup>ð</sup>cjqðD�3Þ<sup>n</sup><sup>Þ</sup> " #exp i N X�1 n¼0 X M m¼1 umnφmn !� �sincðϕcLxÞ � sincðϕsLyÞ þ ψк, ð30Þ sincðxÞ � sin <sup>ð</sup>x<sup>Þ</sup> <sup>x</sup> , <sup>ð</sup>31<sup>Þ</sup>

$$\varphi\_c = kA + K \sum\_{n=0}^{N-1} q^n \sum\_{m=1}^M \mu\_{nm} \cos\left(\frac{2\pi \cdot m}{M}\right), \ Q\_s = kB + K \sum\_{n=0}^{N-1} q^n \sum\_{m=1}^M \mu\_{nm} \sin\left(\frac{2\pi \cdot m}{M}\right). \tag{32}$$

#### 6.2. Average-scattered field

A more convenient parameter for the characterization of scattered field properties is averagescattered field <sup>ψ</sup><sup>~</sup> <sup>р</sup>ð<sup>r</sup> !Þ:

$$
\langle \tilde{\psi}\_{\mathsf{P}}(\overrightarrow{r}) = \langle \psi\_{\mathsf{P}}(\overrightarrow{r}) \rangle\_{s} \tag{33}
$$

Eqs. (32) and (33) are defined as follows:

$$\ddot{\psi}\_{\rm p}(\overrightarrow{r}) = -\frac{iL\_{\rm r}L\_{\rm y}k \cdot \exp(ikr)}{\pi r} 2F(\Theta\_1, \Theta\_2, \Theta\_3) \left[ \prod\_{n=0}^{N-1} f\_0^M(c\_f q^{(D-3)n}) \right] \text{sinc}(kAL\_x) \text{sinc}(kBL\_y) + \psi\_{\rm u} \quad (34)$$

Assume that the outside area �Lx ≤ x<sup>0</sup> ≤ Lxи �Ly ≤ y<sup>0</sup> ≤ Ly surface S is smooth, that is,

$$h(\pm X, \pm Y) \equiv 0,\tag{35}$$

$$\text{where } X > L\_{\text{x}} \text{ } \text{ } Y > L\_{\text{y}}. \tag{36}$$

Then, Eq. (23) can be written as

$$\psi\_{\kappa} = -\frac{ik \cdot \exp(ikr)}{\pi \cdot r} \left(\frac{Aa}{\mathbb{C}} + \frac{Bb}{\mathbb{C}}\right) \lim\_{X \to L\_x^+} X \cdot \text{sinc}(kAX) \cdot \lim\_{Y \to L\_y^+} Y \cdot \text{sinc}(kBY) . \tag{37}$$

#### 6.3. Scattering indicatrixes for field

Scattering indicatrixes for field r<sup>ψ</sup> is defined as

$$\rho\_{\psi} = \frac{\psi\_{\mathbb{P}}(\overrightarrow{r})}{\psi\_{\mathbb{P}^0}(\overrightarrow{r})} \,, \tag{38}$$

where field scattered from perfectly smooth surface ψр<sup>0</sup>ðr !Þ in a specular direction is expressed as

$$\psi\_{\mathbf{p}0}(\overrightarrow{r}) = -\frac{2L\_{\mathbf{x}}L\_{\mathbf{y}}ik \cdot \exp\left(ikr\right)\cos\Theta\_{\mathbf{l}}}{\pi \cdot r}.\tag{39}$$

Average-scattering indicatrix ~r<sup>ψ</sup> can be obtained after normalization:

ψрðr

198 Fractal Analysis - Applications in Physics, Engineering and Technology

� <sup>X</sup> þ∞

� N Y�1 n¼0

N X�1 n¼0 qn X M

ϕ<sup>c</sup> ¼ kA þ K

6.2. Average-scattered field

!Þ:

Eqs. (32) and (33) are defined as follows:

!޼� iLxLyk � expðikr<sup>Þ</sup> πr

Then, Eq. (23) can be written as

6.3. Scattering indicatrixes for field

<sup>ψ</sup><sup>к</sup> ¼ � ik � expðikr<sup>Þ</sup> π � r

Scattering indicatrixes for field r<sup>ψ</sup> is defined as

where field scattered from perfectly smooth surface ψр<sup>0</sup>ðr

scattered field <sup>ψ</sup><sup>~</sup> <sup>р</sup>ð<sup>r</sup>

<sup>ψ</sup><sup>~</sup> <sup>р</sup>ð<sup>r</sup>

u1,0¼�∞

Y M

m¼1

m¼1

unmcos

!޼� iLxLyk � expðikr<sup>Þ</sup> πr

u1,N�1¼�∞

Jumn <sup>ð</sup>cjqðD�3Þ<sup>n</sup><sup>Þ</sup> " #

�sincðϕcLxÞ � sincðϕsLyÞ þ ψк,

… X þ∞

u2, <sup>0</sup>¼�∞

sincðxÞ � sin <sup>ð</sup>x<sup>Þ</sup>

A more convenient parameter for the characterization of scattered field properties is average-

!Þ ¼ 〈ψрð<sup>r</sup>

N Y�1 n¼0 J M

lim X!L<sup>þ</sup> x

<sup>r</sup><sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>рð<sup>r</sup>

!Þ ψр<sup>0</sup>ðr

Assume that the outside area �Lx ≤ x<sup>0</sup> ≤ Lxи �Ly ≤ y<sup>0</sup> ≤ Ly surface S is smooth, that is,

2π � m M � �

<sup>ψ</sup><sup>~</sup> <sup>р</sup>ð<sup>r</sup>

2Fðθ1, θ2, θ3Þ

Aa C þ Bb C � � exp i N X�1 n¼0

… X þ∞

2Fðθ1, θ2,θ3Þ

u2,N�1¼�∞

X M

m¼1

… X þ∞

umnφmn !

> N X�1 n¼0 qn X M

uM,N�1¼�∞

�

m¼1

hð�X, � YÞ � 0, ð35Þ

where X > Lx, Y > Ly: ð36Þ

Y!L<sup>þ</sup> y

!Þ, <sup>ð</sup>38<sup>Þ</sup>

!Þ in a specular direction is expressed as

�

<sup>x</sup> , <sup>ð</sup>31<sup>Þ</sup>

!Þ〉<sup>s</sup> <sup>ð</sup>33<sup>Þ</sup>

sincðkALxÞsincðkBLyÞ þ ψ<sup>к</sup> ð34Þ

Y � sincðkBYÞ: ð37Þ

unmsin <sup>2</sup><sup>π</sup> � <sup>m</sup> M � � ð30Þ

: ð32Þ

… X þ∞

, ϕ<sup>s</sup> ¼ kB þ K

<sup>0</sup> <sup>ð</sup>cf <sup>q</sup>ðD�3Þ<sup>n</sup><sup>Þ</sup> " #

X � sincðkAXÞ � lim

$$
\check{\boldsymbol{\rho}}\_{\psi} = \frac{\bar{\psi}\_{sc}(\mathbf{r})}{\psi\_{sc0}(\mathbf{r})}.\tag{40}
$$

Assume that surface gradients much less than incident angle is θ1, then from Eqs. (30), (37)– (39) we have

$$\begin{split} \tilde{\rho}\_{\psi} &= \frac{F(\Theta\_1, \Theta\_2, \Theta\_3)}{\cos \Theta\_1} \Bigg[ \prod\_{n=0}^{N-1} f\_0^M \left( c\_f q^{(D-3)n} \right) \Bigg] \text{sinc}(k A L\_x) \text{sinc}(k B L\_y) + \\ &+ \frac{1}{2L\_x L\_y \cos \Theta\_1} \left( \frac{A a}{\mathbb{C}} + \frac{B b}{\mathbb{C}} \right) \lim\_{X \to L\_x^+} X \cdot \text{sinc}(k A X) \cdot \lim\_{Y \to L\_y^+} Y \cdot \text{sinc}(k B Y) .\end{split} \tag{41}$$

In specular direction θ<sup>1</sup> ¼ θ2, θ<sup>3</sup> ¼ 0 and coefficients are the A = 0, B = 0, a = 0, b = 0. Using Eqs. (17)–(22), we can write average-scattering indicatrixes ~rψ, which was defined in Eq. (40), as

$$\check{\rho}\_{\psi} = \left[ \prod\_{n=0}^{N-1} f\_0^M (c\_f q^{(D-3)n}) \right] \tag{42}$$

$$\text{where } \mathbf{c}\_f = -2k\boldsymbol{\sigma} \cdot \cos \theta\_1 \left[ \frac{2\left(1 - q^{2(D-3)}\right)}{M\left(1 - q^{2(D-3)N}\right)} \right]^{\frac{1}{2}}.\tag{43}$$

Thus, ~r<sup>ψ</sup> relates to parameters k, σ, θ1, q, D, N, M. If с<sup>f</sup> qðD�3Þ<sup>n</sup> < 1, then in second approximation ~r<sup>ψ</sup> we have

$$\tilde{\rho}\_{\psi} = 1 - 2(k\sigma \cdot \cos\Theta\_1)^2. \tag{44}$$

Eq. (44) shows that in specular direction ~r<sup>ψ</sup> depends on the wavelength of incident radiation, σ of rough surface, and incident angle θ1. This result coincides with conventional results for Gaussian random surfaces [1]. Thus, fractal surfaces have diffraction properties that are similar to the ones of Gaussian random surfaces in a specular direction. This result involves a previous one [26], which was used as main assumption for mean-root-square scattering cross section measurement on this surface with specular ray measurement.

