Radio Propagation and Remote Sensing of the Environment - Chapter 9 pdf

242 Radio Propagation and Remote Sensing of the EnvironmentLet us now consider the power tral density received by the antenna, spec-which is equal to and includes the antenna effective a

Trang 1

Radiation

9.1 RAY INTENSITY

This chapter presents another method to describe wave propagation that is not based

on Maxwell’s equations and the wave concepts that followed from them, but rather

on energy considerations As is becoming evident, such a description is best suitedfor wave propagation that is chaotic in both value and direction We have establishedthat such fields occur, in particular, due to scattering and thermal radiation Naturally,the emission description must be based on the determination of the averaged fieldproperties Coherent components of the field at intensive scattering or upon thermalradiation are extremely scarce or do not exist at all Squared characteristics are theprimary values used to describe regularities of the radiation propagation

Poynting’s vector, S(r,s) is found to be a random function of coordinates r anddirection s Its statistical properties are characterized by the probability density

This, in particular, defines the probability of having a Poynting’svector value in the interval (S, S+dS) and to be inside the solid angle dΩ in thedirection given by vector s In the case being considered here, Poynting’s vector isnot a very suitable energy characteristic because its main value might be equal tozero due to its vector character This does not mean that the field power is also equal

to zero Simply due to the chaotic character of wave propagation, Poynting’s vectorturns out to be, on average, equal to zero; therefore, it is more convenient to regardPoynting’s vector characteristics in a particular direction So, we can define theconcept of ray intensity as:

(9.1)

which is the average value of Poynting’s vector in a single interval of the solid angles

in direction s Let us develop this idea further Because our discussion is about not onlyharmonic oscillations but also those for which the power is distributed in a spectralinterval, we will take the word power to mean spectral density so frequency ω has

to be included in the arguments used to define the values of interest We will not do

so in all cases, however, to avoid overloading the formulae with various sorts ofdesignations It is enough to remember that, apart from noted exceptions, the dis-cussion will concentrate on spectral densities

P1( )S =P1(S,Ω)

I( , )r s = P( ) d

∞

∫S 1S, S2 S,0

Ω

TF1710_book.fm Page 241 Thursday, September 30, 2004 1:43 PM

Trang 2

242 Radio Propagation and Remote Sensing of the Environment

Let us now consider the power tral density received by the antenna,

spec-which is equal to

and includes the antenna effective area

in direction s Thus, it is assumed that

the antenna receiving properties do not

depend on the polarization of the

inci-dent radiation The received spectral

density is a random value due to the

chaotic character of Poynting’s vector:

(9.2)

It might seem odd that the integration over the solid angle is produced within 4π,but integration is performed within the angle given by the antenna directivity pattern Because A e has the dimensionality of a square, another definition of ray intensity,

as the spectral density of the power flow at the unit solid angle, follows from thelast expression This definition is more traditional Let us consider the arbitrarilyoriented elementary area dΣ (Figure 9.1) The orientation direction is determined

by the single vector n The power spectral density passing through the introducedsurface element in the direction of the single vector s is defined by the expression:

(9.3)

The ray intensity differs from the spectral

density of the power flow (Poynting’s

vec-tor values) in that the latter value is

nor-malized per unit square while the ray

intensity is additionally normalized per

unit solid angle This leads to a difference

in the coordinate dependence For

exam-ple, the power flow decreases with

dis-tance r–2 as it recedes from the point

source, while the ray intensity remains

FIGURE 9.1 Oriented elementary area.

FIGURE 9.2 Ray tube element in a

homogeneous, nonabsorbing medium.

