Troyan V., Kiselev Y. Statistical Methods of Geophysical Data Processing

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Models of measurement data 113

problem it is necessary with the maximum attention and carefulness to choose the

model of an experimental material. The development of an optimum model should

be carried out together by geophysicists and mathematicians, it should be included

“art”, intuition and rich a priori information of the interpreter formalized in the

rigorous mathematical form.

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Chapter 4

The functional relationships of sounding

signal ﬁelds and parameters of the

medium

Basic part of remote sounding tasks are the mathematical models of the propagation

of the sounding signals of the diﬀerent physical nature in the media under the

investigation. In the present chapter the laws of elastic, acoustic, electromagnetic

wave ﬁeld propagation are formulated and the ray energy or homoenergetic neutrons

transfer equation are considered.

4.1 Seismology and Seismic Prospecting

As a sounding signal in the seismology and in seismic prospecting the elastic waves,

excited by earthquakes (natural sources of the elastic waves), the shots, vibroplates

and other artiﬁcial sources are used (Aki and Richards, 2002; Jensen et al., 2000;

Karus et al., 1986). The elastic wave ﬁled is registered by the seismograph, carries

the information on parameters of the medium, which determine the propagation

of elastic waves. So, in a Fig. 4.1 the examples of the tomography experiment by

elastic waves are represented ((a) — the scheme of the vertical seismic proﬁling; and

the sources are located on a free surface E; (b) — a section of the vertical seismic

proﬁling in a plane x0z; (c) – the observed data and the sources on a free surface).

The main aim of the seismology and the seismic prospecting consists in the

restoration the Earth internal structure in accordance with signals recorded in the

borehole or at the Earth surface. The most eﬀective method of the seismic prospect-

ing is the reﬂection method, which is prevailing for the exploration activity and the

mineral exploration both in the land and the sea. The method is based on the

registration of the seismic waves reﬂected by some interface, which correspond to

some variation of the wave impedance of the geologic media. Those interfaces are

usually connected with lithological and geologic borders. On carrying out seismic

prospecting in the new area, where the seismic prospecting has never been previ-

ously applied to, it is primary necessary to build the macromodel of the medium,

i.e. to consider the main reﬂected horizons and to determine the average values of

elastic parameters (possibly with their gradients) inside of each geological horizon.

When working out the deposits the macromodel is usually known and goal of seis-

115

116 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

Fig. 4.1 Examples of the tomography sounding by elastic waves.

mic prospecting is to obtain the detailed and reliable information about the elastic

properties of the medium.

The solution of problems of the seismology and seismic prospecting depends on

the connection between the media properties and the corresponding transformation

of the sounding signal, i.e. the propagation equation.

The foundation of the seismic waves propagation problem usually is the model

of the ideal elastic body, built on the linear connection between the strain (change

of a shape or volume of an elastic body) and the backmoving elastic force. This

model contains implicitly the statement about so called ’short-range acting’ between

pieces of solid, i.e. it is assumed that the interacting of two diﬀerent parts of solid

accomplishes only through their interface. It is possible to consider the deformation

of the rock far enough from the center of an earthquake or a shotpoint being small.

Hence, it is possible to use the model of the ideal elastic body.

Let us write the basic relations of the mathematical theory of the elasticity. The

displacement could be induced by a volume force.

df = sdV,

where s = s(x)

∆

= df /dV is a density of the volume force f (or the surface force

df = tdσ). Here t is the density of the surface force, acting to the piece of the

surface dσ of the considered volume dV , which is a called stress:

t = t(x)

∆

dσ

Let us denote the value of the displacement of the solid volume dV (with the mass

density ρ) from the equilibrium state as ϕϕ = ϕ(x) and write the equation of the

motion

∂

∂t

= s + s

, (4.1)

where s

= Kϕ is a density of the backmoving elastic force, induced by the dis-

placement ϕ, which is a result of the integration of the stress vector acting to the

The relationships of sounding signal and medium parameters 117

Fig. 4.2 Distribution of the stress on elementary tetrahedron.

chosen volume

ZZZ

dV =

tdσ. (4.2)

Taking into account that t = t(n), where n is an external (with respect to the

considered volume) normal vector:

t(n) = −t(−n).

