Understanding Wave Field Fluctuations in Random Media

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.

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Reflection Formulas for the Wave Field in a Random Medium over an Ideally Reflective Boundary
Ideally Reflective Flat Surface
Spherical Surface
Trajectory Equations
Moments of the Field
Fluctuations of the Waves in a Random Non-uniform Medium above a Plane with Impedance Boundary Conditions
Comments on Calculation of the LOS Field in the General Situation

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

\sqrt{2 a} (\sqrt{z} + \sqrt{z_{0}}),

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

G (\vec{r}, {\vec{r}}_{0})

can be presented as a superposition of the Green function for a point source

G^{+} (\vec{r}, {\vec{r}}_{0})

situated at the point

{\vec{r}}_{0} = \{0, γ_{0}, z_{0}\}

and that for the mirror-reflected point source

G^{-} (\vec{r}, {\vec{r}}_{0})

located at

{\vec{r}}_{0} = \{0, γ_{0}, - z_{0}\} :

G (\vec{r}, {\vec{r}}_{0}) = G^{+} (\vec{r}, {\vec{r}}_{0}) - G^{-} (\vec{r}, {\vec{r}}_{0}) . E q . 1

The boundary condition at the surface z = 0:

G (\vec{r}, {\vec{r}}_{0}) |_{z = 0} = 0 . E q . 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.

δ ε (\vec{r} (x)) = δ ε (x, γ (x), |z (x)|) .

We will define

{\vec{ρ}}^{\pm} (x),

the trajectories of the waves departing from the sources in the upper and lower half-space. For

G^{\pm} (\vec{r}, {\overset{⇀}{r}}_{0})

we have:

G^{\pm} (\vec{r}, {\overset{⇀}{r}}_{0}) = \int D {\vec{ρ}}^{\pm} (x) e x p \{j k S_{0} [{\vec{ρ}}^{\pm} (x)] + j \frac{k}{2} \int_{0}^{x} d x^{'} δ ε (x^{'}, ρ^{\pm} (x^{'}))\}, E q . 3

We have introduced the notation

{\vec{ρ}}^{\pm} (x) = \{γ^{\pm} (x), z^{\pm} (x)\} .

The action S₀ is given by:

S_{0} [\vec{ρ} (x)] = \int_{0}^{x} d x^{'} [L_{0} (x^{'}, \vec{ρ} (x^{'})) + \frac{1}{2} δ ε (x^{'}, \vec{ρ} (x^{'}))] E q . 4

where

L_{0} (x, \vec{ρ} (x)) = \frac{1}{2} {(\frac{d ρ}{d x})}^{2} .

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:

{\vec{ρ}}^{+} (x = 0) = \{γ_{0}, z_{0}\}, {\vec{ρ}}^{-} (x = 0) = \{γ_{0}, - z_{0}\} .

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

<G (\vec{r}, {\vec{r}}_{0}) G^{*} (\vec{r}, {\vec{r}}_{0})> = <G_{11}> + <G_{22}> - <G_{12}> - <G_{21}> E q . 5

where:

<G_{11}> = \int D {\vec{ρ}}_{1}^{+} (x) \int D {\vec{ρ}}_{2}^{+} (x) \cdot e x p \{j k [S^{+} (1) - S^{+} (2)] - M_{11} [x, {\vec{ρ}}_{1}^{+} (x), {\vec{ρ}}_{2}^{+} (x)]\}, E q . 6

<G_{12}> = \int D {\vec{ρ}}_{1}^{+} (x) \int D {\vec{ρ}}_{2}^{-} (x) \cdot e x p \{j k [S^{+} (1) - S^{-} (2)] - M_{11} [x, {\vec{ρ}}_{1}^{+} (x), {\vec{ρ}}_{2}^{-} (x)]\}, E q . 7

M_{11} = \frac{π k^{2}}{4} \int_{0}^{x} d x^{'} H ({\vec{ρ}}_{1}^{+} (x^{'}) - {\vec{ρ}}_{2}^{+} (x^{'})), E q . 8

M_{12} = \frac{π k^{2}}{4} \int_{0}^{x} d x^{'} H ({\vec{ρ}}_{1}^{+} (x^{'}) - {\vec{ρ}}_{2}^{-} (x^{'})) . E q . 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.