#### 6.4. Average field intensity

Now, let us find scattering indicatrixes for average field intensity r~<sup>I</sup> . The intensity of scattered field is defined as

$$I(\overrightarrow{r}) = \psi\_{\sf{P}}(\overrightarrow{r})\psi\_{\sf{P}}^{\*}(\overrightarrow{r}).\tag{45}$$

The average intensity of scattered field is obtained by Eq. (45) averaging:

$$
\vec{I}(\vec{r}) = \langle I(\vec{r})\rangle\_{\mathbb{S}}.\tag{46}
$$

From Eqs. (30), (45), and (46), we have

$$\begin{split} & \left[ \hat{I}(\vec{r}) = \left[ \frac{L\_{\text{L}} \eta\_{s}}{n \cdot r} 2F(\theta\_{1}, \theta\_{2}, \theta\_{3}) \right]^{2} \times \\ & \times \sum\_{\substack{\omega\_{1} = -\infty \\ \upmu\_{1} = -\infty}}^{+\infty} \dots \sum\_{\substack{\omega\_{1} = 1 \\ \upmu\_{1} = -\infty}}^{+\infty} \dots \sum\_{\substack{\omega\_{2} = -\infty \\ \upmu\_{2} = -\infty}}^{+\infty} \dots \sum\_{\substack{\omega\_{2} = 1 \\ \upmu\_{2} = 1 \\ \upmu\_{2} = \infty}}^{+\infty} \dots \sum\_{\upmu\_{\omega\_{1} \cdot 1} = -\infty}^{+\infty} \times \\ & \times \left[ \prod\_{n = 0}^{N - 1} \prod\_{m = 1}^{M} (c\_{f} q^{(D - 3)n}) \right]^{2} \times \\ & \times \text{sinc}^{2} (q\_{c} L\_{\text{x}}) \cdot \text{sinc}^{2} (q\_{s} L\_{\text{y}}) . \end{split} (47)$$

#### 6.5. Scattering indicatrix for average field intensity

In a similar manner as stated above, here we define scattering indicatrix for average field intensity ~r<sup>I</sup> � g:

$$\lg(\overrightarrow{r}) \equiv \widetilde{\mathfrak{p}}\_I = \frac{\widetilde{I}(\overrightarrow{r})}{I\_0},\tag{48}$$

$$\text{where } I\_0 = \psi\_{\mathbf{p}0}(\overrightarrow{r}) \cdot \psi\_{\mathbf{p}0}^\*(\overrightarrow{r}). \tag{49}$$

Based on the assumptions that were proposed in the beginning of this section, we can write Eq. (48) as

$$\begin{split} s & \approx \frac{F^2(\Theta\_1, \Theta\_2, \Theta\_3)}{\cos^2 \Theta\_1} \Bigg\{ \left[ 1 - \frac{1}{2} (k \mathcal{C} \sigma)^2 \right] \cdot \text{sinc}^2(k A L\_x) \text{sinc}^2(k B L\_y) \\ & + \frac{1}{4} \mathsf{C}\_f^2 \sum\_{n=0}^{N-1} \sum\_{m=1}^M q^{2(D-3)n} \text{sinc}^2 \Bigg[ \left( k A + K q^n \cos \frac{2 \pi \cdot m}{M} \right) L\_x \Big] + \\ & \quad + \text{sinc}^2 \Big[ \left( k B + K q^n \sin \frac{2 \pi \cdot m}{M} \right) L\_y \Big] \Bigg\}, \end{split} \tag{50}$$

where values with the order higher than с<sup>2</sup> <sup>f</sup> q<sup>2</sup>ðD�3Þ<sup>n</sup> in Eqs. (48) and (49) are negligible. Statistical parameter of scattered field σ<sup>I</sup> is defined as

$$\frac{\sigma\_{I} = \hat{I}(\vec{r}) - \hat{\psi}\_{\text{p}}^{2}(\vec{r})}{I\_{0\prime}} \tag{51}$$

that here corresponds to the mean-root-square value of average-scattered field.

Let us compare the view of Eq. (34) with the first term in Eq. (50). It is obvious that the first term in Eq. (50) is equal to the expression for <sup>ψ</sup><sup>~</sup> <sup>2</sup> <sup>р</sup>ðr !Þ that represents specular ray and side lobes. Thus, δ<sup>I</sup> is determined only by the second term in Eq. (50) that relates to scattering by surface roughness. The second moment of scattered field σ<sup>I</sup> can be useful for diffraction studying away from specular direction and also for the determination of the influence of fractal parameters on inverse-scattering pattern. The advantage of such a presentation is that in the consideration it is sufficient to discount only average coefficients. Thus, it is necessary to measure phase components that relate with scattered wave front.

#### 6.6. Results clarification

Iðr

~Iðr

<sup>π</sup>�<sup>r</sup> <sup>2</sup>Fðθ1,θ2,θ3<sup>Þ</sup> h i<sup>2</sup>

> … X þ∞

> > u2, <sup>0</sup>¼�∞

�

In a similar manner as stated above, here we define scattering indicatrix for average field

!Þ � <sup>~</sup>r<sup>I</sup> <sup>¼</sup> <sup>~</sup>Ið<sup>r</sup>

Based on the assumptions that were proposed in the beginning of this section, we can write

<sup>q</sup><sup>2</sup>ðD�3Þ<sup>n</sup>sinc<sup>2</sup> kA <sup>þ</sup> Kqncos

� �

� �)

!Þ � <sup>ψ</sup><sup>~</sup> <sup>2</sup> <sup>р</sup>ðr !Þ

!Þ I0

!Þ � <sup>ψ</sup>� <sup>р</sup><sup>0</sup>ðr

� sinc<sup>2</sup>

M

� �

� �

Ly

<sup>ð</sup>kALxÞsinc<sup>2</sup>

2π � m M

,

<sup>f</sup> q<sup>2</sup>ðD�3Þ<sup>n</sup> in Eqs. (48) and (49) are negligible.

The average intensity of scattered field is obtained by Eq. (45) averaging:

From Eqs. (30), (45), and (46), we have

~Iðr

� N Y�1 n¼0

<sup>g</sup> <sup>≈</sup> <sup>F</sup><sup>2</sup>

þ 1 4 C2 f N X�1 n¼0

where values with the order higher than с<sup>2</sup>

Statistical parameter of scattered field σ<sup>I</sup> is defined as

ðθ1, θ2, θ3Þ cos2θ<sup>1</sup>

> X M

> m¼1

(

intensity ~r<sup>I</sup> � g:

Eq. (48) as

� <sup>X</sup> þ∞

200 Fractal Analysis - Applications in Physics, Engineering and Technology

u1,0¼�∞

6.5. Scattering indicatrix for average field intensity

!Þ ¼ LxLyk

Y M

m¼1

… X þ∞

sinc<sup>2</sup>ðϕcLxÞ � sinc<sup>2</sup>ðϕsLyÞ:

u1,N�1¼�∞

Jumn <sup>ð</sup>cjqðD�3Þ<sup>n</sup><sup>Þ</sup> " #<sup>2</sup>

gðr

where I<sup>0</sup> ¼ ψр<sup>0</sup>ðr

<sup>1</sup> � <sup>1</sup> 2 ðkCσÞ 2

� �

<sup>þ</sup>sinc<sup>2</sup> kB <sup>þ</sup> Kq<sup>n</sup>sin <sup>2</sup><sup>π</sup> � <sup>m</sup>

<sup>σ</sup><sup>I</sup> <sup>¼</sup> <sup>~</sup>Ið<sup>r</sup>

that here corresponds to the mean-root-square value of average-scattered field.

!Þ ¼ <sup>ψ</sup>рð<sup>r</sup>

!Þ ¼ 〈Ið<sup>r</sup>

!Þ<sup>ψ</sup>� <sup>р</sup>ðr

�

… X þ∞

u2,N�1¼�∞

… X þ∞

uM,N�1¼�∞

!Þ: <sup>ð</sup>45<sup>Þ</sup>

!Þ〉S: <sup>ð</sup>46<sup>Þ</sup>

�

, ð48Þ

!Þ: <sup>ð</sup>49<sup>Þ</sup>

ðkBLyÞ

þ

Lx

<sup>I</sup>0, <sup>ð</sup>51<sup>Þ</sup>

ð47Þ

ð50Þ

In Ref. [52], approximate formula of average field intensity for the problem of scattering by fractal phase screen was presented. As it is explained in Ref. [61], this formula includes some errors. Below, details are explained and presented in Ref. [61]. Surface model in Ref. [52] is specified by Weierstrass function (see also expression (6.77) in monograph [7]:

$$\phi(\mathbf{x}) = \frac{\sqrt{2\sigma\_{\phi}[1 - b^{(2D-4)}]^{1/2}}}{[b^{(2D-4)N\_1} - b^{(2D-4)(N\_2+1)}]^{1/2}} \sum\_{n=N\_1}^{N\_2} b^{(D-2)n} \cos(2\pi sb^n \mathbf{x} + \phi\_n) \tag{52}$$

where b is the fundamental spatial frequency; D is the fractal dimension, which varies over interval from 1 to 2; s is the scaling factor; φ<sup>n</sup> is the phase that is distributed uniformly over the [0, 2π]. Number of harmonics in function (Eq. (52)) is determined by N=N2 – N1 + 1.

Average-scattered field intensity is determined by Eq. (22) in Ref. [52] (or by Eq. (6.96) in monograph [7] in the form of weighted array of Bessel functions):

$$\begin{split} \langle I(\mathbf{x}) \rangle &= \frac{L^4}{\lambda^2 z^2} \sum\_{q\_1 = -\kappa}^{\infty} \sum\_{q\_2 = -\kappa}^{\infty} \dots \sum\_{q\_N = -\kappa}^{\infty} f\_{q\_1}^2(\mathbf{C}\_{N\_1}) f\_{q\_2}^2(\mathbf{C}\_{N\_1 + 1}) \dots f\_{q\_N}^2(\mathbf{C}\_{N\_2}) \times \\ &\times \text{sinc}^2 \left[ L \left( \frac{\mathbf{x}}{\lambda\_z} - sq\_1 b - sq\_2 b^{N\_1 + 1} - \dots s q^{N\_2} \right) \right] \text{sinc}^2 \left( \frac{\mathbf{L} \mathbf{x}}{\lambda z} \right) . \end{split} \tag{53}$$
 
$$\text{where } \mathbf{C}\_n = \frac{\sqrt{2 \sigma\_\phi [1 - b^{(2D - 4)} b^{(D - 2)n}]^{1/2}}}{[b^{(2D - 4)N\_1} - b^{(2D - 4)(N\_2 + 1)}]^{1/2}} \tag{54}$$

L is the phase screen size, x, y are the coordinate values in intensity observation plane at a distance of z from the phase screen, and λ is the incident wavelength.

