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Wave Field Fluctuations in Random Media over a Boundary Interface

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Dive into the intricate dynamics of wave field fluctuations in random media over boundary interfaces. This comprehensive article discusses reflection formulas for ideally reflective surfaces, trajectory equations, and the effects of impedance boundary conditions on wave behavior. Discover key insights into the moments of the field and calculations for line-of-sight (LOS) fields in diverse scenarios. Ideal for researchers and students in wave physics and random media analysis.

In this article we study wave propagation over the earth’s surface in a line-of-sight region in the presence of a random component of refractive index. In the absence of fluctuation this problem may be regarded as having been solved many decades ago and numerous publications are available. Nonetheless, it may be noted than even in a classical formulation of the problem, i. e. just a point source of vertical or horizontal polarization above a smooth terrain in the absence of super-refraction, the actual solution is not quite simple and can be described in terms of reflection formulas only in a “true” line-of-sight region, at distances not too close to the horizon. Fock demonstrated that the reflection formulas in a form of superposition of direct and reflected waves are valid at distances of the order of a/m before the horizon
2az+z0,
i. e. outside the “shade cone”. In the case of super-refraction, single reflection formulas are also valid in the range of distances before the horizon, which, in turn, is also modified in the presence of refraction. In particular, in the case of the evaporation duct, there are two separate horizons for the direct and reflected waves. In practical applications, signal strength calculations in a line-of-sight region are performed by means of ray theory that can be applied to a very general profile of refractivity.

The problem becomes even more complicated in the presence of random fluctuations in the refractive index. The number of publications on this topic is rather limited and all studies known to the author are concerned with a plane boundary interface. The applicability of the results discussed in Ref. is limited to “weak” fluctuations of the scattered field. Strong fluctuations in the scattered field are considered in Ref. using the method of “local perturbations”. In Ref. the authors use a perturbation theory for auxiliary functions, treating direct and reflected waves separately. That approach resulted in obtaining an expression for fluctuations in amplitude and phase applicable in the region of the interference minima.

In this article we use path integrals to obtain the second- and fourth order moments for the wave field in a random medium above the plane and spherical boundary interface.

Reflection Formulas for the Wave Field in a Random Medium over an Ideally Reflective Boundary

Ideally Reflective Flat Surface

Let us examine the field in a randomly non-uniform medium above a plane interface. In this case, the Green function
Gr, r0
can be presented as a superposition of the Green function for a point source
G+r, r0
situated at the point
r0=0, γ0, z0
and that for the mirror-reflected point source
Gr, r0
located at
r0=0, γ0, z0:
Gr, r0=G+r, r0Gr, r0.           Eq. 1

The boundary condition at the surface z = 0:

Gr, r0|z=0=0.           Eq. 2
The boundary conditions (2) have the following impact on a continual representation of the Green function: the mirror reflection of the point source requires a mirror reflection of the medium as well, it results in the introduction of the fluctuations in dielectric permittivity to be a function of the modulus of the z-coordinate normal to the surface of separation, z = 0, i. e.
δεrx=δεx, γx, zx.
We will define
ρ±x,
the trajectories of the waves departing from the sources in the upper and lower half-space. For
G±r, r0
we have:
G±r, r0=Dρ±x exp jkS0ρ±x+jk20xdxδεx, ρ±x,           Eq. 3
We have introduced the notation
ρ±x=γ±x, z±x.
The action S0 is given by:
S0ρx=0xdxL0x, ρx+12δεx, ρx             Eq. 4
where
L0x, ρx=12dρdx2.
As we can see from Eq. (3), the expressions for G+ and G are of identical form but differ in terms of the initial conditions for the trajectories:
ρ+x=0=γ0, z0,     ρx=0=γ0, z0.

Consider a calculation for the second moment of the Green function (1). Using Eq. (3), we obtain:

Gr, r0 G*r, r0=G11+G22G12G21             Eq. 5

where:

G11=Dρ1+x Dρ2+x·expjk S+1S+2M11x, ρ1+x, ρ2+x,           Eq. 6
G12=Dρ1+x Dρ2x·expjk S+1S2M11x, ρ1+x, ρ2x,           Eq. 7
M11=πk240xdxHρ1+xρ2+x,              Eq. 8
M12=πk240xdxHρ1+xρ2x.              Eq. 9

Here the subscripts 1 and 2 correspond to the first and second sources, respectively; the symbolic notations S±(1) and S±(2) have the meaning of the actions from the direct source (1) and mirror-reflected source (2). The remaining terms in Eq. (5) can be written in a way similar to Eqs. (6) and (7).

