E. EFIMOVA, V. CHEVERDA 85

Linear Solid (SLS) – superposition of Maxwell medium

and Kelvin-Voigt medium, to represent state equation in

a differential form. The degree of attenuation(see in [3])

of a viscoelastic material is given by a Quality Factor Q

(QF). Quality factor - the number of wavelengths a wave

can propagate through a medium before its amplitude

was decreased in times.

exp( )

It is known that a model that describes real geologic

medium has constant quality factor in dependence of

time frequency over the frequency range [4]. As one can

see SLS does not satisfy this condition, in contrast to the

further considered Generalized Standard Linear Solid

(GSLS) [5] – a combination of several SLSs. Hooke's

law in GSLS rewrites in the form of equations:

,

,

1

//

Lj

j

lllR l

Mtt

where L – is a number of SLS (further L=2);

,

ll

- constants, that are called relaxation times of

stresses and strains respectively; 2

R

M

- deformation modulus,

where

for

P-wave, , for S-wave,

R

M

- Lame parameters .

Using GSLS model quality factor in the frequency

domain can be rewritten as the equation (see [6]):

,

2

22 2

11

2

1

111

(

QLL

ll

l

l

ll

l

L

)

l

where the

is a frequency .

To determine the relaxation times in [6] it is proposed

to use

method by introducing the parameters of

attenuation - variables ,

p

P, S (corresponding to P-,

S-waves), that describe the level of attenuation in the

medium. It should be noted that if we know the parame-

ters of attenuation we have the only way to determine the

quality factor:

2

22

1

2

12, .

11

ps

ll

LL

ps

ll

l

QQ

2

l

So in further considerations we will use the parameters

of attenuation rather than Quality Factor because of sev-

eral advantages of the

method that is listed in [6].

2.2. Linearization

After application of GSLS and the

m

(DB

method system

of equations (1) can be regarded as a nonlinear opera-

tional equation: , where - the parame-

ters of the medium, - observation data, is an

operator from the space of models to the data space. It is

suggested to use Newton's method:

() obs

Bm u

obs

uB

1kk

)k

m mm

where ia a - Frechet derivative of

the operator , is model of the medium on the

k-th step.

)(,

obs k

uBm

BDB

k

m

We mean that the parameters of the medium can be

expressed as the sum of the constant components 0

m

and small perturbations m

:0

mm m

uu ; then the

total wave field will be presented as 0u

, where

0

u

- propagating in a homogeneous medium wave, and

u

- component generated by small perturbations of pa-

rameters. Small quantities of the second order are ig-

nored in the linearization.

3. Coupling Parameters

Coupling between parameters means that in solution of

the problem true heterogeneity of one parameter can be

mistakenly identified as heterogeneity of another pa-

rameter [7]. The presence of coupling indicates wrong

solution. Therefore, we will focus on methods identifying

coupling sets of parameters.

3.1. Diffraction Patterns

To determine the coupling was studied the method of

Tarantola [1986], based on the diffraction patterns [8].

Diffraction patterns show the amplitude of the wave,

scattered from a point target area, as a function of the

scattering angle. If diffraction patterns for different pa-

rameters has similar form it means coupling between

parameters. A good choice of parameters will give dif-

fraction patterns which are as different as possible.

The scattering diagrams were constructed for the sets

of parameters:

1) Density, Lamé parameters, parameters of attenua-

tion ,,{,, }

s

;

2) Density, velocities, parameters of attenuation

{,, },,

sps

VV

;

3) Density, impedances, parameters of attenuation

{},,,,

sps

II

.

The best results were obtained for parameterization

{},,,,

sps

II

.

On Figure 1 are presented diffraction patterns for

parameterization {,,,, }

sps

II

in the case of inci-

dent P-wave and reflected P-,S-wawes. Diffraction pat-

terns for the parameters of attenuation and impedances

corresponding to P-or S-wave are similar in shape, that

indicates their coupling. Coupling may decrease when

considering frequency range, as there is different de-

pendence amplitude of the frequency for various pa-

rameters (Figure 1).

3.2. Singular Value Decomposition

To study the coupling and the possibility of recovery

parameters is also used a method based on singular value

decomposition (SVD) of a compact operator of the prob-

lem. Computation of singular value decomposition for an

arbitrary environment is very complicated and expensive

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