Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015

Abstract

Results of application of theory of fractal and chaos, scaling effects and fractional operators in the fundamental issues of the radio location and radio physic are presented in this report. The key point is detection and processing of super weak signals against the background of non-Gaussian intensive noises and strays. An alternative—the radar range is increased dramatically. The results of researches of spectrum fractal dimensions of lightning discharge in the middle atmosphere at attitudes from 20 to 100 km which are above the majority of clouds are presented. The author has been investigating these issues for exactly 35 years and has obtained results of the big scientific and practical worth. The reader is invited to look at the fundamental problems with the synergetic point of view of non-Markovian micro- and macro systems.

Full text

Chapter 12
Chaos Theory, Fractals and Scaling
in the Radar: A Look from 2015
Alexander A. Potapov
Abstract Results of application of theory of fractal and chaos, scaling effects and
fractional operators in the fundamental issues of the radio location and radio physic
are presented in this report. The key point is detection and processing of super
weak signals against the background of non-Gaussian intensive noises and strays.
An alternative—the radar range is increased dramatically. The results of researches
of spectrum fractal dimensions of lightning discharge in the middle atmosphere at
attitudes from 20 to 100 km which are above the majority of clouds are presented.
The author has been investigating these issues for exactly 35 years and has obtained
results of the big scientific and practical worth. The reader is invited to look at the
fundamental problems with the synergetic point of view of non-Markovian micro-
and macro systems.
12.1 Introduction
The entire current radio engineering is based on the classical theory of an integer
measure and an integer calculation. Thus an extensive area of mathematical analysis
which name is the fractional calculation and which deals with derivatives and
integrals of a random (real or complex) order as well as the fractal theory has been
historically turned out “outboard”(!). At the moment the integer measures(integrals
and derivatives with integer order), Gaussian statistics, Markov processes etc., are
mainly and habitually used everywhere in the radio physics, radio electronics and
processing of multidimensional signals. It is worth noting that the Markov processes
theory has already reached its satiation and researches are conducted at the level of
abrupt complication of synthesized algorithms. Radar systems should be considered
with relation to open dynamical systems. Improvement of classical radar detectors
A.A. Potapov ()
V.A. Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences,
Moscow 125009, Russia
President of Cooperative Chinese-Russian Laboratory of Informational Technologies and Signals
Fractal Processing, Jinan University, Guangzhou, China
e-mail: potapov@cplire.ru
© Springer International Publishing Switzerland 2016
C. Skiadas (ed.), The Foundations of Chaos Revisited: From Poincaré to Recent
Advancements, Understanding Complex Systems,
DOI 10.1007/978-3-319-29701-9_12
195

196 A.A. Potapov
of signals and its mathematical support basically reached its saturation and limit. It
forces to look for fundamentally new ways of solving of problem of increasing of
sensitivity or range of coverage for various radio systems.
At the same time I’d like to point out that it often occurs in science that
the mathematical apparatus play a part of “Procrustean bed” for an idea. The
complicated mathematical symbolism and its meanings may conceal an absolutely
simple idea. In particular the author put forward one of such ideas for the first time
in the world in the end of seventies of twentieth century. To be exact he suggested to
introduce fractals, scaling and fractional calculation into the wide practice of radio
physics, radio engineering and radio location. Now after long intellectual battles my
idea has shown its advantages and has been positively perceived by the majority
of the thoughtful scientific community. For the moment the list of the author’s and
pupils works counts more than 750 papers including 20 monographs on the given
fundamental direction. Nowadays it is absolutely clear that the application of ideas
of scale invariance—“scaling” along with the set theory, fractional measure theory,
general topology,measure geometrical theory and dynamical systems theory reveals
big opportunities and new prospects in processing of multidimensional signals
in related scientific and engineering fields. In other words a full description of
processes of modern signal and fields processing is impossible basing on formulas
of the classical mathematics [1–11].
The work objective is to consider the use of the fractal theory and effects of
physical scaling in development of new informational technologies using examples
of solving of up-to-date basic radar problems. The author has been investigating
these issues in V.A. Kotel’nikov IREE RAS for exactly 35 years.
12.2 On the Theory of Fractional Measure and Nonintegral
Dimension
The main feature of fractals is the nonintegral value of its dimension. A development
of the dimension theory began with the Poincare, Lebesgue, Brauer, Urysohn
and Menger works. The sets which are negligibly small and indistinguishable
in one way or another in the sense of Lebesgue measure arise in different
fields of mathematics. To distinguish such sets with a pathologically complicated
structure one should use unconventional characteristics of smallness—for example
Hausdorff’s capacity, potential, measures and dimension and so on. Application of
the fractional Hausdorff’s dimension which is associated with entropy conceptions,
fractals and strange attractors has turned out to be most fruitful in the dynamical
systems theory [1,3–7,9–11]. This fractional dimension is determined by the
p—dimensional measure with an arbitrary real positive number pproposed by
Hausdorff in 1919. Generally the measure conception is related neither to metric nor
to topology.However the Hausdorff measure can be builtin an arbitrary metric space
basing on its metric and the Hausdorff measure itself is related to the topological
dimension. The Hausdorff–Besicovitch dimension is a metrical conception but