Ω_{1} (x) \supset \{z^{2} (x) > \frac{ς^{2} (x)}{4}\}, Ω_{2} (x) \supset \{z^{2} (x) < \frac{ς^{2} (x)}{4}\} . E q . 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

H (\vec{ρ} (x))

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:

\frac{π k^{2}}{4} \int_{0}^{x} d x^{'} H (\sqrt{\frac{x^{'}}{κ}}) = 0, 73 C_{ε}^{2} k^{7 / 6} x^{11 / 6} ≪ 1 . E q . 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:

\frac{π k^{2}}{8} \int_{0}^{x} d x^{'} {\vec{ρ}}^{2} (x^{'}) \frac{d^{2} H (\vec{ρ})}{d {\vec{ρ}}^{2}} \leq \frac{π k^{2}}{8} x^{2} \int d^{2} {\vec{κ}}_{⊥} Φ_{ε} (0, {\vec{κ}}_{⊥}) {\vec{κ}}_{⊥}^{2} = 0, 037 C_{ε}^{2} k x^{2} l_{0}^{- 1 / 3} ≪ 1 E q . 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

\vec{ρ} (x)

represent solutions to Euler equations and are given by:

{\vec{ρ}}^{\pm} (x^{'}) = \pm {\vec{ρ}}_{0} + (\vec{ρ} - {\vec{ρ}}_{0}) \frac{x^{'}}{x} . E q . 13

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

M_{11} = \frac{π k^{2} x}{4} \int_{0}^{1} d ξ H (\vec{ρ} ξ + {\vec{ρ}}_{0} (1 - ξ)), E q . 14

M_{12} = \frac{π k^{2} x}{4} \{\int_{0}^{ξ_{1}} d ξ H (ς_{0} + ξ (2 z - ς_{0}), γ ξ + γ_{0} (1 - ξ)) + \int_{ξ_{1}}^{1} d ξ H (2 z_{0} (1 - ξ) + ξ ς, γ ξ + γ_{0} (1 - ξ))\}, E q . 15

M_{22} = \frac{π k^{2} x}{4} \{\int_{0}^{ξ_{2}} d ξ H (ς_{0} (1 - ξ) + ξ ς, γ ξ + γ_{0} (1 - ξ)) + \int_{ξ_{2}}^{ξ_{1}} d ξ H (2 (z + z_{0}) ξ - 2 z_{0} ξ, γ ξ + γ_{0} (1 - ξ)) + \int_{ξ_{1}}^{1} d ξ H (ς ξ + ς_{0} (1 - ξ), γ ξ + γ_{0} (1 - ξ))\}, E q . 16

where:

\{γ, ς\} = {\vec{ρ}}_{1} - {\vec{ρ}}_{2}, \{γ_{0}, ς_{0}\} = {\vec{ρ}}_{0}^{'} - {\vec{ρ}}_{0}^{''}, ξ_{1, 2} = \frac{z_{0} \mp \frac{ς_{0}}{2}}{z \mp \frac{ς}{2} \mp z_{0} \mp \frac{ς}{2}}

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:

J (x, z, z_{0}) = 4 π^{2} / k^{2} <{|G (1)|}^{2}>,

when

\vec{ρ} = {\vec{ρ}}_{0} = 0 :

J (x, z, z_{0}) = \frac{2}{x^{2}} [1 - e x p (- M_{12} (x, z, z_{0})) \cos (∆ S (x, z, z_{0}))] E q . 17

where

∆ S (x, z, z_{0}) = 2 k \frac{z z_{0}}{x},

M_{12} = M_{21} = \frac{π k^{2} x}{4} \int_{0}^{1} d ξ H (d_{p} ξ) . E q . 18

The M₁₂ is a mean-square fluctuation in a phase difference between the direct and the reflected waves, d_p = 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 d_p 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:

M_{12} (x, z, z_{0}) = 0, 869 C_{ε}^{2} k^{2} x d_{p}^{5 / 3} . E q . 19

The expression (19) coincides with the structural function of the phase for the base d_p 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, γ_S >> 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:

L_{0} (\vec{ρ} (x)) = \frac{1}{2} {(\frac{d γ}{d x})}^{2} + \frac{1}{2} {(\frac{d z}{d x})}^{2} + \frac{|z (x)|}{a}, E q . 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:

\tilde{x} = \frac{m x}{a}, {\tilde{x}}_{0} = \frac{m x}{a}, γ = \frac{k z}{m}, γ_{0} = \frac{k z}{m} E q . 21

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

S^{+} = \tilde{x} t + \frac{2}{3} {(γ - t)}^{3 / 2} - \frac{2}{3} {(γ_{0} - t)}^{3 / 2} . E q . 22

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

S^{' +} = \tilde{x} - \sqrt{γ - t} + \sqrt{γ_{0} - t} E q . 23

that leads to the solution:

\sqrt{γ_{0} - t} = \frac{γ - γ_{0} - {\tilde{x}}^{2}}{2 x} . E q . 24

The equation for the trajectory

γ^{+} ({\tilde{x}}^{'})

becomes:

or, in physical coordinates,

z^{+} (x^{'}) = z_{0} + \frac{x^{' 2}}{2 a} + \frac{x^{'}}{x} (z - z_{0} - \frac{x^{2}}{2 a}) . E q . 26

For the ray directed downwards from the source we can separate two regions along

{\tilde{x}}^{'}

: before and after the reflection point

{\tilde{x}}_{0} .

