Radio Science

On the solution of Markov's parabolic equation for the second-order spaced frequency and position coherence function

Authors


Abstract

[1] An analytic technique has been developed to construct the asymptotic representation of the spaced position and frequency coherence function in Markov's diffusive approximation. The technique employs the formalism of the quasi-classic complex paths in an extended complex-valued coordinate space. It allows the construction of the coherency for arbitrary realistic models of the structure function of the fluctuations of the refractive index of the medium of propagation. The technique has been employed to obtain explicit analytic asymptotic solutions for some realistic models of the structure function. For the quadratic structure function the method produces the known rigorous solution in an automatic fashion.

1. Introduction

[2] The technique of Markov's parabolic equations for the moments of a stochastic field is one of the classical approaches in the problem of wave propagation in random media [Rytov et al., 1978; Ishimaru, 1978]. Despite many years of exploitation of the method, there still exist unsolved problems in the scope of Markov's approximation. In particular, a general analytic solution has not yet been constructed for the fourth moment, and some questions arise with respect to the two-frequency, two-position coherence function.

[3] As far as the second-order coherence function is concerned, the exhaustive solution has been constructed for the single-frequency (pure spatial) coherence function [Rytov et al., 1978]. The spaced position and frequency coherence function was studied numerically by Lin and Yeh [1975], who investigated this function in a fading ionospheric communication channel numerically for power-law and Gaussian spectra of ionospheric fluctuations. For the two-frequency coherence function the closed form exact analytic solution was only constructed in the case of a plane wave propagating in a random medium with the quadratic structure function of fluctuations [Sreenivasiah et al., 1976]. Knepp [1983] generalized this solution to the case of spherical wave propagation. The technique of separation of variables was employed by Oz and Heyman [1996, 1997a, 1997b, 1997c] to construct the second-order coherence function for an arbitrary structure function of the fluctuations of the properties, but in a medium with a homogeneous background. It implies the expansion of the solution into the series of the transversal eigenfunctions of the problem, and, in this way, requires additional quantification of the number of terms needed to achieve the necessary accuracy for a given distance of propagation and spaced frequency. The series fails to converge for some types of initial conditions (e.g., an incident plane wave) as the distance and difference frequency tend to zero. Additional constraints in the separation variables technique may also arise when considering fluctuations with the structure function tending to a constant as the difference variable tends to infinity. In this case the continuous spectrum may likely occur in the spectrum of the transversal operator of the problem, which makes expansion of the solution in terms of the transversal eigenfunctions much more complicated.

[4] In the present paper the asymptotic technique is developed to construct the solution to Markov's parabolic equation for the two-frequency, two-position coherence function in the case of an arbitrary structure function of the refractive index of fluctuations and an inhomogeneous stratified background medium. The technique employs quasi-classic representation in terms of complex trajectories. It has no constraints pertinent to the initial conditions in the form of an incident plane wave, and is also valid beginning with the zero distance from a boundary surface and zero spaced frequency. In the case of the quadratic structure function it produces the known exact solution of the problem [Sreenivasiah et al., 1976] in automatic fashion.

2. Statement of the Problem

[5] We consider the background medium, stratified in z direction, with local random inhomogeneities embedded, which is characterized by the dielectric permittivity of the form

equation image

Here ε0(z, ω) is the dielectric permittivity of the background medium, and ε1(r, ω, t) represents relative space and time varying fluctuations of the permittivity. Variable t indicates time dependence of fluctuations in quasi-stationary approximation.

