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

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The relationships of sounding signal and medium parameters 123

4.2 Acoustics of the Ocean

The acoustic method has widely application at learning the water mass of the ocean

and the ocean sedimentary cover. During studies were found out, that the severe

ocean has a various bottom topography. The main structural elements of the ocean

bottom topography are: the abyss plains, middle-ocean mountain ridges, the small

shapes (middle-ocean island and shelfs of archipelagoes), large faults. The special

attention and detailed learning of these structure factors to take on the special

signiﬁcance in connection with a progress of geodynamics, and also at the solution

of applied problems (for example, by looking up of ferro-manganese concretions,

oil and gas in a shelf). The presence of a wave guide in the near surface ocean

mass, supplying the long-range propagation of acoustic signals, stipulates unique

possibilities for the sounding of the vast water areas. The analysis of the underwater

propagation of the acoustic signal is founded on the laws of theoretical acoustics

(Brekhovskikh, 1960; Brekhovskikh and Godin, 1998; Brekhovskikh and Lysanov,

1991). The propagation of the acoustic wave is described by the wave equation

with the appropriate boundary conditions on a surface and at the bottom of the

ocean. The problem of the acoustic tomography is reduced to the restoration of a

velocity of the propagation of a sound, i.e. to the restoration of the wave operative

using values of the acoustic pressure in some space points (“receiver or observation

points”).

Let’s write the equation of the motion of a small liquid piece (dV ) in the form

of the second Newton’s law:

ρvdV = S + S

, (4.27)

where S =

sρdV is an external body force, by analogy with (4.1); S

resultant applied force of the surrounding liquid to the considered volume, i.e

= −

∂V

pdσσ = −

∇pdV (4.28)

(p is a pressure). To write down the total time derivative from the left-hand side

of (4.27), let’s remember the connection of the substantial derivative (d/dt is a

trajectory derivative) and the partial derivative (∂/∂t):

f(x(t), t) =

∂

∂t

f +

∂f

∂x

∂f

∂x

∂f

∂x

i. e.

∂

∂t

+ (v · ∇). (4.29)

In the next turn, let us make the expression for the substantial time derivative of the

integral over a small volume V , moving together with a liquid element i.e. consist

124 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

of the same particles

FdV =

FJd

V =

(FJ)d

(FJ)DV =



F + F







∂

∂t

F + (v ·∇∇)F + (∇ · v)F



dV.

Here J is an Jacobian and in the last step uses the equality

∇

∇ · v.

Taking into account (4.29) and (4.28) the left-hand side of (4.27) becomes

(ρv)dV =



ρ + ρ







∂

∂t

ρ + (v · ∇)ρ + (∇ · v)ρ



+ ρ



dV. (4.30)

The equation (4.27) contains the functions p, ρ, v. To obtain the close equation it

needs to add to the equation (4.27) the relations connecting these functions. Taking

into account, that the thermodynamic state equation is generally represented as

p = p(ρ, H) (H is an entropy of the system), shall write the complete system of an

equation of motion as

(ρv) + ∇p = s; (4.31)

for continuity equation:

∂

∂t

ρ + ∇ · (ρv) = 0 (4.32)

following the condition

∂

∂t

ρdV = −

∂V

ρv · dσ = −

(∇ · ρv)dV

∂

∂t

ρ + ρ(

∇

∇ · v) + (v ·

∇

∇)ρ = 0);

for the state equation:

p = p(ρ, H); (4.33)

and for the iso-entropy (adiabatic) equation:

∂

∂t

H +

∇

∇ · (Hv) = 0 (4.34)

The relationships of sounding signal and medium parameters 125

(the liquid has no thermal conductivity).

By virtue of the continuity equation (4.32) the expression in square brackets in

the formula (4.30) will be equal to zero, and the equation of motion (4.31) (the

Euler equation) becomes

∂

∂t

v + (v ·

∇

∇)v +

∇

∇p = s.

