Troyan V., Kiselev Y. Statistical Methods of Geophysical Data Processing
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Ray theory of wave field propagation 153
field at a stratified ocean (see Sec. 5.5):
ϕ(x, y, z, t; α) = A(αx, αy, z; α) exp
i
α
τ(αx, αy, αt)
.
In this representation the phase velocity is equal to c = |∇∇τ|
−1
and we consider the
rays laying in a horizontal plane.
The Lame operator for an isotropic inhomogeneous medium reads as
L = ρ∂
2
t
− (λ + µ)∇(
∇
∇·) − µ∆ − (∇λ)(∇∇·)
− (∇∇µ) × (∇×) − 2(∇µ,∇∇). (5.49)
In order to use the expression (5.5) to build zero approximation of geometrical optics
for the sounding vector field
L
0
(p)A
0
= 0, (5.50)
it needs to obtain the p-representation of the Lame operator (5.49):
exp{−i[(p · x) + p
0
t]}L exp{i[(p · x) + p
0
t]}
∆
= L(p).
Here p = (p
x
, p
y
) is a gradient on horizontal coordinates; p
0
= ∂τ/∂t. In Cartesian
coordinates (x, y, z) the matrix operator L has the next terms:
L
ii
= ρ∂
2
t
−
(λ + µ)∂
2
i
+ µ
3
X
j=1
∂
2
j
+ (∂
i
λ)∂
i
−
i+2
X
j=i+1
(∂
j
µ)∂
j
+ 2
3
X
j=1
(∂
j
µ)∂
j
,
where i =
1 2 3
x y z
with cyclic permutation, for example,
3
z
+ 2 =
2
y
:
L
ij
= −[(λ + µ)∂
i
∂
j
+ ∂
i
λ∂
j
+ (∂
j
µ)∂
i
].
Let us write separately the appropriate components L
ij
(i, j 6= z) in p-
representation, taking into account that τ = τ(x, y, t),
L
ii
(p) = −ρp
2
0
+ (λ + µ)p
2
i
+ µ(p, p) − µ∂
2
z
− ∂
z
µ∂
z
= −ρp
2
0
+ (λ + µ)p
2
i
+ µ(p, p) − ∂
z
µ∂
z
,
L
ij
= (λ + µ)p
i
p
j
,
and components which contain z
L
jz
= −i[(λ + µ)∂
z
+ (∂
z
µ)p
j
],
L
zj
= −i[(λ + µ)p
j
∂
z
+ (∂
z
λ)p
j
],
154 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
L
zz
= −ρp
2
0
− (λ + 2µ)∂
2
z
+ µ(p · p) − (∂
z
(λ + 2µ))∂
z
.
Let us consider the local orthogonal coordinates (ray coordinates) with orts
e
τ
=
∇∇τ
|∇∇τ|
∆
=
p
|p|
, e
z
= e
z
, e
β
= [e
τ
× e
z
].
Then A = (A
τ
, A
β
, A
z
); p = (p
τ
, p
β
), furthermore p
β
≡ 0,
L
ττ
(p) = −ρp
2
0
+ (λ + 2µ)(p, p) − ∂
z
(µ∂
z
),
L
ββ
(p) = −ρp
2
0
+ µ(p, p) − ∂
z
(µ∂
z
),
L
zz
(p) = −ρp
2
0
+ µ(p, p) − ∂
z
((λ + 2µ)∂
z
),
L
τz
(p) = −i
(λ + µ)p
τ
∂
∂z
+ (∂
z
µ)p
τ
,
L
zτ
(p) = −i
(λ + µ)p
τ
∂
∂z
+ (∂
z
λ)p
τ
,
L
βτ
= L
τβ
= L
βz
= L
zβ
= 0.
So, the matrix L(p) has the next shape
L(p) =
L
ττ
0 L
τz
0 L
ββ
0
L
zτ
0 L
zz
,
and the vector equation (5.50) can be represented in the form
L
ββ
A
β
= 0, (5.51)
L
ττ
A
τ
+ L
τz
A
z
= 0,
L
zτ
A
τ
+ L
zz
A
z
= 0. (5.52)
The equations (5.51) and (5.52) should be supplemented with corresponding bound-
ary conditions. If we consider the surface wave propagation inside half space with a
weak horizontal inhomogeneity, then boundary conditions should define at the free
surface Z
0
= Z
0
(τ, β) and over infinity Z −→ ∞.