In Eqs. (B2) and (B3) in Ref. [52], typographical errors were made. In Eq. (53), term sq1b must be: sq1bN<sup>1</sup> , which is clear from expression (6.95) in monograph [7]. In line with Eq. (B2), Eq. (B3) must be

$$\mathcal{C}\_{\mathfrak{n}} = \frac{\sqrt{2\sigma\_{\phi}[1 - b^{(2D-4)}]^{1/2}b^{(D-2)\mathfrak{n}}}}{[b^{(2D-4)N\_1} - b^{(2D-4)(N\_2+1)}]^{1/2}}.\tag{55}$$

Approximate expression for the average intensity is derived from Eq. (22) in Ref. [12] (see also expression (6.97) in monograph [6]):

$$\langle I(\mathbf{x})\rangle = \frac{L^4}{\lambda^2 z^2} \left\{ (1 - \sigma\_{\phi}^2) \text{sinc}^2(\mathbf{L} \mathbf{x}/\lambda \mathbf{z}) + \sum\_{n=-\infty}^{+\infty} (\mathbf{C}\_n^2/4) \text{sinc}^2[\mathbf{L}(\mathbf{x}/\lambda \mathbf{z} - \mathbf{s}b^n)] \right\} \text{sinc}^2(\mathbf{L}y/\lambda \mathbf{z}). \tag{56}$$

After the correction, we have

$$\langle I(\mathbf{x})\rangle = \frac{L^4}{\lambda^2 z^2} \left\{ (1 - \sigma\_{\phi}^2) \text{sinc}^2(\mathbf{L} \mathbf{x}/\lambda z) + \sum\_{n=N\_1}^{N\_2} (\mathbf{C}\_n^2/4) \text{sinc}^2[\mathbf{L}(\mathbf{x}/\lambda z - sb^n)] \right\} \text{sinc}^2(\mathbf{L} y/\lambda z). \tag{57}$$

Accurate derivation of Eq. (57) looks like this [61]:

$$\begin{split} f\_{u\_i}(\mathsf{C}\_n) &= \left(\frac{\mathsf{C}\_n}{2}\right)^u \sum\_{j=-\infty}^{\infty} \frac{(-1)^j (\mathsf{C}\_n/2)^{2j}}{j!(u+j)!} \\ f\_{u\_i}^2(\mathsf{C}\_n) &= \left(\frac{\mathsf{C}\_n}{2}\right)^{2u} \left[\sum\_{j=-\infty}^{\infty} \frac{(-1)^j (\mathsf{C}\_n/2)^{2j}}{j!(u+j)!} \right]^2 \end{split} \tag{58}$$

where ui is the integer and ui ∈ ðq1, …, qNÞ.

Since terms with the order higher than C<sup>2</sup> <sup>n</sup> are negligible, then ui ∈f0, 1g, <sup>X</sup>ui <sup>¼</sup> 0 or 1. Thus

$$\begin{aligned} f\_0(\mathbb{C}\_n) &= \left(\frac{\mathbb{C}\_n}{2}\right)^u \sum\_{j=-\infty}^{\infty} \frac{(-1)^j (\mathbb{C}\_n/2)^{2j}}{j!(u+j)!} \approx 1 - \frac{1}{4} \mathbb{C}\_n^2\\ f\_0^2(\mathbb{C}\_n) &= 1 - \frac{1}{2} \mathbb{C}\_n^2 \end{aligned} \tag{59}$$

$$J\_1^2(\mathbb{C}\_n) \approx \frac{1}{4} \mathbb{C}\_n^2. \tag{60}$$

From Eqs. (53), (59), and (60), we have

$$\begin{split} \langle I(\mathbf{x}) \rangle &= \frac{L^4}{\lambda^2 z^2} \Big[ J\_0^2(\mathbb{C}\_{N\_1}) \dots J\_0^2(\mathbb{C}\_{N\_l}) \text{sinc}^2 \left( \frac{\mathbf{L}\mathbf{x}}{\lambda z} \right) + \\ &+ J\_1^2(\mathbb{C}\_{\text{v}}) J\_0^2(\mathbb{C}\_{N\_1+1}) \dots J\_0^2(\mathbb{C}\_{N\_l}) \text{sinc}^2 \left( \frac{\mathbf{L}\mathbf{x}}{\lambda z} - s b^{N\_l} \right) + \\ &+ J\_0^2(\mathbb{C}\_{N\_1}) J\_1^2(\mathbb{C}\_{N\_1+1}) \dots J\_0^2(\mathbb{C}\_{N\_2}) \text{sinc}^2 \left( \frac{\mathbf{L}\mathbf{x}}{\lambda z} - s b^{N\_l+1} \right) + \\ &+ \dots + J\_0^2(\mathbb{C}\_{N\_1}) \dots J\_1^2(\mathbb{C}\_{N\_2}) \text{sinc}^2 \left( \frac{\mathbf{L}\mathbf{x}}{\lambda z} - s b^{N\_l} \right) \Big] \text{sinc}^2 \left( \frac{\mathbf{L}\mathbf{y}}{\lambda z} \right) . \end{split} \tag{61}$$

where

$$J\_0^2(\mathbb{C}\_{N\_1})\dots J\_0^2(\mathbb{C}\_{N\_2}) \approx \left(1 - \frac{1}{2}\mathbb{C}\_{N\_1}^2\right)\dots \left(1 - \frac{1}{2}\mathbb{C}\_{N\_2}^2\right) \approx 1 - \frac{1}{2}\sum\_{n=N\_1}^{N\_2} \mathbb{C}\_n^2 = 1 - \sigma\_{\phi'}^2\tag{62}$$

$$\begin{aligned} \left. J\_1^2(\mathbb{C}\_{N\_1}) \dots J\_0^2(\mathbb{C}\_{N\_2}) \simeq \frac{1}{4} \mathbb{C}\_{N\_1}^2 \\\\ \left. \begin{array}{c} \\ . \\ . \\ . \\ . \end{array} \right\} . \end{aligned} \tag{63}$$

So, from Eqs. (61)–(63) finally we obtain the expression for average intensity in Fraunhofer zone:

〈IðxÞ〉 <sup>¼</sup> <sup>L</sup><sup>4</sup> λ2

〈IðxÞ〉 <sup>¼</sup> <sup>L</sup><sup>4</sup> λ2

Thus

where

J 2 <sup>0</sup>ðCN<sup>1</sup> Þ…J

<sup>z</sup><sup>2</sup> <sup>ð</sup><sup>1</sup> � <sup>σ</sup><sup>2</sup>

<sup>z</sup><sup>2</sup> <sup>ð</sup><sup>1</sup> � <sup>σ</sup><sup>2</sup>

where ui is the integer and ui ∈ ðq1, …, qNÞ. Since terms with the order higher than C<sup>2</sup>

After the correction, we have

φÞsinc<sup>2</sup>

202 Fractal Analysis - Applications in Physics, Engineering and Technology

φÞsinc<sup>2</sup>

Jui

J 2 ui

<sup>J</sup>0ðCnÞ ¼ Cn

<sup>0</sup>ðCnÞ ¼ <sup>1</sup> � <sup>1</sup>

�

<sup>0</sup>ðCN1þ<sup>1</sup>Þ…J

<sup>1</sup>ðCN1þ<sup>1</sup>Þ…J

2

2 C2 N<sup>1</sup> � �

J 2

〈IðxÞ〉 <sup>¼</sup> <sup>L</sup><sup>4</sup> λ2 <sup>z</sup><sup>2</sup> <sup>J</sup> 2 <sup>0</sup>ðCN<sup>1</sup> Þ…J

From Eqs. (53), (59), and (60), we have

þJ 2 <sup>1</sup>ðCnÞJ 2

þ J 2 <sup>0</sup>ðCN<sup>1</sup> ÞJ 2

þ … þ J 2 <sup>0</sup>ðCN<sup>1</sup> Þ…J

2

<sup>0</sup>ðCN<sup>2</sup> <sup>Þ</sup> <sup>≈</sup> <sup>1</sup> � <sup>1</sup>

2 � �<sup>u</sup>X<sup>∞</sup>

j¼�∞

J 2 <sup>1</sup>ðCn<sup>Þ</sup> <sup>≈</sup> <sup>1</sup> 4 C2

2

2

2 С2 n ð�1Þ j ðCn=2Þ 2j <sup>j</sup>!ð<sup>u</sup> <sup>þ</sup> <sup>j</sup>Þ! <sup>≈</sup> <sup>1</sup> � <sup>1</sup>

2

<sup>1</sup>ðCN<sup>2</sup> <sup>Þ</sup>sinc<sup>2</sup> Lx

… <sup>1</sup> � <sup>1</sup> 2 C2 N<sup>2</sup> � �

<sup>0</sup>ðCN<sup>2</sup> <sup>Þ</sup>sinc<sup>2</sup> Lx

<sup>0</sup>ðCN<sup>2</sup> <sup>Þ</sup>sinc<sup>2</sup> Lx

λz

<sup>0</sup>ðCN<sup>2</sup> <sup>Þ</sup>sinc<sup>2</sup> Lx

λz

λz

� sb<sup>N</sup><sup>2</sup> � �#

λz � � þ

� sbN<sup>1</sup> � �

þ

þ

sinc<sup>2</sup> Ly λz � � ,

> n¼N<sup>1</sup> C2

<sup>n</sup> <sup>¼</sup> <sup>1</sup> � <sup>σ</sup><sup>2</sup>

� sbN1þ<sup>1</sup> � �

> <sup>≈</sup> <sup>1</sup> � <sup>1</sup> 2 X N<sup>2</sup>

Accurate derivation of Eq. (57) looks like this [61]:

<sup>ð</sup>Lx=λzÞ þ <sup>X</sup>

<sup>ð</sup>Lx=λzÞ þ <sup>X</sup>

<sup>ð</sup>CnÞ ¼ Cn 2 � �<sup>u</sup>X<sup>∞</sup>

<sup>ð</sup>CnÞ ¼ Cn 2 � �<sup>2</sup><sup>u</sup> <sup>X</sup><sup>∞</sup>

þ∞

( )

n=4Þsinc<sup>2</sup>

n=4Þsinc2

ð�1Þ j ðCn=2Þ 2j

> ð�1Þ j ðCn=2Þ 2j

j!ðu þ jÞ!

j!ðuþjÞ!

<sup>n</sup> are negligible, then ui ∈f0, 1g,

3 5 2

<sup>½</sup>Lðx=λ<sup>z</sup> � sb<sup>n</sup>Þ�

<sup>½</sup>Lðx=λ<sup>z</sup> � sbnÞ�

9 >>>>>=

>>>>>;

4 С2 n 9 >>>=

>>>;

<sup>n</sup>: ð60Þ

sinc<sup>2</sup>

sinc<sup>2</sup>

, ð58Þ

<sup>X</sup>ui <sup>¼</sup> 0 or 1.

