Reliability of Attenuation Properties Recovery for Viscoelastic Media

doi:10.4236/ojapps.2013.31B1017

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Open Journal of Applied Sciences, 2013, 3, 84-88

doi:10.4236/ojapps.2013.31B1017 Published Online April 2013 (http://www.scirp.org/journal/ojapps)

Reliability of Attenuation Properties Recovery for

Viscoelastic Media

Ekaterina Efimova, Vladimir Cheverda

The laboratory “Numerical methods of inversion of geophysical wavefiel ds”, A.A. Trofimuk Institute of Petroleum

Geology and Geophysics SB RAS. Novosibirsk, Russia

Email: [email protected]

Received 2013

ABSTRACT

The inverse problem of seismology for media with attenuation is considered in this paper. Generalized Standard Linear

Solid is used to describe viscoelastic media. In the numerical solution certain parameterizations can be coupled, it

means that true heterogeneity of the only one of parameters can be restored only as a perturbation of another. This is

why important to investigate reliability of parameters recovery. By using method based on diffraction patterns it is pos-

sible to see whether the parameters are coupled. Singular value decomposition was used to study the possibility of re-

covering the parameters in practice. It was investigated th e possibility of reconstructing of the d ensity, impedances and

attenuation properties. Coupling appears on the attenuation properties and impedances separately corresponding to the

P-wave and S-wave. It is also should be noted that coupling decreases with increasing frequency range and the condi-

tion num ber.

Keywords: Viscoelasticity; Seismic Attenuation; Inverse Theory; Wave Propagation; Si n gul ar Value Decomposi t i on;

Diffraction Patterns

1. Introduction

The main theme of the work is recovery of characteristics

of viscoelastic media. For this purpose was considered

numerical solution of two-dimensional inverse problem

of seismology using the information recorded in the re-

ceivers located on the surface of the Earth. It is known

that a numerical model of viscoelastic media describes

the geological structures, in particular hydrocarbon res-

ervoirs. And the attenuation properties depend on the

composit i on of the fluid.

But there appears coupling of parameters of the me-

dium in the numerical formulation of the problem that

will be discussed in this article. Coupling of parameters

means that the true heterogeneity of only one parameter

would restore as a perturbation of other in the numerical

solution. So if you use coupled parameters, yo ur solution

will be incorrect. Therefore, we need to find uncoupled

set of parameters before developing and implementation

of the algorithm.

Determination the possibility of uncoupled reconstruc-

tion of the parameters of a viscoelastic medium, such as

density, elastic impedances and attenuation properties is

the subject of this research.

This problem was considered in other papers, in par-

ticular, in [1], the problem was solved for the case of

media with a velocity close to a constant value. There

was also shown the inability simultaneous recovery of

the velocity and attenuation properties in viscoelastic

media without additiona l conditions [2].

2. Numerical Description of Viscoelastic

Media

State equation provides the relationship between stresses

and strains at the same time for ideal elastic media. But

this is not true in viscoelastic media, which possess at-

tenuation that caused by memory of the material. For

such media stress state depends on all past states of

strains and the state equation can be expressed with the

use of the generalized Hooke's law. Media with attenua-

tion mathematically can be described by the system of

equations:

/( )/2

(,)(,)/(,) .

kl ijkl

utdiv f

tuu

txGxt







 



 







 











 (1)

2.1. Generalized Standard Linear Solid

Numerical resolution of such integral-differential system

is very troublesome, so it is proposed to use Standard

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:



lllR l

Mtt









 









 

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







- constants, that are called relaxation times of

stresses and strains respectively; 2





- deformation modulus,

where

for

P-wave, , for S-wave,





- Lame parameters .





Using GSLS model quality factor in the frequency

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

22 2

111

(

QLL







)





 

 





















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, 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:

12, .



  















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







(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

1kk

m mm





 

where ia a - Frechet derivative of

the operator , is model of the medium on the

k-th step.