Taking into account the condition

t(n)dσ|

dσ

∆V →0

−→ 0

for the distribution of the stress on an elementary tetrahedron (Fig. 4.2) we obtain:

t(n)∆σ

+ t(−e

)∆σ

+ t(−e

)∆σ

+ t(e

)∆σ

∆σ

+ ∆σ

∆V →0

−→ 0.

We can then derive

t(n) =

t(e

)(e

· n) =

t(e

, (4.3)

The relation (4.3) determines the connection between the density of the surface force

(stress) t, works into the n direction, and the density of the surface force works into

the unit vector e. The relation (4.3) gives us an opportunity to introduce a stress

tensor:

t = ˆτn, (4.4)

i.e. t

, where τ

= t

) i-th component of the stress vector, induced by

the external force of solid pieces laying outside of the outlined volume an acting into

118 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

plane, the normal to the unit vector e

, Not that ˆτ = ˆτ(x), i.e. deformed solid is

described by means of the stress tensor ﬁeld. The symmetrical tensor ˆτ it is possible

to transform to the diagonal form: τ

= τ

, where τ

are principal stress axises,

= 1, i = j, δ

= 0, i 6= j (δ

is Kronecker delta). Using the expression (4.4)

and applying the Gauss theorem, we obtain

t(n)dσ =

ˆτndσ =

ZZZ

(

∇

∇ · ˆτ )dV,

whence for the density of the volume backmoving elastic force (4.2) is found

= (∇ · ˆτ ). (4.5)

In the linear elasticity the connection between the stress and the strain (displace-

ment ϕ) assumed to be linear (the Hooke law), i.e.

ˆτ =

K ˆε, (4.6)

where ˆε is the stress tensor, deﬁned as

dϕ

∆

ξdx

∆

= ˆεdx − ˆηdx. (4.7)

In this relation the general form of the tensor

ξ is represented as a decomposition

into two parts: the symmetric tensor

ˆε = ˆε

, ε



∂ϕ

∂x

∂ϕ

∂x



and the antisymmetric tensor

ˆη = −ˆη, η



∂ϕ

∂x

−

∂ϕ

∂x



The antisymmetric tensor ˆη = (∇×ϕx)/2 describes the rotation under the condition

|∂ϕ

/∂x

|  1 the tensor K is a tensor of 4-th rank of the elastic modulus

ijkl

(i, j, k, l = 1 ÷ 3), (4.8)

determines the elastic properties of the medium and has got the next properties of

the symmetry due to the symmetry of the stress and the strain tensors:

ijkl

= K

jikl

= K

ijlk

= K

jilk

, (4.9)

i.e. the tensor K has 21 independent components.

The model of the isotropic elastic medium with a local invariance relatively to

the rotation is usually used for the solution of the seismic problems. Let us consider

the model with coinciding the main axises for the strain and stress tensors. Let, for

example, the tension stress is coincided with the direction of the homogeneous rod,

which becomes longer longitudinal direction and becomes clenched in transversal

direction. If the ﬁrst Cartesian ort is directed along the rod, then the relation (4.6)

can be written as following

, (4.10)

The relationships of sounding signal and medium parameters 119

= ε

= −

. (4.11)

The coeﬃcient E is called the Young modulus, σ is the Poisson’s ratio, thus E

and σ are constants of a material of a rod describing its elastic properties. All

reductants of a stress tensor, except for τ

, and three components of the strain

tensor ε

, ε

are equal to zero. If to the stress tensor τ

“to add” the tensors

and τ

, then the strain ε

from the expression (4.10) will vary according to

the accepted linear model and the requirement (4.11) as follows:

−

, (4.12)

with τ

= 0 and ε

= 0. We can rewrite the expression (4.12) in the form:

Eε

= (1 + σ)τ

− σ(τ

+ τ

or Eˆε = (1 + σ)ˆτ − σspˆτ

I (4.13)

(

I is the identity matrix). As it is visible, the connection of tensors of the stress and

the strain in case of an isotropic elastic medium is determined by two parameters,

for example, by the Young modulus and the Poisson ratio. By convolving tensors

in the left-hand and right-hand sides of the formula (4.13) we obtain

Espˆε = (1 + σ)spˆτ − 3σspˆτ = (1 − 2σ)spˆτ ,

therefore formula (4.13) can be reduced to

(1 + σ)ˆτ = Eˆε +

σE

1 − 2σ

spˆε

and

ˆτ =

1 + σ

ˆε +

σE

(1 + σ)(1 − 2σ)

spˆε

∆

= 2µˆε + λspˆε

I. (4.14)

The parameters µ = E/[2(1 + σ)] and λ = σE/[(1 + σ)(1 −2σ)] are known as Lame

parameters. The tensor K of the elasticity modulus from the relation (4.6) for the

isotropic medium has a form

ijkl

= λδ

+ µ(δ

+ δ

Let us consider the physical meaning of spˆε from (4.7):

spˆε = (∇ · ϕ) '

(1 + ∂ϕ

/∂x

)dx

−

dV −dV

i.e. spˆε is numerically equal to the relative variation of the volume, named the

dilatation, and it is clear a sense of the Lame parameter λ (λ is a coeﬃcient of the

dilatation).

The free elastic energy F = K ˆε/2 (accurate within terms of the second order)

expresses through Lame parameters as follows

F =

(spˆε)

+ µ

i6=ik

120 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

Fig. 4.3 Shear deformation.

Note, that any strain can be introduced as the sum of strains of the simple shear

and the uniform pressing (ˆε = const

I), we can write

ˆε =



ˆε −

Ispˆε



Ispˆε.

The spur of the tensor which was written to round brackets, obviously it is equal

to zero, therefore F it is possible to introduce in the form

F = µ

i6=ik



−

spˆε



(spˆε)

where k = λ + 2µ/3 is the uniform pressure modulus. Let’s write the equation of

the motion (4.1), using the formula for the backmoving elastic force (4.5) and the

Hooke law (see relation (4.6)):

∂

∂t

= s +

∇

∇ ·

K ˆε = s + ∇

∇

∇ ·

∂ϕ

∂x

, (4.15)

as from formula (4.7) follows

∂ϕ

∂x

= ˆε − ˆη = ˆε −



(∇

∇

∇ ×

ϕ)x



∇∇ · [

K(∇ ×ϕϕ)x] ≡ 0.

Equation (4.15) is called the Lame equation, and the operator

∆

= ρ∂

−∇∇ ·

∂

(4.16)

is called the Lame operator. In the operator form (4.15) reads as

ϕ = s.

We shall name an operator L including boundary and (linear) initial conditions lain

down to ϕ as a generalized Lame operator:

ϕ = s,

ϕ = g.

After this we can write

L =



The relationships of sounding signal and medium parameters 121

The lame equation for the isotropic case:

∂

∂t

− (λ + µ)

∇

∇∇∇· ϕ − µ∆ϕ − ∇λ∇ ·

−

∇

∇µ ×

∇

∇ ×

ϕ − 2(∇

∇

∇µ ·

∇

∇)

ϕ = s. (4.17)

In the case of the uniform isotropic medium (λ = const, µ = const, ρ = const)

formula (4.17) can be written down as

∂

∂t

− (λ + µ)∇∇∇ ·ϕϕ − µ∆ϕϕ = s. (4.18)

Taking into account the equality

∇

∇ ·

∇

ϕ =

∇

∇ ·

ϕ −∇

∇

∇ ×

∇

∇ ×ϕ

ϕ,

we can write down the equation (4.17) in the form

∂

∂t

− (λ + 2µ)∇∇ ·ϕϕ − µ

∇

∇ × ∇ ×

ϕ = s. (4.19)

