Radio Propagation and Remote Sensing of the Environment - Chapter 4 potx

86 Radio Propagation and Remote Sensing of the Environmentthe equations of the zero-order approximation.. Now we can easily show that theeikonal equation may be written down in the form:

Trang 1

of the permittivity is assumed to be similar to that for the Wentzel–Kramers–Brillouin

of permittivity at the wavelength scale This property can be expressed as theinequality:

0 0

TF1710_book.fm Page 85 Thursday, September 30, 2004 1:43 PM

(WKB) approximation carried out in Chapter 3 We assume again a small change

Trang 2

86 Radio Propagation and Remote Sensing of the Environment

the equations of the zero-order approximation Now we can easily show that theeikonal equation may be written down in the form:

(4.4)The value:

(4.5)represents the radiowave phase in the zero approximation, and eikonal ψ (in engi-neering terminology) represents the electrical length passed by the wave We assumethat the phases of components E0 and H0 do not depend on coordinates in theapproximation Furthermore, in this approximation, these vectors are believed to bereal, including the initial wave phase at once in the wavelength Certainly, smalladditions to this phase may be made by calculation of the following items ofexpansion

In the zero-order approximation, the power flow density:

(4.6)

is directed along lines of the eikonal gradient This fact allows us to refer to the zeroapproximation as the geometrical optics approximation, which corresponds to thesmall wavelength conversion (hence the term optics) and allows the wave propaga-tion laws to be formulated in the language of geometry

The validity of the geometrical optics approximation is defined by Equation(4.1) If, as before, the scale of permittivity change is designated Λ, then the Debyeseries is essentially expansion according to the inverse degree of large parameter

kΛ Other conditions will be formulated later

Let us assume, in the beginning, that the permittivity is a real value We candefine the vector of wave propagation by the formula:

(4.7)where s is the unitary vector, which is orthogonal to the equiphase surfaces Thelines orthogonal to the surfaces of the eikonal constant value (to equiphase surfaces)are called rays Vector s is tangential to the rays and describes the wave energypropagation direction

If τ is the length along the ray, then the ray equation has the form dr/dτ = s.Then, ∇ψ = sd/ψ/dτ, and the eikonal equation becomes the common differentialequation:

ψ

Trang 3

Geometrical Optics Approximation 87

the solution of which is written in the form:

(4.9)

where ψ0 is the initial value of the eikonal

equation Let us point out that use of the plus

sign was determined by extraction of the

square root in Equation (4.9) It is important

to note that we are dealing with a direct wave

In the case of a backward wave, the minus sign

should be used It is clear that the eikonal form

as a function of coordinates depends on the

ray along which the integration is provided

We may use the orthogonal unitary vector

system of the normal n and the binormal m (Figure 4.1) Their changes along theray characterize its bending and torsion The Frenet–Serre formulae:

(4.10)

are known from differential geometry.29 The value ρ is the ray curvature radius, and

χ is its torsion The vectors s, n, and m are the basis of the curved-line coordinatesystem formed by the ray ensemble and equiphase surfaces This system is oftenreferred to as the ray coordinates Equation (4.9) is the eikonal equation solution inthe ray coordinates system

Let us use the eikonal equation in Equation (4.8) to calculate ρ and χ We musttake the gradient of both parts to obtain:

.Thus, it follows that:

0

d ,

FIGURE 4.1 The orthogonal

uni-tary vector system.

n

m s

d d

s ε

d d

Trang 4

If angle α between the direction of the ray and the direction of the permittivity isintroduced, then:

(4.15)

On the basis of Equation (4.15) and after some not very complicated calculations,25

we can define the conditions that connect the electrical field components directedalong the normal and along the binormal:

Trang 5

and the equation of torsion:

We may rewrite Equation (4.18) in another form by using Equation (4.7) Then,

in the ray coordinates,

EE

0 0

τ

ψε

Trang 6

direction of the rays (equiphase surface) equals Ddξdη The volume of the ray tubeelement equals Ddξdηdτ Applying vector analysis to Equation (4.8), it is a simplematter to obtain:

(4.23)

We will use this formula for the volume bounded by the ray coordinates and willreduce the volume to zero; moreover, we take into account that (n · s) = 0 at thesides of the tube We then have:

