Radio Propagation and Remote Sensing of the Environment - Chapter 3 ppsx

3 Wave Propagation in Plane-Layered Media3.1 REFLECTION AND REFRACTION OF PLANE WAVES AT THE BORDER OF TWO MEDIA Natural media — the atmosphere, earth, and others — can be supposed to b

3 Wave Propagation in Plane-Layered Media

3.1 REFLECTION AND REFRACTION OF PLANE WAVES

AT THE BORDER OF TWO MEDIA

Natural media — the atmosphere, earth, and others — can be supposed to behomogeneous only within bounded area of space In reality, the permittivity of thesemedia is a function of the coordinates and, in general terms, time, which is ignored,

as a rule, because of the comparative sluggishness of natural processes As usual,the time required for electromagnetic wave propagation in a natural medium is muchless than typical periods of medium property changes

In the first approximation, natural media can be considered to be plane layered;that is, their permittivity changes in only one direction If a Cartesian coordinatessystem is chosen in such a way that one of the coordinates (for example, z) coincideswith this direction, then we may say that permittivity depends on this coordinateonly For the atmosphere of Earth, a concept of spherical-layered media would bemore correct, and this idea is considered in the following chapters; however, weshould point out that the curvature radius of media layers of the Earth is so largethat the concept of plane stratification is sufficient in many cases

In this chapter, we will study only plane wave propagation in plane-layeredmedia We will assume waves as the plane only conventional because surfaces withconstant phase and constant amplitude are not planes in the cases of wave propaga-tion direction inclined to the layers On the whole, wave parameters cannot depend

on only one particular Cartesian coordinate in all cases In most of the cases that

we will consider here, media parameters change little compared to wavelength;therefore, surfaces of equal phase or amplitude have sufficiently large curvature radii

to be considered locally plane It is quite acceptable to talk about plane waves inthese terms

In some situations, medium properties can be changed sharply and on a length scale; however, regions of such great change are usually rather thin layersand the waves are plane, outside of the layers just discussed These layers correspond,

wave-in the extreme, to the areas jumpy changes of permittivity and occur at the boundary

of two media The air–ground boundary is an example of this idea

We turn now to consideration of plane wave reflection and transmission cesses at a plane interface We shall assume for the sake of simplicity that thepermittivity of the medium from where the wave originates is equal to unity Air is

pro-an example of such a medium The problem of a wave incident on the ground fromthe air is investigated in such a way The permittivity in this case is indicated by ε.The problem of wave reflection and transmission in media separated by a plane

TF1710_book.fm Page 53 Thursday, September 30, 2004 1:43 PM

54 Radio Propagation and Remote Sensing of the Environment

interface is well known and has been analyzed in many texts regarding netic waves; therefore, we will not investigate this problem in detail here and willexamine only essential formulae

electromag-The electrical and magnetic fields of the incident wave are described by thevectors:

(3.1)The boundary of the two media will be

assumed to be a plane that is perpendicular

to the z-axis Let θi represent the incidenceangle (see Figure 3.1) in such a way that:

and represent its fields by Er and Hr The wave in the ground is the refracted wave

or transmitted wave, and the fields of this wave are represented by Et and Ht It is

a simple matter to establish that the reflected and refracted waves are also found inthe plane and that their wave vectors lie in the same plane as the vector of theincident wave The wave vector of the reflected wave is:

(3.4)and the wave vector of the refracted wave is:

FIGURE 3.1 Plane wave incidence

at the plane boundary.

Wave Propagation in Plane-Layered Media 55

from which it follows that θr = θi; that is, the angle of reflection equals the angle

of incidence

In the case of complex permittivity ε, vector qr is complex and the transmittedwave is, in general, the inhomogeneous plane wave The refraction angle determinedfrom the equation:

(3.7)

is also complex in the general case It is simple, however, to derive Snell’s lawthrough the formula:

(3.8)

In the case of a weak absorptive medium, when we can neglect the imaginary part

of ε, angle θi is real, and we can fix the propagation direction of the refracted wave.Let us point out for later that:

(3.9)

The problem discussed here can be divided into the two cases of horizontal andvertical polarization In the case of horizontal polarization, the amplitudes of thereflected and refracted waves are connected linearly with the amplitude of theincident wave:

(3.10)where

56 Radio Propagation and Remote Sensing of the Environment

The magnetic and electrical wave components exchange places, in some sense,for the case of vertical polarization In this case, the equations

(3.13)are valid, where the reflection coefficient of the vertically polarized waves is:

(3.14)

and, correspondingly, the coefficient of transmission:

(3.15)

In these equations, the coefficients Fh and Fv are referred to as the Fresnel coefficients

of reflection They are complex values in the general case:

(3.17)and we can obtain the expressions:

Wave Propagation in Plane-Layered Media 57

for the reflected wave A simple expression for the field amplitudes:

We can obtain similar formulae for the transmitted wave field

As was mentioned above, the processes of reflection and refraction are due tochanges in the radiowave amplitude and phase In particular, the tendency is for thereflected and refracted waves to become elliptically polarized by incidence on theinterface the plane wave of arbitrary linear polarization In other words, a change

of the wave polarization takes place As we already know, the polarization charactercan be described by a Stokes matrix The processes of reflection and refraction can

be considered as linear transforms; however, calculation of a Mueller matrix is rather

a complicated procedure in this case It is easier to do direct calculation of Stokes

where the unitary vectors are directed toward the vectors of horizontaland vertical polarization and form the coordinate basis for the coordinate systemrelevant to the incident wave We may use a the similar coordinate basis for thereflected wave and present its field in the form

Now we will establish the relation between the orthogonal amplitude nents of the reflected and incident waves It is easy to do this for the horizontal

vertical components, it is necessary to note that in this case Let us

Stokes parameters of the reflected wave by Simple calculations allow us to setthe relations between two systems of parameters and to establish the Stokes matrixtransformation law for reflection of the wave These relations have the form:

1

r 0

58 Radio Propagation and Remote Sensing of the Environment

Let us also mention the specific relation connecting the reflection coefficients

of vertically and horizontally polarized waves The following relations are obtainedfrom Equation (3.11):

2

11

1

i h i

i F

TF1710_book.fm Page 58 Thursday, September 30, 2004 1:43 PM

Wave Propagation in Plane-Layered Media 59 3.2 RADIOWAVE PROPAGATION IN PLANE-LAYERED MEDIA

Now, we will consider radiowave propagation in a medium whose permittivity is,

in the general case, an arbitrary complex function of Cartesian coordinates; in thiscase, we choose z As has been pointed out, such media are referred to as planelayered (stratified) It was noted, too, that Maxwell equations written in the form ofEquations (1.91) to (1.92) are convenient to use These equations are simplifiedessentially because ε = ε(z) Hence, it should be taken into account that a planewave of any polarization propagates in one plane in this case and can be represented

as the sum of two wave types Let the y0z plane be the wave propagation plane,which means that the field does not depend on the x-coordinate and the operator

∂/∂x = 0 Then, the basic waves are E-waves (or waves of horizontal polarization),for which the electrical field vector is directed perpendicularly to the plane ofpropagation (i.e., it is represented in the form ), and H-waves (or verticalpolarized waves), for which For E-waves, the field is described by theequation:

d d

60 Radio Propagation and Remote Sensing of the EnvironmentThen, the problem is reduced to solution of the common differential equation:

(3.31)

The constant of separation (η) is determined as follows Let us suppose that ε changeswith the z-coordinate beginning from some distance z0, and, at z z0, ε(z) = ε0 =const.Let a plane wave described by exp[ik (y sinθi + zcos θi)] be incident on thedescribed medium from the area z < z0 Thus,

(3.32)because the incident and exited waves are both matched dependent on y

Equation (3.31), in the general case, has no solution in the analytical form It isexpressed through known functions only in some cases with a particular view of thefunction ε(z) We will find one such partial solution in the next section

3.3 WAVE REFLECTION FROM A HOMOGENEOUS

LAYER

To solve the problem of waves in

plane-lay-ered media, let us first study the case of two

media with permittivities ε1 and ε3, separated

by a homogeneous layer of thickness d, and

with permittivity ε2 (Figure 3.2) A layer of

ice floating on water is an example of such a

natural object

It is necessary to analyze two individual

problems for E- and H-waves Let us begin

with the E-wave by assuming that the plane

the reflected wave appears as a result of

inter-action with this layer Let us represent

poten-tial Πe in the form:

FIGURE 3.2 Plane wave

propaga-tion in a homogeneous layer.

Wave Propagation in Plane-Layered Media 61

The first term corresponds to the incident wave, for which the amplitude is assumed

to be equal to unity The second item corresponds to the reflected wave, and F e is

the coefficient of reflection

The function Q e(z) satisfies the equation:

inside the layer 0 < z < d and has the general solution:

Finally, in the third medium (z > d):

,

where T e is the coefficient of transmission The values F e, T e, α, and β can be defined

as solutions of equations derived from the boundary conditions

It is useful next to employ Snell’s law by introducing the angles θ2 and θ3:

(3.34)

If ε1, ε2, and ε3 are real numbers and if ε2 – ε1sin2θi > 0 and ε3 – ε1sin2θi > 0 (i.e.,

no total inner reflection), then these angles characterize the directions of the wave

propagation in the media considered here

The boundary conditions, Equation (1.7), lead to continuity of function Πe and

its first derivative over z at Four algebraic equations are obtained as a result:

62 Radio Propagation and Remote Sensing of the Environment

Solutions to these equations have the following forms:

• For the reflection coefficient from the layer:

is the reflection coefficient of the horizontally polarized waves from the interface of

media 1 and 2, and

(3.38)

is the corresponding coefficient of the reflection from the interface of media 2 and 3

The same results are obtained for H-waves It is necessary, in the previous

formulae, to substitute the reflection coefficients of horizontally polarized waves for

the equivalent ones of vertically polarized waves

We will not analyze the general formulae, as doing so can be rather complicated

Instead, we will confine ourselves to the particular case of a vertical wave incident

on the layer when the reflection coefficients for the E- and H-waves are similar We

will suppose for simplicity that ε1 = 1; that is, assume, for example, that we have a

wave incident on the layered ground from the air Then,

2 3

θθ

2 2

= −

−+

εε

Wave Propagation in Plane-Layered Media 63

Let us calculate the reflection coefficient module using the designations:

(3.40)Thus, we have:

In the case of sufficiently strong absorption inside the layer (when τ >> 1), the reflection

coefficient is |F|2 = |F12|2 , which suggests that a sufficiently thick layer — a layer

with a thickness that is many times greater then the skin depth — is equivalent to

the semispace from the point of view of the wave reflection processes Such layers

are called absorptive

We will refer to layers with moderate absorption inside as half-absorptive In

these cases, the reflective coefficient module oscillates with the frequency change

due to interference of waves reflected from the layer boards The phase shift of the

wave reflected from interfaces 2 and 3 varies with the frequency change, and, as a

result, the waves reflected from the layer boards are by turns summed in phase or

antiphase, which is why the oscillations are dependent on frequency The amplitude

of these oscillations and their quasi-period depend on the layer thickness and its

complex permittivity In particular, the amplitude of oscillations decreases with

increased absorption and tends to zero for the absolute absorptive layer

Let us simplify the problem by considering the case of the dielectric layer The

imaginary part of ε2 is small, and ψ12 = π in this case Then,

64 Radio Propagation and Remote Sensing of the Environment

The frequency dependence of the reflection coefficient is basically determined

by the phase shift value:

The expression reported actually represents the sum of waves reflected

sequen-tially from the layer interfaces For example, the first item, F12, corresponds tothe wave reflected from the interface between the first and the second media;

s is

Wave Propagation in Plane-Layered Media 65

represents the wave transmitted inside the layer through the firstinterface and then reflected from the second interface, finally coming out after thesecond interface to cross the first interface This factor describes the wave decreaserelated to the reflection from the second border, while the first one represents thedecrease caused by the dual wave passing through the first border, and, finally, thethird factor describes the phase shift appearance and corresponding wave attenuationdue to absorption with dual passing of the layer Note that the items in Equation(3.49) describe the waves reflected from the layer interfaces many times

Such representation is especially convenient for describing the reflection of pulseoscillations, for which the spectra have limited bandwidths Let such a spectrum bedescribed by the function In this instance, we will assume that we are dealingwith radar sounding the ground above, illuminating it into the nadir (in ourcalculations, the z-axis is directed downward) The reflected signal form is described

by a function of the form:

If we substitute Equation (3.49) here and assume that the role of the permittivityfrequency dispersion is weak in the frame of the signal bandwidth and that theabsorption in the layer does not depend on the frequency, then the result of calcu-lations will be:

It is logical to establish the definition of an energetic reflective coefficient in thecase of waves with a finite-frequency bandwidth No amplitudes of incident andreflected waves are compared, which is senseless in the given case, but it is more

66 Radio Propagation and Remote Sensing of the Environment

logical compared to their energy flows If the following is the power flow density

of the incident wave:

then the flow density of its energy is:

The Parseval equality18 can be used for conversion from integration over time tointegration with respect to frequency The convenience of introducing the energyflow density is that it does not depend on time For the reflected wave,

Wave Propagation in Plane-Layered Media 67

So, the energetic reflective coefficient for the noise radiation is the squared module

of the common coefficient of the sine waves averaged in the frequency band.Let us now consider the case of the dielectric layer The result of integration inEquation (3.56) gives:

increases) results in a decrease in the amplitude of these oscillations due to theaveraging effect of summation over frequencies This averaging becomes practicallyfull at β2 – β1 = π Then,

of analytical dependencies, p(z); therefore, it is necessary to use approximate

meth-ods of solution of equations such as Equation (3.59), based on same peculiarities ofthe function

One of these methods is the Wentzel–Kramers–Brillouin (WKB) method.18,19 It

can be employed for the case when the scale of function p(z) changes little compared

−

−

21

12 23

12 232 2

τ τ

B

2 0 2

2

11

21

0

z2 + ( )z =

p( ).z