Thus, the coherence function for the field of the point source above the ideally reflective surface is represented by the superposition of correlators between the direct and reflected sources. In the functional space of the trajectories z(x) = 1/2 (z1(x)+z2(x)) and ς(x) = z1(x)-z2(x), we can isolate two regions in the presence of a reflective surface.

Ω1xz2x>ς2x4,     Ω2xz2x<ς2x4.           Eq. 10
In the region Ω1(x) each term in Eq. (5) obeys Green Function for a Parabolic Equation in a Stratified Mediumthis equation and the integrals in Eq. (5) are similar to those in Feynman Path Integrals in the Problems of Wave Propagation in Random Mediathis equation. In the region Ω2(x) instead of
Hρx
we need to use H(γ(x), 2z(x)). In both regions, Ω1(x) and Ω2(x), the criteria for applicability of the Markov approximation (Feynman Path Integrals in the Problems of Wave Propagation in Random Mediain this equation) and (Feynman Path Integrals in the Problems of Wave Propagation in Random Mediain this equation) remain the same as for an unbounded medium.

Subsequent calculations are performed here for partially saturated fluctuations, engendered by atmospheric turbulence. While the mean-square value of the phase fluctuations is large the following two conditions have to be satisfied.

First we assume the smallness of the mean-square value of the fluctuations in the phase difference at the base to be of the order of the Fresnel zone size:

πk240xdxHxκ=0,73Cε2k7/6x11/61.           Eq. 11

The inequality (11) means that even in the presence of phase fluctuations the Fresnel zone remains a characteristic region in the plane x = const from which the rays arrive in phase. As a consequence, no stochastic multipath occurs and integration in Eq. (5) is carried along a single ray-tube limited by a Fresnel zone volume.

The second condition is to require a small fluctuation of the arrival angle compared with the angular size of the Fresnel zone. In path integral representation this requirement is equivalent to:

πk280xdxρ2xd2Hρdρ2πk28x2d2κΦε0, κκ2=0,037Cε2kx2l01/31           Eq. 12

where l0 is an internal scale of turbulence. Therefore, when inequalities (11) and (12) hold, the integration in Eq. (5) can be performed along non-perturbed trajectories. For turbulence in the atmosphere and radio frequencies above 10 GHz, the inequalities (11) and (12) hold at distances x ≤ 300 km.

Non-perturbed trajectories
ρx
represent solutions to Euler equations and are given by:
ρ±x=±ρ0+ρρ0xx.             Eq. 13

Introducing the sum- and difference-coordinates of the corresponding points we obtain:

M11=πk2x401dξHρξ+ρ01ξ,            Eq. 14
M12=πk2x40ξ1dξHς0+ξ2zς0, γξ+γ01ξ+ξ11dξH2z01ξ+ξς, γξ+γ01ξ,            Eq. 15
M22=πk2x40ξ2dξHς01ξ+ξς, γξ+γ01ξ+ξ2ξ1dξH2z+z0ξ2z0ξ, γξ+γ01ξ+ξ11dξHςξ+ς01ξ, γξ+γ01ξ,            Eq. 16

where:

γ, ς=ρ1ρ2,    γ0, ς0=ρ0ρ0,   ξ1,2=z0ς02zς2z0ς2

are the distances, expressed in units of xalong the surface from the source to the point of reflection. The equation for M21 is similar to Eq. (15) while ξ1 is replaced with ξ2.

A similar, but rather simple, equation follows for the intensity of the wave field from the point source:

Jx, z, z0=4π2/k2G12,
when
ρ=ρ0=0:
Jx, z, z0=2x21exp M12x, z, z0 cos Sx, z, z0              Eq. 17
where
Sx, z, z0=2kzz0x,
M12=M21=πk2x401dξHdpξ.             Eq. 18

The M12 is a mean-square fluctuation in a phase difference between the direct and the reflected waves, dp = zz0/(z+z0) is the maximum possible separation in the vertical plane between the trajectories of the direct wave and the wave reflected from the surface, in fact dp is the height at which the direct ray passes the point of mirror-reflection for the reflected wave. In the case of turbulence fluctuations in de we obtain:

M12x, z, z0=0,869Cε2k2xdp5/3.             Eq. 19

The expression (19) coincides with the structural function of the phase for the base dp with accuracy to the numerical coefficient.