For

{\tilde{x}}^{'} < {\tilde{x}}_{0}

we obtain:

S^{-} = {\tilde{x}}^{'} t + \frac{2}{3} {(γ_{0} - t)}^{3 / 2} - \frac{2}{3} {(γ ({\tilde{x}}^{'}) - t)}^{3 / 2} E q . 27

and the stationary value of the phase is given by:

S^{' -} = {\tilde{x}}^{'} + \sqrt{γ ({\tilde{x}}^{'}) - t} - \sqrt{γ_{0} - t} = 0 . E q . 28

This leads to an equation for the ray trajectory in the region

{\tilde{x}}^{'} < {\tilde{x}}_{0}

γ ({\tilde{x}}^{'}) = γ_{0} + {\tilde{x}}^{'} \sqrt{γ_{0} - t} . E q . 29

Assume now that t = t₀, the solution to Eq. (27) for fixed γ, γ₀ and

\tilde{x} .

In that case

γ ({\tilde{x}}^{'} = {\tilde{x}}_{0}) = 0,

where

{\tilde{x}}_{0} \equiv {\tilde{x}}_{0} (t = t_{0}),

and:

γ ({\tilde{x}}^{'}) = γ_{0} + {\tilde{x}}^{' 2} - \frac{x^{'}}{x_{0}} (x_{0}^{2} + γ_{0}) E q . 30

becomes the equation for the ray trajectory in the region x^′ < x₀, when

{\tilde{x}}_{0}

is used as a stationary parameter instead of t₀.

The equation for

{\tilde{x}}^{'} \geq {\tilde{x}}_{0}

can be obtained in a similar way using the boundary condition

γ ({\tilde{x}}^{'} = \tilde{x}) = γ :

γ ({\tilde{x}}^{'}) = {({\tilde{x}}^{'} - {\tilde{x}}_{0})}^{2} + \frac{{\tilde{x}}^{'} - {\tilde{x}}_{0}}{\tilde{x} - {\tilde{x}}_{0}} (γ - {(\tilde{x} - {\tilde{x}}_{0})}^{2}) . E q . 31

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

z (x^{'}) = z_{0} + \frac{x^{' 2}}{2 a} - \frac{x^{'}}{x_{0}} (\frac{x_{0}^{2}}{2 a} + z_{0}), E q . 32

z^{-} (x^{'} > x_{0}) = \frac{{(x^{'} - x_{0})}^{2}}{2 a} + \frac{x^{'} - x_{0}}{x - x_{0}} (z - \frac{{(x - x_{0})}^{2}}{2 a}) . E q . 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^{-'} = x - 2 \sqrt{γ_{0} - t_{0}} + 2 \sqrt{- t_{0}} + δ t (\frac{1}{\sqrt{γ_{0} - t_{0}}} - \frac{1}{\sqrt{- t_{0}}}) - δ γ \frac{1}{2 \sqrt{γ_{0} - t_{0}}} = 0 . E q . 34

Equating the terms of the same order we obtain:

\sqrt{- t_{0}} = \frac{4 γ_{0} - x^{2}}{4 x} = \frac{γ_{0}}{x} - \frac{x}{4}, E q . 35

δ t = \frac{δ γ}{2} \frac{\sqrt{- t_{0}}}{(\sqrt{- t} - \sqrt{γ_{0} - t})} = - δ γ \frac{4 γ_{0} - x^{2}}{4 x^{2}} . E q . 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:

δ x = - δ γ \frac{x}{4 γ_{0} + x^{2}} . E q . 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:

{\tilde{x}}_{0} = \frac{\tilde{x}}{2} - (γ - γ_{0}) \frac{\tilde{x}}{4 γ_{0} + {\tilde{x}}^{2}} E q . 38

and:

x_{0} = \frac{x}{2} - (z - z_{0}) \frac{x}{4 z_{0} + \frac{x^{2}}{2 m}} . E q . 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):

J (x, z, z_{0}) = \frac{2}{x^{2}} [1 - e x p (- M_{12}^{s p h} (x, z, z_{0})) \cos (∆ S^{s p h} (x, z, z_{0}))] E q . 40

where:

M_{12}^{s p h} = \frac{π k^{2} x}{4} \int_{0}^{1} d ξ H (d_{s p h} ξ), E q . 41

∆ S^{s p h} = k [- \frac{x^{3}}{32 a} + \frac{{(z - z_{0})}^{2}}{8 x} + \frac{(z + z_{0}) x}{4 a} - \frac{z^{2}}{2 (x - x_{0})} - \frac{z_{0}^{2}}{2 x_{0}}] . E q . 42

The parameter d_sph is given by:

d_{s p h} (x) = z_{0} - \frac{x_{0}^{2}}{2 a} = z - \frac{{(x - x_{0})}^{2}}{2 a} . E q . 43

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

x \to x_{m} = \sqrt{2 a} (\sqrt{z} + \sqrt{z_{0}}) .