, ð59Þ

ð61Þ

<sup>φ</sup>, ð62Þ

ðLy=λzÞ: ð56Þ

ðLy=λzÞ: ð57Þ

n¼�∞ ðC2

N<sup>2</sup>

( )

j¼�∞

2 4

j¼�∞

n¼N<sup>1</sup> ðC2

$$\langle I(\mathbf{x})\rangle = \frac{L^4}{\lambda^2 z^2} \left\{ \left(1 - \sigma\_{\phi}^2\right) \text{sinc}^2(L\mathbf{x}/\lambda z) + \sum\_{n=N\_1}^{N\_2} \left(\mathbf{C}\_n^2/4\right) \text{sinc}^2[(L(\mathbf{x}/\lambda z - s b^n))] \right\} \text{sinc}^2(L\mathbf{y}/\lambda z). \tag{64}$$

## 7. Results of the theoretical investigations of scattering indicatrixes in MW range

In Figures 6–80, we present a thorough array of typical kinds of dispersing fractal surfaces with the basis of Weierstrass function, and also 3D-scattering indicatrixes and their cross sections that

Figure 6. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 0�: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 7. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 8. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 9. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 7. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 5: (a) fractal surface for

Figure 8. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 10: (a) fractal surface for

D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 10. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 11. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 12. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

On the Indicatrixes of Waves Scattering from the Random Fractal Anisotropic Surface http://dx.doi.org/10.5772/intechopen.68187 207

Figure 13. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 11. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 0: (a) fractal surface for

Figure 12. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 5: (a) fractal surface for

D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 14. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 15. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 16. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 17. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 15. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 20: (a) fractal surface for

Figure 16. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 0: (a) fractal surface for

D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 18. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 19. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 20. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 21. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 19. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 15: (a) fractal surface for

Figure 20. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 20: (a) fractal surface for

D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 22. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 23. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 24. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 25. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 23. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 10: (a) fractal surface for

Figure 24. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 15: (a) fractal surface for

D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 26. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 27. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 28. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 29. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 27. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 5: (a) fractal surface for

Figure 28. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 10: (a) fractal surface for

D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 30. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 2.2 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 31. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 32. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 33. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 31. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 0: (a) fractal surface for

Figure 32. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 5: (a) fractal surface for

D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 34. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 35. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 36. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 37. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 35. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 20: (a) fractal surface for

Figure 36. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 0: (a) fractal surface for

D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 38. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 39. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 40. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 41. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 39. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 15: (a) fractal surface for

Figure 40. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 20: (a) fractal surface for

D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 42. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 43. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 44. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 45. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 43. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 10: (a) fractal surface for

Figure 44. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 15: (a) fractal surface for

D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 46. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 47. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 48. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 49. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 50. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 48. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 10: (a) fractal surface for

Figure 47. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 5: (a) fractal surface for

D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

224 Fractal Analysis - Applications in Physics, Engineering and Technology

D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 51. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 52. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 53. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 54. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 52. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 5: (a) fractal surface for

Figure 51. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 0: (a) fractal surface for

D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

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D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 55. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 56. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 57. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 58. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 56. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 0: (a) fractal surface for

Figure 55. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 8.6 mm and θ<sup>1</sup> = 20: (a) fractal surface for

D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

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D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 59. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 60. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 61. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 62. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 60. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 20: (a) fractal surface for

Figure 59. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 15: (a) fractal surface for

D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

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D = 2.2; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 63. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 64. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 65. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 66. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 64. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 15: (a) fractal surface for

Figure 63. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 10: (a) fractal surface for

D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

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D = 2.2; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 67. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 68. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 69. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 70. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 68. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 10: (a) fractal surface for

Figure 67. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 5: (a) fractal surface for

D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

234 Fractal Analysis - Applications in Physics, Engineering and Technology

D = 2.5; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 71. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 72. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 73. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 74. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 72. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 5: (a) fractal surface for

Figure 71. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 0: (a) fractal surface for

D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

236 Fractal Analysis - Applications in Physics, Engineering and Technology

D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 75. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 76. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 0: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 77. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 5: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 78. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 10: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 76. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 0: (a) fractal surface for

Figure 75. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 20: (a) fractal surface for

D = 2.5; N = M = 20; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

238 Fractal Analysis - Applications in Physics, Engineering and Technology

D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 79. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 15: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

Figure 80. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 20: (a) fractal surface for D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

were calculated in the summer of 2006 for the wavelengths λ ¼ 2:2 mm, λ ¼ 8:6 mm and λ ¼ 3:0 cm for the different values of fractal dimension D and different scattering geometry, respectively. It is significant to note that in this work there is only part of all of our theoretical results obtained for these courses. Some of results for this course that relates to "Fractal Electrodynamics" (this conception appeared for the first time in the USA in the monographs [66, 67]; see also native monographs [6, 7]) were published by us earlier in works [8, 15, 40].

## 8. Conclusion

Figure 80. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 20: (a) fractal surface for

Figure 79. The fractal surface and the scattering indicatrix g(θ2, θ3) when λ = 30 mm and θ<sup>1</sup> = 15: (a) fractal surface for

D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

D = 2.8; N = M = 10; q = 2.7; (b) g(θ2, θ3); (c) g(θ2, θ3), top view; (d) g(θ2, θ3), side view.

240 Fractal Analysis - Applications in Physics, Engineering and Technology

Now on the basis of large scattering characteristics data array, we can arrive at some significant conclusions. When D has small value, the main part of energy is scattered in the specular direction. Side lobes appear due to Bragg scattering. The number of side lobes and their intensity increases with an increased value of fractal dimension D of the dispersive surface. Angular range of the side lobes also increases with an increase of D when higher spatial frequencies become significant. Radio wave that interacts with a fractal can be viewed as a yardstick to probe rough surfaces by means of spatial frequencies selection on the basis of Bragg diffraction conditions [6, 7]. In the case of small D values, classical and fractal approaches for scattered field solution coincide with each other. In practice, sizes of illuminated area must be at least two times greater than the main period of a surface structure in order to obtain fractal parameters information from scattering patterns.

Undoubtedly, fractal describing of the wave-scattering process [5–10, 15, 63, 72, 73, 77] will result in establishing new physical laws in the wave theory. Author is sure that the use of fractal theory and determined chaos jointly with formalism of the apparatus of fractional operators in the just considered problems allows to generate more valid radio physical and radar models that decrease significantly discrepancies between theory and measurements.

This work reviews in detail a variety of modern wave-scattering problems that appear in theoretical and applied areas of radio physics and radiolocation when the theory of integer and fractional measuring is used in general case. In other words, the use of dissipative system dynamics formalism (fractality, fractional operators, non-Gaussian statistics, distributions with heavy tales, mode of determined chaos, existing of strange attractors in the phase space of reflected signals, their topology, etc.) allows us to expect that classical problem of wave scattering by random mediums will be area of productive investigations in the future as before.

All results presented here are the priority ones in the world, and it is a basis material for the further development and foundation of practical application of fractal approaches in radio location, electronics, and radio physics and also for generating fundamentally new fractal elements/devices and fractal radio systems [5–48, 62, 63, 68–83]. These results can be applied widely for fractal antennas modeling, fractal frequency-selective structures modeling, solidstate physics, physics of nanostructures, and for the synthesis of nano-materials.

## Acknowledgements

This work was supported in part by the project of International Science and Technology Center No. 0847.2 (2000–2005, USA), Russian Foundation for Basic Research (projects No. 05-07- 90349, 07-07-07005, 07-07-12054, 07-08-00637, 11-07-00203) and also was supported in part by the project "Leading Talents of Guangdong Province," No. 00201502 (2016–2020) in the JiNan University (China, Guangzhou).

## Author details

Alexander A. Potapov

Address all correspondence to: potapov@cplire.ru

1 Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Moscow, Russia

2 JNU-IRE RAS Joint Laboratory of Information Technology and Fractal Processing of Signals, JiNan University, Guangzhou, China

## References


[9] Podosenov SA, Potapov AA, Sokolov AA. Pulse Electrodynamics of Wide-Band Radio Systems. In: Potapov AA, editor. Moscow: Radiotekhnika; 2003. p. 720

Acknowledgements

University (China, Guangzhou).

Address all correspondence to: potapov@cplire.ru

242 Fractal Analysis - Applications in Physics, Engineering and Technology

SM, editor. Moscow: Nauka; 1978. p. 464

[dissertation]. Moscow: IREE RAS; 1994. p. 436

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sity Library; 2005. p. 848

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Author details

Moscow, Russia

References

Alexander A. Potapov

This work was supported in part by the project of International Science and Technology Center No. 0847.2 (2000–2005, USA), Russian Foundation for Basic Research (projects No. 05-07- 90349, 07-07-07005, 07-07-12054, 07-08-00637, 11-07-00203) and also was supported in part by the project "Leading Talents of Guangdong Province," No. 00201502 (2016–2020) in the JiNan

1 Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences,

2 JNU-IRE RAS Joint Laboratory of Information Technology and Fractal Processing of Signals,

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## **Chapter 10**

## **Fractal Geometry and Porosity**

Oluranti Agboola, Maurice Steven Onyango, Patricia Popoola and Opeyemi Alice Oyewo

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68201

#### Abstract

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tute of Electronics (CIE); 2016. pp. 585-587

tute of Electronics (CIE); 2016. pp. 692-696

Chinese Institute of Electronics (CIE); 2016. pp. 799-803

A fractal is an object or a structure that is self-similar in all length scales. Fractal geometry is an excellent mathematical tool used in the study of irregular geometric objects. The concept of the fractal dimension, D, as a measure of complexity is defined. The concept of fractal geometry is closely linked to scale invariance, and it provides a framework for the analysis of natural phenomena in various scientific and engineering domains. The relevance of the power law scaling relationships is discussed. Fractal characteristics of porous media and the characteristic method of the porous media are also discussed. Different methods of analysis on the permeability of porous media are discussed in this chapter.

Keywords: fractal geometry, fractal structure, fractal dimension, porous media, permeability

#### 1. Introduction

Fractal is one of the subjects, which recently attracted attention in natural science and social science. A fractal is defined as a geometric object whose fractal dimension is larger than its topological dimension. Many fractals also have a property of self-similarity; within the fractal lies another copy of the same fractal, smaller but complete. Mandelbrot [1] referred to fractals as structures consisting of parts that, in some sense, are similar to integers; fractals are of a fine (non-integer) dimension (D) that is always smaller than the topological dimension. In the past 40 decades, fractal theory has significantly contributed to the characterization of the distribution of physical or other quantities with a geometric support. In science and engineering, fractal geometry provides a wide range and powerful theoretical framework that is used to describe complex systems, which have been successfully applied to the quantitative description of microstructures such as surface roughness and amorphous metal structure [2].

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Typically, microstructure elements can be explained using the Euclidean dimension (d). With respect to point defects (e.g., vacancies and interstitial atoms), d ¼ 0; with respect to linear defects (dislocations), d ¼ 1; with respect to planar defects (twins), d ¼ 2; and with respect to threedimensional (3-D) formations, d ¼ 3. Nonetheless, the Euclidean dimension cannot be used to illustrate structural elements differing from standard ones (e.g., points or straight lines). Thus, a well-known grain boundary, being the most significant element of the microstructure, is curvilinear, and this form can be described by the fractal dimension (D) correlating to 1 ≤ D ≤ 2. Surface defects may also be illustrated using the fine dimension that will commensurate to the range 2 ≤ D ≤ 3 [3]. Therefore, fractal theory introduces a new quantitative parameter-fractal dimension for illustrating structures, which, because of its universal nature, is appropriate for illustrating structures in systems types. With a system such as a deformed solid, the fractal concept provides the possibility of quantitatively illustrating the elements of the initial microstructure (e.g., phases, grain, boundaries, etc.) and the structures formed during deformation [3]. Fractal theory thus provides a new and effective method for characterizing complex structure of the engineering materials. The theory of fractals is considered a basis for quantitative description by means of the fractal dimension of various structures.