The linear equations of motion can be obtained, using a few simplifying as-

sumptions: incompressibilities of liquid (∇

∇

∇ · v = 0); its barotropies (p = p(ρ(t)));

small perturbations of the stationary steady-state ﬂow (p = p

+ p

, ρ = ρ

+ ρ

v = v

+ v

). Let us replace the initial system of equations (4.31) —(4.34) by

another one, using above mentioned assumptions

∂

∂t

v + (v ·∇∇)v +

∇∇p = s

+ εs

(4.35)

(we shall consider, that the solution of system (4.31) — (4.34): p

= 0, ρ

= 0, v

0 corresponds to a hydrostatic state and is aroused by an action of the mass force s

for example, by the gravity ﬁeld g ≡ s

, and the presence of perturbations p, ρ, v

is a result of the perturbing force εs

containing a small parameter). The equation

∂

∂t

ρ + (v · ∇)ρ = 0 (4.36)

corresponds to the equation (4.32) under the condition of incompressibility (∇·v =

0). But there would be a system like

p = p(ρ). (4.37)

Then the system of equations for the perturbed medium becomes (with loss of index

1: p

⇒ p, ρ

⇒ ρ, v

⇒ v, s

⇒ s):

∂

∂t

v −

∇p

∇p = s; (4.38)

∂

∂t

ρ + ρ

(∇ · v) + (v · ∇)ρ

= 0; (4.39)

p =

dρ

∆

= c

ρ. (4.40)

As a rule, in problems of acoustics of ocean the measured value is the pressure.

To derive of the closed equation relative to p we turn down in last system (4.38),

(4.39) the terms containing

∇

∇p

and ∇ρ

(neglect an action of volume forces such

as gravity). Diﬀerentiate on time the left-hand side of the equation (4.39):

∂

∂t

ρ + ρ

(

∇

∇ ·

∂

∂t

v) = 0

and the substituting expression from (4.38) instead of ∂/∂tv, and expression from

(4.40) (ρ = c

−2

p) instead of ρ, we obtain

∆p −

∂

∂t

p =

∇

∇ · s. (4.41)

126 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

So, the wave equation (4.41) describes a wave propagation aroused by the

perturbing force (s), thus the velocity of the acoustic wave propagation is c =

(∂p/∂ρ)

1/2

Let’s note, that by using the known solution (4.41) p = p(x, t) it is possible

by the integration of the equation (4.38) at ∇∇p

= 0 to obtain a ﬁeld of velocities

v = v(x, t):

v(x, t) = v(x, t

) −

∇

p(x, τ )dτ.

The law of the propagation of an acoustic signal (4.41) can be introduced now in

the operator form, similarly to the law of elastic waves propagation (4.16): Lϕ = s,

here L ⇒ ∆ − [c

(x)]

−1

∂

, ϕ ⇒ p, s ⇒ ∇∇ · s.

4.3 Wave Electromagnetic Fields in Geoelectrics and Ionospheric

Sounding

The electromagnetic methods of the prospecting are based on the inﬂuence of the

medium parameters on the propagation of electromagnetic ﬁelds. The primary

electromagnetic ﬁelds can be excited, passing an alternating current through a loop

composed of many coils of a wire, or large loop. An occurrence of secondary electro-

magnetic ﬁelds is an echo of the medium. The arising ﬁelds can be registered with

the help of alternating currents induced by them in a receiving loop under inﬂuenc-

ing of an electromagnetic induction. The primary electromagnetic ﬁeld is diﬀused

from a generating loop to the receiver loop both above and under the ground sur-

face. These ﬁelds at the propagating in the presence of a homogeneous medium

have a small diﬀerence. But at the presence of a conductive body of a magnetic

component of the electromagnetic ﬁeld, penetrating on the medium, induces in the

body the eddy currents. The eddy currents generate a natural secondary electro-

magnetic ﬁeld, coming to a measuring device, which one ﬁxes the signals of primary

and secondary ﬁelds. These signals diﬀer both on a phase, and on an amplitude

from a signal produced by only the primary ﬁeld. Such distinctions between trans-

mitted and received signals of electromagnetic ﬁelds are conditioned by the presence

of a conductor and carry the information on its geometrical and electric properties

(Zhdanov et al., 1988).