The phase difference ΔS^sph → 0 and the field intensity J(x, z, z₀) → 0 when x → x_m, in accordance with the laws of geometric optics. The numerical value of Eq. (41) is given by Eq. (19) where the parameter d_p is replaced by d_sph (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:

<J^{2}> = <G_{11}^{2}> + <G_{22}^{2}> + 2 <G_{11} G_{22}> + <G_{12}^{2}> + <G_{21}^{2}> + 2 <G_{12} G_{21}> - 2 <G_{11} G_{12}> - 2 <G_{11} G_{22}> - 2 <G_{21} G_{11}> - 2 <G_{21} G_{22}> . E q . 44

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

<G_{11}^{2}> = \int D {\vec{R}}^{'} (x) \int D {\vec{R}}^{''} (x) \int D {\vec{ρ}}^{'} (x) \int D {\vec{ρ}}^{''} (x) \cdot e x p \{i k \int_{0}^{x} d x^{'} [\frac{d {\vec{R}}^{'}}{d x^{'}} \frac{d {\vec{p}}^{'}}{d x^{'}} + \frac{d {\vec{R}}^{''}}{d x^{'}} \frac{d {\vec{ρ}}^{''}}{d x^{'}}]\} \cdot e x p \{- \frac{π k^{2}}{4} \int_{0}^{x} d x^{'} [H ({\vec{ρ}}^{'} (x^{'})) + H ({\vec{ρ}}^{''} (x^{'}))]\} \cdot e x p \{- \frac{π k^{2}}{2} \int_{0}^{x} d x^{'} \int d^{2} {\vec{κ}}_{⊥} Φ_{ε} (0, {\vec{κ}}_{⊥}) e x p (j {\vec{κ}}_{⊥} ({\vec{R}}^{'} (x^{'}) - {\vec{R}}^{''} (x^{'}))) \cdot [\cos ({\vec{κ}}_{⊥} \frac{{\overset{⇀}{ρ}}^{'} (x^{'}) - {\overset{⇀}{ρ}}^{''} (x^{'})}{2}) - \cos ({\vec{κ}}_{⊥} \frac{{\overset{⇀}{ρ}}^{} (x^{'}) + {\overset{⇀}{ρ}}^{''} (x^{'})}{2})]\} E q . 45

where:

{\vec{ρ}}^{'} (x) = {\vec{ρ}}_{1}^{+} (x) - {\vec{ρ}}_{2}^{+} (x), {\vec{ρ}}^{''} (x) = {\vec{ρ}}_{3}^{+} (x) - {\vec{ρ}}_{4}^{+} (x), {\vec{R}}^{'} (x) = 1 / 2 ({\vec{ρ}}_{1}^{+} (x) + {\vec{ρ}}_{2}^{+} (x)), {\vec{R}}^{''} (x) = 1 / 2 ({\vec{ρ}}_{3}^{+} (x) + {\vec{ρ}}_{4}^{+} (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

σ_{J}^{2} = <J^{2}> - {<J>}^{2} :

σ_{J}^{2} = \frac{2}{x^{2}} [1 - e x p (- 2 M_{12} (x, z, z_{0}))] \cdot [1 - e x p (- 2 M_{12} (x, z, z_{0}) \cos (2 ∆ S (x, z, z_{0})))] E q . 46

and for the scintillation factor

β_{J}^{2} = \frac{σ_{J}^{2}}{{<J>}^{2}}

respectively:

β_{J}^{2} = \frac{1}{2} \frac{[1 - e x p (- 2 M_{12} (x, z, z_{0}))] [1 - e x p (- 2 M_{12} (x, z, z_{0}) \cos (2 ∆ S (x, z, z_{0})))]}{{[1 - e x p (- M_{12} (x, z, z_{0}) \cos (2 ∆ S (x, z, z_{0})))]}^{2}} E q . 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 x_min and x_max at a distance x where the unperturbed field has a maximum and minimum respectively. These points can be derived from the equations ΔS(x_max, z, z₀) = π(2n+1) and ΔS(x_min, z, z₀) = 2πn. The value of the scintillation factor

β_{J}^{2}

at those points we define as

β_{J}^{2} (x = x_{m a x}) \equiv β_{J m a x}^{2}

and

β_{J}^{2} (x = x_{m i n}) \equiv β_{J m i n}^{2}

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

β_{J m a x}^{2} = β_{J m i n}^{2} = 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.

β_{J m a x}^{2} = 1 / 2 M_{12}^{2}

and

β_{J m i n}^{2} = 2;

and

σ_{J}^{2} (x_{m i n}) = 8 M_{12}^{2}

while

{<J (x_{m i n})>}^{2} = 4 M_{12}^{2} .

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 x_min are contributed by the fluctuation of the phase difference between the direct and reflected waves:

δ A ~ ∆ ϕ_{12} = δ S^{+} (1) - δ S^{-} (2) and <{(δ A)}^{2}> ~ <∆ ϕ_{12}^{2}>; <J^{2} (x_{m i n})> ~ <∆ ϕ_{12}^{4}> .