Trang 3

Transfer Equation of Radiation 243

(9.4)

which expresses the ray intensity constancy along the ray In differential form, thisproperty of the ray intensity can be written as the transfer equation:

(9.5)

in a homogeneous, nonabsorbing medium

Let us now study the changes in ray intensity with wave reflection and refraction

on the interface of transparent media The energy conservation law requires the

The sense of the subscripts introduced here is the same as that used for the wave

θr = θi and (Snell’s law), as well as the relations δΩr = δΩi = ,, and The differentiation result follows from Snell’s

(9.8)

expression is generalized by the equation for the value:

(9.9)

I( )r2 =I( )r1 ,

dI

ds= ⋅∇( )s I =0

I icosθi dΣ Ωd i=I rcosθr dΣ Ωd r+Itcosθtd dΣ Ωt

ε1sinθi= ε2sinθt sinθ θ ϕi d d i i

reflection problems in Chapter 3 Further, we will take into account that

Trang 4

along the ray for a medium with permittivity that changes slowly (at the wavelengthscale) Note that the stated constancy follows from the geometrical optics approxi-mation Let us state once more that we are discussing transparent media — moreaccurately, the imaginary part, for which the permittivity is much less than for thereal part The transparency of the type of medium considered here is particularlyemphasized by the fact that the reflection angle is implicitly assumed to be a realvalue for calculation of the solid angle elements

Note that invariant Equation (9.9) takes place when the geometrical opticsconditions are not observed but the medium or bordering media are in a state of thethermal equilibrium (see, for example, Bogorodsky and Kozlov57 or Aspresyan andKravztov58)

9.2 RADIATION TRANSFER EQUATION

Ray intensity satisfies an equation known as the radiation transfer equation, which

is of the integer–differential type It derives from consideration of the energy balancewithin an elementary volume, represented as a cylinder of length ds with unit square

of transverse cross section Equation (9.5) is an example of such an equation, and

it is correct for a homogeneous nonabsorbing and nonscattering medium Theseprocesses, together with changes in the ray tube cross section, must be taken intoconsideration The change in ray tube cross section is described by Equation (9.9),which, in this case, is convenient to rewrite in the form of the differential relation

It describes amplification or attenuation of the ray intensity on theelementary segment of ray ds dependent on the permittivity gradient sign Theradiation will be weakened over the same distance due to absorption in the mediumitself (for example, in air) and due to absorption and scattering by particles enteringthe medium (for example, drops of clouds) The change in radiation intensity caused

by volume sources has to be addressed Thus, we obtain the following differentialequation:

(9.10)

Here, Γ is the absorption coefficient of the medium, N is the particle concentration,and E is the volume density of radiation sources

Let us regard two processes that are important radiation sources One of them

is the process of rescattering due to particles of radiation coming upon the consideredvolume from other (side) directions The density of the flow incident within the solidangle element in the direction s′ is equal to Si = I Multiplying by the

distance R from the particle We can obtain the value of the ray intensity reradiated

in direction s by integrating over all directions and summing up the contribution ofall the particles inside the volume being considered:

dI=I d(ln ).ε

dI d

Trang 5

The second source of volume radiation is the proper thermal emission of the medium

We will establish the spectral density value by a rather unusual method First, wewill consider the thermal radiation of the particles that are in the medium Thecorresponding spectral density is equal to:

on the basis of Equation (8.34) The combined wave power of both polarizationswas taken into account when writing this expression, so the polarization effects arenot discussed here The appearance of permittivity of the medium as a factor israther unexpected; however, it is an expression of Clausius’ law, which connects theequilibrium intensity in a transparent isotropic medium with the same in a vacuum

It is now more convenient to examine the spectral density in the oscillation frequencyscale f In this case, 2π has to be reduced (recall that ω = 2πf)

Now, it is easy to think about how to calculate the radiation of the medium itself.For this purpose, it is enough to replace the product with absorption coefficient

Γ in the previous formula This substitution is a reflection of Kirchhoff’s law forequilibrium radiation in the ray definition Thus, we have the ratio:

Here, is the universal function of temperature and frequency We substitutedthe permittivity for the refractive index, emphasizing the ray character of Equation(9.11) The subscript with the frequency is introduced to note the spectral character

of the given relation The universal function indicated here, which is independent

of the physical nature of the substance, is expressed through the Planck or

Ray-we obtain the equation for the value defined by Equation (9.9):

(9.12)

The subscript f is introduced here in order to emphasize again that we are discussingspectral density on the scale of the oscillation frequency but not the circular one;however, the subscript will be omitted from now on We note also that λ is theradiation wavelength in a vacuum

Let us express the differential cross section in terms of the scattering indicatrixwith the help of the relation:

leigh–Jeans function for the case of radio frequencies (see Chapter 8) As a result,

Trang 6

The transfer equation can be rewritten as:

.(9.14)

It is necessary at this point to make some remarks The transfer equation wederived obviously has a geometric–optical character that reveals itself in the inclusion

of the dependence of all parameters on the only coordinate counted over the ray, inthe inclusion of the propagation medium permittivity, and in the label itself — theray intensity of the main energetic value By the way, other labels have been usedfor value I in the literature: intensity, spectral brightness, simple brightness, energeticbrightness, etc The deduction itself is not based on the wave and statistical concept

of radiation propagation In particular, wave interference due to scattering by anassembly of particles is not considered, but we have introduced, by the simplesummation of the scattered power, the idea of a scattered wave incoherence More-over, the waves coming to any point from different directions are also considered

to be incoherent The integral term and the extinction component coefficient inEquation (9.14) are written, obviously, using the single scattering approach; conse-quently, the equation concerns the case of rarefied media The situation is ratherimproved by the substitution of the product for the total cross section per unitvolume; however, it is difficult to determine up to what level of media density thegiven substitution works In the case of small particles (compared to the wavelength),for which dipole scattering is the primary type, the matter is reduced to the intro-duction of dielectric permittivity based on the Lorentz–Lorenz formula This formulatakes into account the mutual polarization of particles as the first approximationwith respect to the density, and the scattering itself is described as scattering by thedensity fluctuations

The most logical deduction of the transfer equation is based on the analysis ofspatial spectral properties of the coherency function In particular, the ray intensity

is defined as a Fourier transform of this function over the differential coordinates

We will not examine this problem in detail but refer interested readers to theappropriate literature.58 The validity of the transfer equation has been proven by thefact that the solutions of many problems based on it agree with the experimental data Let us point out, as well, that the transfer equation is analogous to the Boltzmannkinetic equation for the stationary case Particularly, its integral term is similar tothe collision integral and describes the scattering of light particles (photons) byheavy particles It explains why in our case the collision integral is linear with respect

to the ray intensity Obviously, many methods developed to solve the kinetic equationcan be used to solve radiation transfer problems.61

Let us now introduce the optical thickness using the formula:

Trang 7

with the following definition:

(9.16)Then, the transfer equation can be written down in a dimensionless view:

(9.17)

For transfer problems, we do not raise the question of emission polarization.This problem demands a special approach that considers scattering effects Thepolarization of scattered radiation can differ from the polarization for incidentradiation, and, generally speaking, it should be taken into account by adding theintensities of differently polarized waves with the same weight, as was done withEquation (9.12) The situation gets even more complicated when radiation transfer

in anisotropic plasma is taken into account In this case, it is possible to have partialtransformation of waves of one type to the other (for example, ordinary into extraor-dinary) upon scattering In order to describe this, we would have to put together asystem of linked transfer equations, including pointed transformations We will not

go into the details of this problem but instead refer readers to the appropriateliterature.58 Here, we will note only that polarization phenomena do not play a crucialrole in our further discussions; therefore, we will consider the formulated equationsabove to be quite sufficient for our aims

9.3 TRANSFER EQUATION FOR A PLANE-LAYERED

MEDIUM

The approach for a plane-layered medium is appropriate for many of the cases wewill examine further We will suppose that all the medium properties (absorption,scattering) depend on the z-coordinate We will also assume that all rays are straight

at an angle θ to the z-axis For the atmosphere of Earth, for example, this meansthat we can ignore its spherical stratification and neglect the ray bending due torefraction We are entitled to assume that and to write:

σΓ

2 4

Trang 8

We also represented the coordinate dependence of the scattering indicatrix, sizing the possible types of particle changes and their shape from the layer to layer Let us now consider a simple example of solving the transfer equation — anabsorbing atmosphere with no scattering particles The albedo is equal to zero, so

empha-c = 0 We assume that the radiation comes from the semispace toward a receivingpoint on Earth (z = 0) In this case, the transfer equation can be written in the form:

(9.20)

The minus before the first summand on the left-hand side is here because we areconcerned with radiation propagating in the direction of negative values of z Thesolution of this equation at the formulated boundary condition is:

(9.21)