The anomalous areas with a high conductivity produce strong secondary elec-

tromagnetic ﬁelds. Since 1970 years for electromagnetic sounding of the Earth the

heavy-lift impulsive hydromagnetics oscillators are used. One hydromagnetics oscil-

lator, allows geophysical mapping in vast territory and to reach penetration depths

about the ﬁrst tens kilometers in areas with high resistivity.

The exploring of the the ionosphere by means of electromagnetic methods has

also a great signiﬁcance. With the help of these methods the basic patterns of

The relationships of sounding signal and medium parameters 127

relationships of the structure of ionosphere are found.

The mathematical model of the electromagnetic signal propagation can be de-

rived on the basis of the Maxwell equations. Let us denote the basic physical laws

underlie the Maxwell system of equations.

From the Coulomb law

E = (q/|r|

)r,

using the superposition principle (E is a vector of an electric intensity; q is a quantity

of an electric charge) we can obtain:

∂V

E · d

σ = 4π

qdV. (4.42)

Applying the Gauss theorem

∂V

E · dσσ =

∇ · EdV,

we can write down the relation of (4.42) in the diﬀerential form:

∇ · E = 4πq. (4.43)

The experimental conﬁrmation of magnetic charges being absent (H is a curl

ﬁeld):

H · d

σ ≡ 0, (4.44)

and its diﬀerential representation:

∇ · H = 0. (4.45)

The generalized Faraday law (connection of an electromotive force in a loop

subtending a surface S, with the variation of a magnetic ﬂux through this surface);

in a Fig. 4.4 the diﬀerentials of vector quantities in the Stokes formula are shown:

E · dl = −

∂

∂t

H · dσ. (4.46)

In accordance with the Stocks theorem

E·dl =

(∇×E)·dσ, and we can overwrite

Fig. 4.4 The diﬀerentials of vector quantities from the Stokes formula.

128 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

(4.46) in the next

(∇ × E) · dσσ = −

∂

∂t

H · dσσ, (4.47)

and the diﬀerential form of the above expression is:

∇

∇ × E = −

∂

∂t

H. (4.48)

The Maxwell hypothesis consists in the assumption, that the circulation of a

magnetic ﬁeld is connected with the variation of the ﬂux (by analogy to the formula

(4.47)):

∇

∇ × H =

∂

∂t

E. (4.49)

The Oersted law (the connection of the charger’s motion with the originating of

a magnetic ﬁeld in a loop which enclose a current):

H · dl =

4π

j · dσ =

4π

J, (4.50)

and its diﬀerential shape:

∇

∇ × H =

4π

j. (4.51)

If to accept the Maxwell hypothesis, the analysis of expressions (4.49) and (4.51)

demonstrates, that the circulation of a magnetic ﬁeld is connected as to currents j,

and with the variation of E (“displacement current”)

∇ × H =

∂

∂t

E +

4π

j. (4.52)

Let’s note, that the Maxwell hypothesis (4.49) can be validated with applying of a

continuity equation (“electric charges do not arise and do not disappear”), i.e. the

variation of a charge inside the chosen volume V is connected only with a charge

stream through a covered surface of V :

∂

∂t

qdV = −

∂V

(qv)d

∆

= −

∂V

j · dσ = −

∇ · jdV, (4.53)

and in the diﬀerential form:

∂

∂t

q + ∇ · j = 0. (4.54)

If to use for the connection of the circulation of a magnetic ﬁeld and a current only

the Oersted law (4.50), that applying the operator of the divergence to the right

and left-hand sides of this expression, we write down:

∇ ·

∇

∇ × H =

4π

∇∇ · j.