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

σ_{J}^{2} (x_{m i n}) = 2 <∆ ϕ_{12}^{2}> = 2 {<J (x_{m i n})>}^{2},

and consequently

β_{J m i n}^{2} = 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 E_z = u·e^jkx. The slow varying field amplitude u is governed by the equation:

2 j k \frac{\partial u}{\partial x} + ∆_{⊥} u + δ ε (\vec{r}) u = 0 E q . 48

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

\frac{\partial u}{\partial z} + j q u = 0 E q . 49

and initial condition (at the source):

u (x = 0, γ, z) = (2 π / k) δ (\vec{ρ} - {\vec{ρ}}_{0}),

where:

\vec{r} = \{x, \vec{ρ}\}, \vec{ρ} = \{γ, z\}, {\vec{ρ}}_{0} = \{γ_{0}, z_{0}\}, ∆_{⊥} = \partial^{2} / \partial γ^{2} + \partial^{2} / \partial z^{2}, q = κ / \sqrt{ε_{g}}, k = 2 π / λ

λ is the wavelength, ε_g is the effective dielectric permittivity of the lower half-space (z < 0),

δ ε (\vec{r})

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

δ ε (\vec{r})

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 = u_{0}^{+} (x, γ, z) + u_{0}^{-} (x, γ, z) + u_{q} (x, γ, z) E q . 50

where:

u_{q} (x, γ, z) = - 2 j k \cdot e^{j q z} \int_{\infty \cdot e^{j π / 4}}^{z} d ς \cdot e^{j k ς} u_{0}^{-} (x, γ, - ς) . E q . 51

Here

u_{0}^{+}

is the field of the incident wave, and

u_{0}^{-}

is the field of the wave generated by the mirrored source. The

u_{0}^{+}

and

u_{0}^{-}

fields determine the solution of the boundary problem (1), (2) when

|ε_{g}| \to \infty,

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 ε_g which we can refer to the field of the impedance source.

Let us examine the average intensity

I (\vec{r})

of the scattered field at the point

\vec{r} = \{x, 0, z\},

excited by a vertical electric dipole situated at the point

{\vec{r}}_{0} = \{0, 0, z_{0}\} :

I (\vec{r}) = <{|u (\vec{r})|}^{2}> = I_{0} (\vec{r}) + <u_{q} (\vec{r}) u_{0}^{+} {(\vec{r})}^{*}> + <u_{q} (\vec{r}) u_{0}^{-} {(\vec{r})}^{*}> + c . c . + <{|u_{q} (\vec{r})|}^{2}> . E q . 52

Here,

I_{0} (\vec{r})

is the average intensity of the wave field above the ideally reflective surface:

I_{0} (\vec{r}) = <{|u_{0}^{+} + u_{0}^{-}|}^{2}> .

The fields

u_{0}^{\pm}

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

I_{0} (\vec{r})

given by:

I_{0} (\vec{r}) = \frac{1}{x^{2}} \{1 + e x p [- M_{12} (x, z, z_{0})] \cos (∆ S (x, z, z_{0}))\} E q . 53

where

∆ S = 2 k z z_{0} / 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:

M_{12} (x, z, z_{0}) = M_{12} (x, d) = \frac{π k^{2} x}{4} \int_{0}^{1} d \cdot H (ξ d) E q . 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:

\frac{π k^{2}}{4} \int_{0}^{x} H (\frac{\sqrt{x^{'}}}{k}) d x^{'} ≪ 1; E q . 55

\frac{π k^{2} x^{2}}{8} \int_{0}^{x} Φ_{ε} (0, κ_{⊥}) κ_{⊥}^{2} d^{2} κ_{⊥}^{2} ≪ 1; E q . 56

where Φ_ε(κ) is a three-dimensional spectrum of fluctuations in the dielectric permittivity

δ ε (\vec{r}), κ = \{κ_{x}, κ_{⊥}\}

is the wave vector of the fluctuations,

κ_{⊥} = \{κ_{γ}, κ_{z}\} .

Let us examine the correlation function for the field of the impedance source u_q and the direct wave

u_{0}^{+} .