Let us write the operator Γ of the boundary condition at the free surface t =
τn = 0:
ΓA = Γ
ij
A
j
,
Γ
ii
= n
i
(λ + 2µ)∂
i
+
X
k6=i
n
k
µ∂
k
,
Γ
ij
= n
i
λ∂
j
+ n
j
µ∂
i
.
We remind that
n = (1 + (∂
τ
Z
0
)
2
+ (∂
β
Z
0
)
2
)
−1/2
(∂
τ
Z
0
, ∂
β
Z
0
, −1),
Ray theory of wave field propagation 155
then we can write the p-representation of the operator Γ (Γ(p) = e
−i(p,x)
Γe
(p,x)
,
p = (p
τ
, p
β
)):
Γ(p) = −
µ∂
z
0 iµp
τ
0 µ∂
z
0
iλp
τ
0 (λ + 2µ)∂
z
.
The complete solvability condition for the equation Lϕ = 0, Γϕ = 0 or Lϕ = 0
has a form L
0
(p)A
0
= 0, i.e. in this case the equation for the β-component of the
amplitude A (L
ββ
A
β
= 0, ΓA
β
= 0) transforms to the Sturm–Liouville problem
∂
z
(µ∂
z
)A
β
− (λ
2
n
− ρp
2
0
)A
β
= 0,
∂
z
A
β
|
Z
0
= 0, A
β
z→0
−→ 0. (5.53)
Here λ
2
n
= µ(p, p), i.e. the eikonal equation (and respectively the ray geometry of
the surface wave) determines by the mode (λ
n
) and oscillation of the component
A
β
along z axis.
We should note, that in the problem (5.53) the shear modulus µ appears only.
The polarization vector is orthogonal to the propagation direction and lies in hori-
zontal plane. The such wave is the well known Love wave (Levshin et al., 1989).
The reminder of the boundary conditions at the free surface for a linear combi-
nation is described by the vectors with polarizations e
z
and e
τ
:
µ∂
z
A
τ
− iµp
τ
A
z
= 0,
iλp
τ
A
τ
+ (λ + 2µ)∂
z
A
z
= 0.
Supplemented by conditions A
τ
z→∞
−→ 0 and A
z
z→∞
−→ 0 the above equations and
equations (5.52) describe the Rayleigh wave (Aki and Richards, 2002; Levshin et al.,
1989; Yanovskaya, 1996).
5.7 Ray Approximation of Electromagnetic Fields
The description of the processes of the propagation in the nonuniform medium
can be based on the ray theory of electromagnetic fields. The ray method can
be considered as a basis for obtaining the transport equations and to establish a
connection between the statistical characteristics of the medium and the parameters
of the phenomenological transport theory of an electromagnetic radiation.
Let us write the equations for an electric intensity vector using the formulas
(4.63), (4.64):
∇∇ × E = −µ
1
c
∂
∂t
H,
∇
∇ × H =
4π
c
j +
ε
c
∂
∂t
E,
156 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
If the medium is described by the inductivity tensor ˆε and the magnetic conductivity
µ ≈ 1, and the extraneous currents (j = 0) are missed, we can write:
ˆ
LE =
∇ × ∇x − c
−2
∂
2
∂t
2
ˆε
E = 0.
Using the general approach from Sec. 5.1, to write the asymptotic expansion of
the vector E in the next form
E = A(y, α) exp
i
α
τ(y)
,
where y = (y
0
, y
1
, y
2
, y
3
)
∆
= (αt, αx, αy, αz). The operator L in the p-representation
(p
∆
= (p
0
, p
1
, p
2
, p
3
)
∆
= (p
0
, p), p
0
= ω) has the next form:
L(p) = e
−i(p,x)
ˆ
Le
i(p,x)
= −p × px −
ω
2
c
2
ˆε.
Let us represent the operator L(p) as L = L
0
+ αL
0
, where αL
0
∆
=
∆
= (−ω
2
/c
2
)(ˆε −
ˆ
IRe sp ˆε/3) describes an anisotropy and a nonconservativity of the media, L
0
=
−p × px − (ω
2
/c
2
)ε
0
ˆ
I, ε
0
is real and equal to Re sp ˆε/3. The equation for a zero
approximation of the ray method in the local coordinates
e
1
=
p
|p|
, e
2
=
∂
s
e
1
|∂
s
e
1
|
, e
3
= e
1
× e
2
looks as follows
ˆ
L
0
A =
ω
2
c
2
ε
0
0 0
0 −(p, p) +
ω
2
c
2
ε
0
0
0 0 −(p, p) +
ω
2
c
2
ε
0
·
A
1
A
2
A
3
= 0, (5.54)
and in the canonical representation:
ˆ
L
0
(p) = (e
2
e
T
2
+ e
3
e
T
3
)
ω
2
c
2
ε
0
− (p, p)
+ e
1
e
T
1
ω
2
c
2
ε
0
.
