Klyatskin V.I. Lectures on Dynamics of Stochastic Systems
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356 Lectures on Dynamics of Stochastic Systems
plays here the role of time), i.e., its solution satisfies the relationship
δu(x, R)
δε(x
0
, R
0
)
= 0 for x
0
< 0, x
0
> x. (13.3)
The variational derivative at x
0
→ x − 0 can be obtained according to the standard
procedure,
δu(x, R)
δε(x − 0, R
0
)
=
ik
2
δ
R − R
0
u(x, R). (13.4)
Consider now the statistical description of the wavefield. We will assume that ran-
dom field ε(x, R) is the homogeneous and isotropic Gaussian field with the parameters
h
ε(x, R)
i
= 0, B
ε
(x −x
0
, R − R
0
) =
ε(x, R)ε(x
0
, R
0
)
.
As was noted, field u(x, R) depends functionally only on preceding values of field
ε(x, R). Nevertheless, statistically, field u(x, R) can depend on subsequent values
ε(x
1
, R) for x
1
> x due to nonzero correlation between values ε(x
0
, R
0
) for x
0
< x and
values ε(ξ, R) for ξ > x. It is clear that correlation of field u(x, R) with subsequent val-
ues ε(x
0
, R
0
) is appreciable only if x
0
− x ∼ l
k
, where l
k
is the longitudinal correlation
radius of field ε(x, R). At the same time, the characteristic correlation radius of field
u(x, R) in the longitudinal direction is estimated approximately as x (see, e.g., [33,34]).
Therefore, the problem under consideration has small parameter l
k
/x, and we can use
it to construct an approximate solution.
In the first approximation, we can set l
k
/x → 0. In this case, field values u(ξ
i
, R)
for ξ
i
< x will be independent of field values ε(η
j
, R) for η
j
> x not only functionally,
but also statistically. This is equivalent to approximating the correlation function of
field ε(x, R) by the delta function of longitudinal coordinate, i.e., to the replacement
of the correlation function B
ε
(x, R) whose three-dimensional spectral function is
8
ε
(q
1
, q) =
1
(
2π
)
3
∞
Z
−∞
dx
Z
dR B
ε
(x, R)e
−iq
1
x−iqR
(13.5)
with the effective function B
eff
ε
(x, R),
B
ε
(x, R) = B
eff
ε
(x, R) = δ(x)A(R), A(R) =
∞
Z
−∞
dxB
ε
(x, R) (13.6)
Using this approximation, we derive the equations for moment functions
M
mn
(x;R
1
, . . . , R
m
;R
0
1
, . . . , R
0
n
) =
*
m
Y
p=1
n
Y
q=1
u
x;R
p
u
∗
x;R
0
q
+
. (13.7)
In the case of m =n, these functions are usually called the coherence functions of
order 2n.
Caustic Structure of Wavefield in Random Media 357
Differentiating function (13.7) with respect to x and using Eq. (13.1) and its com-
plex conjugated version, we obtain the equation
∂
∂x
M
mn
(x;R
1
, . . . , R
m
;R
0
1
, . . . , R
0
n
)
=
i
2k
m
X
p=1
1
R
p
−
n
X
q=1
1
R
0
q
M
mn
(x;R
1
, . . . , R
m
;R
0
1
, . . . , R
0
n
)
+ i
k
2
*
m
X
p=1
ε(x, R
p
) −
n
X
q=1
ε(x, R
0
q
)
m
Y
p=1
n
Y
q=1
u
x;R
p
u
∗
x;R
q
+
.
(13.8)
To split the correlator in the right-hand side of Eq. (13.8), we use the Furutsu–
Novikov formula together with the delta-correlated approximation of medium para-
meter fluctuations with effective correlation function (13.6) to obtain the following
relationships
ε(x, R)u
x;R
p
=
1
2
Z
dR
0
A(R − R
0
)
*
δu
x;R
p
δε(x − 0, R
0
)
+
,
D
ε(x, R)u
∗
x;R
0
q
E
=
1
2
Z
dR
0
A(R − R
0
)
*
δu
∗
x;R
0
q
δε(x − 0, R
0
)
+
.