An extremely disordered morphology, such as surface roughness and porous media having the self-similarity property, is scrutinized by fractal geometry. This implies that the morphology stays similar in magnification over a broad range. Another significant attribute of natural fracture is that their formation needs the supply of a large amount of energy externally [3]. If microstructure formation is preferentially caused by a phenomena taking place outside of thermodynamic equilibrium, they are also characterized by fractal property. This implies that the description of highly disordered microstructures on the basis of conventional approach is not sufficient [3]. Thus, most of the objects that occur in nature are disordered and irregular, and they do not follow the Euclidean illustration because of the scale-dependent measures of length, area, and volume [4, 5]. Examples of such objects are the surfaces of mountains, coastlines, microstructure of metals, and so on. These objects are termed fractals and are illustrated by a non-integral dimension known as fractal dimension [1]. The fractal property is a physical property expressed at the super-molecular level, at a microscopic scale, and at a macroscopic scale.

The phenomenon of fractal is ubiquitous in a wide array of materials, such as the fracture of nanoparticle composites [6–8], the growth of crystal [9–12], the quasicrystal structure [13], the fracture of martensite morphology [14, 15], the porous materials [16–19], and the deposited film [20–25]. These materials are of uncommon class of disordered materials and usually show complex microstructures. Fractal theory has been widely used in many fields of modern science since it was presented by Mandelbrot [1] in 1982. It has been used in studying permeability of porous media [17, 26–28], dual-porosity medium [29, 30], evaluating dislocation structure [31], simulating the failure of concrete [32–35], analyzing fracture surfaces or network [36, 37], and thermal conductivity performance [38]. Fractal has also been used to establish the morphology of highly irregular objects imbedded onto two- and three-dimensional spaces and is defined as two- and three-dimensional fractal dimensions [39].

## 2. Fractal structure

Typically, microstructure elements can be explained using the Euclidean dimension (d). With respect to point defects (e.g., vacancies and interstitial atoms), d ¼ 0; with respect to linear defects (dislocations), d ¼ 1; with respect to planar defects (twins), d ¼ 2; and with respect to threedimensional (3-D) formations, d ¼ 3. Nonetheless, the Euclidean dimension cannot be used to illustrate structural elements differing from standard ones (e.g., points or straight lines). Thus, a well-known grain boundary, being the most significant element of the microstructure, is curvilinear, and this form can be described by the fractal dimension (D) correlating to 1 ≤ D ≤ 2. Surface defects may also be illustrated using the fine dimension that will commensurate to the range 2 ≤ D ≤ 3 [3]. Therefore, fractal theory introduces a new quantitative parameter-fractal dimension for illustrating structures, which, because of its universal nature, is appropriate for illustrating structures in systems types. With a system such as a deformed solid, the fractal concept provides the possibility of quantitatively illustrating the elements of the initial microstructure (e.g., phases, grain, boundaries, etc.) and the structures formed during deformation [3]. Fractal theory thus provides a new and effective method for characterizing complex structure of the engineering materials. The theory of fractals is considered a basis for quantitative description

An extremely disordered morphology, such as surface roughness and porous media having the self-similarity property, is scrutinized by fractal geometry. This implies that the morphology stays similar in magnification over a broad range. Another significant attribute of natural fracture is that their formation needs the supply of a large amount of energy externally [3]. If microstructure formation is preferentially caused by a phenomena taking place outside of thermodynamic equilibrium, they are also characterized by fractal property. This implies that the description of highly disordered microstructures on the basis of conventional approach is not sufficient [3]. Thus, most of the objects that occur in nature are disordered and irregular, and they do not follow the Euclidean illustration because of the scale-dependent measures of length, area, and volume [4, 5]. Examples of such objects are the surfaces of mountains, coastlines, microstructure of metals, and so on. These objects are termed fractals and are illustrated by a non-integral dimension known as fractal dimension [1]. The fractal property is a physical property expressed at the super-molecular level, at a microscopic scale, and at a macroscopic

The phenomenon of fractal is ubiquitous in a wide array of materials, such as the fracture of nanoparticle composites [6–8], the growth of crystal [9–12], the quasicrystal structure [13], the fracture of martensite morphology [14, 15], the porous materials [16–19], and the deposited film [20–25]. These materials are of uncommon class of disordered materials and usually show complex microstructures. Fractal theory has been widely used in many fields of modern science since it was presented by Mandelbrot [1] in 1982. It has been used in studying permeability of porous media [17, 26–28], dual-porosity medium [29, 30], evaluating dislocation structure [31], simulating the failure of concrete [32–35], analyzing fracture surfaces or network [36, 37], and thermal conductivity performance [38]. Fractal has also been used to establish the morphology of highly irregular objects imbedded onto two- and three-dimen-

sional spaces and is defined as two- and three-dimensional fractal dimensions [39].

by means of the fractal dimension of various structures.

250 Fractal Analysis - Applications in Physics, Engineering and Technology

scale.

Fractal structure is a structure that is characterized with self-similarity, that is, it is composed of such fragments whose structural motif is repeated if the scale changes. Fractal structure outlined the degree of occupancy of a structure in a space (dimension), which is not an integer value. Therefore, n-dimensional fractal occupies an intermediate position that lies between the n-dimensional and (n þ 1)-dimensional objects. Recursive functions are used to construct a fractal object. An important characteristic of fractal structure is the scale independence [40]. Thus, fractal structures do not have a single length scale, and fractal processes (e.g., time series) cannot be characterized by a single time scale [41]. Fractal structures are associated to rough or fragmented geometric structures [42]. The complexity of a fractal structure is described by its fractal dimension; this is greater than the topological dimension. It is much easier to obtain fractal dimension from datasets by using fractal analysis, for example, digital images, obtained from the investigation of natural phenomena, and from theoretical models. Different techniques to perform fractal analysis include box-counting, lacunarity analysis, multifractal analysis, and mass methods. An interesting application of fractal analysis is the description of fractured surfaces [43]. Mandelbrot et al. [42] have shown that fractured surfaces are fractal. Zhang [44] reported a quadratic polynomial relationship between the rock burst tendency and fractal dimension of fracture surface. A fractal dimension threshold of ^df was found, and there was a positive correlativity between the rock burst tendency index and the fractal dimension when df <sup>≤</sup> ^df , an inverse correlativity when df <sup>≤</sup> ^df . In the investigation of fractured surfaces, Liang and Wu investigated the relationship between the fracture surface fractal dimension and the impact strength of polypropylene nanocomposites. A strong correlation was observed, and it indicated that the fracture surface of the composites was fractal, and the relationship between the impact strength and fractal dimension of the composites obeyed roughly exponential function [7]. Lung et al. have also demonstrated that there is a relation between the roughness and the fractal dimension of the surface [45].

## 3. Fractal analysis

Fractal analysis is defined as a contemporary method of applying non-traditional mathematics to patterns that defy understanding with traditional Euclidean concepts. It means assessing the fractal description of data, and it is a common technique to study a variety of problems. It consists of different methods assigned to a fractal dimension and other fractal characteristics to a dataset. It, in essence, measures complexity using fractal dimension. In fractal analysis, other different parameters can also be assessed [43], for instance, lacunarity and succolarity, and can be used to classify and segment images [46]. Whatever type of fractal analysis has to be done, it always rests on some kind of fractal dimension. In fractal analysis, complexity is a change in detail with change in scale. The simplest form of fractal dimension is described using the relation in Eq. (1).

$$N = \frac{1}{S^D} \tag{1}$$

where N is the number of self-similar "pieces," S is the linear scaling factor (sizes) of the pieces to the whole, D is the dimension that characterizes the (invariant) relationship between size and number. Rearranging the elements in Eq. (1), one can solve for D.

$$D = -\left(\frac{\log \text{N}}{\log \text{S}}\right) \tag{2}$$

D is an algebraic equation, that is, Eq. (1) can give a dimension, which is the concept of geometry, not algebra. Let's say, one-dimensional line is cut into pieces, each of which is a fraction (S) of the original line, like making S ¼ 1/4. For example, one-dimensional line can be cut into pieces such that each one-fourth will be the size of the original line, then N will be equal to four little lines. Then one can say that Eq. (1) gives the line a fractal dimension D ¼ 1, because <sup>N</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> <sup>1</sup> 4 <sup>1</sup> . If a two-dimensional square area is cut into pieces, the side of which is one-fourth the size of the original square, then N ¼ 16 little squares. Eq. (1) will then tell that the area of the square has a fractal dimension <sup>D</sup> <sup>¼</sup> 2, because <sup>N</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> <sup>1</sup> 4 <sup>2</sup> . If a three-dimensional cube volume is cut into pieces, such that the side of which is one-fourth the size of the original cube, then N will be equal to 64 little cubes. Eq. (1) then tells that the volume of the cube has a fractal dimension of <sup>D</sup> <sup>¼</sup> 3, because <sup>N</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> <sup>1</sup> 4 <sup>3</sup> . No matter the value of S, N will still be found as 1=SDpieces when one-, two-, and three-dimensional objects have been cut into pieces. Thus, Eq. (1) gives the correct fractal dimension for one-dimensional line, two-dimensional area, and three-dimensional volume [47].

#### 3.1. Concepts of the fractal dimension

The ratio that gives statistical index of intricacy and compares how detailed a shape (fractal pattern) changes with the scale at which it is measured is called fractal dimension. It is sometimes identified by a measure of the space filling volume of a pattern that states how a fractal scale is different from the space it is rooted in; a fractal dimension is not always an integer [48–50]. There are several different concepts of the fractal dimension of a geometrical configuration [5].