As observed from Eq. (17) the interference-like structure of the field is kept until the direct and reflected waves are correlated in phase, M12 ≪ 1. It can be noted that the mean square of the phase fluctuations in either direct or reflected waves may not necessarily be small, γS >> 1. As the correlation between the direct and the reflected waves diminishes, the first term in Eq. (17) predominates, yielding a field intensity that is twice as large as that in a free space.

Spherical Surface

The second moment for the field over a spherical surface is determined by Eqs. (5) to (9) where the Lagrangian is:

L0ρx=12dγdx2+12dzdx2+zxa,             Eq. 20

a is the curvature radius of the spherical surface, z is the height above the surface. In fact, the sphere is replaced by a cylinder with an infinitely long generatrix parallel to the γ-axis. The difference between the given problem and that examined in the previous section is found in the presence of the potential term |z|/a in the Lagrangian (20). This term takes into account the spherical boundary surface in a parabolic approximation. The presence of this term leads to the introduction of two segments on the trajectory z(x) of the reflected wave (departing from the imaginary mirror-reflected source), and these two segments are separated by the reflection point x0.

Trajectory Equations

Following Fock, let us introduce the dimensionless coordinates:

x~=mxa,   x~0=mxa,   γ=kzm,   γ0=kzm             Eq. 21

and consider the ray in the direction upwards from the source. The phase is given by:

S+=x~t+23γt3/223γ0t3/2.             Eq. 22

The derivative over t gives the equation for the stationary value of t:

S+=x~γt+γ0t             Eq. 23

that leads to the solution:

γ0t=γγ0x~22x.             Eq. 24
The equation for the trajectory
γ+x~
becomes:

or, in physical coordinates,

z+x=z0+x22a+xxzz0x22a.              Eq. 26
For the ray directed downwards from the source we can separate two regions along
x~
: before and after the reflection point
x~0.
For
x~<x~0
we obtain:
S=x~t+23γ0t3/223γx~t3/2              Eq. 27

and the stationary value of the phase is given by:

S=x~+γx~tγ0t=0.              Eq. 28
This leads to an equation for the ray trajectory in the region
x~<x~0
γx~=γ0+x~γ0t.              Eq. 29
Assume now that t = t0, the solution to Eq. (27) for fixed γ, γ0 and
x~.
In that case
γx~=x~0=0,
where
x~0x~0t=t0,
and:
γx~=γ0+x~2xx0x02+γ0                Eq. 30
becomes the equation for the ray trajectory in the region x < x0, when
x~0
is used as a stationary parameter instead of t0.
The equation for
x~x~0
can be obtained in a similar way using the boundary condition
γx~=x~=γ:
γx~=x~x~02+x~x~0x~x~0γx~x~02.                Eq. 31

In physical coordinates, Eqs. (30) and (31) take the form:

zx=z0+x22axx0x022a+z0,               Eq. 32
zx>x0=xx022a+xx0xx0zxx022a.               Eq. 33

Now we need to find a reflection point x0 from the equations for the stationary phase (28).

The general case solution is obtained by Fock and is provided in Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applications“Parabolic Approximation to the Wave Propagation”. Let us consider here approximate formulas for small δγ = γ-γ0, |δγ| ≪ 1. Introduce t = t0+δt and expand Eq. (28) into series over δγ and δt. Leaving only first order terms we obtain:

S=x2γ0t0+2t0+δt1γ0t01t0δγ12γ0t0=0.               Eq. 34

Equating the terms of the same order we obtain:

t0=4γ0x24x=γ0xx4,               Eq. 35
δt=δγ2t0tγ0t=δγ4γ0x24x2.               Eq. 36

The stationary value t0 corresponds to the case of equal heights of the transmitting and receiving antennas above the surface, δγ = 0, in this case the reflection point x0 is apparently x0 = x/2. The correction term to t0 is given by Eq. (36) and substitution of δt into Eq. (28) will lead to the correction term δx to the distance x0 to reflection point:

δx=δγx4γ0+x2.               Eq. 37

Similar approximations can be obtained from the general solution (Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applications“Parabolic Approximation to the Wave Propagation”) when the inequality δγ x/ρ3 ≪ 1 holds, which in turn is equivalent to δγ ≪ 1 since x and ρ3 are terms of the same order of magnitude. Finally, we have:

x~0=x~2γγ0x~4γ0+x~2               Eq. 38

and:

x0=x2zz0x4z0+x22m.               Eq. 39

Moments of the Field

Under the above conditions of Eqs. (11) and (12) we can use the equations for unperturbed trajectories of direct and reflected waves (26), (32), (33) in the calculation of the field’s moments.