(13.9)
If we additionally take into account Eq. (13.4) and its complex conjugated version,
then we arrive at the closed equation for the wavefield moment function
∂
∂x
M
mn
(x;R
1
, . . . , R
m
;R
0
1
, . . . , R
0
n
)
=
i
2k
m
X
p=1
1
R
p
−
n
X
q=1
1
R
0
q
M
mn
(x;R
1
, . . . , R
m
;R
0
1
, . . . , R
0
n
)
−
k
2
8
Q(R
1
, . . . , R
m
;R
0
1
, . . . , R
0
n
)M
mn
(x;R
1
, . . . , R
m
;R
0
1
, . . . , R
0
n
),
(13.10)
where
Q(R
1
, . . . , R
m
;R
0
1
, . . . , R
0
n
) =
m
X
i=1
m
X
j=1
A
R
i
− R
j
− 2
m
X
i=1
n
X
j=1
A
R
i
− R
0
j
+
n
X
i=1
n
X
j=1
A
R
0
i
− R
0
j
. (13.11)
358 Lectures on Dynamics of Stochastic Systems
An equation for the characteristic functional of random field u(x, R) also can be
obtained; however it will be the linear variational derivative equation.
Draw explicitly equations for average field
h
u(x, R)
i
, second-order coherence
function
0
2
(x, R, R
0
) =
γ
2
(x, R, R
0
)
, γ
2
(x, R, R
0
) = u(x, R)u
∗
(x, R
0
),
and fourth-order coherence function
0
4
(x, R
1
, R
2
, R
0
1
R
0
2
) =
u(x, R
1
)u
(
x, R
2
)
u
∗
(x, R
0
1
)u
∗
(x, R
0
2
)
,
which follow from Eqs. (13.10) and (13.1) for m = 1, n = 0; m = n = 1; and
m = n = 2. They have the forms
∂
∂x
h
u(x, R)
i
=
i
2k
1
R
h
u(x, R)
i
−
k
2
8
A(0)
h
u(x, R)
i
,
h
u(0, R)
i
= u
0
(R), (13.12)
∂
∂x
0
2
(x, R, R
0
) =
i
2k
(
1
R
− 1
R
0
)
0
2
(x, R, R
0
) −
k
2
4
D(R − R
1
)0
2
(x, R, R
0
),
0
2
(0, R, R
0
) = u
0
(R)u
∗
0
(R
0
), (13.13)
∂
∂x
0
4
(x, R
1
, R
2
, R
0
1
, R
0
2
)
=
i
2k
1
R
1
+ 1
R
2
− 1
R
0
1
− 1
R
0
2
0
4
(x, R
1
, R
2
, R
0
1
, R
0
2
)
−
k
2
8
Q(R
1
, R
2
, R
0
1
, R
0
2
)0
4
(x, R
1
, R
2
, R
0
1
, R
0
2
),
0
4
(0, R
1
, R
2
, R
0
1
, R
0
2
) = u
0
(R
1
)u
0
(
R
2
)
u
∗
0
(R
0
1
)u
∗
0
(R
0
2
), (13.14)
where we introduced new functions
D(R) = A(0) − A(R),
Q(R
1
, R
2
, R
0
1
, R
0
2
) = D(R
1
− R
0
1
) + D(R
2
− R
0
2
) + D(R
1
− R
0
2
)
+ D(R
2
− R
0
1
) − D(R
2
− R
1
) − D(R
0
2
− R
0
1
)
related to the structure function of random field ε(x, R).
