Radio Propagation and Remote Sensing of the Environment - Chapter 6 pps

In particular, the electric field on the surface approx-is written as: 6.25 Here, wi is the wave vector of the incident wave on the plane z = 0, and Having found the field component of the

Rough Surfaces

6.1 STATISTICAL CHARACTERISTICS OF A SURFACE

Many natural surfaces, such as the soil surface or water surface of the ocean, can

be regarded as smooth only in certain circumstances In general, these surfacesshould be considered to be rough, and their interaction with radiowaves should beseen as a scattering process Whether or not we assume that the surface is roughgenerally depends on the problem formulation and, particularly, on the ratio of theroughness scales and the wavelength The nature of the roughness varies depending

on the type of surface Sea surface roughness is a result of the interaction of thewind with the water surface This interaction has a nonlinear character A greatnumber of waves with different frequencies and wave numbers are generated as aresult, and their mixture leads to oscillations of the sea surface height according tothe stochastic function of coordinate and time However, the velocity of the seasurface movement is small compared to the velocity of light, so time dependencecannot be taken into account in the first approximation Soil roughness can form as

a result of wind erosion, urban activity, and other causes The soil roughness is also

a random function of coordinates Again, a dependence on time is not considered

in the beginning and we are dealing generally with wave scattering by randomsurfaces The specific surface will be described by a random function of the elevation

The average value so it is assumed that the averagesurface is given by the plane The function is supposed to be statisticallyhomogeneous It simplifies the problem substantially, as the statistical homogeneity

of the real natural rough surfaces take place in the restricted cases The correlationfunction:

(6.1)

depends, in this case, on the coordinate difference of the points involved In manycases, the surface may be assumed to be statistically isotropic Then, the correlationfunction depends only on the module (i.e., on the distance between thepoints of correlation) The correlation function has significant value within thecorrelation radius which is often defined by the relation:

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158 Radio Propagation and Remote Sensing of the Environment

It is necessary to represent the correlation scales along the main axis of the anisotropyellipse in the case of statistical anisotropy

If we suppose that roughness occupies the bounded surface Σ for whose measure

is much larger than the radius of correlation, then the following Fourier expansion

is correct:

The spatial spectrum:

(6.4)

is a random function with zero mean Its correlation function is written as:

By introducing the “gravity center” coordinate and the difference

, the last integral can be written in the form:

Let us represent the elevation fluctuation spectrum of the examined surface as:

(6.5)

according to the Wiener–Chintchin theorem The integral in this expression can bespread over infinite limits because the size of the chosen surface was set much largerthan the correlation scale Thus,

K qζ Kζ

π( )= ˆ ( ) − ⋅

′ = ′′ =

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Wave Scattering by Rough Surfaces 159

(6.7)

where we have simplified the expression by not distinguishing between the value ofthe surface and its square If then the integral in Equation (6.6) can be alsotaken to infinite limits by maintaining the condition As a result,

On the other hand, it is easy to determine from Equation (6.2) that

0

2

3 4

2 K
( )0 q q3 8 K
( ).0

d l

l .

=i∫q
Kζ( )q e iq s⋅ ′− ′′ ( s)d q

.2 TF1710_book.fm Page 159 Thursday, September 30, 2004 1:43 PM

160 Radio Propagation and Remote Sensing of the Environment

And, finally, we can calculate the value:

The last expression is easily transformed to the form:

(6.13)

The previous discussion was concerned with smooth and differentiable surfaces

It is often convenient to eliminate the requirement of differentiability when ing natural surfaces (sea, soil, etc.) To illustrate, we will analyze the structurefunction of properties of the surface:

The differentiability of the surface is understood in such sense

Many natural surfaces have a fractal character.78 Their structure function satisfiesthe equation:

Wave Scattering by Rough Surfaces 161

The value obtained lies within the range , which leads to an index interval

of in Equation (6.17) It would not be correct, in this case, to address thedifferentiability of the surface The maximum value of α leads to Equation (6.15),which is typical for smooth surfaces The value , which follows fromthe theory of Brownian motion, corresponds to the Brownian fractal