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 197
there is its fundamental association with topological dimension dim E,whichwas
established by L.S. Pontryaginand L.G. Shnirelman who introduced a conception of
the metrical order in 1932: the greatest lower bound of the Hausdorff–Besicovitch
dimension for all the metrics of compact Eis equal to its topological dimension
dim E˛.E/. One of much used methods for estimation of sets Hausdorff
dimension known as the mass distribution principle was proposed by Frostman in
1935.
Sets whose Hausdorff–Besicovitch dimension is a fractional number are called
fractal sets or fractals. More strictly, set Eis called fractal (a fractal) in the wide
sense (in the B. Mandelbrot sense) if its topological dimension is not equal to the
Hausdorff–Besicovitch dimension, to be exact ˛0.E/>dim E. For example, set E
of all the surd points [0; 1] is fractal in the wide sense since ˛0.E/D1,dimED0.
Set Eis called fractal (a fractal) in the narrow sense if ˛0.E/is not integer. A fractal
set in the narrow sense is also fractal in the wide sense.
12.3 Measuring of Fractal Dimension and Fractal Signatures
Fractal methods can function on all signal levels: amplitude, frequency, phase and
polarized. The absolute worth of Hausdorff–Besicovitch dimension is the possibility
of experimental determining [3–10]. Let’s consider some set of points N0in d—
dimensional space. If there are N(")—dimensional sample bodies (cube, sphere)
needed to cover that set with typical size ",atthat
N."/ 1="D;"!0(12.1)
is determined by the self-similarity law.
The practical implementation of the method described above faces the difficulties
related to the big volume of calculations. It is due to the fact that one must measure
not just the ratio but the upper bound of that ratio to calculate the Hausdorff–
Besicovitch dimension. Indeed, by choosing a finite scale which is larger than two
discretes of the temporal series or one image element we make it possible to “miss”
some peculiarities of the fractal. Building of the fractal signature [4–7] or estimates
dependence (1) on the observation scale helps to solve this problem. Also the fractal
signature describes the spatial fractal cestrum of the image. In IREE RAS we
developed various original methods of measuring the fractal dimension including
methods: dispersing, singularities accounting, on functionals, triad, basing on the
Hausdorff metric, samplings subtraction, basing on the operation “Exclusive OR”
andsoon[4–7]. During the process of adjustment and algorithms mathematical
modeling our own data were used: air photography (AP) and radar images (RI)
on millimeter waves [9]. Season measurements of scattering characteristics of the
earth coverings were already naturally conducted on wavelength 8.6 mm by the
author in co-operation with representatives of Central Design Bureau “Almaz” from
a helicopter MI-8 in the 1980s of twentieth century.