[6] We consider an incident plane wave propagating in the positive z direction. The field random realization is searched for in the following form:

equation image

provided that ε0(z, ω) is finite and not equal zero at any z, ω. In equation (2)k = ω/c is the vacuum wave number corresponding to the circle frequency ω, and U(z, ρ, t, ω) is the random complex amplitude of the field. The two-frequency, two-position, two-time coherence function of the field is defined as

equation image

Employing a standard averaging procedure of Markov's technique for a medium described by equation (1) results in the following parabolic equation for the coherence function Γ:

equation image

Here k1 and k2 are the vacuum wave numbers corresponding to the frequencies ω1 and ω2 in the background medium, and ∇12 and ∇22 are the Laplacians with respect to coordinates ρ1 and ρ2. Equation (4) was derived in the approximation of the fluctuations delta-correlated in the direction of propagation. This means that functions Amn are determined through the following relationship for the correlation functions of the relative dielectric permittivity:

equation image

m, n = 1,2, so that

equation image

Here z = (z1 + z2)/2, z′ = z1z2 are longitudinal central and difference variables.

[7] Introducing also central and difference transversal variables R = (ρ1 + ρ2)/2, ρ = ρ1ρ2, as well as difference time t = t1t2 in the assumption of the stationarity of fluctuations in time, and substituting

equation image

results in the following equation for function Γ1

equation image

with

equation image

s and ∇d are the operators of gradient with respect to the sum and difference coordinates R and ρ.

[8] Until now, the assumption of the statistical homogeneity of the fluctuations of the dielectric permittivity in space was not implied. When introducing fluctuations of the dielectric permittivity in the form of a product in the second item in equation (1) (e.g., as given by Sreenivasiah et al. [1976]), it is convenient to consider the relative fluctuations of the dielectric permittivity ε1 as a stationary and homogeneous zero mean random function. In the case where the background medium is homogeneous and nondispersive and the relative fluctuations are statistically homogeneous, equations (4) and (8) become the same as those treated by Sreenivasiah et al. [1976] and Oz and Heyman [1996, 1997a, 1997b, 1997c]. Alternatively, for the cold ionospheric plasma the statistically homogeneous relative fluctuations of the electron density are introduced (e.g., as given by Lin and Yeh [1975] or Knepp [1983]) instead of homogeneous fluctuations of the relative dielectric permittivity. In this case, equations (4) and (8) lead to those considered by Knepp [1983]. We shall follow our formulation of the problem (equations (2), (4), (7), (8)).

[9] To complete the statement of the problem, equation (8) should be complemented by the boundary condition at z = 0

equation image

which is determined by the incident field.

3. Solution to the Plane Wave

[10] Here we confine the consideration by the case where the incident field is a plane wave propagating in the z direction. In this case, Γ0 = 1, and equation (8) can be simplified. Indeed, the coefficients and boundary conditions do not depend on the central variable R; therefore the solution Γ1 should not depend on R, and a simplified equation can be considered as follows:

equation image

When considering this simplified equation, the technique of solving based on quasi-classic representation in terms of complex trajectories can be most transparently outlined. The extension of the technique to the general case of a full equation (8) and the incident field not necessarily propagating along the z axis is the next step that will be considered separately.

[11] To construct the asymptotic solution to equation (10), first, the dimensionless variables should be introduced. Let us denote as lε the scale of the random inhomogeneities in the z direction and substitute z = ζlε and ρ = rlε. In new dimensionless variables (ζ, r) the last equation can be rewritten as follows:

equation image

where equation image and in the equation (6) for Amn transfer to the dimensionless variables was performed, equation image. ∇r2 is the transversal Laplacian written in the dimensionless difference variables, and K = k1k2lε2 is the dimensionless parameter, which is assumed to be the large parameter of the problem. Physically, this means that random inhomogeneities of a medium are of large spatial scale in terms of vacuum wavelengths for both frequencies ω1 and ω2. At the same time this is one of the limitations of Markov's diffusive approximation.

[12] Formally, provided K → ∞, the solution of equation (11) is sought for in the form of the following asymptotic series:

equation image

The dependencies of the functions newly introduced here, ψ and Un, on frequencies ω1, ω2 are not indicated explicitly in representation (12).