At z = 0, the spectral density of the ray intensity that we obtained should be multiplied

by the effective area of the antenna directed at angle θ to the zenith and integratedover all possible directions The antenna itself is assumed to have a narrow pattern,which permits us to neglect the µ change during the integration As a result, weobtain the spectral density of power and then the brightness temperature In fact, itwill be the antenna temperature value, but both temperature values, as was alreadymentioned, practically coincide at a rather acute antenna pattern Thus,

(9.22)

Let us now study the radiation of a layer of thickness d situated in the altitude

and we will also assume an absence of absorption in the layer environment of thefrequency waves being studied It follows from Equation (9.22) that, in this case:

z

0 0

Trang 9

where the optical thickness is defined by the integral within the radiating layer:

This equation agrees with Equation (8.32) and conveys the fact that the reflectioneffects on the interfaces are neglected

Let us investigate further the radiation of a semispace by considering a soil withconstant temperature radiating into an absorbing atmosphere To simplify theproblem, we will concentrate on the analysis of radiation propagation in the verticaldirection relative to the boundary; that is, we will assume that µ = 1 Equation (9.21)can be used to determine the radiation ray intensity of the soil at the surface; however,

we should not ignore the fact that the soil permittivity differs substantially fromunity, which is why it is necessary to derive an equation for value J that describesthe radiation inside the soil Thus, we will have:

(9.24)

Equations (9.24) and (9.8) define the boundary condition for the ray intensity

of atmospheric radiation The transfer equation gives us the solution:

us the opportunity to regard z →∞ in our formulae, and, accordingly, z→z∞ Thecorresponding expression for the brightness temperature gives us:

z

0 0

Tg

g g

Trang 10

The first summand describes the soil radiation attenuated by absorption in theatmosphere, and the second one describes the sum of the proper radiation of theatmospheric layer, also attenuated by absorption; however, the represented formula

is not complete, as it excludes the atmospheric summand directed downward towardthe soil border, reflected upward, and then attenuated by absorption in the atmo-sphere The intensity of descending radiation on the interface is easily defined fromEquation (9.21) as follows:

Now, it is easy to take into account the resulting contribution in the brightnesstemperature and to obtain as a result:

(9.27)

The known equation is obtained for the soil brightness temperature in the absence

of atmospheric absorption The solution to the total problem could be obtained atonce if we were to use the more complete boundary condition:

(9.28)

instead of Equation (9.26)

9.4 EIGENFUNCTIONS OF THE TRANSFER EQUATION

Let us now consider a medium in which particles put into it influence predominantlyradiation propagation This means that the absorption in the medium itself is rathersmall (i.e., assume Γ = 0) To simplify the problem, we also assume a small variance

in permittivity from unity; hence, it is sufficient to assume that ε = 1 In this case,the function is a constant value which means that the scattering particlesare invariable in all space and the albedo does not depend on spatial coordinates

So, the transfer equation can now be written as:

dI

k T b

+ −Aˆ∫ ( )s s′, ( ),s′ Ω′ = −( )1 Aˆ 2 .4

Trang 11

Equation (9.29) is an integer–differential equation in which the right-hand sideplays the role of an impressed force Let us now consider the question: What arethe eigenfunctions of the homogeneous equation (i.e., the equation with an impressedforce equal to zero)? Naturally, these functions will depend on the scattering indi-catrix type To simplify the problem, we will assume the simplest case of isotropicscattering indicatrix; that is,

(9.30)

In this case, the ray intensity will depend on only the z-coordinate and angle θ orvalue µ Then, it is possible to perform the integration in Equation (9.29) with respect

equation can be written in the form:

Trang 12

The eigenvalues themselves are defined from the condition of the normalization,

Equation (9.33), and we obtain the following equation:

This other view of the equation form is more transparent for our analysis:

Because the albedo value is less than unity and based on the hyperbolic tangent

behaviors, we can state that this equation has two real roots differing only in sign

Let us designate them as ±ν0 We exclude from our consideration the roots on the

imaginary axis because doing so would require a more detailed analysis.60

It is a simple matter to estimate the values of the roots in extreme cases If

, then ξ >> 1, and

(9.38)

The value ξ is small in the other extreme case of the albedo approaching unity

(1 – << 1) The following equation is obtained from the hyperbolic tangent

expansion:

Also, we have established the existence of two modes:

(9.40)

from the discrete spectrum of eigenfunctions The eigenvalues of this spectrum are

situated on the real semi-axis [–1,–] and [+1,+]

The situation becomes rather more complicated when we include [–1,+1] in the

discussion First, Equation (9.34) cannot be divided by ν – µ in all cases Second,

we must take into account the functional equality xδ(x) = 0, which allows us to add

continuous spectrum modes from the class of generalized functions61,62 are added

ˆln

Aν νν

A

Λ( ) (ν δ ν µ− )

Trang 13

to the already found functions of the discrete spectrum In general, they can be

written in the form:

(9.41)

where P represents the principal value Let us recall that generalized functions have

meaning not per se but under the integral sign; it is important to understand the

principal value sign and delta-function in this regard The normalization condition

leads to the definition:

(9.42)

Let us point out that the principal representation of function Λ(ν) corresponds

to the representation of the second summand via the integral Its representation

through the logarithm of Equation (9.42) is related to the case when ν lies inside

the integration interval; in this case, we should use the rules of principal value

calculations In other cases, it is necessary to introduce the substitution:

which means that

These eigenfunctions possess the properties of orthogonality and completeness

with weight µ; that is,

(9.43)

the value:

(9.44)

Calculation of the norm in the case of a continuous spectrum becomes more

complex due to the peculiarities of eigenfunctions, particularly because they are not

ννΛ( )ν0 =0

N(−ν0)= −N(ν0)

νν

0

0 2 0

Trang 14

functions with the integrated square For completeness, any “good” function can be

represented as:

The components of the discrete spectrum are easily found For example, the first of

them is defined from the obvious equality:

(9.46)

By analogy, the spectral density has to satisfy the relation:

(9.47)

Further calculations are more complex due to the fact that the result of integration

depends on the order of integration due to the singularity of the integrands

Substi-tution in Equation (9.47) of the obvious expressions of the eigenfunctions provided

in Equation (9.41) gives us:

The Poincare–Bertrand formula now becomes very useful:63

Here, L is a smooth contour on the plane of complex variable t The points t0 and

t1 are on this contour This formula now gives us:

f( )µ = f+0φν ( )µ + f−0φ− ν ( )µ + f( ) ( )ν φ µ′ ν ′ dν′

−−∫

1 1

2

4

1ˆ

1 1

L

, 11

Trang 15

(9.49)

9.5 EIGENFUNCTIONS FOR A HALF-SEGMENT

The eigenfunctions of the transfer equation discussed previously apply to the case

of infinite space; that is, we considered the case when the radiation propagation

direction could be any (–1 µ 1) However, the case of semispace occurs frequently

when the direction of radiation propagation is restricted by the limits (0 µ 1) The

opposite case of negative directions is similar The transfer to the half-segment (0

µ 1) demands, generally speaking, reworking the problem because the eigenfunctions

examined in the previous part were defined on a full segment (–1 µ 1) Luckily, we

can prove that the functions defined by us have the property of completeness for the

half-segment, too.61 Hence, it is not necessary to introduce new functions, but the

orthogonality problem does require some special attention It is necessary to

intro-duce another weight function W(µ), which satisfies the singular equation:

ˆ

Trang 16

The theory of singular integral equations is covered in detail in the well-knownmonograph of Musheleshvily.63

We can now see that, according to our method, the solution has the form:

0

1

ˆ( ) .

10

1

πΘ

0 2 0 2

1

µ

Trang 17

is determined as the function of complex variable z (compare with Equation (9.42)),and

(9.61)

Let us point out that the first expression in Equation (9.59) is valid on the entirecomplex plane z with the section along the real axis on the segment from zero tothe unity It can be written as:

0 0

φ µ φ µ−ν( ) ν′( ) ( )W µ µd = Aˆν φ ν ν ν′ −ν( )′ ( + )X(−ν)

0 1

∫

Tiêu đề	Transfer Equation of Radiation
Trường học	CRC Press
Chuyên ngành	Radio Propagation and Remote Sensing
Thể loại	sách
Năm xuất bản	2005
Thành phố	Boca Raton

Định dạng
Số trang	34
Dung lượng	622,98 KB