198 A.A. Potapov
A significant advantage of dispersing dimension is its implementation simplicity,
processing speed and calculations efficiency. In 2000 it was proposed to calculate
a fractal dimension using the locally dispersing method (reference for example [4–
7,9–11]). In the developed algorithms they use two typical windows: scale and
measuring. The scale window defines the necessary scale of measurements which
the scaling is observed in. That is why the scale window serves for selection of the
object to be recognized and its following description in the framework of fractal
theory. An image brightness or image intensity local variance is determined by
the measuring window. The locally dispersing method of the fractal dimension
Dmeasurements is based on measuring a variance of the image fragments inten-
sity/brightness for two spatial scales:
Dln 2
2ln 2
1
ln ı2ln ı1
:(12.2)
In formula (2) 1;
2—root-mean-squares on the first ı1and second ı2scales of
image fragment, respectively. Accuracy characteristics of the locally dispersing
method were investigated in [4,5,7]. It is proved [7] that in the Gaussian
case the dispersing dimension of a random sequence converges to the Hausdorff
dimension of corresponding stochastic process. The essential problem is that any
numerical method includes a discretization (or a discrete approximation) of the
process or object under analysis and the discretization destroys fractal features.
The development of special theory based on the methods of fractal interpolation
and approximation is needed to fix this contradiction. Various topological and
dimensional effects during the process of fractal and scaling detecting and multi-
dimensional signals processing were studied in [4–11].
12.4 Textural and Fractal Measures in Radio Location
During the process of radio location the useful signal from target is a part of the
general wave field which is created by all reflecting elements of observed fragments
of the target surrounding background, that is why in practice signals from these
elements form the interfering component.
It is worthwhile to use the texture conception to create radio systems for the
landscape real inhomogeneous images automatic detecting [4–6,9]. A texture
describes spatial properties of earth covering images regions with locally homoge-
nous statistical characteristics. Target detecting and identification occurs in the case
when the target shades the background region at that changing integral parameters
of the texture. Many natural objects such as a soil, flora, clouds and so on reveal
fractal properties in certain scales [4–6].
The fractal dimension Dor its signature D(t,f,!
r) in different regions of the
surface image is a measure of texture, i.e., properties of spatial correlation of
radio waves scattering from the corresponding surface regions. At already far first

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 199
steps the author initiated a detailed research of the texture conception during the
process of radio location of the earth coverings and objects against its background.
Further on a particular attention was paid to development of textural methods
of objects detecting against the earth coverings background with low ratios of
signal/background [4].
12.5 Fractal Signal and Image Processing in the Interference
The author was the first who shows that the fractal processing excellently does for
solving modern problems when processing the low-contrast images and detecting
super weak signals in high-intensity noise while the modern radars does not
practically function [4–7,9–11]. The author’s developed fractal classification was
approved by B. Mandelbrot during the personal meeting in USA in 2005. It
is presented on Fig. 12.1 where the fractal properties are described, D0—is a
topological dimension of the space of embeddings.
FRACTALS
A Infinite Number
оf Scales and Self -
Similarity (Scaling)
The Hausdorf
Fractal Dimension
D > D0
The Number of
Iteration
n → ∞
Mathematical Physical
The Hausdorf
Fractal Dimension
D ≥ D0
Finite Number of
Iteration
n
Fractional
Derivates
and Integrals
A Finite Number of
Scales and Self -
Similarity (Scaling)
A Piecewise
Differentiable
Function
Fig. 12.1 The author’s classification of fractal sets and signatures

200 A.A. Potapov
Fig. 12.2 Textural and
fractal methods of processing
low-contrast images and
super weak signals in
high-intensity non-Gaussian
noise
TEXTURES and FRACTALS for
SIGNALS and IMAGES
PROCESSING
Patterns Recognition
,
1987 + 1997
Morphologic Processing
,
1987 + 1997
Images and Signals
Analysis,
1987
Textural and Fractal
Signatures,
1987
Textural and Fractal
Characteristics
Selection,
1987
Conversion from
Gaussian Statistics
to Power Laws,
1980
Contours Selection
,
1987 + 1997
Radio Signals
Fine Structure,
1983
Images Filtering
, 1987
Images Fractal
Synthesis,
1996
Images Segmentation
,
1987
Textural and Fractal
Characteristics
Dictionary,
1987 + 2003
Sampling Topology
,
2000
Images Superposition
,
1988
Images Clustering
,
1987+1997
Textural Images
Autoregressive
Synthesis,
1987
Terrain Etalons
Synthesis,
1988+2006
Histograms
Modification,
1987
The textural and fractal digital methods under author’s development (Fig. 12.2)
allow to overcome a prior uncertainty in radar problems using the sampling
geometry or topology (one- or multidimensional). At that topological peculiarities
of the sampling and also the scaling hypothesis and stable laws with heavy “tails”
get important as opposed to the average realizations which frequently have different
behavior [4–7,9–11].
12.6 Development of “Fractal Ideology” in Radio Physics
A critical distinction between the author’s proposed fractal methods and classical
ones is due to fundamentally different approach to the main components of a signal
and a field. It allowed to switch over the new level of informational structure of
the real non-Markov signals and fields. Thus this is the fundamentally new radio
engineering. For 35 years of scientific researches my global fractal scaling method