[13] Series (12) is almost the traditional Debye series for constructing the high-frequency asymptotic. The distinction is that the exponent function does not obey the imaginary unity i in its power and that the expansion is carried out into inverse powers of the real parameter K rather than in powers of (iK). The reason to do so is that in the case of a single frequency the equation (11) for the pure space coherency, evidently, has the solution in the form of a real exponential function (it will be shown that in this case U0 = 1, Uj = 0, j > 0). Additionally, in the general case of a spaced frequency we shall be dealing with complex “eikonals” ψ, so that it is no matter whether or not it is introduced with i in the exponent's power in (12).

[14] The standard asymptotic procedure of substituting (12) into equation (11) results in the following “eikonal” equation for ψ

equation image

and transport equations for amplitudes Uj

equation image
equation image

[15] To solve equations (13), (14), and (15), the general method of characteristics can be employed. Equation (13) is a Hamilton-Jacobi type equation, so that the appropriate Hamilton equations may be written in the following form:

equation image
equation image
equation image
equation image

In this set, three-dimensional vectors (ζ, r) and (pζ, p) are coordinates and moments, respectively, along the complex trajectory, which is parameterized by variable τ. Due to equation (16) it may be accepted τ = ζ, where ζ is a real variable.

[16] Equations (16)(19) determine complex trajectories r = r(ζ), pζ = pζ(ζ), p = p(ζ), which arrive at real points of observation (ζ, r) and are subject to the initial conditions (from Γ0 = 1 on the initial surface ζ = 0):

equation image
equation image

Value r(0) = r0 is a complex coordinate of the point in the initial plane ζ = 0 that should be determined to provide a complex trajectory to arrive at the real point of observation (ζ, r).

[17] Once equations (16)(19) with proper initial conditions (20) and (21) have been solved and complex trajectories have been determined, the function ψ in representation (12) can then be calculated as the integral

equation image

As for solving the transport equations (14) and (15), the solution of the main equation is given by

equation image

where Jacobian J(ζ) is calculated as follows:

equation image

with r = (x1, x2) and r0 = (x01, x02). Coordinates r0 are two-dimensional orthogonal complex trajectories transversal to the ζ axis; J(0) = 1. To calculate the determinant in equation (24), quantities αlk = ∂x1/∂x0k should be calculated directly after the ray equations have been solved, or additional differential equations for αlk may be obtained. In particular, in the case of the homogeneous background medium, the following equations may be derived for αlk differentiating (17) in variables r0 = (x01, x02) and making use of (19):

equation image

l, k, q = 1,2. In this way, the general scheme of constructing the asymptotic solution of Markov's parabolic equation (11) in terms of the representation (12) is completed. It is clear that this scheme in its most general form, when the background medium is inhomogeneous and dispersive, can only be realized numerically. When doing this, some general problems arise. In particular, when solving ray equations (17)(19), the homing problem to construct the complex trajectories will be the crucial point. Additionally, the problem of the analytic continuation of the correlation function of fluctuations into the complex domain of its argument is also a nontrivial one. It should be considered independently for each accepted model of the correlation function of fluctuations.

[18] To demonstrate more transparently how the developed technique works, below the problem is discussed under the simplifying assumption of a homogeneous background medium characterized by ε0 = 1. This is just the problem treated by Sreenivasiah et al. [1976] and Oz and Heyman [1996, 1997a, 1997b]. Here we can proceed further analytically.

4. Plane Wave Propagating in a Homogeneous Background Medium With Fluctuations

[19] In this case, equation image does not depend on ζ and, according to equations (13) and (18), equation image (equation image, if a medium is nondispersive). Provided, additionally, that the fluctuations are isotropic in the transversal planes equation image and cylindrical coordinates are introduced to express vector p, the explicit relationship for a complex trajectory can be written (ray or Hamilton equations can be solved analytically) as follows:

equation image

At the same time, equation (26) is the transcendent equation, which allows us to find the initial complex point r0 = r0(ζ, r), where the complex trajectory comes out at ζ = 0, to arrive at the real point of observation (ζ, r).