Let us now turn our attention to the properties of rough surfaces Generally, wecannot expect to develop an exact technique to solve the problem of radiowavediffraction on stochastic surfaces; instead, we must rely on approximation methodsthat, as a rule, are found effective for asymptotic cases In our case, the roughness

is small compared to the wavelength, which is the opposite of the case of largeinhomogeneities The method of small perturbation is effective in our case, and theKirchhoff approach is best suited for the second case Recently, some attempts havebeen made to find analytical solutions of the latter problem on the basis of an integralequation model (IEM) for electromagnetic fields79; however, only some refinement

of results have been reported, and the IEM is undergoing improvement.80

6.2 RADIOWAVE SCATTERING BY SMALL INHOMOGENEITIES AND CONSEQUENT APPROXIMATION SERIES

Let us assume that the described surface separates into two media The upper mediumhas permittivity equal to unity The permittivity of the lower medium is equal to ε

We assume that a plane wave of single amplitude isincident from the side of the z-coordinate positive values To find the scattered field

it is necessary to solve Maxwell’s equations for both media while maintaining theboundary conditions:

(6.19)

on the surface The numbers 1 and 2 indicate, respectively, the fields over and underthe examined surface If ez is the single normal to the average plane z = 0, then thesingle normal to the surface ζ(s) satisfies the equation:

162 Radio Propagation and Remote Sensing of the Environment

This inequality means that the probability is extremely low that deviations of theroughness from the average plane can be more than the wavelength We also assumethat the considered rough surface slopes are small, or << 1 Equation (6.11)allows us to assume that the correlation radius of such a surface is much larger thanthe height; that is,

We can now apply the method of sequential approximations Let us representthe unknown fields in the form of series:

(6.22)

Here, j = 1, 2, and the sum terms represent expansion over the growing degree of

Naturally, the sum terms of the same small size satisfy Maxwell’sequations

The fields, however, are expanded into a Taylor series of the form:

Taking into account the approximation , the borderconditions in Equation (6.19) are transferred from surface ζ(s) on the plane z = 0.Further, the corresponding expansions are continued until the second order of small-ness is obtained Let us set the term of the same order of smallness equal to zero

to obtain the boundary conditions for fields of a different order These boundaryconditions for the field of the zero order have the form:

0 2 0

0 2 0 0

1 0 2 0

1 1 2 1

( ) ( )

( ) ( ) ζζ ∂

∂ε

1

1 2 1

Wave Scattering by Rough Surfaces 163

Analogous expressions can be obtained for the second-order fields We will notprovide them here but refer the reader to Armand,52 where the second approach isanalyzed in detail

Further actions deal with using Maxwell’s equations to solve for every imation at given boundary conditions In doing so, the incident plane wave (moreexactly, the source of the radiated wave) is the source for the field of the zeroapproximation The fields of subsequent approaches are excited by the precedingfields So, a system of constrained fields is obtained for which it is necessary tosolve a succession of Maxwell’s equations for fields of different orders We shouldbear in mind when doing so that the boundary conditions for the normal to theaverage plane field components are odd in some sense, but we must keep them inorder to minimize calculations when they are indirectly presented Note that theseboundary conditions are analogous to Equations (1.93) and (1.94), which means thatour problem is reduced to a problem of fields excited by surface currents The formalsolution of this problem is Equation (1.111) which will be used from here on.Let us begin with the zero-arch approximation It does not require any specialconsideration as it is reduced to a problem of plane wave reflection by the planethe formulae in a more convenient form In particular, the electric field on the surface

approx-is written as:

(6.25)

Here, wi is the wave vector of the incident wave on the plane z = 0, and

Having found the field component of the zero approximation, let us now computethe fields of the first approximation of the perturbation theory The first step is torewrite the equivalent surface currents and charges of Equation (6.23) with a morecompact right side We refer the reader to Bass48 and Armand52 for more details onthe procedure, as we provide only the results here The first expression of Equation(6.23) can be expressed as:

1 01

= c −  × ∇ ⋅( )π

ε

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interface This problem has been examined in Chapter 3, but here we will rewrite

164 Radio Propagation and Remote Sensing of the Environment

The second boundary condition is rewritten as:

(6.32)

The electric current is represented as:

(6.33)where vector i is defined as:

1 04

1 0

1 01

Wave Scattering by Rough Surfaces 165

inte-gral of the first term is transformed into such over the boundary of the surface Σ,and we can set it equal to zero, as the roughness is zero, or the incident field is small

on the border of the radiating antenna footprint The integral of the second summand

is transformed to:

(6.36)according to Equation (6.4) Similarly, for the electric current:

and introduce the vector:

(6.41)and we obtain:

+

ε

k c

166 Radio Propagation and Remote Sensing of the Environment

Here, the subscript s indicates the direction of scattering, and

Computation of the field scattered into the lower medium is done in the same way.The formulae for scattered fields allow us to compute easily the scatteringamplitudes into upper and into lower semispaces and to determine accordingly thecross section of the scattering It is necessary to take into account that the scatteringamplitude is a stochastic value with a mean value equal to zero The squared module

of the scattering amplitude is also an occasional quantity; therefore, an importantdefinition of the cross section is It is appropriate to take into accountEquation (6.7), which leads us to the conclusion that the scattering section isproportional to the square of the illuminated surface So, it is reasonable to introduce

a definition of the scattering cross section per unit of surface σ0, the so-calledreflectivity It is a dimensionless value that characterizes the scattering properties ofthe surface regardless of size:

(6.44)

Thus, the intensity of the wave scattering is proportional in the first approximation

to the power of the surface spatial component with wave vector Itsabsolute value is:

(6.45)

The subscript i represents values related to the incident wave The result obtainedindicates that the incident wave interacts most effectively with only one of the spatialharmonics of the surface This effect is said about to be a resonance one

The spatial spectrum of the surface is a rather acute function of angles, so theangle dependence of the scattering intensity (scattering indicatrix) is generallydefined by the properties of the function Particularly, we can assert thatmaximum scattering occurs in the direction of the specular reflection when

We must pay special attention to the very important case of backscattering.Particular interest is raised here by the fact that the radar images are formed against

a background of wave backscattering In this case, , and the followingexpression is obtained for the backscattering reflectivity:

Wave Scattering by Rough Surfaces 167

(6.46)Here,

168 Radio Propagation and Remote Sensing of the Environment

One can see that the second approximation changes the coherent component of thefield This change proceeds at the second order of smallness and, as a rule, cannot

be taken into consideration in mean field calculation

Another situation appears during calculation of the second moment: average powerflow Here, keeping the summand not greater than the second order of smallness,

(6.51)

where the interference term:

(6.52)

is analogous to the second summand according to the value order, which corresponds

to the power flow density of the first approximation waves We can now concludethat we cannot neglect waves of the second approach order in the calculation ofenergetic values

We will not fully calculate the second approximation fields at permittivity values

of a medium with a rough interface An example of such a calculation can be found

in Armand.52 The analysis given there indicates that we cannot use the perturbationmethod in the common form to calculate second approximation fields at high per-mittivity values of the scattering medium To calculate the field inside the medium,limitation of expansion by the first term of the series is possible by maintaining thecondition:

Thus, the extent of roughness must be smaller than the wavelength in the medium(or skin depth) This requirement may not be valid in some cases — for example,when solving the problem of wave scattering by ripples For high permittivity values,the problem must be analyzed, from the very beginning, based on the assumptionthat the field inside the scattering medium is equal to zero, as occurs in the case ofideal conductivity We can use the Shchukin–Leontovich boundary conditions.67Now we will address the coherent component of the second approachfield, As was shown in Leontovich,67 this component has a plane wave formand propagates in the direction of the specular reflection, thereby interfering withthe reflected wave of the zero approximation Having said this, we will focusprimarily on the vertical incident of the original wave and will analyze this problembriefly for ideal conductivity of the scattering surface Assume we have the followingboundary condition:

Wave Scattering by Rough Surfaces 169

By repeating the procedure for the previous expansions, it is easy to obtain theboundary conditions system on the plane z = 0 These boundary conditions have aform analogous to Equations (6.23) and (6.24), where the fields in the medium areassumed to be equal to zero We can show that the field on the plane z = 0 is:

These approximations lead to development of a formula describing the power flow

of the radiation scattered by a single area:

(6.59)

Let us turn now to calculation of the second-order field (we are interested only

in its mean value) We can show that its value on the averaged plane is:

(6.60)

E( )1 s g

0 2z= = ikζ( ) i

2π

( )

2 2w

2

2π
K kζ cos θ sinθ ϕ θ , k s

ππ π

∫

∫0 2

170 Radio Propagation and Remote Sensing of the Environment

This expression represents the amplitude of the plane wave describing, on average,the reflected field of the second order

When calculating the interference term, we also must take into account that thereal part of Poynting’s vector is important, so:

(6.61)

The integration limit inside of a circle of radius w = k corresponds to extraction ofthe real part of the integral in Equation (6.61)

Let us now turn to integration in the cylindrical coordinate system:

It is a simple matter to see that the power flow of the interference summand withthe minus sign is equal to the power flow absolute value of scattered waves So, wehave established by a simple example that the energy of scattered waves is theinterference component, which reduces the energy of coherently reflected waves bythe corresponding value To be more exact, we should point out that the equality ofboth power flows is correct with a precision on the order of We can obtainthis result using the Shchukin–Leontovich boundary conditions for the waves of allorders

Finally, let us suppose that the topic of interest is not the stochastic but thedetermined surface with the profile:

42

2 0

w w w

K

π π

ez
K ( )cos sink 2 0

2 0

 ( ) = −  ×  sin

sin2

0 2

Wave Scattering by Rough Surfaces 171

In the right side of the boundary conditions, we have left only the first nonvanishingterm in order to declare a particular circumstance, thus avoiding complete analysis ofthe scattering process It is a simple matter to find that the second-order field on the

k means that these spectral components scatter most intensively in direction es, whichsatisfies the Bragg conditions for first-order diffraction This spectral component hasthe wave number due to backward scattering

6.4 WAVE SCATTERING BY LARGE

INHOMOGENEITIES

Gravitational sea waves are higher, as a rule, than wavelengths of the microwavesregion We will consider sea waves to be surfaces with large inhomogeneities andwill examine the small slopes of these surfaces In doing so, we also assume thatthe curvature radius of the considered surfaces is much larger than the wavelength,which allows us to use Kirchhoff’s approximation to solve the problem of diffractionthe surface is not considered for this approximation, which allows us to reduce thescattering process at each point to a process of local reflection from the plane surface Let us consider a plane wave incident on a surface with large roughness tocalculate the field of scattered waves Based on Equations (1.82–1.84), the scatteringamplitude can be expressed in the form:

k1=ksinθi+2Kandk2=ksinθi−2K

sinθ1 sinθ , sinθ2 sinθ

K k

K k

Ei Hi

, gi, [ei×gi]TF1710_book.fm Page 171 Thursday, September 30, 2004 1:43 PM

by large inhomogeneities It was pointed out in Chapter 1 that the facet model of

172 Radio Propagation and Remote Sensing of the Environment

indicate that we are dealing with a plane wave of single amplitude The reflectedfield is a function of the local incident angle, whose variations from point to point

of the surface are determined by the its slope change The normal to the surfacechanges together with slope variations according to Equation (6.20) Thus, vector

B is a function of gradient

It is convenient now to turn our attention from integration over the stochasticsurface to integration with respect to the mean plane In this case, the surfaceelements are connected by the relations Further,

we should take into account that Therefore,

(6.66)

Here, are projections of vectors on the plane z = 0

Let us now calculate the average value of the scattering amplitude whichdescribes the coherent component of the scattered field We should take into accountthe lack of correlation between the degree of roughness and the slopes measured atthe same point Due to this, vector B and exponent in the integral of Equation (6.66)are statistically independent As a result:

Wave Scattering by Rough Surfaces 173

,

assuming a lack of correlation between the slopes along the orthogonal coordinates.Being able to ignore the second term in this expansion follows from our previousestimation of slope values which means that should be takeneverywhere, regarding that the normal is the outer one here The last equality is theresult of multiplication of the entire expression by the delta-function, which gives us:

(6.68)

Coherent scattering occurs in the direction of the plane wave reflected by the interfaceplane (us = ui) Because the value of the reflected field, , is proportional to thecoefficient of reflection at the interface of two media, coherent scattering takes placewith the following effective coefficient of reflection:

=
ζ(2 cosθ ).

P1

2

2 21

ζ

π ζ

ζζ

P k i e

1

22

2 2 2cosθ ζ cos θ

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174 Radio Propagation and Remote Sensing of the Environment

which allows us to state that the coherent component is practically absent in thescattering of microwaves by natural surfaces

Thus, the incoherent component of scattering is the primary one, the energycharacteristics of which will be the focus of our further calculations First, we want

to find the squared module of the scattering amplitude This value is equal to:

(6.73)

con-cerns the local normal vector The exponent under the integrals is a fast oscillatingfunction because the degree of roughness is assumed to be large; therefore, the mainarea of integration is concentrated near the point and we can use the expan-sion Further, we will integrate with respect to and over s′ The first integration gives us the delta-function, so:

(6.74)

We will consider the cross section of the scattering to be the average value of thescattering amplitude squared module To average Equation (6.73) it is necessary tomultiply the scattered amplitude squared module by the distribution function

of the slopes and to integrate over the entire region of the change Due to thedelta-function properties, we can obtain the value of the distribution function by theargument:

(6.75)

The points where the gradient has this value correspond to the surface areas wherespecular reflection takes place (so-called specular or bright points) The scatteringvector es coincides with the local vector of reflection at these points Therefore,

where is understood to be the local angle of incidencecorresponding to Equation (6.74) As a result,

(6.76)

cosθi< 1 k 2 ζ2 ,

216

E2

0 1cos i cos

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Wave Scattering by Rough Surfaces 175

The final formula for the specific cross section is written as:

0

2 4 21

F P

σ

θζ0

2

2 20

176 Radio Propagation and Remote Sensing of the Environment

Applying Equation (6.11), we obtain:

(6.82)

It is apparent that the cross section of backward scattering due to a large degree ofroughness depends rather strongly on the angle of incidence with a normal heightdistribution

Let us note that we did not take into consideration the shadow effect on someareas of the surface by others If this effect exists, not all of the areas are ofimportance in the process of scattering, and the established formulae must beadjusted accordingly We do not investigate this problem here, instead referringreaders to Ishimaru,49 who has performed such a special study

Now, we shall regard the case of wave scattering with regard to vertical incidence

of the plane wave For this purpose, let us use Equation (6.78) In this case, and In order to simplify the problem, we will restrict ourselves toassuming statistical isotropy of the roughness, and then the slope probability willdepend only on Hence,

(6.83)

Our aim is to calculate the power flow scattered by the surface of the singlesquare As is already known, this flow is defined by the integral with respect to thesolid angle We must take into account, when integrating overthe scattering angle , the small probability of the scattering covering the largeangles because the slopes are small This gives us the opportunity to factor out theintegration sign, as the reflecting coefficients at the incident angle value are equal

to zero So, the scattered power flow is equal to:

The probability that the slope value is greater than unity is extremely low due tothe assumptions we have made; therefore, the upper limit in the last integral can beexpanded to infinity, which gives us:

coss

2

0( )∫ ( )

∫π

S8

s= c F

π

20( ) TF1710_book.fm Page 176 Thursday, September 30, 2004 1:43 PM

Wave Scattering by Rough Surfaces 177

The value on the right, however, is none other than the power flow density of thereflected wave in the absence of roughness Thus, the coherently reflected wavepractically disappears in the presence of considerable roughness and its energy ispumped over the scattered wave energy