In the case of the equality of the eigenvalues λ
2
= λ
3
= (ω
2
/c
2
)ε
0
− (p, p), it is
possible to make the concluding about the degeneration of the polarisation of the
shear waves ((e
2
· p) = 0, (e
3
· p) = 0). From the equation (5.9) in this case it
follows |p|
2
= (ω
2
/c
2
)ε
0
, i.e. the dispersion relation has the form
ω − ω(p, x) = ω − |p|cε
−1/2
0
= 0.
The group velocity v
gv
= (∂/∂p) ω(p, x) is equal to v
gv
= cε
−1/2
0
e
1
in this case.
Instead of the ray parameter s we introduce an arc length l, which is connected
with the ray parameter by the next relation
(dl)
2
=
∂H
∂p
2
ds
2
,
where
H =
1
2
(p, p) −
ω
2
c
2
ε
0
. (5.55)
Ray theory of wave field propagation 157
From here it follows that dl = |p|ds.
Let us write the ray equations using arc length of the space projection of the
ray dl:
dx
dl
=
1
|p|
dx
ds
=
1
|p|
p =
v
g
|v
g
|
= e
1
, (5.56)
dt
dl
=
1
|p|
dt
ds
=
ω
|p|
=
1
|v
g
|
, (5.57)
dp
dl
=
1
|p|
dp
ds
= −
1
|p|
∂H
∂x
=
|p|
|p|
ε
1/2
0
c
∂
x
ω = v
−1
g
∂
x
ω, (5.58)
dω
dl
=
1
|p|
dω
ds
= −
1
|p|
∂H
∂t
=
1
|p|
ω
c
2
ε
0
∂
t
ω(p, x) = v
−1
g
∂
t
ω. (5.59)
Taking into account the equations (5.56), (5.59) it is possible to write equation for
kinematics of a point belonging to the ray
dx
dt
= v
gv
.
So, the point motion describes by the group velocity, which for the isotropic medium
is coincided with the vector p.
The equations (5.58), (5.59) allow to obtain the connection between the wave
vector and the spatial gradient ω:
dp
dt
= ∂
x
ω.
At last, from the relations (5.56) – (5.59) follow, that total time derivative of ω is
equal to the local derivative:
dω
dt
= ∂
t
ω.
The ray tomography can be realized not only using phase measurements, but
also using the polarization vector.
Let us consider the derivation of the law of rotation of the polarization vector
(Kravtsov and Apresyan, 1996). For this purpose we write the relation (5.6) for the
amplitude of a zero approximation of the ray method E
0
:
∂
p
∂
x
L
0
(p)E
0
(x) = L
0
(p)E
0
(x), (5.60)
where
L
0
(p) = −
ω
2
c
2
ˆε −
1
3
ˆ
IRespˆε
.
We will use a doubled Hamilton function H (5.55) instead of L
0
(p) and rewrite the
formula (5.60) as follows
∂
p
0
H∂
t
E
0
+ ∂
p
H∂
x
E
0
=
1
2
L
0
(p)E
0
(x).
158 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
To change the derivatives of the Hamilton function:
∂
p
0
H =
dt
ds
, ∂
p
H =
dx
ds
and to pass to the parameter l (arc length of the spatial ray), we obtain
∂t
∂l
∂
t
E
0
+
dx
dl
∂
x
E
0
=
1
2
|p|L
0
E
0
. (5.61)
Let us note, that in the left hand side of (5.61) there is the first total derivative
of E
0
on length of the nonstationary ray, i.e.
d
dl
E
0
=
ω
2l
√
ε
0
L
0
E
0
.
Writing the equation with zero approximation for E
0
on the dispersion surface, in
other words, taking into account that in accordance with the equation (5.54)
E
0
= A
2
e
2
+ A
3
e
3
,
we project the equation (5.61) to a plane which is orthogonal to
P
1
d
l
E
0
=
ω
2c
√
εP
1
L
0
E
0
, (5.62)
where
P
1
= I −e
1
· e
T
1
.