After averaging, trajectory integration and expanding the phase of the derived equation in a series over ν = z-ς, we obtain:

<u_{q} u_{0}^{+} *> = \frac{2 j q}{x^{2}} e x p [j ∆ S (x, z, z_{0}) - M_{12} (x, d)] \cdot \int_{C}^{0} e x p \{j k \frac{ν^{2}}{2 x} - j q ν (1 + \sqrt{ε_{g}} \tan (ψ)) - k v α - k v^{2} ϑ_{s}^{2} (x, d)\} d v E q . 57

where the contour C emanates from infinity along the ray

e^{j 5 π / 4}, \tan (ψ) = (z + z_{0}) / x, ψ

is the angle of reflection. The coefficient

ϑ_{s}^{2} (x, d) = C_{ε}^{2} x d^{- 1 / 3}

has the meaning of the angular width of the scattered field for a base equal to d, the parameter

α = ϑ_{s}^{2} (x, 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

δ ε (\vec{r})

are caused by turbulence and d > l₀ where l₀ is an internal scale of the turbulence, and for the structure function

H (\vec{ρ})

we will assume

H (\vec{ρ}) = C_{ε}^{2} ρ^{5 / 3},

where C_ε is a structure constant for the fluctuations δε. For coefficients

α, ϑ_{s}^{2} (x, d)

and M_{12(x, d)} we have:

α = 1, 46 k x C_{ε}^{2} [2^{5 / 3} (ξ_{1}, ξ_{2}) (z^{2 / 3} ξ_{1}^{5 / 3} + z_{0}^{2 / 3} ξ_{2}^{5 / 3}) - \frac{5}{8} ({(2 z)}^{2 / 3} ξ_{1}^{8 / 3} + {(2 z_{0})}^{2 / 3} ξ_{2}^{8 / 3}) + {(2 z_{0})}^{2 / 3} ξ_{2}^{5 / 3} (1 - 2 ξ_{1}^{2})] E q . 58

ϑ_{s}^{2} (x, d) = 1, 46 k x C_{ε}^{2} [\frac{11}{6} 2^{5 / 3} (z^{- 1 / 3} ξ_{1}^{8 / 3} ξ_{2}^{2} + z_{0}^{- 1 / 3} ξ_{2}^{2 / 3} ξ_{1}^{3}) (ξ_{1} - \frac{6}{11}) - \frac{5}{3} ({(2)}^{2 / 3} (z^{- 1 / 3} ξ_{1}^{8 / 3} ξ_{2} + z_{0}^{- 1 / 3} ξ_{2}^{2 / 3} ξ_{1}^{3}) + {(2 z_{0})}^{2 / 3} ξ_{2}^{8 / 3}) + \frac{5}{24} ({(2 z)}^{- 1 / 3} ξ_{1}^{8 / 3} + {(2 z_{0})}^{- 1 / 3} ξ_{2}^{8 / 3} (1 + \frac{4}{5} ξ_{2}))] E q . 59

M_{12} (x, d) = 0, 869 C_{ε}^{2} k^{2} x d^{5 / 3} E q . 60

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

ξ_{1} = \frac{x_{0}}{x} = \frac{z_{0}}{z + z_{0}} and ξ_{2} = \frac{x - x_{0}}{x} = \frac{ς}{z + z_{0}} .

When d < l₀ the structure function can be approximated by

H (ρ) \infty C_{ε}^{2} ρ^{2} l_{0}^{- 1 / 3}

and the coefficient

α = 0, ϑ_{s}^{2} (x, d) \equiv σ_{s}^{2} (x),

where

σ_{s}^{2} (x)

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:

<u_{q} u_{0}^{+}> = \frac{1}{x^{2}} e x p \{- M_{12} (x, d) + j ∆ S (x, z, z_{0})\} \cdot \{- \frac{2}{(1 + j g) (1 + \sqrt{ε_{g}} \tan (ψ))} + \frac{2 j g (1 + j β)}{k x {(1 + j g)}^{3} {(1 + \sqrt{ε_{g}} \tan (ψ))}^{3}} + \frac{6 ε_{g}^{2}}{{(k x)}^{2}} \frac{1}{{(1 + j g)}^{5} {(1 + \sqrt{ε_{g}} \tan (ψ))}^{5}}\} E q . 61

Parameter

β = 2 k x ϑ_{s}^{2}

determines the ratio of the width of the angular spectrum of the scattered field

ϑ_{s}^{2}

to the angular size of the Fresnel zone

θ_{F}^{2} = 1 / k x .

In the light of the validity of inequality (56) we have β ≪ 1. The parameter:

g = \frac{α}{(\frac{1}{\sqrt{ε_{g}}} + \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 / \sqrt{ε_{g}} + \tan (ψ) .

The correlation function of the impedance-source field u_q with reflected field

u_{0}^{-}

is calculated similarly to Eqs. (57)–(61). Assume that the following inequalities hold true:

m a x \{k l_{0} ≫ |\sqrt{ε_{g}}|, k l_{0} \tan (ψ) ≫ 1\} . E q . 62

In this case the trajectories determining the principal contribution to the trajectory integrals, u_q and

u_{0}^{-}

actually coincide, and for the structure function H(ρ) we can use the quadratic approximation

H (ρ) = C_{ε}^{2} ρ^{2} l_{0}^{- 1 / 3} .