Here the left-hand side of the equation is identically equal to zero, and the right-

hand side is not equal to zero. In accordance with the equation of the continuity

The relationships of sounding signal and medium parameters 129

(4.54), we should add the expression ∂/∂t q to the right-hand side of this equation.

So we have

∇

∇ ·

∇

∇ × H =

4π

∇

∇ · j +

∂

∂t

q =

4π

∇

∇ · j +

∂

∂t

4π

∇

∇ · E.

Here we have used the Coulomb law (4.42) for the latest transformation. The most

simple dynamic equations of electromagnetic ﬁelds can be introduced with the use

of the ﬁeld potential. Owing to (4.45) and the identity ∇ · ∇ × A ≡ 0, ∀A, vector

H can always be represented as:

H =

∇

∇ × A, (4.55)

where A is the vector potential. When A is used, (4.48) has a form

∇

∇ × E = −



∇

∇ ×

∂

∂t



, (4.56)

i.e.

∇∇ ×



E +

∂

∂t



= 0.

Taking into account the last equality and the identity ∇×∇∇ϕ = 0, ∀ϕ we can write

the next representation

E +

∂

∂t

A = −∇ϕ, (4.57)

where ϕ is the scalar potential.

We use formulas (4.43) and (4.52) for the determination A and ϕ. Substituting

the expression for E from (4.57) into (4.43), we obtain

∆ϕ = −4πq −

∂

∂t

∇∇ · A. (4.58)

Substituting (4.55) and (4.57) into the formula (4.52), we (by analogy) obtain:

∇

∇ ×∇

∇

∇ × A = −

∂

∂t

A −

∇

∂

∂t

ϕ +

4π

j. (4.59)

Using the identity ∇ × ∇ × A = ∇ · ∇ · A − ∆A we rewrite the formula (4.59) in

the form

∆A

∂

∂t

A = −

4π

j +

∇



∇

∇ · A +

∂ϕ

∂t



. (4.60)

At last, using a freedom in a choice of A and ϕ, we shall accept a following require-

ment:

∇

∇ · A +

∂ϕ

∂t

= 0

is the “Lorentz gauge”. In this calibration the expression (4.60) becomes a vector

wave equation:

∆A −

∂

∂t

= −

4π

130 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

and the expression (4.58) becomes a scalar wave equation:

∆ϕ −

∂

∂t

ϕ = −4πq.

Let us write the Maxwell equations (4.43) — (4.52) for an isotropic medium:

∇ · εE = 4πq, (4.61)

∇

∇ · µH = 0, (4.62)

∇

∇ × E = −µ

∂

∂t

H, (4.63)

∇

∇ × H =

4π

j + ε

∂

∂t

E. (4.64)

Here ε is a dielectric permittivity and µ is a magnetic permeability. It is a common

practice to supply the equations of (4.61)–(4.64) by the Ohm law

j = σE, (4.65)

where σ is an electrical conductivity.

In the uniform isotropic medium (σ = const, ε = const, µ = const). At the

absence of extraneous currents (j) and free charges (q) the ﬁelds E and H satisfy

the homogeneous telegraph equations

∆E − ε

∂

∂t

E − σ

∂E

∂t

= 0, (4.66)

∆H − µ

∂

∂t

H − σ

∂H

∂t

= 0. (4.67)

In connection with the problems of geoelectrics it is possible to represent quasi-

stationary areas (ﬁeld) (∂

/∂t

H  ∂/∂t H, ∂

/∂t

E  ∂/∂t E), so the telegraph

equations transform to the homogeneous diﬀusion equation

∆E − σ

∂

∂t

E = 0, (4.68)

∆H −

∂

∂t

H = 0. (4.69)

Let’s remind, as from the equations (4.49) and (4.52), it is possible to obtain the

connection between the energy and the vector of the energy ﬂux (Poynting’s vector).