There are several ways of measuring length-related fractal dimensions. Mandelbrot [51] first proposed the concept of a fractal dimension to describe structures, which look the same at all length scales. His concept takes into account the measuring of the perimeter of an object with several lengths of rulers (spans or calipers) (using a trace method). For a fractal object, the plot of the log of the perimeter against the log of the ruler lengths will give a straight line with a negative slope S. This plot will then result to D ¼ 1 – S [52]. Although this is mainly mathematical concept, many examples in nature that can be closely approximated to fractal objects are available for only over a particular range of scale. The likes of these objects are generally named self-similar in order to indicate their scale invariant structure. The common attribute of such objects is that their length (for a curve object, otherwise it could be the area or volume) mainly rests on the length scale used for measuring it, and the fractal dimension provides the exact nature of this reliance [53]. Fractal dimensions (D) are numbers used to quantify these properties [5]. In fractal geometry [1], the fractal dimension, D, is given as:

<sup>N</sup> <sup>¼</sup> <sup>1</sup>

where N is the number of self-similar "pieces," S is the linear scaling factor (sizes) of the pieces to the whole, D is the dimension that characterizes the (invariant) relationship between size

> <sup>D</sup> ¼ � log<sup>N</sup> logS

D is an algebraic equation, that is, Eq. (1) can give a dimension, which is the concept of geometry, not algebra. Let's say, one-dimensional line is cut into pieces, each of which is a fraction (S) of the original line, like making S ¼ 1/4. For example, one-dimensional line can be cut into pieces such that each one-fourth will be the size of the original line, then N will be equal to four little lines. Then one can say that Eq. (1) gives the line a fractal dimension D ¼ 1,

one-fourth the size of the original square, then N ¼ 16 little squares. Eq. (1) will then tell that

cube volume is cut into pieces, such that the side of which is one-fourth the size of the original cube, then N will be equal to 64 little cubes. Eq. (1) then tells that the volume of the cube has a

as 1=SDpieces when one-, two-, and three-dimensional objects have been cut into pieces. Thus, Eq. (1) gives the correct fractal dimension for one-dimensional line, two-dimensional area, and

The ratio that gives statistical index of intricacy and compares how detailed a shape (fractal pattern) changes with the scale at which it is measured is called fractal dimension. It is sometimes identified by a measure of the space filling volume of a pattern that states how a fractal scale is different from the space it is rooted in; a fractal dimension is not always an integer [48–50]. There are several different concepts of the fractal dimension of a geometrical

There are several ways of measuring length-related fractal dimensions. Mandelbrot [51] first proposed the concept of a fractal dimension to describe structures, which look the same at all length scales. His concept takes into account the measuring of the perimeter of an object with several lengths of rulers (spans or calipers) (using a trace method). For a fractal object, the plot of the log of the perimeter against the log of the ruler lengths will give a straight line with a negative slope S. This plot will then result to D ¼ 1 – S [52]. Although this is mainly mathematical concept, many examples in nature that can be closely approximated to fractal objects are available for only over a particular range of scale. The likes of these objects are generally named self-similar in order to indicate their scale invariant structure. The common attribute of such objects is that their length (for a curve object, otherwise it could be the area or volume)

4 <sup>3</sup>

. If a two-dimensional square area is cut into pieces, the side of which is

and number. Rearranging the elements in Eq. (1), one can solve for D.

252 Fractal Analysis - Applications in Physics, Engineering and Technology

the area of the square has a fractal dimension <sup>D</sup> <sup>¼</sup> 2, because <sup>N</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> <sup>1</sup>

because <sup>N</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> <sup>1</sup>

configuration [5].

4 <sup>1</sup>

three-dimensional volume [47].

3.1. Concepts of the fractal dimension

fractal dimension of <sup>D</sup> <sup>¼</sup> 3, because <sup>N</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> <sup>1</sup>

SD <sup>ð</sup>1<sup>Þ</sup>

4 <sup>2</sup>

. No matter the value of S, N will still be found

. If a three-dimensional

ð2Þ

$$D = \lim\_{r \to 0} \frac{\log(N\_r)}{\log(1/r)}\tag{3}$$

This is a statistical quantity that shows how a fractal totally fills the space when viewed at finer scales.

The second concept was proven by Pentland on the basis that the image of a fractal object is also a fractal [54], which has made scientific investigations on the methods of estimating the fractal dimensions of images. Many researchers have put great efforts into this field of fractal geometry, and many methods for estimating fractal dimensions of certain objects have been proposed since the establishment of fractal geometry theory. Typical methods of this concept involve the use of spectral analysis and box-counting. Usually, spectral analysis method applies fast Fourier transformation (FFT) to image in order to obtain the coefficients and mean spectral energy density. The fractal dimension can be evaluated by analyzing the power law reliance of spectral energy density and the square size [55]. The box-counting method is the widely used method for calculating fractal dimensions in the natural sciences; this is called box-counting dimension. It is a method based on the concept of "covering" the border, it is also known as the grid method. Sets of square boxes (i.e., grids) are used here in order to cover the border. Each set is represented by a box size. The number of boxes essential to cover the border is considered a function of the box size. Figure 1 is an example of the log of the number of covering boxes of each size times the length of a box edge plotted against the log of the length of a box edge. Furthermore, a straight line with slope S which is equal to the dimension D will be obtained [52]. The slope is defined as the amount of change along the Y-axis, divided by the amount of change along the X-axis. Any result with a steeper slope shows that the object is more "fractal," which means it gains in complexity as the box size reduces. Any result with a lower-valued slope shows that the object is closer to a straight line, less "fractal," and that the amount of detail does not grow as quickly with an increase in magnification. Again, the 3-D space containing a specific object, partitioned into boxes of a certain size and how many boxes could fill up the object, is also accounted for. With the use of ratio r in Eq. (1), in order to decide the box size, the box-counting method will account for the total number of boxes (i.e., Nr of Eq. (1)) that are needed to form the object. The fractal dimension D of Eq. (1) can then be estimated from the least square linear fit of log(Nr) versus log(1/r) by counting Nr for different scaling ratio r [56].

Several traditional box-counting methods have been used for the calculation of the fractal dimensions of images, this includes differential box-counting (DBC) method [57], Chen et al.'s approach [58], the reticular cell counting method [59], Feng's method [60], and so on. DBC method has been proved to have better performance than other methods [61]. Many analyses have been done in order to improve the DBC method [62–64].

A third concept was developed by Flook [65], and the method of this concept is called the dilation method. Dilation, in this case, means a widening and smoothing of the border. This

Figure 1. Example of fractal dimension of a material.

can be accomplished by convolution operation with a binary disk, that is, all the non-zero components of all the convolution kernels have a (Boolean) unitary value. The result is a thickened, but grey-scale border. All non-zero pixels are thresholded to a Boolean one when returning this border to Boolean one values. The speed at which the total surface area of the border raises as a function of the diameter of the convolution kernel relies on the dimension D. The log of each resulting area, divided by the kernel diameter, is plotted against the log of the kernel diameter [52]. A straight line results with a negative slope S, and D can be further estimated.

#### 3.2. Power law scaling relationship

A functional relationship between two quantities is known as power law. This relationship takes place when a relative change in one quantity results in a proportional change in the other quantity and independent of the initial size of those quantities; thus, one quantity varies as power law to another. The characteristic of fractals is known as power law scaling. Therefore, a relationship, which yields a straight line on log-log coordinates, can often identify an object or phenomena as fractal. Although not all power law relationships are due to fractals, an observer needs to consider the existence of such relationship in order to know if the system is self-similar [66]. Self-similarities indicate the existence of scaling relationship which implies the type of a relationship called "power law." Thus, self-similarities give rise to the power law scaling. The power law scalings are shown as a straight line when the logarithm of the measurement is plotted against the logarithm of the scale at which it is measured. Fractal dimension is based on self-similarities; thus, power law scaling can be used to determine the fractal dimension. For a set to achieve the complexity and irregularity of a fractal, the number of self-similar pieces must be related to their size by power law [47]. The power law scaling describes how the property L(r) of the system depends on the scale r at which it is measured using the relation in Eq. (4).

$$L(r) = Ar^{\alpha} \tag{4}$$

The fractal dimension describes how the number of pieces of a system depends on the scale r, using the relation in Eq. (5).

$$N(r) = Br^{-D} \tag{5}$$

where B is a constant. The similarity between Eqs. (4) and (5) means that one can determine the fractal dimension D from the scaling exponent α if one knows how the measure property L(r) depends on the number of pieces N(r). For example, for each little square of sides of an object with two-dimensional area, the surface area is proportional to r 2 . Thus, one can determine the fractal dimension of the exterior of such an object by showing that the scaling relationship of the surface area depends on the scale r. For example, to determine the fractal dimension D from the scaling exponent is to derive the function of the dimension f(D), such that the property measured is proportional to r <sup>f</sup>(D) [66]. If the experimentally determined scaling of the measured property is proportional to r<sup>α</sup>, then the power of the scale r can be equated to the relation in Eq. (6):

$$f(D) = \alpha \tag{6}$$

Then, one can solve for D.

can be accomplished by convolution operation with a binary disk, that is, all the non-zero components of all the convolution kernels have a (Boolean) unitary value. The result is a thickened, but grey-scale border. All non-zero pixels are thresholded to a Boolean one when returning this border to Boolean one values. The speed at which the total surface area of the border raises as a function of the diameter of the convolution kernel relies on the dimension D. The log of each resulting area, divided by the kernel diameter, is plotted against the log of the kernel diameter

A functional relationship between two quantities is known as power law. This relationship takes place when a relative change in one quantity results in a proportional change in the other quantity and independent of the initial size of those quantities; thus, one quantity varies as power law to another. The characteristic of fractals is known as power law scaling. Therefore, a relationship, which yields a straight line on log-log coordinates, can often identify an object or phenomena as fractal. Although not all power law relationships are due to fractals, an observer needs to consider the existence of such relationship in order to know if the system is self-similar [66]. Self-similarities indicate the existence of scaling relationship which implies the type of a relationship called "power law." Thus, self-similarities give rise to the power law scaling. The power law scalings are shown as a straight line when the logarithm of the measurement is plotted against the logarithm of the scale at which it is measured. Fractal dimension is based on self-similarities; thus, power law scaling can be used to determine the fractal dimension. For a set to achieve the complexity and irregularity of a fractal, the number of self-similar pieces must be related to their size by power law [47]. The power law scaling describes how the property L(r) of

[52]. A straight line results with a negative slope S, and D can be further estimated.

the system depends on the scale r at which it is measured using the relation in Eq. (4).

The fractal dimension describes how the number of pieces of a system depends on the scale r,

<sup>L</sup>ðrÞ ¼ Ar<sup>α</sup> <sup>ð</sup>4<sup>Þ</sup>

3.2. Power law scaling relationship

Figure 1. Example of fractal dimension of a material.

254 Fractal Analysis - Applications in Physics, Engineering and Technology

using the relation in Eq. (5).

#### 4. Fractal characteristics of porous media

Porous media include many manmade as well as natural materials. All solid substances are in fact porous either to some degree or at some length scale [67]. A porous medium is a randomly multi-connected medium with channels randomly obstructed. The quantity that measures how "holed" the medium is due to the presence of these channels, and it is called the porosity of the medium. A pore network description can represent the porous medium as an ensemble of pores and throats of different geometries and sizes that can take values from appropriate distributions [68]. Therefore, fractal theory gives a favorable layer of structures of different models that will address the complexity of the disordered, heterogeneous, and hierarchical porous media like soil, materials with fracture networks. Theoretically, Yu et al. [69] provided an overview of the physical properties of ideal fractal porous media and explained how natural heterogeneous materials can exhibit both mass- and pore-fractal scaling. Cihan et al. [70] reported new analytical models for predicting the saturated hydraulic conductivity based on the Menger sponge mass fractal. They tested their model predictions against lattice Boltzmann simulations of flow performed in different configurations of the Menger sponge.