Introducing new variables
R → R +
1
2
ρ, R
0
→ R −
1
2
ρ,
Caustic Structure of Wavefield in Random Media 359
Eq. (13.13) can be rewritten in the form
∂
∂x
−
i
k
∇
R
∇
ρ
0
2
(x, R, ρ) = −
k
2
4
D(ρ)0
2
(x, R, ρ),
0
2
(0, R, ρ) = γ
0
(R, ρ) = u
0
R +
1
2
ρ
u
∗
0
R
0
−
1
2
ρ
.
(13.15)
Equations (13.12) and (13.15) can be easily solved for arbitrary function D(ρ) and
arbitrary initial conditions. Indeed, the average field is given by the expression
h
u(x, R)
i
= u
0
(x, R)e
−
γ
2
x
, (13.16)
where u
0
(x, R) is the solution to the problem with absent fluctuations of medium para-
meters,
u
0
(x, R) =
Z
dR
0
g(x, R − R
0
)u
0
(R
0
).
The function g(x, R) is free space Green’s function for x > x
0
g
x, R;x
0
, R
0
= exp
i(x −x
0
)
2k
1
R
δ
R − R
0
=
k
2πi(x − x
0
)
exp
ik
R − R
0
2
2(x −x
0
)
!
(13.17)
and quantity γ =
k
2
4
A(0) is the extinction coefficient.
Correspondingly, the second-order coherence function is given by the expression
0
2
(x, R, ρ) =
Z
dqγ
0
q, ρ − q
x
k
exp
iqR −
k
2
4
x
Z
0
dξD
ρ − q
ξ
k
, (13.18)
where
γ
0
(
q, ρ
)
=
1
(
2π
)
2
Z
dRγ
0
(R, ρ)e
−iqR
.
The further analysis depends on the initial conditions to Eq. (13.1) and the fluctu-
ation nature of field ε(x, R). Three types of initial data are usually used in practice.
They are:
l
plane incident wave, in which case u
0
(R) = u
0
;
l
spherical divergent wave, in which case u
0
(R) = δ(R); and
360 Lectures on Dynamics of Stochastic Systems
l
incident wave beam with the initial field distribution
u
0
(R) = u
0
exp
(
−
R
2
2a
2
+ i
kR
2
2F
)
, (13.19)
where a is the effective beam width, F is the distance to the radiation center (in the case of
free space, value F = ∞ corresponds to the collimated beam and value F < 0 corresponds
to the beam focused at distance x = |F|).
In the case of the plane incident wave, we have
u
0
(R) = u
0
= const, γ
0
(R, ρ) = |u
0
|
2
, γ
0
(
q, ρ
)
= |u
0
|
2
δ(q),
and Eqs. (13.16) and (13.18) become significantly simpler
h
u(x, R)
i
= u
0
e
−
1
2
γ x
, 0
2
(x, R, ρ) = |u
0
|
2
e
−
1
4
k
2
xD
(
ρ
)
(13.20)
and appear to be independent of plane wave diffraction in random medium. Moreover,
the expression for the coherence function shows the appearance of new statistical scale
ρ
coh
defined by the condition
1
4
k
2
xD
ρ
coh
= 1. (13.21)
This scale is called the coherence radius of field u(x, R). Its value depends on the
wavelength, distance the wave travels in the medium, and medium statistical parame-
ters.
In the case of the wave beam (13.19), we easily obtain that
γ
0
(q, ρ) =
|u
0
|
2
a
2
4π
exp
(
−
1
4
"
ρ
2
a
2
+
kρ
F
− q
2
a
2
#)
.
Using this expression and considering the turbulent atmosphere as an example of ran-
dom medium for which the structure function D(R) is described by the Kolmogorov–
Obukhov law (see, e.g., [33, 34])
D(R) = NC
2
ε
R
5/3
(R
min
R R
max
),
where N = 1.46, and C
2
ε
is the structure characteristic of dielectric permittivity fluc-
tuations, we obtain that average intensity in the beam
h
I(x, R)
i
= 0
2
(x, R, 0)
is given by the expression
h
I(x, R)
i
=
2|u
0
|
2
k
2
a
4
x
2
g
2
(x)
∞
Z
0
dtJ
0
2kaRt
xg(x)
exp
(
−t
2
−
3πN
32
C
2
ε
k
2
x
2a
g(x)
5/3
t
5/3
)
,
Caustic Structure of Wavefield in Random Media 361
where
g(x) =
s
1 + k
2
a
4
1
x
+
1
F
2
and J
0
(t) is the Bessel function. Many full-scale experiments testified this formula
in turbulent atmosphere and showed a good agreement between measured data and
theory.
