One-Dimensional Gaussian Random Rough Surface

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For the sake of simplicity, let us consider a one-dimensional random rough surface as shown in Fig. 4.6.1. The surface profile is described by a height function j(x), which is a random function of coordinate x. The coordinates of a point on the surface are denoted by (x, j(x)). The surface heights assume values z = j(x), with a Gaussian probability density function p(z) as

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p(z) =

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1 (72V2ii exp ( -

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(Z-17)2)

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2(72

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(4.6.1)

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where (7 is the standard deviation or root-mean-square (7ms) height, and 17 is the mean value of the surface height which is usually assumed to be zero. The joint probability density function PZ IZ2(Zl, Z2) of two Gaussian random variables, Zl and Z2, is given by [Papoulis, 1984]

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180 4 CHARACTERISTICS OF DISCRETE SCATTERERS AND ROUGH SURFACES

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(x,f(x))

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Figure 4.6.1 One-dimensional random rough surface.

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where 1]1 and 1]2 are the respective mean values of Z1 and Z2, and o-f and o-~ their variances. In (4.6.2), C is the correlation coefficient. Let 1]1 = 1]2 = 0, and let 0-1 = 0-2 = 0-; in this case, the covariance of the two random variables Z1 and Z2 is

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(Z1 Z 2)

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dZ1 dZ2 Z1 Z2

PZIZ2(Z1,Z2)

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/00 dZ 2 Z2 exp (_ z~2) -00 20Z1 { C2]

00 d

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Z1 0- J27f

[1 _

[Z1 - Z2 C j2 } 20- 2 [1 - C2]

00 ~ -00 dZ 2 z~ C exp (- 20- = 0-2C z~2) o-v 27f

(4.6.3)

where the inner integral over dZ1 is equal to Z2C, which is the average value of the random variable Z1 for a normal density with mean Z2C. The correlation coefficient of two random variables Z1 and Z2 is generally defined as the ratio between their covariance and the product of their standard deviations 0-10-2 [Papoulis, 1984]. Note that ICI :s; 1. If C = 0, then Z1 and Z2 are independent random variables.

6 Gaussian Rough Surface and Spectral Density

Characteristic Function

The characteristic function of a random variable z is defined as the average value of exp( ikz)

<I>(k)

= (exp(ikz)) =

dz p(z) exp(ikz)

(4.6.4)

where k is the Fourier transform variable. Equation (4.6.4) states that the characteristic function is equal to the Fourier transform of the probability density function p(z). This function is maximum at the origin, 1<I>(k) I :s; <I>(O) = 1. For a Gaussian random variable z with the probability density function of (4.6.1), the characteristic function is

<I>(k)

k2u2) = exp(ik1]) exp ( --2-

(4.6.5)

The characteristic function of joint Gaussian random variables Zl and with the joint probability density function (4.6.2) is equal to (exp [i(klz l

+ k 2z 2)])

= exp

[-~ (krur + 2CUW2klk2 + U~k~)]

(4.6.6)

Correlation Function

The correlation function of the random process of surface height f(x) is defined as

(4.6.7)

It is a measure of the correlation of surface profile f(x) at two different loca-

tions Xl and X2. For the case of Gaussian height distribution with zero mean and rms height u, we can obtain, from (4.6.3) and (4.6.7), the correlation function Rf(XI' X2) = U2C(Xl, X2) (4.6.8) where C(XI, X2) is a function of Xl and X2. The correlation function is often assumed to be Gaussian:

C(XI' X2) = exp

~2 X2)2)

(4.6.9)

where 1 is known as the correlation length. As IXI - x21 1, C(XI, X2) tends to be zero, and functions f(xI) and f(X2) become independent, which means that when two points on a rough surface are separated by a distance much larger than the correlation length, the function values at these two points are

4 CHARACTER/STICS OF DISCRETE SCATTERERS AND ROUGH SURFACES

independent. Other types of descriptions, such as exponential and fractal, have also been used for the rough surface correlation functions. For a stationary random process f (x), the correlation function depends only on the separation Xl - X2 : (4.6.10) For example, the correlation function for a stationary Gaussian height distribution with zero mean and rms height u, by using (4.6.3) and (4.6.9), is given by

R f (X1, X2) =