Fractal models have been used to describe the solid volume, the pore volume, or the interface between the two phases of porous media. In the past three decades, fractal models of pore space were developed and used in the petroleum physics with application in hydraulic system and in engineering communities with the application electrical conductivity [71, 72]. Turcotte [73] proposed a fractal fragmentation model, which identified a physical basis for the existence of fractal soils in the scale invariance of the fragmentation of soil particles. Hence, his model elucidated a mechanism in which scale-independent fragmentation processes could form fractal distribution of particles, giving theoretical legitimacy to the study of fractal models on porous media. Fragmentation can be viewed as the chief mechanism of physical weathering [67].

#### 4.1. Characteristic method of porous media

Hunt [67] stated that model characteristics are defined so that the porosity and water retention functions are identical to those of the discrete and explicit fractal model of Rieu and Sposito [74] (called hereafter the RS model). They began with a description of virtual pore size fractions in a porous medium that permits a facile foundation that conceptualized the fractal of solid matrix and pore space. These concepts resulted in equations used in solving the porosity and bulk density of both the size fractions and the porous medium in terms of a characteristic fractal dimension, D.

A porous medium with a porosity from a broad range of pore sizes was considered, the porosity decreases in mean (or median) diameter from <sup>p</sup><sup>0</sup> � pm�<sup>1</sup>ð<sup>m</sup> <sup>≥</sup> <sup>1</sup>Þ. A bulk element of the porous medium has the volume V0, massive enough to contain all sizes of pore; it has porosity φ and the dry bulk density σ0. They divided the pore-volume distribution of V<sup>0</sup> mathematically into m virtual pore size fractions, with the ith virtual size fraction defined by:

$$P\_i \equiv V\_i - V\_{i+1} \ (i = 0, \dots, m-1) \tag{7}$$

where Pi represents the volume of pores entirely made of size pi contained in Vi which is the ith partial volume of the porous medium, which itself has all pores of size ≤ pi . The partial volume Viþ<sup>1</sup> is therefore incorporated in Vi and the partial volume Vm�<sup>1</sup> is then incorporated in the smallest pore-size fraction Pm�1, together with the residual solid volume symbolized by Vm. They stated that the solid material whose volume is Vm will not be chemically or mineralogically homogeneous. Its mass density, symbolized by σm, represents an average "primary particle" density. Eq. (7) gives the bulk volume of the porous medium which can mathematically be represented as the summation of m increments of the basic pore-size fraction P<sup>0</sup> � Pm�<sup>1</sup> added to a residual solid volume Vm:

$$V\_0 = \sum\_{i=1}^{m-1} P\_i + V\_m \tag{8}$$

The porosity of the medium can then be given as [74]:

$$\phi \equiv \frac{(V\_0 - N^m V\_m)}{V\_0} \tag{9}$$

$$\mathbf{1} = \mathbf{1} - (\mathbf{1} - \boldsymbol{\Gamma})^{\mathsf{m}} \tag{10}$$

Going by the fractal dimension of Eq. (3), proposed by Mandelbrot [1], the fractal dimension is related closely to the pore coefficient, Γ [74].

$$
\Gamma = 1 - Nr^3 \tag{11}
$$

which, with Eq. (3), results to:

$$
\Gamma = 1 - r^{3-D} \text{ ( $\Gamma < 1$ ,  $r < 1$ )}\tag{12}
$$

It was shown from Eq. (12) that in a fractal porous medium where pore sizes are scaled by the same ratio r < 1, the fractal dimension increases with the decrease in the magnitude of the pore coefficient [74]. Thus, the relation between the porosity and the fractal dimension from Eqs. (10) and (12) gives:

4.1. Characteristic method of porous media

256 Fractal Analysis - Applications in Physics, Engineering and Technology

fractal dimension, D.

size fraction defined by:

added to a residual solid volume Vm:

The porosity of the medium can then be given as [74]:

related closely to the pore coefficient, Γ [74].

which, with Eq. (3), results to:

Hunt [67] stated that model characteristics are defined so that the porosity and water retention functions are identical to those of the discrete and explicit fractal model of Rieu and Sposito [74] (called hereafter the RS model). They began with a description of virtual pore size fractions in a porous medium that permits a facile foundation that conceptualized the fractal of solid matrix and pore space. These concepts resulted in equations used in solving the porosity and bulk density of both the size fractions and the porous medium in terms of a characteristic

A porous medium with a porosity from a broad range of pore sizes was considered, the porosity decreases in mean (or median) diameter from <sup>p</sup><sup>0</sup> � pm�<sup>1</sup>ð<sup>m</sup> <sup>≥</sup> <sup>1</sup>Þ. A bulk element of the porous medium has the volume V0, massive enough to contain all sizes of pore; it has porosity φ and the dry bulk density σ0. They divided the pore-volume distribution of V<sup>0</sup> mathematically into m virtual pore size fractions, with the ith virtual

where Pi represents the volume of pores entirely made of size pi contained in Vi which is the

volume Viþ<sup>1</sup> is therefore incorporated in Vi and the partial volume Vm�<sup>1</sup> is then incorporated in the smallest pore-size fraction Pm�1, together with the residual solid volume symbolized by Vm. They stated that the solid material whose volume is Vm will not be chemically or mineralogically homogeneous. Its mass density, symbolized by σm, represents an average "primary particle" density. Eq. (7) gives the bulk volume of the porous medium which can mathematically be represented as the summation of m increments of the basic pore-size fraction P<sup>0</sup> � Pm�<sup>1</sup>

> <sup>φ</sup> � <sup>ð</sup>V<sup>0</sup> � <sup>N</sup>mVm<sup>Þ</sup> V0

¼ 1 � ð1 � ΓÞ

Going by the fractal dimension of Eq. (3), proposed by Mandelbrot [1], the fractal dimension is

ith partial volume of the porous medium, which itself has all pores of size ≤ pi

V<sup>0</sup> ¼ m X�1 i¼1

Pi � Vi � Viþ<sup>1</sup> ði ¼ 0, …, m � 1Þ ð7Þ

Pi þ Vm ð8Þ

<sup>m</sup> <sup>ð</sup>10<sup>Þ</sup>

<sup>Γ</sup> <sup>¼</sup> <sup>1</sup> � Nr<sup>3</sup> <sup>ð</sup>11<sup>Þ</sup>

. The partial

ð9Þ

$$
\phi = 1 - (r^{\mathbf{3} - D})^m \tag{13}
$$

For a given value of the exponent m, the porosity of a fractal porous medium decreases as the fractal dimension increases. Further, Eq. (13) shows that the fractal dimension of a porous medium must be < 3.

Moreover, integration over the continuous pore size distribution between qr and r, where q is the ratio of radii of successive pore classes in fractal soil, r is the pore radius, q < 1 is an arbitrary factor, yields the contribution to the porosity from each size class obtained by RS model. The distribution of pore sizes is defined by the following probability density function [75]:

$$\mathcal{W}(r) = \frac{3 - D\_p}{r\_m^{3 - D\_p}} r^{-1 - D\_p} r\_0 \le r \le r\_m \tag{14}$$

where r<sup>0</sup> and rm refer explicitly to the minimum and maximum pore radii, respectively. The power law distribution of pore sizes is bounded by a maximum radius, rm, and truncated at the minimum radius, r0. Eq. (14), as written, is compatible with a volume, r3, for a pore of radius r and Dp describes the pore space. The result for the total porosity derived from Eq. (14) is given in [75] as:

$$\phi = \frac{3 - D\_p}{r\_m^{3 - D\_p}} \int\_{r\_0}^{r\_m} r^3 r^{-1 - D\_p dr} = 1 - \left(\frac{r\_0}{r\_m}\right)^{3 - D\_p} \tag{15}$$

Eq. (15) is exactly as in RS (Eq. (13)). If a particular geometry for the pore shape is envisioned, it is possible to change the normalization factor to maintain the result for the porosity and also maintain the correspondence to RS [67].

#### 4.2. Fractal analysis on the permeability of porous media

The fluid flow through porous media is governed by geometrical properties, such as porosity, properties of the flowing fluid, the connection and the tortuosity of the pore space.

The transport phenomena in porous media, that include single-phase and multiphase fluid flow through porous media, electrical and acoustical transport in porous media, and heat transfer in porous media, are focused on common interests and have emerged as a separate field of study [76–79]. A matrix of a porous medium combined with fractured networks is called the dual-porosity medium. In the dual-porosity media, fracture and matrix are generally considered as different media, each with its own property. Thus, gas flow through these dualporosity media could consist of two physically distinguished migration processes: one is associated to the movement of gas through the larger-scale fractures, that is, a permeability flow, which can be described by Darcy law, the other is related to the movement of gas inside matrix blocks, that is, diffusion processes, which may be involved in several different mechanisms, subject to the pore size [30, 74, 80].

In reality, surfaces of capillaries are rough and have great impact on fluid flow behavior and permeability of a porous medium. Analytically, permeability expression is a function of the relative roughness, the tortuosity fractal dimensions, capillaries sizes, and surface roughness, together with the microstructural parameters (such as the characteristic length, the maximum and minimum pore diameters, and the fractal dimensions) [19].

#### 4.3. Methods of fractal analysis on the permeability of porous media

Fractal, multifractal, Gaussian, and log-normal models have been initiated, perhaps in all scale range. The validation of unchanging theoretical framework used in calculating transport properties, at least at some scales, has the capacity of eliminating much confusion regarding the appropriate theoretical approaches used and the appropriate model to choose [67]. Investigation on gas flow through a dual-porosity medium, for example, a flow domain made up of matrix blocks (with low permeability) implanted in a network of fractures, is not common. Physical and computer modeling are commonly used for permeability of porous media. Different methods of analysis on the permeability of porous media will be discussed in this section.

Zheng and Yu [30] studied the permeability of a gas with the use of matrix porous media embedded with randomly distributed fractal-like tree networks. The scientific expression for gas permeability in dual-porosity media was obtained based on the pore size of matrix and the mother channel diameter of embedded fractal-like tree networks having fractal distribution. It was discovered that gas permeability was a function of structural parameters, which includes the fractal dimensions for pore area and tortuous capillaries, porosity and the maximum diameter of matrix, the length ratio, the diameter ratio, the branching levels, and angle of the embedded networks used for dual-porosity media. The model that was initiated does not contain any empirical constant. The model predictions were validated with the available experimental data and simulating results, a fair agreement among them was found. An analysis of the influences of geometrical parameters on the gas permeability in the media was done.