Equation (13.14) for the fourth-order coherence function cannot be solved in the
analitic form; the analysis of this function requires either numerical, or approximate
techniques. It describes intensity fluctuations and reduces to the intensity variance for
equal transverse coordinates.
In the case of the plane incident wave, Eq. (13.14) can be simplified by introducing
new transverse coordinates
e
R
1
= R
0
1
− R
1
= R
2
− R
0
2
,
e
R
2
= R
0
2
− R
1
= R
2
− R
0
1
.
In this case, Eq. (13.14) assumes the form (we omit here the tilde sign)
∂
∂x
0
4
(x, R
1
, R
2
) =
i
k
∂
2
∂R
1
∂R
2
0
4
(x, R
1
, R
2
) −
k
2
4
F(R
1
, R
2
)0
4
(x, R
1
, R
2
),
(13.22)
where
F(R
1
, R
2
) = 2D(R
1
) + 2D(R
2
) − D(R
1
+ R
2
) − D(R
1
− R
2
).
We will give the asymptotic solution to this equation in Sect. 13.3.1, page 371.
13.2 Wavefield Amplitude–Phase Fluctuations. Rytov’s
Smooth Perturbation Method
Here, we consider the statistical description of wave amplitude–phase fluctuations.
We introduce the amplitude and phase (and the complex phase) of the wavefield by
the formula
u(x, R) = A(x, R)e
iS(x,R)
= e
φ(x,R)
,
where
φ(x, R) = χ(x, R) + iS(x, R),
χ(x, R) = ln A(x, R) being the level of the wave and S(x, R) being the wave ran-
dom phase addition to the phase of the incident wave kx. Starting from parabolic
362 Lectures on Dynamics of Stochastic Systems
equation (13.1), we can obtain that the complex phase satisfies the nonlinear equa-
tion of Rytov’s smooth perturbation method (SPM)
∂
∂x
φ(x, R) =
i
2k
1
R
φ(x, R) +
i
2k
[
∇
R
φ(x, R)
]
2
+ i
k
2
ε(x, R). (13.23)
For the plane incident wave (in what follows, we will deal just with this case), we
can set u
0
(R) = 1 and φ(0, R) = 0 without loss of generality.
Separating the real and imaginary parts in Eq. (13.23), we obtain
∂
∂x
χ(x, R) +
1
2k
1
R
S(x, R) +
1
k
[
∇
R
χ(x, R)
] [
∇
R
S(x, R)
]
= 0, (13.24)
∂
∂x
S(x, R) −
1
2k
1
R
χ(x, R) −
1
2k
[
∇
R
χ(x, R)
]
2
+
1
2k
[
∇
R
S(x, R)
]
2
=
k
2
ε(x, R).
(13.25)
Using Eq. (13.24), we can derive the equation for wave intensity
I(x, R) = e
2χ(x,R)
in the form
∂
∂x
I(x, R) +
1
k
∇
R
[
I(x, R)∇
R
S(x, R)
]
= 0. (13.26)
If function ε(x, R) is sufficiently small, then we can solve Eqs. (13.24) and (13.25)
by constructing iterative series in field ε(x, R). The first approximation of Rytov’s
SPM deals with the Gaussian fields χ(x, R) and S(x, R), whose statistical characte-
ristics are determined by statistical averaging of the corresponding iterative series.