In conclusion, let us turn to an analysis of wave scattering by fractal surfaces

In the expression under the integral of Equation (6.73), the values depend

on slopes but the exponent depends on the roughness spectrum The roughnessspectrum is mainly concentrated in the interval of large scales of inhomogeneities

in contrast to the slope spectrum, which gravitates toward the small-scale area (refer

to Equation (6.10)) Therefore, the roughness and slopes are found to be weaklycorrelated for any separation of the correlation points We have reason, then, toassume a complete absence of the mentioned correlation between roughness andslopes Moreover, due to the tendency toward the small-scale part of the spectrum,the correlation radius of the slopes is on the order of the inner scale, while the largescales dominates the mechanism of scattering It is important to remember thatKirchhoff’s approximation used here in our research is applicable only in the case

of wave scattering by inhomogeneities that are large compared to the wavelength Assuming isotropy and Gaussian statistics of roughness, we will now introduce

in Equation (6.73) differential coordinates of integration and coordinates of thegravity center, as we have repeatedly done before Then,

(6.85)

We particularly emphasize the dependence of vector B on the direction of scattering.Because the smallness of the wavelength relative to the surface outer scale isassumed, the wave number k appears as a large parameter (the geometrical opticsapproximation); therefore, the main area of integration in Equation (6.85) is con-centrated close to zero This gives the opportunity to use the approximation ofEquation (6.17) for the structure function to obtain:

28

178 Radio Propagation and Remote Sensing of the Environment

In the marginal case when α = 2 (the differentiable surface), we have Equation(6.81), if we assume:

(6.88)

Equation (6.88) is correct in all cases independent of the fractal dimension value.Thus, we must always consider wave scattering due to the large-scale part of theroughness spatial spectrum (i.e., wave numbers that satisfy the condition q < k) Theconcept of specular points is correct in this case, and we have to assume that

in the expression for So, we obtain:

(6.89)

In the specific case of Brownian fractals (α = 1), the integral is tabulated,44 and

we obtain Hagfor’s formula:81

E l

2 2 22

14

Wave Scattering by Rough Surfaces 179

Let us designate the specific cross section of large roughness as and thecorresponding cross section of scattering by small roughness as Then, thecombined cross section of scattering is equal to:

(6.91)

The cross section of scattering by small roughness can be written as a function ofthe local normal because of the change of slope, revealing its dependence on coor-dinates on the scattering surface It is necessary, in this case, to distinguish betweenthe surface of the mean plane and the large-scale rough area under considerationwhich leans against it Let us mention that the given formula is true not only for arandom surface but also for another one (not random)

The second term in Equation (6.91) is a stochastic value for casual surfaces;therefore, we must perform the second averaging relative to the slope and write:

(6.92)

Here, we have introduced the designation Let us consider in detail the case

of backward scattering by the surface with ideal conductivity ( ) In this case,

(6.93)

To analyze the second term we can designate it as ; specifically, we willassume a Gaussian distribution of altitudes of large-scale roughness and, correspond-ingly, a Gaussian distribution of slopes Thus, we will neglect the quadratic values

in Equation (6.93), as doing so will not cause any misunderstanding because of thelow probability of large slopes We will use the model correlation function:

(6.94)

for small roughness which supposes statistical isotropy of the surface Equation(6.94) is rather often used for tentative computations The corresponding spatialspectrum is presented in the form:

σmac 0

σmic 0

n r( ) d r

Σ0

σ0 σ0 2σ0

1 21

l

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180 Radio Propagation and Remote Sensing of the Environment

(6.95)

To distinguish the parameters of large and small roughness, we will add the subscripts

mac and mic, respectively The roughness, as already noted above, will be assumed

to be statistically isotropic

Bearing in mind these assumptions, we can write

and Here, γ is the angle between the vectors

ui and v The second term of Equation (6.93) now has the form:

vv

0 0

K( )v,γ = 4v θi 3θi−2( z⋅ i) ( )i⋅ 

4

sin coscos

0

0

2 2 2 2

mic

i mac

Wave Scattering by Rough Surfaces 181

polar-ization (horizontal or vertical) In the result, the expression of slope dispersion via

dispersion of elevations and the correlation radius was as defined in the formula