As a less cumbersome case of calculation of the second moment, let us consider the intensity of the field. In the line-of-sight region we obtain an equation similar to Eq. (17):

Jx, z, z0=2x21exp M12sphx, z, z0 cos Ssphx, z, z0               Eq. 40

where:

M12sph=πk2x401dξHdsphξ,              Eq. 41
Ssph=kx332a+zz028x+z+z0x4az22xx0z022x0.               Eq. 42

The parameter dsph is given by:

dsphx=z0x022a=zxx022a.               Eq. 43

As observed from Eq. (43) and the geometry of the problem, parameter dsph(x) → 0 when the distance x between corresponding points approaches the horizon

xxm=2az+z0.

The phase difference ΔSsph → 0 and the field intensity J(x, z, z0) → 0 when x → xm, in accordance with the laws of geometric optics. The numerical value of Eq. (41) is given by Eq. (19) where the parameter dp is replaced by dsph (x).

Let us examine the fluctuations in the field intensity < J2 > above either plane or spherical boundaries with ideal reflection. Using representation (1) we obtain:

J2=G112+G222+2G11G22+G122+G212+2G12G212G11G122G11G222G21G112G21G22.               Eq. 44

As an example, the fourth-order correlator for a direct wave has the form:

G112=DRxDRxDρxDρx·expik0xdxdRdxdpdx+dRdxdρdx·expπk240xdxHρx+Hρx·expπk220xdxd2κΦε0, κ exp jκRxRx·cosκρxρx2cosκρx+ρx2               Eq. 45

where:

ρx=ρ1+xρ2+x,     ρx=ρ3+xρ4+x,     Rx=1/2ρ1+x+ρ2+x,     Rx=1/2ρ3+x+ρ4+x.

The remaining terms in Eq. (44) have a form similar to Eq.(45).

Under the same conditions defined by inequalities (11) and (12) we can obtain the mean-square value of the fluctuations in intensity of the wave field
σJ2=J2J2:
σJ2=2x21exp 2M12x, z, z0·1exp 2M12x, z, z0 cos 2Sx, z, z0               Eq. 46
and for the scintillation factor
βJ2=σJ2J2
respectively:
βJ2=121exp 2M12x, z, z0 1exp 2M12x, z, z0 cos 2Sx, z, z01expM12x, z, z0 cos 2Sx, z, z02              Eq. 47

Here the phase difference ΔS and the mean-square of the fluctuations in a phase difference between the direct and reflected waves M12 are determined by the respective formulas for a plane or spherical interface, either Eqs. (17) and (18) or (41) and (42).

Given the values of z and z0 let us introduce the points xmin and xmax at a distance x where the unperturbed field has a maximum and minimum respectively. These points can be derived from the equations ΔS(xmax, z, z0) = π(2n+1) and ΔS(xmin, z, z0) = 2πn. The value of the scintillation factor
βJ2
at those points we define as
βJ2x=xmaxβJmax2
and
βJ2x=xminβJmin2
respectively.
As observed from Eq. (47), if M12 ≫ 1 and, therefore, the direct and reflected waves are completely uncorrelated, the fluctuations in intensity are uniform in space. In this case the scintillation factor
βJmax2=βJmin2=1/2
at both the maximum and minimum of the unperturbed field. In another limiting case when M12 ≪ 1, the values of the scintillation factor at the minimum and maximum of the unperturbed field are significantly different.
βJmax2=1/2 M122
and
βJmin2=2;
and
σJ2xmin=8M122
while
Jxmin2=4M122.

Such behaviour of the field fluctuation is of a general nature. When the only phase fluctuates significantly, which is the case, the amplitude fluctuations δA of the total field at the minimum xmin are contributed by the fluctuation of the phase difference between the direct and reflected waves:

δA~ϕ12=δS+1δS2   and   δA2~ϕ122;   J2xmin~ϕ124.

With normal distribution of the fluctuations in the phase it follows that:

σJ2xmin=2ϕ122=2Jxmin2,
and consequently
βJmin2=2.