For example, the second moments (including variances) of these fields are determined
from the linearized system of equations (13.24) and (13.25), i.e., from the system
∂
∂x
χ
0
(x, R) = −
1
2k
1
R
S
0
(x, R),
∂
∂x
S
0
(x, R) =
1
2k
1
R
χ
0
(x, R) +
k
2
ε(x, R),
(13.27)
while average values are determined immediately from Eqs. (13.24) and (13.25). Such
amplitude–phase description of the wave filed in random medium was first used by
A. M. Obukhov more than 50 years ago in paper [70] (see also [71]) where he pio-
neered considering diffraction phenomena accompanying wave propagation in ran-
dom media using the perturbation theory. Before this work, similar investigations were
based on the geometrical optics (acoustics) approximation. The technique suggested
by Obukhov has been topical up untill now. Basically, it forms the mathematical appa-
ratus of different engineering applications. However, as it was experimentally shown
Caustic Structure of Wavefield in Random Media 363
later in papers [72, 73], wavefield fluctuations rapidly grow with distance due to the
effect of multiple forward scattering, and perturbation theory fails from a certain dis-
tance (region of strong fluctuations).
The liner system of equations (13.27) can be solved using the Fourier transform
with respect to the transverse coordinate. Introducing the Fourier transforms of level,
phase, and random field ε(x, R)
χ
0
(x, R) =
Z
dqχ
0
q
(x)e
iqR
, χ
0
q
(x) =
1
(
2π
)
2
Z
dRχ
0
(x, R)e
−iqR
;
S
0
(x, R) =
Z
dqS
0
q
(x)e
iqR
, S
0
q
(x) =
1
(
2π
)
2
Z
dRS
0
(x, R)e
−iqR
;
ε(x, R) =
Z
dqε
q
(x)e
iqR
, ε
q
(x) =
1
(
2π
)
2
Z
dRε(x, R)e
−iqR
,
(13.28)
we obtain the solution to system (13.27) in the form
χ
0
q
(x) =
k
2
x
Z
0
dξε
q
(ξ) sin
q
2
2k
(x −ξ ),
S
0
q
(x) =
k
2
x
Z
0
dξε
q
(ξ) cos
q
2
2k
(x −ξ ).
(13.29)
For random field ε(x, R) described by the correlation and spectral functions (13.6)
and (13.5), the correlation function of random Gaussian field ε
q
(x) can be easily
obtained by calculating the corresponding integrals.
Indeed, in the case of the delta-correlated approximation of random field ε(x, R),
the relationship between the correlation and spectral functions has the form
B
ε
(x
1
− x
2
, R
1
− R
2
) = 2πδ(x
1
− x
2
)
Z
dq8
ε
(0, q)e
iq(R
1
−R
2
)
. (13.30)
Multiplying Eq. (13.30) by e
−i
(
q
1
R
1
+q
2
R
2
)
, integrating the result over all R
1
, R
2
and
taking into account (13.28), we obtain the desired equality
ε
q
1
(x
1
)ε
q
2
(x
2
)
= 2πδ(x
1
− x
2
)δ(q
1
+ q
2
)8
ε
(0, q
1
). (13.31)
If field ε(x, R) is different from zero only in layer (0, 1x) and ε(x, R) =0 for x >
1x, then Eq. (13.31) is replaced with the expression
ε
q
1
(x
1
)ε
q
2
(x
2
)
= 2πδ(x
1
− x
2
)θ(1x − x)δ(q
1
+ q
2
)8
ε
(0, q
1
). (13.32)
364 Lectures on Dynamics of Stochastic Systems
If we deal with field ε(x, R) whose fluctuations are caused by temperature turbulent
pulsations, then the three-dimensional spectral density can be represented for a wide
range of wave numbers in the form
8
ε
(q) = AC
2
ε
q
−11/3
(q
min
q q
max
), (13.33)
where A = 0.033 is a constant, C
2
ε
is the structure characteristic of dielectric permit-
tivity fluctuations that depends on medium parameters. The use of spectral density
(13.33) sometimes gives rise to the divergence of the integrals describing statistical
characteristics of amplitude–phase fluctuations of the wavefield. In these cases, we
can use the phenomenological spectral function
8
ε
(q) = 8
ε
(q) = AC
2
ε
q
−11/3
e
−q
2
/κ
2
m
, (13.34)
where κ
m
is the wave number corresponding to the turbulence microscale.