(6.11)

It is easy to show that the influence of large surface inhomogeneities affects

wave scattering by small roughness at the condition From this,

in accordance with the accepted approximations , we have the

condition So, large inhomogeneities have an effect only in the case

when the scattering is large scale (meaning that their scale is greater than the

wavelength) The inhomogeneities discussed here have a scattering indicatrix with

an angle spread on the order of

It is easy to conclude from this that the discussed effect reveals itself at the

condition:

Let us now compare the backscattering cross sections of large and small

rough-ness First, however, we can improve the formula for the small-roughness spatial

spectrum The correlation function and the spectrum may be written as:

(6.100)

The advantage of these formulae is their nondimensionality Let us point out also

Note that at the nadir radiation , and

So, at small zenith angles the intensity of wave scattering from large inhomogeneities

exceeds that for small roughness, which seems rather natural; however, due to the

smallness of the slope angle dispersion, the angle dependence of the cross section

is a rather pointed function, and the intensity of the quasi-specular scattering

decreases fast with increase in the zenith angle The angle dependence from small

roughness is not so sharp; therefore, beginning with any zenith angle value, the

Λh=4tanθiandΛv= −2sin2θi (1+sin2θi)

πk4 ζ2mac lmic4 ≥lmac2

182 Radio Propagation and Remote Sensing of the Environment

intensity of this scattering mechanism becomes dominant Let us estimate the zenith

angle value at which the change of the scattering mechanism occurs The

corre-sponding equation can be written as:

(6.101)

Here, for horizontal polarization and for

ver-tical polarization The second summand to the right is a slowly changed function of

angle, so, in the first approximation, it can be assumed to have the value , and

(6.102)

6.6 IMPULSE DISTORTION FOR WAVE SCATTERING

BY ROUGH SURFACES

So far we have investigated the problem of harmonic wave scattering It is obvious

that the scattered field will be an aggregate of sine waves at the sine wave incident

on the rough surface If we turn to the problem of complicated form wave scattering,

we will call them impulses for short so the picture will change little Every spectrum

component of the impulse is being scattered at its own individual amplitude and

phase So, the sum of the scattered components of the spectrum will form an impulse

of such a shape that, generally speaking, differs from the initial impulse incident on

the surface The situation here is similar to the one which we encountered during

The spectrum of the scattered impulse is equal to:

(6.103)

Here, is the complex amplitude of the corresponding spectral component of

the incident impulse In the scattering amplitude f, only its frequency dependence

is assigned, which is the main dependence in this case It is easy to see that the

scattering amplitude plays the role of a frequency filter The filtering approaches of

the scattering amplitude are the cause of impulse deformation The form of the

scattered impulse is set by the equality:

TF1710_book.fm Page 182 Thursday, September 30, 2004 1:43 PM

consideration of the phenomena of frequency dispersion in Chapter 2

Wave Scattering by Rough Surfaces 183

The scattering amplitude is a random function and follows the random shapes of

the scattered impulse We shall regard the average value of the scattering amplitude

to be equal to zero As we have determined, this assumption is true in the first

approximation of the perturbation theory for wave scattering by small-scale

rough-ness and is also correct for large-scale roughrough-ness on a scale greater than the

wave-length By virtue of this, the average value of vector is equal to zero, and

the primary object of interest is the mean power:

(6.105)

where the stroked functions depend on ω′ and ω′′

Now it is necessary to calculate the frequency correlation function of the

scat-tering amplitude We will focus on the case of large roughness, as it is the most

interesting one It follows from Equation (6.66) that:

Here,

and perform expansion of the function ζ as we have already done when

we derived Equation (6.73) Then,

Repeating the method of our previous calculations, we obtain:

2 216