If an anisotropy of the absorption is absent, then L
0
= 0, and the relation (5.62)
can be rewritten in a form P
1
d
l
E
0
= 0, i.e. we obtain the system of two equations
for A
2
and A
3
. Denoting the first derivative on l by a dot, we write
˙
A
2
+ A
2
(e
2
, e
3
) = 0,
˙
A
3
+ A
3
(e
3
, e
2
) = 0. (5.63)
Multiplying the first equation of (5.63) by A
3
, the second equation by A
2
, and after
summing up, we obtain
A
3
˙
A
2
− A
2
˙
A
3
= A
2
2
(e
3
,
˙
e
2
) − A
2
3
(e
2
,
˙
e
3
). (5.64)
Since d
l
(e
2
, e
3
) = 0, then the equation (5.64) is possible to rewrite in a form
(e
2
,
˙
e
3
) =
A
3
˙
A
2
− A
2
˙
A
3
A
2
2
+ A
2
3
.
Let us note, that in the right-hand side of the above equality there is the derivative
of arctgA
2
/A
3
= θ, and the scalar product (e
2
,
˙
e
3
) is equal to |
˙
e
3
|. In turn, |
˙
e
3
| is
rotating of curve (ray) by definition. Having designated value of rotating as T , we
shall write the law of rotating of the polarization vector as
d
l
θ = T.
Ray theory of wave field propagation 159
5.8 Statement of Problem of the Ray Kinematic Tomography
Reviewed in the previous sections, the ray approximation is a natural basis for the
statement problem of a ray kinematic tomography. All ray approximations in an
explicit form contain a phase factor e
iωτ
, where the value τ in a spatial point x is
determined by the value of τ in the some point x
0
(for example, a source point) and
by the integration of the (ray) refraction index along the ray from the point x
0
to
a point x. Therefore if the experimental data allow to trace a phase of signal, then
the connection of properties of the medium and the travel time of a sounding signal
can be recognized. The same concept allows to institute a field of the attenuation
factor of the medium.
As an example of the interpretation of the ray approximation, let as consider
the case of an anisotropic reference medium. Let a scalar wave field ϕ(x, t) satisfies
the equation
Lϕ = 0, (5.65)
where
L = ∂
2
t
−
X
ij
A
ij
(x)∂
i
∂
j
, i, j = 1 ÷ 3,
a matrix A is positive defined independently of a point x, i.e. the operator L
is hyperbolic everywhere in an applicable domain. The characteristic equation is
written in this case as
(p,
ˆ
Ap) = 1 (5.66)
and it can be considered as an eikonal equation; the vector p is equal to ∇∇τ. To
equate gradient of the characteristic surface to zero, we obtain a differential equation
(in a vector form) of the first order relatively to the derivative p:
2(
∇
∇ · p)
T
ˆ
Ap + (p, (∇∇
ˆ
A)p) = 0. (5.67)
From this equation we obtain the characteristic equations:
dx
ds
=
ˆ
Ap,
dp
ds
= −
1
2
p, (∇ ·
ˆ
A)p
. (5.68)
The last equality is the characteristic equation (5.66).
The characteristic equations (5.68) determine bicharacteristics of the wave equa-
tion (5.65), and for their building it is necessary to establish the initial conditions
x
s=0
= x
(0)
, p
s=0
= p
(0)
,
where p
(0)
: (p
(0)
, Ap
(0)
) = 1.
The ray direction at the point s = 0 we determine using the first equation of
the system (5.68):
dx
ds
s=0
=
ˆ
A
x
0
p
(0)
.
160 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
Having noted, that p =
ˆ
A
−1
dx/ds, we write the equation (5.66) in a form
dx
ds
,
ˆ
A
−1
dx
ds
= 1 . (5.69)
Parameter s accurate within additive constant is equal to eikonal τ, because
dτ
ds
=
∇
∇
x
τ,
dx
ds
= (p, Ap) = 1.
Therefore from the equation (5.69) we obtain
ds
2
= dτ
2
= (dx,
ˆ
A
−1
dx) ,
and we can interpret the characteristic equations (5.68). Taking into account the
relation
τ = τ
0
+
Z
L
(dx,
ˆ
A
−1
dx)
1/2
, (5.70)
we conclude that the characteristic equations (5.68) determine the parametric de-
termination of geodesic curves in the space with the metric dτ = = (dx, A
−1
dx).
Using the equality |dx|
2
= dl
2
(dl is a length of the ray arc), it is possible to write
dx = n (x)dl (n(x) is a unit vector with direction coinciding with the direction of
the signal propagation in the point x). The expression for the (slowness) can be
represented as
dτ
dl
= v
−1
(x, n) = (n, A
−1
n) .