Then

<u_{q} u_{0}^{- *}>

can be written as follows:

<u_{q} u_{0}^{- *}> = \frac{1}{x^{2}} \{- \frac{2}{(1 + \sqrt{ε_{g}} \tan (ψ))} + \frac{2 j ε (1 + j g_{s})}{k x {(1 + \sqrt{ε_{g}} \tan (ψ))}^{3}} + \frac{6 ε_{g}^{2}}{{(k x)}^{2}} \frac{{(1 + j g_{s})}^{2}}{{(1 + \sqrt{ε_{g}} \tan (ψ))}^{5}}\} E q . 63

where

g_{s}^{2} = σ_{s}^{2} (x) / θ_{F}^{2} .

The parameter

σ_{s}^{2} (x) = 0, 82 C_{ε}^{2} l_{0}^{- 1 / 3} x

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 <|u_q|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:

<{|u_{q}|}^{2}> = \frac{4 {|q|}^{2}}{x^{2}} \int_{0}^{\infty e^{j π / 4}} d u \int_{- 2 u}^{2 u} d v e x p \{j k \frac{ν^{2}}{2 x} - j ν (\frac{k u}{x} - R e (q) - k \tan (ψ)) + 2 ν I m (q) - k ν^{2} σ_{s}^{2}\} d v . E q . 64

Let us assume here and below that Im ε_g = 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 ε_g comes from displacement currents and Im ε_g ≪ Re ε_g. For large distances |qx| ≫ 1 we obtain:

<{|u_{q}|}^{2}> = \frac{1}{x^{2}} \{- \frac{4}{{(1 + \sqrt{ε_{g}} \tan (ψ))}^{2}} + \frac{8 ε_{g} σ_{s}^{2}}{{(1 + \sqrt{ε_{g}} \tan (ψ))}^{2}} \cdot (1 + \frac{1}{{(1 + \sqrt{ε_{g}} \tan (ψ))}^{2}}) - \frac{20 ε_{g}^{2}}{{(k x)}^{2}}\} . E q . 65

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

I (x, z, z_{0}) = \frac{1}{x^{2}} \{1 + {|R_{0}|}^{2} + 2 e x p [- M_{12} (x, d)] R e [R_{s} e^{j ∆ S}] + 2 e x p [- M_{12} (x, d)] \cdot R e \{e^{j ∆ S} [\frac{2 j ε_{g} (1 + j β)}{k x {(1 + j g)}^{3} {(1 + \sqrt{ε_{g}} \tan (ψ))}^{3}} + \frac{6 ε_{g}^{2}}{{(k x)}^{2}} \frac{1}{{(1 + j g)}^{5} {(1 + \sqrt{ε_{g}} \tan (ψ))}^{5}}]\} + \frac{12 ε_{g}^{2}}{(k x^{2}) {(1 + \sqrt{ε_{g}} \tan (ψ))}^{5}} + \frac{8 σ_{s}^{2} ε_{g}}{{(1 + \sqrt{ε_{g}} \tan (ψ))}^{2}} (1 - \frac{1}{1 + \sqrt{ε_{g}} \tan (ψ)} + \frac{1}{{(1 + \sqrt{ε_{g}} \tan (ψ))}^{2}}) - \frac{20 ε_{g}^{2}}{{(k x)}^{2}}\} E q . 66

Here:

R_{0} = \sqrt{ε_{g}} \tan (ψ) - 1 / \sqrt{ε_{g}} \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:

R_{s} = \frac{\sqrt{ε_{g}} \tan (ψ) - 1 + j g (\sqrt{ε_{g}} \tan (ψ) + 1)}{((\sqrt{ε_{g}} \tan (ψ) + 1)) (1 + j g)} E q . 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 / \sqrt{ε_{g}}), R_{0} = 0,

, and the coefficient R_s is defined by the ratio of the angular width of the scattered field to the width of the interference lobe.

R_{s} = \frac{j g}{(1 + j g)} \approx j \frac{ϑ_{s}^{2} (x, d)}{2 γ} \sqrt{ε_{g}} . E q . 68

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

k x \tan (ψ) \approx k d ≫ \sqrt{ε_{g}} E q . 69

we can drop the terms of the order of ε_g/kx and (ε_g/kx)². Then:

I (x, z, z_{0}) = \frac{1}{x^{2}} [1 + {|R_{0}|}^{2} + 2 R_{0} \cos (∆ S)] . E q . 70

For sliding angles with

\sqrt{ε_{g}} \tan (ψ) ≪ 1

and:

k d ≪ \sqrt{ε_{g}} E q . 71

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

I (x, z, z_{0}) = I (x) = \frac{1}{x^{2}} {|w (x)|}^{2} . E q . 72

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

Let us examine Eq. (66) in the presence of fluctuations of the refractive index,

δ ε (\vec{r}) \neq 0 .