[20] Once r0 was found from equation (26), function ψ in the asymptotic representation (12) is expressed through

equation image

When calculating the main amplitude Uo according to equations (23)(25) after introducing cylindrical coordinates for isotropic fluctuations, equation (25) is reduced to a simpler form

equation image

with α = ∂r/∂r0. Then instead of a general set of equations (25), it is now one equation for the divergence α, as follows:

equation image

This way, the general scheme of constructing the asymptotic solution to the second-order Markov's equation for the space-frequency coherency has now, in the conditions of a homogeneous background medium with isotropic fluctuations, been reduced to a very simple procedure of considering equations (26)(29). This allows the construction of the asymptotic solution for an arbitrary given correlation (or structure) function of fluctuations.

[21] Some results may even be obtained without specifying a model of the structure function of fluctuations. In particular, for the case of pure frequency coherency (no spaced position and time, r = 0, t = 0) equation (26) yields r = r0; therefore ψ = 0, as follows from equation (27). Finally, equation (29) for the divergence α = αω is simplified to the form

equation image

so that

equation image

and the main term of the high-frequency asymptotic solution for the two-frequency coherence function is given by

equation image

This is quite a fundamental result. Actually, it is straightforward to obtain the two-frequency coherence function in the approximation of geometrical optics, which only describes the case of weak fluctuations (unsaturated regime). The second factor in (32) just stands for the geometrical optics approximation. A nontrivial thing is the derivation of the first factor, which may only account for a possible saturated regime of propagation (strong fluctuations). It was first obtained when the strict coherence function was constructed for the parabolic structure function of fluctuations [Sreenivasiah et al., 1976]. The same type of results were also obtained for the quadratic structure function in the scope the technique of path integrals (functional integrals) [Dashen, 1979; Flatte, 1983]. We have obtained (32) for an arbitrary model of the structure function. This was just a particular result of the general theory being developed here.

[22] It should also be shown how the technique works to produce well-known results for a single-frequency case equation imaged = 0. In this case, according to equation (17), r0 = r, and according to (28) and (29), J = 1. As to the main amplitude U0 in representation (12), this results in U0 = const. As far as the higher-order amplitudes Uj, (j = 1,2…) are concerned, Uj = 0, (j = 1,2…) may be chosen because of the zero initial conditions and homogeneous equations (15) in the case of U0 = const. Finally, integrating equation (13) or (27) along straight lines r0 = r yields the known solution to the single-frequency problem (see equation (45.20) from Rytov et al. [1978]).

[23] Below the general case of a spaced frequency and position coherency in the medium with a structure function of fluctuations other than quadratic will be considered. Before doing this, it will be shown in the next subsection how equations (26)(29) produce the exact solution for the space-frequency coherence function in the case of the quadratic structure function of fluctuations obtained by Sreenivasiah et al. [1976].

4.1. Quadratic Structure Function

[24] For the structure function of the form

equation image

integral in equation (26) is calculated analytically to yield

equation image

This expression explicitly relates the initial complex point r0 (at ζ = 0) and a real point of observation (ζ, r). Alternatively, when r0 was determined for a given real point (ζ, r), the equation explicitly describes a complex trajectory arriving at a real given point in the form r = r(ζ, r0).

[25] Once a complex trajectory has been defined by (34), integration in equation (27) along this complex ray gives the following complex eikonal:

equation image

Equation (29) for the divergence becomes of the form

equation image

which allows the explicit analytic solution, satisfying necessary initial conditions, as follows:

equation image

Finally, putting together relationships (7), (12), (23), (28), (34), (35), and (37), the following representation for the space-frequency coherence function U0 can be written

equation image

In corresponding notations, this is exactly the function derived by Sreenivasiah et al. [1976]. It is worth pointing out that in our technique the higher-order transport equations (15) give identically zero solutions for the quadratic structure function of fluctuations (33). As a result, in this case the asymptotic theory produces the rigorous solution.