Let’s implement a scalar multiplication of the equation (4.49) on H and the equation

(4.52) on E. And after the summation of these equations, we shall write

E ·

∂

∂t

E + H ·

∂

∂t

H = cE ·∇

∇

∇ × H − cH ·∇

∇

∇ × H − E · 4πj. (4.70)

Taking into account that E ·

∇

∇×H −H ·∇×E ≡ −c∇∇·(E ×H), we integrate both

parts of the expression (4.70) over the volume:

∂

∂t

(E · E) + (H · H)

8π

dV =

∇ ·



4π

(E × H)



dV −

j · EdV. (4.71)

The relationships of sounding signal and medium parameters 131

The equation (4.71) becomes the continuity equation for the conserved quantity E,

which has the physical sense of the energy:

∂

∂t

dE = −

∇ · RdV −

j · EdV,

where

dE =

+ H

8π

dV ; R =

4π

(E × H)

is an energy ﬂux vector (owing to

∇

∇ · RdV = −

∂V

(R · dσ)). The last expression

has a physical sense of losses on the Joule heat (at j = qv Coulomb force F = qE,

and the work in a unit time is equal F · v = qE · v = j · E).

Considering (4.65), we get in the similar way from (4.63), (4.64) the expression

for an isotropic medium:

∂

∂t

εE

+ µH

8π

dV = −

∂V

(R · d

σ) −

σE

dV,

i.e. the energy density is described by

εE

+ µH

8π

E · D + H · B

8π

Here the Poynting vector has the same form, as in the vacuum: R =

[c/(4π)] , E × H; the loss power on the Joule heat is given by:

j · E = (σE) · E = σE

The expressions introduced in this section, form the basis mathematical model of

the propagation of the electromagnetic wave ﬁelds in the vacuum and the isotropic

medium.

4.4 Atmospheric Sounding

The remote sensing is a signiﬁcant tool for the investigation of a structure and

dynamics of the ionosphere. So, in Fig. 4.5 the scheme of the sounding of the atmo-

sphere using a satellite (the geometry of the observation of a limb) is represented.

The received information is used for the improvement of the quality of a weather

forecast and development of methods of the artiﬁcial aﬀecting on it, for the study

of atmospheric pollution. The remote sensing of the atmosphere is grounded on

the phenomenon of the carrying of the solar radiation or the heat radiation, gener-

ated by the atmosphere, which has electromagnetic nature. The phenomenological

transport theory can be considered as an extreme case of the rigorous statistical

theory basing on the stochastic wave equation (Kravtsov and Apresyan, 1996). The

transport theory operates with the so-called photometric concept (the authors of

(Kravtsov and Apresyan, 1996) point out that the photometric quantities with the

132 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

Fig. 4.5 The schema of sounding of an atmosphere using satellite.

equal success can be called acoustic-metric) quantities). The applicability of the

phenomenological transport theory is ensured with following requirements: 1) the

wave ﬁeld appears as ray one, i.e. the requirements of the applicability of the geo-

metrical optics are carried out; 2) a complete incoherence of the rays (the additivity

of the transferred ﬁeld), i.e. the interference is excluded; 3) the observed value is a

time average and a space average process (i.e. the square quantities of wave ﬁelds

are considered); 4) the radiating is supposed stationary and ergodic.

Let us consider a radiant ﬂux dP at the point X, passing in the bodily cone dΩ

through the area element dσ in the direction nn (Fig. 4.6):

Fig. 4.6 Radiant ﬂux dP through area element dσ.

dP = ϕ(x, n)(n · d

σ)dΩ.

The quantity ϕ(x, n) is called the intensity or the brightness, i.e.:

ϕ(x, n) =

dΩdσ cos(

σ)

Let’s consider the physical bases for the obtaining of the integro-diﬀerential

transport equation. The variation of the radiation intensity (∆ϕ) on a small length