Therefore the integral from the formula (5.70) over a geodesic curve L
Z
L
(dx, A
−1
dx)
1/2
=
Z
L
dl
v(x, n)
,
has a sense of a minimum propagation time of the signal along a curve connected two
spatial points. The explicit form of the velocity dependence v(x, n) on the direction
of propagation of the signal (n) contains a specific character of the anisotropic
model.
The restoration of coefficients A
ij
(x) using the known lengths of geodesic curves
L, which joining the pairs of arbitrary points x and x
0
is a substance of the inverse
kinematic problem. In the standard statements of such problems, the pairs of
the points x and x
0
are considered laying on the boundary of the explored area.
Let’s consider a statement of the linearized solution of a three-dimensional inverse
kinematic problem. Let, the basic reference medium is characterized by a matrix
A
0
(x), and the problem is to restore A(x):
ˆ
A(x) =
ˆ
A
0
(x) + α
ˆ
A
1
(x) . (5.71)
Here α is a small parameter;
ˆ
A
0
and
ˆ
A
1
are the positive defined matrices. To
represent the eikonal as the next expansion
τ(x
0
, x) =
∞
X
n=0
α
n
τ
n
(x
0
, x) . (5.72)
Ray theory of wave field propagation 161
By substituting the expressions (5.71) and (5.72) into the characteristic equation,
we obtain
(p
0
+ αp
1
, (A
0
+ αA
1
)(p
0
+ αp
1
)) = 1 .
Here, in expansion on p =
∇
∇τ ≈
∇
∇τ
0
+ α
∇
∇τ
1
∆
= p
0
+ αp
1
we keep only the linear
on α terms. Taking into account that
(p
0
,
ˆ
A
0
p
0
) = 1 , (5.73)
and keeping the terms of order α, we obtain the equation
2(p
1
,
ˆ
A
0
) p
0
+ (p
0
,
ˆ
A
1
p
0
) = 0 . (5.74)
The equation (5.73) determines the geodetic L
0
(x
0
, x). Taking into account that
ds = dτ
0
we can write
(p
1
, A
0
p
0
) =
p
1
,
dx
0
dτ
0
=
dτ
1
dτ
0
,
and from equation (5.74) we get the expression for the deduction τ
1
:
τ
1
(x, x
0
) = −
1
2
Z
L
0
(x
0
,x)
(p
0
,
ˆ
A
1
(x)p
0
)dτ
0
. (5.75)
This expression is a basis for solution the linearized inverse kinematic problem.
Let’s note, that
ˆ
A
1
(x) should satisfies the condition (A
1
(x) + A
0
(x) > 0).
At the remote sensing problem the kinematic statement, as a rule, connects with
the isotropic reference medium, therefore
ˆ
A
0
(x) = a
0
(x) = v
2
0
(x) ,
ˆ
A
1
(x) = a
1
(x) = v
2
1
(x) ,
(p
0
, A
1
p
0
) = a
1
(x)(p
0
, p
0
) =
a
1
(x)
a
0
, dτ
0
=
dl
0
v
0
,
and an appropriate constraint equation to correct the propagation time from the
point x
0
to a point x looks like follows
τ
1
(x, x
0
) = −
1
2
Z
L
0
(x
0
,x)
v
2
1
(x)dl
0
v
3
0
(x)
∆
=
Z
L
0
(x
0
,x)
θ(x)dl
0
. (5.76)
Here dl
0
is the length of ray arc L
0
(x
0
, x), which joins together the points x
0
and
x in the reference medium with the velocity v
0
(x).
The representation (5.76) is a basis for the statement of the ray tomography
problems. It will be demonstrated in Sec. 11.7, the characteristic equation is de-
termined by higher derivatives of the propagation operator L only. Therefore the
ray concept is easily transferred on restoration of the attenuation factor, which is
connected with the first derivatives, which are included in the propagation operator
L:
ln
|Φ(ω)|
|Φ
0
(ω)|
= −
Z
L
0
β
ω
(x)dl
0
,
where (Φ(ω) is the Fourier transform of the signal ϕ
ϕ
ϕ observed at the point x; Φ
0
(ω)
is the Fourier transform of a theoretical sounding signal, exited in the point x
0
,
which propagates inside the reference medium; the ray L
0
(x
0
, x) joins together the
source point and the observation point; β
ω
(x) is a ray attenuation coefficient for
the frequency ω.
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