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

ϑ_{s}^{2} (x, d)

at the base d to the angular width of the interference lobe γ:

M_{12} (x, d) = ϑ_{s}^{2} (x, d) / γ^{2} . E q . 73

Let us define three characteristic distances on the propagation path: x_s, x_g and x_c from the following relationships:

g (x_{2}, d) = 1, M_{12} (x_{c}, d) = 1, σ_{s}^{2} (x_{s}) = 1 . E q . 74

Then:

x_{s} = k^{- 1 / 2} C_{ε}^{- 1} l_{0}^{1 / 8}, x_{g} = {[k d^{2 / 3} C_{ε}^{2} \sqrt{ε_{g}}]}^{- 1}, x_{c} = {[k^{2} d^{5 / 3} C_{ε}^{2}]}^{- 1} . E q . 75

One finds that x_c 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; x_s is the distance where the variance of the fluctuation of the angle of incidence

σ_{s}^{2}

becomes equal to the square of the angular size of the Fresnel zone

θ_{F}^{2};

and x_g 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) / \sqrt{ε_{g}} .

Because of the validity of inequality (56), the range of distances under consideration is such that x < x_s. To analyse the value of the parameters we will define the ratios:

\frac{x_{s}}{x_{g}} = k^{1 / 2} d^{2 / 3} C_{ε} \sqrt{ε_{g}} l_{0}^{1 / 6}, E q . 76

\frac{x_{s}}{x_{c}} = k^{3 / 2} d^{5 / 3} C_{ε} \sqrt{ε_{g}} l_{0}^{1 / 6}, E q . 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

C_{ε}^{2} = 10^{- 14} c m^{- 2 / 3}, |ε_{g}| \approx 10^{2},

the ratio x_s/x_g ≪ 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 x_c/x_s < 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 < x_s.

Let us assume that x_c > x_s, in this case M₁₂(x, d) < 1, g ≪ min

\{1, \sqrt{ε_{g}} \tan (ψ)\},

and R_s ≈ R₀. In the interference region (inequality (69)) the field intensity can be approximated as follows:

I (x, z, z_{0}) = \frac{1}{x^{2}} [1 + {|R_{0}|}^{2} + 2 R_{0} \cos (∆ S) e x p [- M_{12} (x, d)]] . E q . 78

In the case where

\sqrt{ε_{g}} \tan (ψ) ≪ 1,

when inequality (71) holds, R_s ≈ R₀ ≈ -1. In this case there are two small parameters M₁₂(x, d) and ε_g/kx. Let us introduce yet another characteristic distance x_M, where M12(x_M, d) = ε_g/kx_M:

x_{M}^{2} = \frac{ε_{η}}{k^{3} C_{ε}^{2} d^{5 / 3}} . E q . 79

When inequality (62) holds, x_M < x_s. Assuming x < x_M 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:

I (x, z, z_{0}) = \frac{1}{x^{2}} [{|w (x)|}^{2} + 8 ε_{g} σ_{s}^{2} (x)] . E q . 80

In the range of distances x_s < x < x_M, the main contribution to the sliding angle propagation mechanism is provided by scattering on the non-uniformity of the refractive index,

δ ε (\vec{r}),

and the field intensity in that region is given by:

I (x, z, z_{0}) = \frac{1}{x^{2}} [{(k d)}^{2} ϑ_{s}^{2} (x, d) + 8 ε_{g} σ_{s}^{2} (x)] . E q . 81

In the regime of non-coherent composition of the direct and reflected waves we have M₁₂(x, d) ≫ 1, and x_c < x_s. 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:

I (x, z, z_{0}) = \frac{1}{x^{2}} [{|w (x)|}^{2} + 8 σ_{s}^{2} (x)] . E q . 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

<u_{q} u_{0}^{- *}>

and

<{|u_{q}|}^{2}> .

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:

J (x, z, z_{0}) = \sum_{n = 1}^{N} \sum_{m = 1}^{N} A_{n} A_{m}^{*} e x p [j k (S_{n} - S_{m}) - M_{n m}] E q . 83

where A_n, S_n are the amplitude and phase of the nth ray, M_nm is a structure function of the phase difference of the phases along the ray’s trajectories:

M_{n m} = 0, 73 C_{ε}^{2} k^{2} \int_{0}^{x} d s {|r_{n} (s) - r_{m} (s)|}^{5 / 3} E q . 84

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

S_{n} = j \frac{k}{2} \int_{0}^{x} d s [\frac{d^{2} r_{n}}{d s^{2}} + ε_{m} (r_{n} (s))] E q . 85

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

\frac{d^{2} r_{n}}{d s^{2}} = \frac{d ε_{m}}{d r_{n}} E q . 86

with boundary conditions:

r_{n} (0) = z_{0}, r_{n} (x) = z . E q . 87

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

A_{n} = A_{n} (x) = {[\frac{D_{n} (0)}{D_{n} (x)}]}^{1 / 2} E q . 88

where D_n(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 A_n is modulated by a reflection coefficient:

A_{n} = A_{n} (x) \cdot R (θ_{n}) E q . 89

where θ_n 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

Olga Nesvetailova

Freelancer

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Footnotes