Fluctuations of the Waves in a Random Non-uniform Medium above a Plane with Impedance Boundary Conditions

Let us introduce the Cartesian coordinate system (x, y, z) and the attenuation factor u by equation Ez = u·ejkx. The slow varying field amplitude u is governed by the equation:

2jkux+u+δεru=0             Eq. 48

with the impedance boundary conditions at the surface separating two media:

uz+jqu=0                Eq. 49

and initial condition (at the source):

ux=0, γ, z=2π/k δρρ0,

where:

r=x, ρ,   ρ=γ, z,   ρ0=γ0, z0,   =2/γ2+2/z2,    q=κ/εg,   k=2π/λ
λ is the wavelength, εg is the effective dielectric permittivity of the lower half-space (z < 0),
δεr
is the random component of the medium’s dielectric permittivity in the upper half-space (z > 0), < * > = 0, and the angle brackets denote averaging over the ensemble of the
δεr
realisations.

Following the representation introduced by Malyuzhinetz, the boundary problem (48), (49) reduces to a problem with ideal conditions of reflection via introduction of the Malyuzhinetz transformation:

u=u0+x, γ, z+u0x, γ, z+uqx, γ, z             Eq. 50

where:

uqx, γ, z=2jk·ejqz ·ejπ/4z dς·ejkς u0x, γ, ς.            Eq. 51
Here
u0+
is the field of the incident wave, and
u0
is the field of the wave generated by the mirrored source. The
u0+
and
u0
fields determine the solution of the boundary problem (1), (2) when
εg,
i. e., the field of the vertical electrical dipole in the case of a randomly non-uniform medium above an infinitely conducting plane. The last term in Eq. (50) takes into consideration the corrective factor for the finite value of εg which we can refer to the field of the impedance source.
Let us examine the average intensity
Ir
of the scattered field at the point
r=x, 0, z,
excited by a vertical electric dipole situated at the point
r0=0, 0, z0:
Ir=ur2=I0r+uqru0+r*+uqru0r*+c.c.+uqr2.               Eq. 52
Here,
I0r
is the average intensity of the wave field above the ideally reflective surface:
I0r=u0++u02.
The fields
u0±
can be represented in the form of Feynman trajectory integrals. A trajectory integration is described above for the boundary problem with ideal reflective conditions, and
I0r
given by:
I0r=1x21+exp M12x, z, z0 cos Sx, z, z0              Eq. 53
where
S=2kzz0/x
represents the phase difference between the direct and reflected waves. The variance of the fluctuations in phase difference between the direct and reflected waves M12(x, z, z0) is given by:
M12x, z, z0=M12x, d=πk2x401d·Hξd              Eq. 54

where H(ρ) is a structure function of the fluctuations δε, d = zz0/z+z0 is the height at which the direct beam passes above the surface at the point of mirror reflection. Expression (53) and the subsequent formulas of this section have been derived using the approximation of smooth perturbations, i. e. on the validity of the inequalities:

πk240xHxk dx 1;              Eq. 55
πk2x280xΦε0, κκ2d2κ21;              Eq. 56
where Φε(κ) is a three-dimensional spectrum of fluctuations in the dielectric permittivity
δεr, κ=κx, κ
is the wave vector of the fluctuations,
κ=κγ, κz.
Let us examine the correlation function for the field of the impedance source uq and the direct wave
u0+.
After averaging, trajectory integration and expanding the phase of the derived equation in a series over ν = z-ς, we obtain:
uqu0+*=2jqx2expjSx, z, z0M12x, d·C0expjkν22xjqν1+εg tan ψkvαkv2ϑs2x, d dv                  Eq. 57
where the contour C emanates from infinity along the ray
ej5π/4, tan ψ=z+z0/x, ψ
is the angle of reflection. The coefficient
ϑs2x, d=Cε2xd1/3
has the meaning of the angular width of the scattered field for a base equal to d, the parameter
α=ϑs2x, d/γ
determines the ratio of the angular spectrum width to the width of the interference lobe γ = 1/kd.
Let us assume that fluctuations
δεr
are caused by turbulence and d > l0 where l0 is an internal scale of the turbulence, and for the structure function
Hρ
we will assume
Hρ=Cε2ρ5/3,
where Cε is a structure constant for the fluctuations δε. For coefficients
α, ϑs2x, d
and M12(x, d) we have:
α=1,46kxCε225/3ξ1, ξ2z2/3ξ15/3+z02/3ξ25/3582z2/3ξ18/3+2z02/3ξ28/3+2z02/3ξ25/312ξ12                Eq. 58
ϑs2x, d=1,46kxCε211625/3z1/3ξ18/3ξ22+z01/3ξ22/3ξ13 ξ16115322/3z1/3ξ18/3ξ2+z01/3ξ22/3ξ13+2z02/3ξ28/3+5242z1/3ξ18/3+2z01/3ξ28/31+45ξ2                Eq. 59
M12x, d=0,869Cε2k2xd5/3               Eq. 60