4.2. Structure Function of Fluctuations Other Than Quadratic

[26] We have developed the method of constructing the space-frequency coherency, enabling investigation of the problem with the models of the structure function of fluctuations more realistic than those considered in the literature. Namely, both the quadratic structure function [Sreenivasiah et al., 1976] and the structure function of the form rν, investigated by Oz and Heyman [1996, 1997a, 1997b] do not allow limiting transition to the case of moderate or weak fluctuations, at least, in the case of spaced position. Both tending to infinity as r → ∞ are good models in the case of strong fluctuations, whereas for the opposite case the structure function should tend to a positive constant as r → ∞. As already mentioned in section 1, the method of separation of variables employed by Oz and Heyman [1996, 1997a, 1997b] is also formally valid for this type of structure function. However, it will likely face additional constraints, stipulated by a more complicated structure of the spectrum of the transversal operator of the problem. When the structure function is a constant at infinity, the continuous spectrum may likely occur in the spectrum of a transversal operator. This makes expansion of the solution in terms of the transversal eigenfunctions much more complicated. We use another type representation of the coherence function, and our approach is free of the mentioned difficulties.

[27] The central point of the paper is to demonstrate how the technique developed works for realistic structure functions of fluctuations others than quadratic and, in this way, to investigate both cases of weak and strong fluctuations together. Some models of the structure function (correlation function) allow explicit evaluation or direct calculation of the integrals in equations (26)(28). Below we shall consider two model cases. Before doing this, it should be pointed out that the key point in realizing the developed technique is the analytic continuation of the model correlation function into the complex domain of its argument. As was already mentioned, for each model chosen this should be particularly discussed, but any model should possess the property of positive Fourier spectrum as representing the energy spectrum of fluctuations.

4.2.1. Inverse Power Law Correlation Function

[28] This is the case where the correlation function of fluctuations is modeled by

equation image

Recalling the definition of the dimensionless variable r = ρ/lε, (39) means that the correlation function, in some sense, has the effective spatial scale lεα−1 in the plane perpendicular to the z axis. In particular, in the case of 3-D isotropic fluctuations, one obtains α = 1. The variance of fluctuations is σε2. Model (39) is smooth at r = 0 in the sense that equation image, if ν > 1. Additionally, parameter ν should be chosen to provide the property of positive Fourier spectrum of the correlation function.

[29] The structure function

equation image

corresponds to the correlation function (39). As r increases, equation image tends to the positive constant 2σε2 (if ν > 1).

[30] For the correlation function (39), equation (26) has the form

equation image

The integral in the last equation can be expressed asymptotically through the full hyper-geometric function with different sets of parameters for the cases β ≪ 1 and β ≫ 1. The comprehensive analysis of equation (41) for an arbitrary ν is not trivial. In particular, when speaking of the analytic continuation of the function (39) into the complex domain r, it should be noted that, on the one hand, it is straightforward, because the function is given explicitly, and the complex-valued r can be employed. On the other hand, all the singularities of this analytic function should be carefully accounted for, which are the appropriate poles and possible cut at r = 0, and the manifold Riemann surface should be properly introduced [see, e.g., Shabat, 1976] with a finite, or, possibly, infinite number of folds (if ν is an irrational fraction).

[31] However, the transparent explicit asymptotic results may be obtained in the particular case of ν = 2. For this case the spatial spectrum of (39) can be easily calculated to be positive. As far as its analytical continuation into the complex r-domain is concerned, it is defined on a single-fold Riemann surface and it is the even function of its argument on the real axis. It has no branch points, but only two poles at r = ±iα−1, α > 0. The circle |r| = α−1 separates the complex plane r into two domains |r| < α−1 and |r| > α−1. The appropriate Loran series in the domain |r| < α−1 coincides with the Tailor series of function (39) as follows:

equation image

In the domain |r| > α−1, the Loran series for (39) is given by the expansion into inverse powers of r:

equation image

The explicit formula (39) just shows how the series for |r| < α−1 should be analytically continued into the domain |r| > α−1, and vice versa.