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 201
Fig. 12.3 A sketch of author’s new informational technologies development basing on fractals,
fractional operators and scaling effects for nonlinear physics and radio electronics
has justified itself in many applications—Fig. 12.3.Thisisa challenge to time in
a way. Here only the facts say! Slightly exaggerating one can say that the fractals
formed a thin amalgam on the powerful frameworkof science of the end of twentieth
century. In the modern situation attempts of underestimating its significance and
basing only on the classical knowledge came to grief in an intellectual sense.
In fractal researches I always rest upon my three global theses:
1. Processing of information distorted by non-Gaussian noise in the fractional
measure space using scaling and stable non-Gaussian probabilistic distributions
(1981)—Figs. 12.1,12.2,and12.3.
2. Application of continuous nondifferentiable functions (1990)—Fig. 12.1.
3. Fractal radio systems (2005)—Figs. 12.3 and 12.4 [4–7,9–11].

202 A.A. Potapov
Fig. 12.4 The author’s conception of fractal radio systems, devices and radio elements
A logic aggregation of the problems triad described above into the general
“fractal analysis and synthesis” creates a basis of fractal scaling method (2006)
and a unified global idea of the fractal natural science and fractal paradigm (2011)
which were proposed and are investigated by the author now [4–7,9–11]. Basing
on the matter reviewed above next we will proceed to description of the fractal
radar conception and also issues of its scale-invariant principles application in other
systems of radio monitoring. In fact the question is about a fundamentally new type
of radio location: fractal scale or scale-invariant radio location.
12.7 Principles of Scale-Invariant or Fractal Scaling Radio
Location and Its Applications
At the moment world investigations on fractal radio location are exclusively
conducted in V.A. Kotel’nikov IREE RAS. Almost all the application points
of hypothetic or currently projectable fractal algorithms, elements, nodes and
processes which can be integrated into the classical radar scheme are represented
on Fig. 12.5. The ideology of proceeding to the fractal radar is based on the fractal
radio systems conception—Fig. 12.4.
In particular a multifrequency work mode is typical for the fractal MIMO-system
[11–13] proposed by the author earlier since fractal antennas can radiate several
waves lengths at the same time. Building of a tiny fractal radar with fractal elements
and modern parametrons is possible for unmanned aerial vehicles (UAV).
At the same time the fractal processing at the point of control of UAV transmitted
information will allow to improve sharply and automatize the processes of detecting,
clustering and identification of targets and objects. Moreover UAV fractal coating
will sharply reduce the probability of its detecting in flight.

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 203
Fig. 12.5 The points of application of fractals, scaling and fractional operators for proceeding to
the fractal radar
12.8 Fractal Detection of Objects on Images from SAR
and UAV
The base data for digital fractal processing of radar images were obtained by satellite
radar with the synthetic aperture (SAR) PALSAR of L-range (Japan). PALSAR is
a space SAR at wavelength 23 cm with spatial resolution of about 7 m which is
developed by Japanese agency JAXA and which was successfully working on orbit
from 2006 till 2011.
A radar image of Selenga estuary in Transbaikalia obtained in the FBS high
resolution mode on the coherent horizontal polarization on 7 August 2006 is
presented on Fig. 12.6 as an example.
The shooting zone of about 6050 km includes the forest covered mountainous
area Hamar-Daban (at the bottom, it is reproducedby a brighter tone with the typical
“crumpled” structure), the flat area of Selenga estuary (in the middle of the top
image part, it is reproduced by darker tones) and the smooth water surface of the
lake Baikal (the black segment in the left upper corner of the image). The banded
structures are seen in the flat part of the image, these are the bounds of agricultural
fields. Also the clusters of bright objects are seen, these are the strongly reflecting

204 A.A. Potapov
Fig. 12.6 Selenga estuary on
the P´£ PALSAR photo
from 7 August 2006
Fig. 12.7 The result of
fractal processing of the P´£
PALS A R
elements of buildings and other constructions in the range of settlements. The long
twisting dark lines on the plain are the multiple arms of Selenga.
The fields of local values of dispersing fractal dimension Dwere measured at the
first stage of radar images fractal processing by a SAR (Fig. 12.7). Next the empiric
distribution of values of the instant fractal dimension Dwas obtained Fig. 12.8.
Below the examples of fractal clustering over Dare presented (Figs. 12.9 and
12.10). The selected image fragment with fractal dimension D2.2 nearby the first
big peak (Fig. 12.8) is presented on Fig. 12.9. The selected image fragment with
fractal dimension D2.5 (Brownian surface) nearby the third and fourth big
peak (Fig. 12.8) is shown on Fig. 12.10.