where the parameters ξ1 and ξ2 correspond to the distance of the mirror reflection point x0 from the source and the receiver respectively:

ξ1=x0x=z0z+z0   and   ξ2=xx0x=ςz+z0.
When d < l0 the structure function can be approximated by
HρCε2ρ2l01/3
and the coefficient
α=0, ϑs2x, dσs2x,
where
σs2x
is a variance of the fluctuations in the angle of incidence in the vertical plane.

Let us calculate the integral (57) for large numerical distances, i. e., when |qx| ≫ 1. Integrating by parts, we obtain:

uqu0+=1x2expM12x, d+jSx, z, z0·21+jg1+εg tan ψ+2jg1+jβkx1+jg3 1+εg tan ψ3+6εg2kx211+jg51+εg tan ψ5                 Eq. 61
Parameter
β=2kxϑs2
determines the ratio of the width of the angular spectrum of the scattered field
ϑs2
to the angular size of the Fresnel zone
θF2=1/kx.
In the light of the validity of inequality (56) we have β ≪ 1. The parameter:
g=α1εg+tan ψ
determines the ratio of the angular spectrum width of the scattered field to the product of the angular width of the interference lobe γ and the effective grazing angle of the reflected wave
1/εg+tan ψ.
The correlation function of the impedance-source field uq with reflected field
u0
is calculated similarly to Eqs. (57)–(61). Assume that the following inequalities hold true:
maxkl0εg, kl0 tan ψ1.               Eq. 62
In this case the trajectories determining the principal contribution to the trajectory integrals, uq and
u0
actually coincide, and for the structure function H(ρ) we can use the quadratic approximation
Hρ=Cε2ρ2l01/3.
Then
uqu0*
can be written as follows:
uqu0*=1x221+εg tan ψ+2jε1+jgskx1+εg tan ψ3+6εg2kx21+jgs21+εg tan ψ5                   Eq. 63
where
gs2=σs2x/θF2.
The parameter
σs2x=0,82Cε2l01/3x
is a variance of the fluctuations in the arrival angle of the incident wave.

Let us examine the average intensity of the impedance-source field <|uq|2>. Using an approach similar to the calculation of Eq. (57) we carry out averaging and continuous integration under the condition of validity of the inequalities (55) and (56). Expanding the phase of the expression under the integral sign into a series over the distance from the upper limit zand retaining the quadratic terms, we obtain:

uq2=4q2x20ejπ/4du2u2udv exp jkν22xjνkuxReqk tan ψ+2ν Imqkν2σs2dv.            Eq. 64

Let us assume here and below that Im εg = 4πσ/ω = 0, σ is the conductivity of the lower medium (z < 0) and ω = 2πf is a circular frequency of radiation. This condition does not limit the generality of the results since the solution in quadra- tures for the average intensity (52) has already been determined by Eqs. (53), (57) and (64). In the meanwhile such an approximation significantly simplifies the asymptotic expressions for the total field. In the case of radio frequencies above 10 GHz and with the sea surface assumed to be a boundary between the two media, the main contribution to εg comes from displacement currents and Im εg ≪ Re εg. For large distances |qx| ≫ 1 we obtain:

uq2=1x241+εg tan ψ2+8εgσs21+εg tan ψ2·1+11+εg tan ψ220εg2kx2.                Eq. 65

Collecting all components of Eq. (52) we obtain:

Ix, z, z0=1x21+R02+2 exp M12x, d Re RsejS+2 exp M12x, d·ReejS2jεg1+jβkx1+jg31+εg tan ψ3+6εg2kx211+jg51+εg tan ψ5+12εg2kx21+εg tan ψ5+8σs2εg1+εg tan ψ2111+εg tan ψ+11+εg tan ψ220εg2kx2                 Eq. 66

Here:

R0=εg tan ψ1/εg tan ψ+1

is a Fresnel coefficient of reflection in the parabolic approximation for the wave field polarized in a plane of incidence. The coefficient:

Rs=εg tan ψ1+jgεg tan ψ+1εg tan ψ+11+jg              Eq. 67
also has the meaning of the reflection coefficient with provision for wave scattering in the media above the surface. When the angle of incidence ψ becomes equal to a Brewster angle
ψ=a tan 1/εg, R0=0,
, and the coefficient Rs is defined by the ratio of the angular width of the scattered field to the width of the interference lobe.
Rs=jg1+jgjϑs2x, d2γεg.             Eq. 68

Let us consider some limiting cases. One of the limiting cases is when δε = 0. We assume in Eq. (66) that σs, g, β = 0. In the area of applicability of reflection formulas derived by:

kx tan ψkdεg             Eq. 69

we can drop the terms of the order of εg/kx and g/kx)2. Then:

Ix, z, z0=1x21+R02+2R0 cos S.             Eq. 70
For sliding angles with
εg tan ψ1
and:
kdεg            Eq. 71

the field intensity (66) can be expressed in terms of the familiar attenuation function w(x) = 2jεg/kx:

Ix, z, z0=Ix=1x2wx2.            Eq. 72

In this limiting case, R0 ≈ -1, ΔS ≪ εg/(kx) ≪ 1.

Let us examine Eq. (66) in the presence of fluctuations of the refractive index,
δεr0.
The variance of the phase difference between the direct and reflected waves can be written in the form of a ratio of the angular width of the scattered field
ϑs2x, d
at the base d to the angular width of the interference lobe γ:
M12x, d=ϑs2x, d/γ2.             Eq. 73

Let us define three characteristic distances on the propagation path: xs, xg and xc from the following relationships:

gx2, d=1,   M12xc, d=1,    σs2xs=1.             Eq. 74

Then:

xs=k1/2Cε1l01/8,    xg=kd2/3 Cε2 εg1,    xc=k2d5/3 Cε21.            Eq. 75
One finds that xc is the distance where the variance of the phase difference at the base d reaches the order of unity, or the angular spectrum of the scattered field becomes wider than the interference lobe; xs is the distance where the variance of the fluctuation of the angle of incidence
σs2
becomes equal to the square of the angular size of the Fresnel zone
θF2;
and xg is the distance from which the angular spectrum of the scattered field becomes wider than the product of the width of the interference lobe γ and the effective grazing angle of the reflected wave
tan ψ/εg.
Because of the validity of inequality (56), the range of distances under consideration is such that x < xs. To analyse the value of the parameters we will define the ratios:
xsxg=k1/2 d2/3 Cεεgl01/6,                  Eq. 76
xsxc=k3/2 d5/3 Cεεgl01/6,                  Eq. 77
from which it follows that in the range of radio frequencies when k ≈ 1 cm-1 and under conditions of the earth’s atmosphere
Cε2=1014 cm2/3, εg102,
the ratio xs/xg ≪ 1. Given the fact that we have used the method of smooth perturbations (inequalities (8) and (56)), we have always g ≪ 1, at the same time xc/xs < 1, when d > 103 cm.

In the area of sliding angle propagation when inequality (71) is satisfied, the parameter M12(x, d) < 1 for all x < xs.

Let us assume that xc > xs, in this case M12(x, d) < 1, g ≪ min
1, εg tan ψ,
and Rs ≈ R0. In the interference region (inequality (69)) the field intensity can be approximated as follows:
Ix, z, z0=1x21+R02+2R0 cos S exp M12x, d.              Eq. 78
In the case where
εg tan ψ1,
when inequality (71) holds, Rs ≈ R0 ≈ -1. In this case there are two small parameters M12(x, d) and εg/kx. Let us introduce yet another characteristic distance xM, where M12(xM, d) = εg/kxM:
xM2=εηk3Cε2d5/3.                  Eq. 79

When inequality (62) holds, xM < xs. Assuming x < xM the field intensity can be approximated by a composition of two terms: one is an attenuation factor w(x) and the other is a weighted variance of the angle fluctuation of the scattered field:

Ix, z, z0=1x2wx2+8εgσs2x.                Eq. 80
In the range of distances xs < x < xM, the main contribution to the sliding angle propagation mechanism is provided by scattering on the non-uniformity of the refractive index,
δεr,
and the field intensity in that region is given by:
Ix, z, z0=1x2kd2ϑs2x, d+8εgσs2x.                 Eq. 81