Khlaifat et al. [80] experimentally studied a single-phase gas flow through fractured porous media of tight sand formation of Travis Peak Formation under different operation conditions. Their study enhanced gas recovery from low permeability reservoirs by the creation of a single fracture perpendicular to the flow direction. The porous medium sample that was taken into account was a slot-pore-type tight sand from the Travis Peak Formation with permeability in the microdarcy range and a porosity of 7%. A number of single-phase experiments that include water and gas were performed at different pressure drops conditions ranging from 100 to 600 psig and at overburden pressures of 2000, 3000, and 4000 psig, respectively. It was shown from their results that the sample used was very sensitive to overburden pressure. Again, it was shown from the experimental data that the presence of a fracture in a low permeability porous media is the main factor responsible for reinforcing the gas recovery from tight gas reservoirs. The presence of a fracture reinforces the gas flow, due to the increase in overall permeability and the creation of different flow patterns, which locally shifted the two-phase flow away from capillary force domination region. Furthermore, the fracture aperture played a significant role in enhancing flow due to both reconfigurations of connecting pores and joining of the nonconnecting pores to the flow network.

called the dual-porosity medium. In the dual-porosity media, fracture and matrix are generally considered as different media, each with its own property. Thus, gas flow through these dualporosity media could consist of two physically distinguished migration processes: one is associated to the movement of gas through the larger-scale fractures, that is, a permeability flow, which can be described by Darcy law, the other is related to the movement of gas inside matrix blocks, that is, diffusion processes, which may be involved in several different mecha-

In reality, surfaces of capillaries are rough and have great impact on fluid flow behavior and permeability of a porous medium. Analytically, permeability expression is a function of the relative roughness, the tortuosity fractal dimensions, capillaries sizes, and surface roughness, together with the microstructural parameters (such as the characteristic length, the maximum

Fractal, multifractal, Gaussian, and log-normal models have been initiated, perhaps in all scale range. The validation of unchanging theoretical framework used in calculating transport properties, at least at some scales, has the capacity of eliminating much confusion regarding the appropriate theoretical approaches used and the appropriate model to choose [67]. Investigation on gas flow through a dual-porosity medium, for example, a flow domain made up of matrix blocks (with low permeability) implanted in a network of fractures, is not common. Physical and computer modeling are commonly used for permeability of porous media. Different methods of analysis on the permeability of porous media will be discussed in

Zheng and Yu [30] studied the permeability of a gas with the use of matrix porous media embedded with randomly distributed fractal-like tree networks. The scientific expression for gas permeability in dual-porosity media was obtained based on the pore size of matrix and the mother channel diameter of embedded fractal-like tree networks having fractal distribution. It was discovered that gas permeability was a function of structural parameters, which includes the fractal dimensions for pore area and tortuous capillaries, porosity and the maximum diameter of matrix, the length ratio, the diameter ratio, the branching levels, and angle of the embedded networks used for dual-porosity media. The model that was initiated does not contain any empirical constant. The model predictions were validated with the available experimental data and simulating results, a fair agreement among them was found. An analysis of the influences of geometrical parameters on the gas permeability in the media was done. Khlaifat et al. [80] experimentally studied a single-phase gas flow through fractured porous media of tight sand formation of Travis Peak Formation under different operation conditions. Their study enhanced gas recovery from low permeability reservoirs by the creation of a single fracture perpendicular to the flow direction. The porous medium sample that was taken into account was a slot-pore-type tight sand from the Travis Peak Formation with permeability in the microdarcy range and a porosity of 7%. A number of single-phase experiments that include water and gas were performed at different pressure drops conditions ranging from 100 to 600 psig and at overburden pressures of 2000, 3000, and 4000 psig, respectively. It was shown from

nisms, subject to the pore size [30, 74, 80].

258 Fractal Analysis - Applications in Physics, Engineering and Technology

this section.

and minimum pore diameters, and the fractal dimensions) [19].

4.3. Methods of fractal analysis on the permeability of porous media

A well-testing technique for Devonian shale gas reservoirs characterized by a low storage and high flow-capacity natural fractures fed by a high storage, low flow-capacity rock matrix was developed by Kucuk and Sawye [81] by using analytical methods and numerical simulator. They developed analytical solutions in order to analyze the basic fractured reservoir measurable factors that influence well productivity. These measurable factors are fracture system porosity and permeability, matrix porosity and permeability, and matrix size. They found that the traditional way of testing the well does not usually work for fractured Devonian shale gas reservoirs. Most of the time, the two parallel straight lines with a vertical separation are not shown in the semi-log plot of the drawdown and build-up data. They further found that the inter-porosity flow parameter is not the only parameter, which characterizes the nature of semi-log straight line.

A permeability model assumed to be comprised of a bundle of tortuous capillaries whose size distribution and roughness of surfaces follow the fractal scaling laws has been derived for porous media [19]. The proposed model includes the effects of the fractal dimensions for size distributions of capillaries, for tortuosity of tortuous capillaries, and for roughness of surfaces on the permeability. The proposed model is given by Eq. (16):

$$K\_{R} = \frac{\pi L\_{0}^{1-D\_{T}} D\_{f} \lambda\_{\text{max}}^{3+D\_{T}}}{128A(3+D\_{T}-D\_{f})} (1-\varepsilon)^{4} \tag{16}$$

where KR denotes the permeability for flow in porous media with roughened surfaces. Eq. (16) indicates that the permeability is a function of the relative roughness ε, the fractal dimensions DT (the fractal dimension for tortuosity of tortuous capillaries) and Df (the fractal dimension for pore space), as well as the structural parameters A (cross-sectional area), L<sup>0</sup> (the representative length or straight line along the flow direction of a capillary), and λmax (maximum capillary diameter). Eq. (16) also shows that the higher the relative roughness, the lower the permeability value; this can be explained by saying that the flow resistance is increased with the increase in roughness. This is consistent with a physical situation [19].

The proposed Eq. (16) was found to be a function of the relative roughness ε, the fractal dimension DT for tortuosity of tortuous capillaries, and structural parameters A, L0, and λmax. The ratio of the permeability for rough capillaries to that for smooth capillaries follows the quadruplicate power law of (1 � ε) given by Eq. (17). That is, Eq. (17) indicated that the decrease of permeability for porous media with roughened surfaces in capillaries follows the quadruplicate power law of (1 � ε). The authors concluded that the permeability of porous media with roughened capillaries will be drastically decreased with the increase in relative roughness, and the proposed model can reveal more mechanisms that affect the flow resistance in porous media than conventional models [19]. K in Eq. (17) is the permeability of porous media with smooth capillaries.

$$
\eta = \mathbf{K}\_{\mathbb{R}} / \mathbb{K} = (1 - \varepsilon)^4 \tag{17}
$$

Zinovik and Poulikakos [82] evaluated the relationships between porosity and permeability for a set of fractal models with porosity approaching unity and a finite permeability. Prefractal tube bundles generated by finite iterations of the corresponding geometric fractals can be used as a model porous medium where permeability-porosity relationships are derived analytically as explicit algebraic equations. Their investigation showed that the tube bundles generated by finite iterations of the corresponding geometric fractals can be used to model porous media where the permeability-porosity relationships are derived analytically. It was further shown that the model of prefractal tube bundles can be used to obtain fitting curves of the permeability of high porosity metal foams and to provide insight on permeability-porosity correlations of the capillary model of porous media.

All the methods discussed here have shown that the permeability of a porous media is strongly affected by its local geometry and connectivity, the matrix size of the material, and the pores available for flow. All the methods gave concept and knowledge of fractal geometry in relation to the characterization of the porous structure with respect to the permeability of the porous media.

## 5. Conclusion

Fractal is considered a self-similar system. It has been confirmed that the fractal technique is a powerful technique that has been successfully used in the characterization of the geometric and structural properties of fractal surfaces and pore structures of porous materials. It gives an understanding on how the geometry affects the physical and chemical properties of systems since their complex patterns are better described in terms of fractal geometry if the selfsimilarity is satisfied. It also builds a bridge between micro-morphology and macro performance. This chapter shows that the structural and functional characters of porous materials depend on the pore structure, which can be described effectively by the fractal theory.

## Author details

Oluranti Agboola1,2\*, Maurice Steven Onyango2 , Patricia Popoola<sup>2</sup> and Opeyemi Alice Oyewo2

\*Address all correspondence to: funmi2406@gmail.com

1 Department of Chemical Engineering, Covenant University, Nigeria

2 Department of Chemical, Metallurgical, and Materials Engineering, Tshwane University of Technology, South Africa

## References

roughness, and the proposed model can reveal more mechanisms that affect the flow resistance in porous media than conventional models [19]. K in Eq. (17) is the permeability of

η ¼ KR=K ¼ ð1 � εÞ

Zinovik and Poulikakos [82] evaluated the relationships between porosity and permeability for a set of fractal models with porosity approaching unity and a finite permeability. Prefractal tube bundles generated by finite iterations of the corresponding geometric fractals can be used as a model porous medium where permeability-porosity relationships are derived analytically as explicit algebraic equations. Their investigation showed that the tube bundles generated by finite iterations of the corresponding geometric fractals can be used to model porous media where the permeability-porosity relationships are derived analytically. It was further shown that the model of prefractal tube bundles can be used to obtain fitting curves of the permeability of high porosity metal foams and to provide insight on permeability-porosity correlations

All the methods discussed here have shown that the permeability of a porous media is strongly affected by its local geometry and connectivity, the matrix size of the material, and the pores available for flow. All the methods gave concept and knowledge of fractal geometry in relation to the characterization of the porous structure with respect to the permeability of the porous media.

Fractal is considered a self-similar system. It has been confirmed that the fractal technique is a powerful technique that has been successfully used in the characterization of the geometric and structural properties of fractal surfaces and pore structures of porous materials. It gives an understanding on how the geometry affects the physical and chemical properties of systems since their complex patterns are better described in terms of fractal geometry if the selfsimilarity is satisfied. It also builds a bridge between micro-morphology and macro performance. This chapter shows that the structural and functional characters of porous materials

depend on the pore structure, which can be described effectively by the fractal theory.

2 Department of Chemical, Metallurgical, and Materials Engineering, Tshwane University of

<sup>4</sup> <sup>ð</sup>17<sup>Þ</sup>

, Patricia Popoola<sup>2</sup> and Opeyemi Alice Oyewo2

porous media with smooth capillaries.

260 Fractal Analysis - Applications in Physics, Engineering and Technology

of the capillary model of porous media.

5. Conclusion

Author details

Technology, South Africa

Oluranti Agboola1,2\*, Maurice Steven Onyango2

\*Address all correspondence to: funmi2406@gmail.com

1 Department of Chemical Engineering, Covenant University, Nigeria


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