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 205
Fig. 12.8 An empiric distribution of values of the instant fractal dimension D
Fig. 12.9 A fragment with D2.2
Previously invisible (hidden) peculiarities (for example earth coverings distant
probing clustering data [4–6]) along with a stable distribution by earth coverings
types are registered after fractal processing of surface images. It allows speaking of
application of fractal recognition methods for the identification of image parts which
are “invisible” when using classical methods of clusterization over the brightness
field.

206 A.A. Potapov
Fig. 12.10 A fragment with
D2.5
12.9 Fractal Characteristics of the High-Altitude Discharges
in Ionosphere
Four million lightnings draw the sky every 24 h and about 50 lightnings draw
the sky every second. And over the lead thunderheads, a light show of “unreal
lightnings” is developing in the upper atmosphere: azure jets, red-purple sprites,
red rings of highly soaring elves. These are discharges of very high energy which
do strike the ionosphere and not the ground! Thus high-altitude electrical discharges
(20–100 km) subdivide into several basic types: elves, jets, sprites, halo and so on—
Fig. 12.11 (This is the first colour image captured of one by NASA aircraft in 1994).
A history brief: a significant event occurred in the Earth study history in the night of
5 to 6 July 1989. Retired professor and 73 years old NASA veteran John Randolph
Winkler pointed an extremely sensitive camera recorder to thunderstorm clouds and
then he detected two bright blazes during inspecting the record frame by frame. The
blazes go up to the ionosphere in contrast to lightning’s which should go down to
the ground. This way the sprites were discovered. The sprites are the biggest high-
altitude discharges in the Earth atmosphere. After these publications NASA had not
already been able to disregard the potential threat to space vehicles and they started
a comprehensive research of high-altitude discharges.
The most short-lived high-altitude discharges are elves. They arise in the lower
ionosphere at altitudes 80–100 km. The luminescence arise in the center and
expands to 300–400 km for less than a millisecond and then it goes out. The
elves are born in 300 s after a strong lightning stroke from a thunderstorm
cloud to the ground. It gets altitude 100 km for 300 s where it “arouse” a red

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 207
Fig. 12.11 Dynamical fractal
structures in the atmosphere
(copyright: Abestrobi
(Wikipedia))
luminescence of nitrogen molecules. The most enigmatic high-altitude discharges
are azure jets. These are also a luminescence of nitrogen molecules in the ultraviolet-
blue band. They look like an azure narrow inverse cone which “starts” from the
upper edge of a thunderstorm cloud. Sometimes jets reach altitude 40 km. Their
propagation speed varies from 10 up to 100 km/s. Their occurrence is not always
due to lightning discharges. Besides azure jets they mark out “azure starters” (they
propagate up to altitudes 25 km) and “giant jets” (they propagate up to altitudes of
the lower ionosphere about 70 km). Sprites are very bright three-dimensional blazes
with duration around milliseconds. They arise at altitude 70–90 km and descend
down 30–40 km. Their width reaches tens of kilometers in the upper part. Sprites
blaze up in the mesosphere in about 100th part of a second after the discharge of
powerful lightnings “cloud–ground.” Sometimes it occurs at a distance of several
tens kilometers horizontally from the lightning channel. The red-purple colour of
sprites as well as elves is due to the atmosphere nitrogen. The frequency of sprites
occurrence is about several 1000 events per 24 h over the entire globe. The fine
structure of the lower sprites part is characterized by dozens of luminous channels
with cross sectional dimensions from tens to hundreds meters. Sprites occurrence is
related with formation of high electrical dipole moment of uncompensated charge
after especially powerful lightning discharges cloud–ground with usually positive
polarity.
Dynamical spatial-temporal singularities and morphology of sprites can be
particularly explained by the discharges fractal geometry and percolation [14]. Here
we have one more example of a self-organized criticality when the system (a high-
altitude discharge in this case) dynamics is determined by reaching the threshold
of the so called directed percolation which characterizes a formation of branchy