In the regime of non-coherent composition of the direct and reflected waves we have M12(x, d) ≫ 1, and xc < xs. In this range the wave parameter kd2/x must be sufficiently large kd2/x ≫ 1. The interference structure of the wave field in this region is entirely distorted by the large fluctuation of the phase difference between the direct and reflected waves and the intensity of the field is composed of two terms:

Ix, z, z0=1x2wx2+8σs2x.                 Eq. 82
In conclusion, one can note that the limitation caused by the quadratic approximation of the structure function H(ρ) (inequality (63)) is not a fundamental one and serves only to simplify the derivation of the asymptotic expansion for the moments
uqu0*
and
uq2.
The analytical solution for the coherence function and the moments of higher order can be obtained similarly to the solution for intensity provided in this section, however, the final expressions are exceedingly cumbersome.

Comments on Calculation of the LOS Field in the General Situation

From the above study it is apparent that in the general case of the stratified tropo- sphere filled with random fluctuations in the refractive index, the line-of-sight field can be calculated in a similar way, applying the ray theory. This approach can, in principle, be applied to calculation of the field in the tropospheric duct at distances of up to a few hops. In the general case of refractivity, the intensity of the line-of-sight field can be presented in the form:

Jx, z, z0=n=1N m=1N AnAm* exp jkSnSmMnm                 Eq. 83

where An, Sn are the amplitude and phase of the nth ray, Mnm is a structure function of the phase difference of the phases along the ray’s trajectories:

Mnm=0,73Cε2k2 0xdsrnsrms5/3                 Eq. 84

and z, z0 are the heights of the receiving and transmitting antennas respectively, x is the distance between them. The phase Sn along the nth ray is given by:

Sn=jk20xdsd2rnds2+εmrns      Eq. 85

and the trajectories rn(s) are given by a solution to the Euler equation:

d2rnds2=dεmdrn      Eq. 86

with boundary conditions:

rn0=z0, rnx=z.      Eq. 87

Amplitude An accounts for the divergence (or convergence) of the rays:

An=Anx=Dn0Dnx1/2      Eq. 88

where Dn(s) is a cross-section of the ray tube at a distance s along the nth ray. For the waves reflected from the sea surface, the amplitude An is modulated by a reflection coefficient:

An=Anx·Rθn     Eq. 89

where θn is the angle of incidence of the nth wave in the point of reflection. In most cases, Eqs. (83)-(87) can be solved by applying computer-based algorithms. Apparently, the above equations are not applicable with small grazing angles in the case of impedance boundary conditions and should be modified before being applied in the vicinity of caustics.

Author
Author photo - Olga Nesvetailova
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Literature
  1. Fock, V.A. Electromagnetic Diffraction and Propagation Problems, Pergamon Press, Oxford, 1965.
  2. Hitney, H.V., Richter, J.H., Pappert, R.A., Anderson, K.D. and Baumgartner, G.B. Tropospheric radio propagation assessment, Proc. IEEE, 1985, 73 (2), 265–283.
  3. Belobrova, M.V., Ivanov, V.K., Kukushkin, A.V., Levin, M.B. and Fastovsky, J.A. Prediction system on UHF radio propagation conditions over the sea, Institute of Radio Astronomy, Ukrainian Acad. Sci., Preprint No 31, 1989, 39 pp.
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  5. Puzenko, A.A., Chaevsky, E.V. Function of mutual coherence in a problem of small grazing angle wave propagation in random medium over boundary interface, Radiophys. Quantum Electron., 1976, 19 (2), 228–239.
  6. Kostenko, N.L., Puzenko, A.A. and Chaevsky, E.V. Correlation of the amplitude and phase of the scattered field in case of propagation through turbulent medium over boundary interface, Preprint of the Institute of Radiophysics and Electronics, Ukrainian Academy of Science, No 153, 1980, 36 pp.
  7. Dashen, R. Path Integrals for waves in random media, J. Math. Phys., 1979, 20 (5), 894–918.
  8. Tatarskii, V.I. The Effects of the Turbulent Atmosphere on Wave Propagation, IPST, Jerusalem, 1971.
  9. Feinberg, E.L. Radio Wave Propagation along the Earth’s Surface, Nauka, Moscow, 1961.
  10. Feynman, R.P. and Hibbs, A.R. Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.
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