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3 The Applications of Chaos and Fractal in EM
Scattering
3.1 The Application of Chaos and Fractal in Modeling of Rough
Surfaces
The multiple scattering theory for waves in mediums containing random scat-
terers was presented by Flody, Twersdy and Lax originally. Then others began
researches on scattering of random turbulence medium. As the only Gaussian
process that has self-similarity inherent, FBM (Fractional Brownian Motion) is
continuous everywhere but can not be derived. So it was used to analyze wave
scattering and propagation in fractals by Berry in the year 1979[7].
Numerical accuracy of the representation of a Weierstrass structure function
based on FBM was examined by M.F.M Sanghaasa et al in 1994.The results
showed that this representation was satisfying for a large range of parameters.
With spectrum being a reliable approximation of FBM and performing well in
predictability and controllability, WM(WeierstrassCMandelbrot) function was
introduced to synthesize FBM[8]. As chaos and fractal theories could character-
ize natural surfaces with a few parameters, much attention was paid to rough
surfaces modeling based on them. In the year 2008, Ren Xin-Cheng et al used a
normalized two-dimensional band-limited Weierstrasss fractal function to model
rough surfaces. The function shows a combination of both deterministic peri-
odic structure and random rough structure. A general solution for the scattered
field based on the KA(Kirchhoff Approximation) was given to dielectric rough
surface. The influences of fractal dimension, the patch size of surface and the
fundamental wavenumber on the scattering field were discussed theoretically and
numerically. It was concluded that diffracted envelopes of scattering pattern can
be approximated as a slope of linear equation in the near forward direction or
the right of specular direction. This conclusion could be applicable for solving
the inverse problem of reconstructing rough surface and remote sensing[9].
3.2 The Application of Chaos and Fractal in Studies of Sea Surfaces
The theories of chaos and fractal are widely used in the studies of electromag-
netic scattering at fractal sea surface. Based on the 2D band-limited Weierstrass
function, Hou Deting et al developed the model of rough sea surface. The ef-
fects of fractal dimension, frequency, amplitude factor and incidence angle were
discussed in the research. The analytic expression of scattering coecient was
deduced using Helmholts integral and KA on the condition of the simulated
fractal surface. The numberical simulations were carried out at the same time.
Results showed that microwaves may be shaded by the rough sea surface, so the
shadow function was introduced to amend the scattering coecient at low gazing
angle. Furthermore, the effects of fractal dimension, frequency, amplitude factor
and incidence angle on the electromagnetic and scattering were discussed. Con-
clusions can be made that the peaks of scattering coecient are more uniformly
distributed with the increasing of the fractal dimension or the frequency and
 
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