Up to this point, it was assumed that permittivity is the real value Let us nowturn to the more realistic case of weak absorption in the media and, related to this,complex permittivity We can still use Equation (4.4), but the eikonal itself mustnow be complex (i.e., ψ = ψ′ + iψ′′) It is apparent that the eikonal imaginary partdescribes wave attenuation due to absorption and, being multiplied by the wavenumber, is equal to the coefficient of extinction The separation of real and imaginaryparts in Equation (4.4) leads to a pair of equations concerning ∇ψ′ and ∇ψ′′ It isdifficult to find the solution of these equations, particularly because it is necessary

to know the angle between ∇ψ′ and ∇ψ′′

∇2ψd3 = ∇ ⋅ ∇ψ d3 = ∫ ( ⋅∇ψ)d2 = ∫ ε( )⋅ d2

S S

particular, the turning point, which we mentioned in Chapter 3 with regard to the

Trang 7

We will now consider a simple but common case of small absorption in thesense that ε′′ << ε′ and (∇ψ′′) << (∇ψ′)2 The pair of equations then acquires theform:

(4.26)The first equation is solved as before and the second one is transformed to the form:

iono-(4.28)

These waves, generally speaking, can be considered to be independent if the length

of the beating between them is much smaller than the scale of the medium geneities The beating length is estimated by the value:

inhomo-(4.29)

The substitution of specific values (f = 108 Hz, N = 2 · 106 cm3, H0 = 0.5 Oersted)gives the estimation l≈ 10 km It would seem that the independence of ordinaryand nonordinary waves can be broken with increasing frequency and, correspond-ingly, with increasing beating length This is not so, however, because in this casethe wave relation coefficient decreases with increases in frequency;28 therefore,ordinary and extraordinary waves are practically always independent for the ultra-high-frequency and microwave bands for the ionosphere of Earth

The ray trajectories of ordinary and extraordinary waves practically coincide,because their refractive indexes differ little in the range of waves being consideredhere, which allows us to develop a formula to calculate the polarization angle rotationvalue due to the Faraday effect:

εε2

Trang 8

(4.30)

Finally, we will calculate the Doppler frequency shift for wave propagation in an

inhomogeneous medium For this purpose, let us refer back to Equation (2.97) and

rewrite it as follows:

(4.31)

It is easy to see that the Doppler shift value is proportional to the velocity component

directed along the ray (ray velocity)

4.2 RADIOWAVE PROPAGATION IN THE

ATMOSPHERE OF EARTH

The atmosphere of Earth can be considered, in the first approximation, as a

spher-ically layered medium where the permittivity is a function of the radius beginning

at the center of the Earth We do not include in our consideration here the changes

in atmospheric parameters along the surface of the Earth that take place at the

transition from day to night (light to shadow), along frontal zones with significant

changes of air temperature, and so on It should be supposed that ε = ε(R), ∇ε =

R/R(dε/dR), etc It is believed, that, on average, 1/m in the

troposphere near the surface of Earth; therefore, the geometrical optics

approxima-tion is highly accurate for the given wave range The vertical gradient in the

iono-sphere is even smaller, so applying the geometrical optics approximation is still

appropriate Although the permittivity of air does differ from unity, the difference

is insignificant and we can assume it to be equal to unity without causing problems

We can prove rather easily the permanency of vector along the ray

trajectory; hence, we can make the statement that in the case of a spherically layered

medium the ray trajectories are plane curves The product:

(4.32)

is invariant along the ray, where the constant η is determined from the initial

conditions If, for example, a ray left Earth at angle α0, then , where

the radius of the Earth a≅ 3

established on the ray prolongation until the surface of the Earth Equation (4.32)

is often referred to as Snell’s law for spherically layered media

We will now consider the situation when a ray passes by the surface of Earth

ε

→ 1, and Rsinα→p, where

p is the aimed distance (a term borrowed from the theory of particle scattering) In

this case, a ray turning point occurs at R = Rm, where α(Rm) = π/2

( ) ′

∫

0 0

6.4 · 10 km (see Figure 4.2) The starting point may be

(Figure 4.3) Far from the atmosphere of Earth, (R)

Trang 9

It is convenient, in our case, to write the eikonal equation for the sphericalcoordinate system with the center coinciding with the center of the Earth Thissystem can be chosen in such a way as to take into account the plane character ofthe ray trajectories so that the eikonal will not depend on the azimuthal angle ϕ.The eikonal equation can be written as:

(4.36)

FIGURE 4.2 Radio propagation from the

surface of the Earth.

FIGURE 4.3 Propagation radio wave along

the surface of the Earth.