208 A.A. Potapov
Fig. 12.12 The original sprite image (USA, NASA http://science.compulenta.ru/701264/)
Fig. 12.13 Results of fractal filtering of a sprite image: (a) a pattern of fractal dimension with the
mean value DD2.3; (b)2.8;(c)3.0
(fractal) conductive channels overlapping all the sprite length. A different situation
arises with issues of data statistical processing.
Here the classical methods are used by tradition. It does not allow to extract
all the information about such newest atmospherically structures. Selected exam-
ples of our fractal processing of sprite profiles (Fig. 12.12) are presented on
Fig. 12.13a–c. Examples of fractal processing of a jet (Fig. 12.14a) are presented
on Fig. 12.14b, c.
The fractal-scaling methodology which was used for describing the morphology
of jets, sprites and elves can be successfully used to estimate their parameters
and dynamics of their evolution [14]. Then the mathematical physics problems are
solved.

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 209
Fig. 12.14 Results of fractal filtering of a giant jet image (the photos were taken in China August
12, 2010) (a) the jet image [15], (b)and(c)profilesofDestimates
12.10 Fractal Signal Detectors in Radiolocation
Classical detectors and their mathematical supply have virtually reached its satu-
ration and limit. It causes searching principally new ways of solving the problem.
Principally, fractals and fractional operators are not possible one without the other.
We showed for the first time that fractal processing is suitable as well as possible for
solving modern problems of the low-contrast images identification and ultra weak
signal detection in the presence of intensive non-Gaussian noises, when modern
radars can not operate. One of our main conclusions is that working on the pointed
evaluation of the fractal dimension Dleads to absurd results. At the same time
almost all the authors who begins using the fractal signal processing give absolutely
accurate meanings even with the RMS deviation! In our works we introduced fractal
signatures and fractal kepsters [4–7,9,16]. Therefore the accuracy problems in
digital fractal processing in real-time mode are solved.
The series of principally new fractal signal detectors (FSD) not mentioned by
me in press is shown below as an example of effective operation of the global
fractal methodology and the conception of radio systems and devices created by
the author. The main principles of fractal detection were proposed by us for the first
time as early as in 1989 works. At the same time a working model of the fractal
non-parametric radar signals detector (FNRSD—Fig. 12.3) was created. The high
accuracy of fractal detecting was proved. The main kinds of FSD proposed by us
during 2011–2012 are shown at Fig. 12.15.
Figures 12.16,12.17,and12.18 show selected results of fractal nonparametric
filtering of low-contrast objects. Aircraft images were masked by an additive
Gaussian noise. In this case, the signal/noise ratio (SNR) q2
0D–3 dB. It is seen
in the figures that all desired information is hidden in the noise.
The optimum mode of filtering of necessary contours or objects is chosen by
the operator using the spatial distribution of fractal dimensions Dof a scene. This
distribution is determined automatically and is shown in the right panel of the
computer display [4–7,9].

210 A.A. Potapov
Fig. 12.15 The main kinds of new dynamical FSD proposed by author
Fig. 12.16 Real image

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 211
Fig. 12.17 Source image and noise q2
0–3 dB
Fig. 12.18 Results of fractal filtration Fig. 12.17

212 A.A. Potapov
12.11 Wave Scattering by Fractal Surface
In many works it has been 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. This fact is result of self-similarity of fractal surface. In our work band-
limited Weierstrass function was used. For the scattered field analysis we use
Kirchhoff approach [17].
The most convenient function which both describes fractals well and is easy for
using in calculations is the modified 2D band-limited Weierstrass function. It has a
view:
W.x;y/Dcw
N1
X
nD0
q.D3/n
M
X
mD1
sin Kqnxcos 2m
MCysin 2m
MC'nm 
(12.3)
where cw—the constant, that provides unit normalization; q> 1—the fundamental
spatial frequency; D—the fractal dimension (2 < D<3); K—is the fundamental
wave number; Nand M—number of tones; ®nm—an arbitrary phase that has a
uniform distribution over the interval [; ].
Since the natural surfaces are neither purely random nor periodical and are often
anisotropic [2,4] then function that was proposed above is a good candidate for
characterizing of natural surfaces. Figure 12.19 shows us examples of band-limited
Weierstrass function for different scales. It is also important that function (12.3)
describes the mathematical fractals only if Mand Ngo to infinity. It is clear from
Fig. 12.19 that the function proposed possesses the self-similarity and multi-scale.
Fig. 12.19 W(x,y) for (a)—ND2, MD3, DD2.01, qD1.01; (b)—ND5, MD5, DD2.5,
qD3; (c)—ND10, MD10, DD2.99, qD7