[32] Employing series (42) for a small spaced position results in the coherence function, coinciding with that for the quadratic model of the structure function. When using the second series (43) for the large spaced position in equations (26)(28) and (41), this yields the following explicit asymptotic form of the complex path

equation image

and the following coherence function [Bitjukov et al., 2001]

equation image

where r0(ζ, r) is given by equation (44). Equations (44) and (45) describe the coherency for all spaced frequencies and large transversal spaced position r. In contrast to (38), where in the case of equation imaged = 0 and any finite ζ function Γ(ζ, r) → 0 as r → ∞, coherency (equations (44) and (45)) yields in the single-frequency case

equation image

This is exactly what the single-frequency two-position coherence function should be [see Rytov et al., 1978; equation (45.20)] as r → ∞ in the case of the structure function of fluctuations (40) with ν = 2. For finite ζ it gives a nonzero constant as r tends to infinity. The value of the constant depends on the intensity of fluctuations of dielectric permittivity σε2.

4.2.2. Exponential Correlation Function

[33] Finally, one more model of the correlation function of fluctuations, allowing analytic assessment, is the exponential correlation function as follows:

equation image

Here the consideration is confined by the isotropic model of fluctuations with the correlation radius lε (recall r = ρ/lε). The spatial spectrum of (47) is positive. When formally continuing (47) evenly to the negative r, equation image(r) = σε2er if r < 0 should be accepted. Therefore, this model has the finite-step first derivative at r = 0 and turns out to be nonanalytic. This results in the fact that the analytic continuation is not possible in the vicinity of r = 0, and no proper solution for the coherency can be constructed for small spaced positions, including the pure frequency coherence function, for the model (47).

[34] On the other hand, when considering the large positive r the analytic continuation of (47) can be performed into a limited complex domain around the real axis for large r, which provides the explicit form to equation (26) as follows:

equation image

This is valid for large r and any equation imaged, so that the coherency can be constructed for the large value of the spaced position and any value of the mistuning frequency equation imaged. This is of importance in the case of the inverse power law spatial spectrum of fluctuations, when the spatial correlation function, expressed through the modified Bessel function, has the exponential asymptotic at large r.

[35] Employing (48), the appropriate analysis of equations (27) and (28) can then be performed and the representation for the spaced position (large r) and frequency (any equation imaged) coherence function can be written. It is fairly space consuming and is not presented here, but in the limiting case of equation imaged = 0 (i.e. for a single-frequency spaced position coherency) it yields the following expected [Rytov et al., 1978, equation (45.20)] rigorous result:

equation image

If comparing (46) and (49), both functions tend to the same constant when the spaced position increases, but the rate of decay is different as the different models were employed in these two cases.

[36] To conclude this section, for both power law (39) with ν = 2 and exponential model (47) of the correlation function of fluctuations considered here, the general solutions constructed for the space-frequency coherency in the limiting case of single frequency (pure spatial coherency) produce in automatic fashion the results, which are in agreement with the general theory of the pure spatial coherence functions [Rytov et al., 1978].

5. Conclusion

[37] An analytic technique has been developed to construct the asymptotic representation of the two-position, two-frequency coherence function in Markov's diffusive approximation. The technique employs the formalism of the quasi-classic complex paths. It allows the construction of the coherency for a wide range of realistic models of the structure function of fluctuations of dielectric permittivity of the medium of propagation, which tend to a finite value as the spaced transversal variable tends to infinity. For some models the final result can even be achieved analytically; others need numerical calculations. For the quadratic structure function the method produces the known rigorous solution in an automatic fashion.

Acknowledgments

[38] The work was performed under the financial support of the Russian Ministry for Education, grant E00-3.5-138, and the UK EPSRC Visiting Fellowships (V.E.G. and N.N.Z.) GR/R37517/01.

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