2 2 0

Rd dR

Trang 10

and

(4.37)

The rule for choosing the appropriate sign is the same as for the previous case As

a result, we now have:

(4.38)

It follows from Equation (4.38) that:

Taking into account Equation (4.21), the equality can be derived as:

2 0

dR R

12ψ

R R p

R R dR R

R R

m m

0

π,

When the ray passes through point R = R (Figure 4.3) it comes into contact

Trang 11

The important ray parameter is the

angle of refraction characterizing the

degree of its bending The differential of

this angle is defined as the angle between

the ray direction in infinity nearby points τ

and τ + dτ (Figure 4.4) The differential

value (let us represent it as ξ) is determined

–msin(dξ) Let us now use the expansion:

FIGURE 4.4 Refraction angle

differen-tials and changing optical density

21

0 0

m

for the rays shown in Figure 4.3 In particular, if R and R are sufficiently large, the

Trang 12

in determination of the angle position of these sources This error in determination

of the zenith angle can be written as:

(4.45)

By determining this error from data measured at different values of angle α0, wecan define the altitude profile of the atmospheric permittivity using the inverseproblems technique

Let us point out that Equation (4.45) can be rewritten as:

2 2 0

ε ςς

0

2 0 2

0 2 0 0

∞

∫

sin ( )α ς = ε ε ς0 ( ) sinα0

ξ α( )0 =arcsin( ε0sinα0)−α0

Trang 13

Let us point out that this result is easily obtained using Snell’s law for plane-layeredmedia The small deviation of the permittivity from unity allows the use of the Taylorexpansion to obtain:

(4.49)

As previously mentioned, the Doppler frequency shift must be considered in thecase of moving sources:

(4.50)

according to Equation (4.31) Here, vR = dR/dt is the radial velocity and Ωθ = dθ/dt

is the angular velocity The Doppler frequency shift depends not only on movementparameters but also on the atmospheric characteristics, because angle α is determined

by the refraction phenomenon

4.3 NUMERICAL ESTIMATIONS OF ATMOSPHERIC

EFFECTS

Let us now turn to numerical estimation of atmospheric effects that occur duringradio propagation of ultra-high-frequency and microwave bands Let us begin withthe troposphere The altitude dependence of atmospheric air permittivity can begenerally described by:31

(4.51)

Here, as before, ς is the altitude above land, ε0 – 1 ≅ 6 · 10–4 defines the near-land

value of the air permittivity, and HT ≅ 8 km is the frequently used troposphere height.The parameters of this exponential model of the troposphere depend on meteoro-logical conditions and, particularly, on geographical location So, the parametersused here should be considered only as reference ones that are close to average,although the real values are not very different from these values Estimations showthat the integral of refraction for this model can be represented in the form:32

( )= − tan Z s( ),

Trang 14

e T

12

HT

Trang 15

which corresponds to the plane-layered atmosphere approximation When s0 << 1,

(4.58)

The maximum value is ∆ψ ≅ 102 m for the parameters used here

Now, we turn our attention to calculation of the frequency Doppler shift Indoing so, we will be interested in that part of the frequency change that is determinedusing a case typical for planet occultation observations;32 the method presented herehas been used for measurement of troposphere parameters.35 Let the transmitter be

at point T, the receiver at infinity, and the ray come to the receiver at sighting distance

p Angle α in Equation (4.50) is measured, in this case, at point At this point,

, and if radius RT is large enough; ξ = 0 in the absence ofatmosphere; and the corresponding frequency shift is:

When the radiation reception occurs on Earth, we must refer to Equation (4.50),which we can rewrite to take into account the trajectory equation, Equation (4.32):

Trang 16

100 Radio Propagation and Remote Sensing of the Environment

Expansion when parameters ε0 – 1 and ξ are small leads to the formula:

(4.63)

We have supposed so high altitude of the radiation source that we can assume

If now we use formula (4.52) for medium refraction the result will be

(4.64)

For the common example of an artificial satellite on a circular orbit around Earth,the radial velocity may be expected to be equal to zero Further, we can assume that

aΩθ = 8 km/sec, a0 = π/2, and ∆ωT/ω ≅ 8 · 10–9 at these parameters

For the analysis of ionospheric effects, we will work primarily with spacecraftorbits that are above the ionospheric electron concentration maximum Again, let usfirst estimate the refraction angle value In this case, knowledge of the angle betweenthe ray direction at the point of reception on Earth and the sight line to the radiationsource is more important than knowledge of the refraction angle itself Let us indicatethis angle as δ We can show that, approximately,32

(4.65)

where R is the spacecraft altitude (distance from the center of Earth), Rm is the

function:

(4.66)

is the electron content at the altitude of the radiation source Let us suppose for

numerous estimations that R >> a, Rm = a + zm, and zm = 300 km We can propose

δπ

a R A R N R

A R A R a

2 2

0 3

Định dạng
Số trang	25
Dung lượng	561,27 KB