Within the framework of the first approximation of Rytov’s SPM, statistical cha-
racteristics of amplitude fluctuations are described by the variance of amplitude level,
i.e., by the parameter
σ
2
0
(x) =
D
χ
2
0
(x, R)
E
.
In the case of the medium occupying a layer of finite thickness 1x, this parameter can
be represented by virtue of Eqs. (13.29) and (13.31) as
σ
2
0
(x) =
Z Z
dq
1
dq
2
χ
0
q
1
(x)χ
0
q
2
(x)
e
i(q
1
+q
2
)R
=
π
2
k
2
1x
2
∞
Z
0
dqq8
ε
(q)
1 −
k
q
2
1x
sin
q
2
x
k
− sin
q
2
(x −1x)
k
. (13.35)
As for the average amplitude level, we determine it from Eq. (13.26). Assuming
that incident wave is the plane wave and averaging this equation over an ensemble of
realizations of field ε(x, R), we obtain the equality
h
I(x, R)
i
= 1.
Rewriting this equality in the form
h
I(x, R)
i
=
D
e
2χ
0
(x,R)
E
= e
2
h
χ
0
(x,R)
i
+2σ
2
0
(x)
= 1,
we see that
h
χ
0
(x, R)
i
= −σ
2
0
(x)
in the first approximation of Rytov’s SPM.
Caustic Structure of Wavefield in Random Media 365
The range of applicability of the first approximation of Rytov’s SPM is restricted
by the obvious condition
σ
2
0
(x) 1.
As for the wave intensity variance called also flicker rate, the first approximation
of Rytov’s SPM yields the following expression
β
0
(x) =
D
I
2
(x, R)
E
− 1 =
D
e
4χ
0
(x,R)
E
− 1 ≈ 4σ
2
0
(x). (13.36)
Therefore, the one-point probability density of field χ(x, R) has in this approxima-
tion the form
P(x;χ) =
s
2
πβ
0
(x)
exp
(
−
2
β
0
(x)
χ +
1
4
β
0
(x)
2
)
.
Thus, the wavefield intensity is the logarithmic-normal random field, and its one-
point probability density is given by the expression
P(x;I) =
1
I
√
2πβ
0
(x)
exp
−
1
2β
0
(x)
ln
2
Ie
1
2
β
0
(x)
. (13.37)
The statistical analysis considers commonly two limiting asymptotic cases.
The first case corresponds to the assumption 1x x and is called the random phase
screen. In this case, the wave first traverses a thin layer of fluctuating medium and then
propagates in free space. The thin medium layer adds to the wavefield only phase fluc-
tuations. In view of nonlinearity of Eqs. (13.24) and (13.25), the further propagation
in free space transforms these phase fluctuations into amplitude fluctuations.
The second case corresponds to the continuous medium, i.e., to the condition
1x = x.
Consider these limiting cases in more detail assuming that wavefield fluctuations
are weak.
13.2.1 Random Phase Screen (1 x x)
In this case, the variance of amplitude level is given by the expression following from
Eq. (13.35)
σ
2
0
(x) =
π
2
k
2
1x
2
∞
Z
0
dq q8
ε
(q)
1 − cos
q
2
x
k
. (13.38)
If fluctuations of field ε(x, R) are caused by turbulent pulsations of medium, then
spectrum 8
ε
(q) is described by Eq. (13.33), and integral (13.35) can be easily calcu-
lated. The resulting expression is
σ
2
0
(x) = 0.144C
2
ε
k
7/6
x
5/6
1x, (13.39)