Using the Helmholtz theorem, the ﬁeld of the vector ϕ can be introduced as the

sum of the potential (ϕ

) and the solenoidal (ϕϕ

) ﬁeld of vectors:

∆

= ϕ

= −∇Φ + ∇ × A, (4.20)

where A : ∇ · A = 0; Φ is a scalar potential; A is a vector potential. Substituting

ϕϕ =

+ ϕ

and s = s

+ s

to (4.19), and applying operator ∇×(·) and ∇·(·) to

its left hand side and right-hand side, we obtain the system of linear equations for

ϕϕ

∇

∇ × ϕ

= 0 ⇒

∂

∂t

−

λ + 2µ

∇∇∇ ·

∆



∂

∂t

− v

∆



= s

(4.21)

and for ϕ

∇ ×

= 0 ⇒

∂

∂t

−

∇∇∇ ·

∆



∂

∂t

− v

∆



ϕϕ

= s

. (4.22)

Substituting the partial solution of the systems (4.21), (4.22) in the form

ϕϕ

= e

(n · x − v

t), (4.23)

where |e

p,s

| = 1, and taking into account that conditions

∇

∇ × ϕ

≡ 0 follows

[n ×e

= 0, and the condition ∇

∇

∇·

≡ 0 leads to [n ×e

= 0, we can interpret

as the longitudinal wave with the velocity of the propagation v

= ((λ+2µ)/ρ)

1/2

moreover polarization vector e

is codirectional with the unit vector n, which is a

normal to the wave front. We can interpret ϕ

as a shear wave with the velocity

= (µ/ρ)

1/2

. The displacement vector e

of such wave is orthogonal to direction

of its propagation. Because of conditions ∇∇ · ϕ

= spˆε and

∇

∇ · ϕϕ

≡ 0 the shear

wave

is not connected with the change of the volume. At the same time valid

∇∇×

≡ 0, the longitudinal wave is a wave of the dilatation. It is necessary to note,

122 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

that the representation about the longitudinal and shear waves is a consequence of

the translation (on time and space) invariance.

Let us consider the case of the uniform isotropic medium (Petrashen, 1978, 1980;

Petrashen and Kashtan, 1984). The tensor K of the elastic module is able to be

represented in the form of the block symmetric matrix due to symmetry relations

(4.8), (4.9):

K → K

αβ



. . . K

. . . . . .

. . . K



, (4.24)

αβ

= K

βα

and, if between pairs ij and kl from relations (4.8), (4.9) and α, β respectively the

links is introduced, for example: i, i ↔ i; (2, 3) = (3, 2) ↔ 4; (3, 1) = (1, 3) ↔ 5;

(1, 2) = (2, 1) ↔ 6. It is seen from (4.24) that tensor

K has no more than 21

independent components in general case. By the virtue of translation invariance

three degrees of freedom are determined by a choice of the orientation of a coordinate

frame, therefore 18 components are independent.

In the notation (4.24) the tensor

K of the isotropic medium has a form:

αβ



λ + 2µ λ λ 0 0 0

λ λ + 2µ λ 0 0 0

λ λ λ + 2µ 0 0 0

0 0 0 µ 0 0

0 0 0 0 µ 0

0 0 0 0 0 µ



Let us write down the motion equation without the source (s = 0):

∂

∂t

k,l,j

ijkl

∂

∂x

∂

∂x

. (4.25)

Substituting into (4.25) the partial solution in the form of the plane wave: ϕ =

ef(t − (p · x)), where p is a slowness vector (gradient of eikonal), we obtain the

condition:

det



ρδ

−

j,l

ijkl



= 0. (4.26)

The general solution of the equation (4.26) for an arbitrary anisotropic medium

misses.

In an isotropic medium the equation (4.26) is reduced to three quadratic equa-

tions for the slowness components: a longitudinal wave and two shear waves.