## 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.

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 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.

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 time-spatial continuum [5]. Therefore in the late 1970s of the ?? 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” ≠ ”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

It is significant to note that Gaussian model is parabolic near the angle of incidence

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-mean-square 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 band-limited 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

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

where *c*
_{
w
} 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; ϕ_{
nm
}- 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 ρ(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 ρ(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

where

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

If *σ* = 1, then Eq. (3) is as follows:

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

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 *W*
_{
u
}(*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 ρ(τ) and correlation length Г. By definition

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

The average spatial autocorrelation coefficient

where

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

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

From Eqs. (10)–(12), one can find relationships between average correlation length
*D*, and also *q*. There are dependences
*q* and *D* shown in
**Figures 3**
and
**4**
, respectively. It is shown that with an increased value of *D*,
*q*. It is shown in
**Figure 4**
that the value of
*D* value. However,
*q* = 1.01.

Consequently, the mean correlation length
*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*.

## 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*
_{0} is constant for this surface, and surface scales are much greater than incident wavelength.

### 6.1. Scattered field

Scattering geometry is presented in
**Figure 5**
. Then, *scattered field*
*S* of

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

(23) |

Component

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

(25) |

From expression (Eq. (25)) and by taking into account expansion (Eq. (26)) and relationship (Eq. (27)):

we obtain

(28) |

Eq. (28) can be written as

(29) |

As result from Eqs. (13)–(23) and (29) **
field
**

**from finite site**

*scattered**S*is

### 6.2. Average-scattered field

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

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

Assume that the outside area
*S* is smooth, that is,

Then, Eq. (23) can be written as

### 6.3. Scattering indicatrixes for field

Scattering indicatrixes for field

where field scattered from perfectly smooth surface

Average-scattering indicatrix

Assume that surface gradients much less than incident angle is

(41) |

In specular direction
*A* = 0, *B* = 0, *a* = 0, *b* = 0. Using Eqs. (17)–(22), we can write average-scattering indicatrixes

Thus,
*k*, σ, θ_{1}, *q, D, N, M*. If

Eq. (44) shows that in specular direction
_{1}. 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
*. The intensity of scattered field is defined as

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

### 6.5. Scattering indicatrix for average field intensity

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

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

(50) |

where values with the order higher than

Statistical parameter of scattered field *σ*
_{
I
} is defined as

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
_{
I
} is determined only by the second term in Eq. (50) that relates to scattering by surface roughness. The second moment of scattered field *σ*
_{
I
} 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

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]:

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*
_{
n
} is the phase that is distributed uniformly over the [0, 2*π*]. Number of harmonics in function (Eq. (52)) is determined by *N = N*
_{
2
} *– N*
_{
1
} *+* 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):

(53) |

*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 *sq*
_{1}
*b* must be:

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

After the correction, we have

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

where *u*
_{
i
} is the integer and

Since terms with the order higher than

Thus

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

(61) |

where

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

## 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 were calculated in the summer of 2006 for the wavelengths

## 8. Conclusion

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, solid-state physics, physics of nanostructures, and for the synthesis of nano-materials.