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 213
Fig. 12.20 Function Q
dependence on Dfor various values of q
In this section of our work a statistical parameter is introduced for estimation
of the fractal dimension Dinfluence and other fractal parameters influence on the
surface roughness. Such parameter as the correlation length is conventionally used
for numerical characterization of rough surface [4,18,19]:
Q
./ Dh./
isD"1q2.D3/
1q2.D3/N#N1
X
nD0
q2.D3/nJ0.Kqn/ (12.4)
There are Q
dependences on qand Din Fig. 12.20 and Fig. 12.21 respectively.
It is shown that with increased value of D,Q
decreases more rapidly for the same
variation of q. It is shown in Fig. 12.20 that value of Q
reduces steadily with the
increase of Dvalue. However Q
does not change when qD1.01.
As mentioned above the Kirchhoff approach has been already used for analysis
of wave scattering by fractal surfaces [18,19]. Conventional conditions of the
Kirchhoff approach applicability 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, an incident wave is plane and
monochromatic, there are no points with infinite gradient on the surface, the Fresnel
coefficient V0is constant for this surface, surface large scales are much greater than
incident wave length. Shading effects will be taken into account in the following our
investigations and studies.

214 A.A. Potapov
Fig. 12.21 Function Q
dependence on qfor various values of D
Scattering indicatrix for average field intensity and two-dimensional surface [18,
19]:
gF2.1;
2;
3/
cos2111
2.kC/2sinc2.kALx/sinc2kBLyC1
4C2
f
N1
X
nD0
M
X
mD1
q2.D3/n
sinc2kA CKqncos 2 m
MLxsinc2kB CKqnsin 2 m
MLy (12.5)
We have got a data base of scattering indicatrixes for various fractal scattering
surfaces [19–21]. Also in terms of Weierstrass function (12.1) for one-dimensional
fractal scattering surface we obtained scattering field absolute value dependences
on incident angle and surface fractal dimension D. In subsequent computer calcula-
tions, we used the above expression for the coherence function
‰kDhEs.k1/Es.k2/i(12.6)
of the fields scattered by the fractal surface [19–21].
We can show that the tail intensity of signals reflected by a fractal surface is
described by power functions:
I.t/1=t03D(12.7)

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 215
Result (12.7) is very important because, for standard cases, the intensity of a
reflected quasi-monochromatic signal decreases exponentially. Thus, the shape of a
signal scattered by a fractal statistically rough surface substantially differs from the
shape of a scattered signal obtained with allowance for classical effects of diffraction
by smoothed surfaces [20,22].
The results obtained can be widely applied for designing various modern radio
systems in the microwave, optical, and acoustic bands.
12.12 Personal Meetings with Benois Mandelbrot
As it’s seen from above the author uses the “fractal” term almost everywhere. In
conclusion I’d like to share my impressions about the meeting with B. Mandelbrot
with readers. The meeting occurred in his house near New York in December
2005. At that time I was responsible for the international project. I had to visit
America frequently. My personal meeting with the founder of the fractal geometry
B. Mandelbrot occurred on Friday 16 December 2005. Before this the intensive
correspondence was going on both when I was still in Moscow and when I
flew across USA from south to north with my lectures on the results of 5 years
international project. Mandelbrot himself was in an extensive trip but his secretary
phoned and told that the maitre would come back home ad hoc. He was extremely
interested to meet at home and talk with the Russian physicist who dealt with various
“fractal” experiments and applications of the fractal theory in radio physics and
radio electronics.
I and my translator got to the New York Central railroad terminal by taxi and then
we got to an electric train leaving at 9.30 local time. After a while we went down
on a small station and went to the B. Mandelbrot’s house by taxi. As approaching
to the house we saw a silhouette of a high strong grayish man with glasses on a lace
appeared behind the door. He dressed in home clothing. While we are getting off
the car he’s already opened the front glassed-in door. Mandelbrot’s looking at me,
smiling, holding out a hand first. Then he’s suggesting us to undress and all of us are
going to his room. He’s asking me to seat down in front of him explaining that this
way is better for him to talk and there is a more comfortable arm chair for me. There
is his world-famous fundamental book “Mandelbrot B.B. The Fractal Geometry of
Nature” on the table. I am taking out my monograph on fractals and presenting
to Mandelbrot. At that I am telling about scientific work and my results on fractal
applications in radio physics, answering his questions. Mandelbrot is listening with
a keen interest and very attentively. It is a surprise for him that there is such a
success on fractals’ applications in Russia and there is already the direct approach
to the fractal technologies. He became very interested in my proposed conception
of fractal radio systems and in designing an essentially new fractal elemental base.

216 A.A. Potapov
Hi is well familiar with fractal antennas. Suddenly he’s spoken that sometime
the matter would be in producing a fractal capacitor! I reply with enthusiasm that
I am already “caught” with this idea for a lot of years and a big paper about
physical modeling of fractal impedances, fractional operators and production of
fractal capacitors is ready for the press. This is incredibly: we are thinking about
realizing the same idea on the opposite sides of the world! Mathematical questions
are less interesting for Mandelbrot. He is getting more and more interested in
disciplines created in IREE RAS: the fractal radio physics and the fractal radio
electronics, its development.
Human simplicity, openness, interest in the surrounding world and wisdom—
these particular properties are peculiar to B. Mandelbrot. Sometime they tell about
B. Mandelbrot’s arrogance. I can assert only the reverse. He did not make me feel
the difference between our statuses during all the conversation. He first inquired
about all fractal developments.
Forty minutes later Mandelbrot stands up and after apologizing and going out to
other room he comes back with a pile of his books. He asks me if I already have
some of these books. Mandelbrot says that he likes my works. He inquires when
and how my book was written. I reply that I prepared the first version as early as
in the beginning of nineties of XX. Then the search for publishers began and at
the same time improving and significant rework of the monograph text was going
on. Mandelbrot says that now he has two books on the go: the one is in Italy and
the other is in America. With a smile he admits that he writes slowly, thoroughly
using all his old works. Our conversation’s been lasting for almost 2 h. Tempus
fugit. At a certain moment he is called by the phone. He suggests to give us a lift
to the railway station on his car. Mandelbrot drives the car on his own. We are at
the railway terminal already at noon. I tell goodbye to B. Mandelbrot and we are
waving to each other. This is the unforgettable meeting. There are all the minutest
details of the meeting with the great scientist in my memory.

12 Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015 217
12.13 Conclusions
The fractal problem in radio location, radio physics and radio engineering is indeed
immense. Here I illustrate only fundamental initial issues. It is always hard and
even impossible to recede from habitual standards ::: But the author has good
reasons to think that the extensive and valuable material he already obtained and
the results of further researches will be used in advanced radio systems. The fractal
radio physics, fractal radio engineering and fractal radio location are peculiar radio
sciences. They are suffused with a spirit and ideas of the classical radio physics
and radio engineering but at the same time they are fundamentally new areas of
focus. The results of conducted researches oriented to enhancing the interference
immunity of work of radio systems on a radio channel with high-intensity noise and
distortion showed opportunities of the approach on the basis of using textural and
fractal-scaling methods of detecting and processing random signals and fields.
The author raised these questions back in 1980, and for 35 years has been
successfully working on their resolution [4–6]. Fractal methods similar to ones
presented in this work can be applied when considering wave and oscillatory
processes in optics, acoustics and mechanics. Results and conclusions obtained by
the author and his pupils have great innovative potential. We think that its realization
will resolve a number of current problems of radio physics, radio engineering,
radio location, communication and operation and also will allow to provide a new
quality for detecting and recognition systems and also development of the new
informational technologies.
Many important stages in fractal directions development including the stage of
this science field formation have been already passed. However many problems are
still to be solved. Results and specific solutions are not of so greatest value like the
solution method and its approach are. The method is created by the author [4–14,
16,19–21,23–25]. It is necessary to put it all into practice!
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