)(,

obs k

uBm



BDB



We mean that the parameters of the medium can be

expressed as the sum of the constant components 0



and small perturbations m



:0

mm m





 

uu ; then the

total wave field will be presented as 0u





 

, where



- propagating in a homogeneous medium wave, and





- 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 ,,{,, }









;

2) Density, velocities, parameters of attenuation

{,, },,

sps





;

3) Density, impedances, parameters of attenuation

{},,,,

sps





The best results were obtained for parameterization

{},,,,

sps





On Figure 1 are presented diffraction patterns for

parameterization {,,,, }

sps





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

E. EFIMOVA, V. CHEVERDA

(in terms of computer resources) problem. Considering of larger frequency range allows minimize

coupling. As well as increase of the condition number

reduces coupling (Figures 4-6).

SVD analysis was performed for the environment

model, which allows to determine the coupling between

parameters (Figure 2). To construct the matrix represen-

tation of the operator target area was covered with a grid.

Frequency range and positions of the receivers also were

divided into finite segments. As the basis were used

functions equal to one in the cell and zero outside. As a

result, finite sums approach the integrals in the operator,

that allows to get matrix approximation.

Because of the compactness of the operator it is pro-

posed to consider the properties of solutions, using the

truncated singular value decomposition [9]: it is consid-

ered r-solutions which are projections of the desired so-

lution to the linear combin ation to th e right sing ular vec-

tors. The number of r involved singular vectors controls

conditionality of the problem and allows you to build a

solution with acceptable accuracy. Figure 1. Diffraction patterns for the incident P-wave for

frequencies 25 Hz, 15 Hz, 10 Hz (blue, red, black c o lor s).

To construct the singular numbers of the operator we have

been considered different frequency ranges and different

grid spacing. Convergence to zero of singular values

(Figure 3) confirms the fact that the matrix is an ap-

proximation of a compact operator. With increasing of

frequency range curves become flatter as you can see

from the graphs, presented if the Figure 3. This indicates

that in the case of a larger frequency range, we will have

more vectors, to construct r-solutions, and thus increase

the accuracy of the results.

To analyze coupling of parameterization we have con-

structed projection on the singular vectors that corre-

spond to the largest singular numbers. Figures 4-6 show

that the density couldn’t be recovered, for other parame-

ters only the amplitude of the gap on th e boundary of the

area with the perturbed parameter. Coupling occurs be-

tween the parameters,



, and the parameters,



Figure 2. Mathematical model of the medium. Target area

is a square. The red part has perturbation of the S-wave

impedance, in blue – P-wave impedance; second case:

S-wave attenuation - in red and P-wave attenuation - in

blue.

Number of singular value, r

Figure 3. Singular values in the logarithmic scale for different grids(in metres) and frequency ranges (in Hz).

E. EFIMOVA, V. CHEVERDA 87

Figure 4. R-solutions for the model shown in Figure 2. In the upper layer - perturbation of dencity, the condition number is

10000 and a range of frequencies is (5.45) Hz.

Figure 5. R-solutions for the model shown in Figure 2. In the upper layer - perturbation of S-wave impedance, in the lower –

of P-wave impedance. In the first line of the condition number is 100 and the frequency range (5.30) Hz in the second - the

condition number is 1000 and a frequency range is (5.45) Hz.

Figure 6. R-solutions for the model shown in Figure 2. In the upper layer - perturbation of S-wave attenuation, in the lower –

of P-wave attenuation. In the first line of the condition number is 100 and the frequency range (5.30) Hz in the second - the

condition number is 1000 and a frequency range is (5.45) Hz.

E. EFIMOVA, V. CHEVERDA

Figure 7. The scaled trace recorded in the first receiver in Fig. 2. Assumed the presence of P-wave attenuation: Q = 200, 60,

40, 20, and also in the linearized case. Right picture is zoomed rectangle from the left picture.

The presented in Figure 7 seismogram tracks shows

that the graph for the linearized case with initial ap-

proximation acquisitions 0

0,0





 (the first ap-

proximation step of the viscoelastic medium) is most

similar to the graph for the quality factor Q = 200. This

indicates that even a linear formulation gives good results.

4. Conclusions

To study the linearized operator of the dynamic theory of

elasticity for the viscoelastic media were used methods

based on diffraction patterns [8] and on the SVD-analysis

[9]. The target area was chosen as a homogeneous me-

dium with two areas, each has perturbation of only one

parameter. This option provides an obvious determina-

tion of coupling o f parameterizat i o n .

Coupling between the elastic impedances and the

quality factors (corresponding to the P- and S-waves

separately) was found.

By using diffraction patterns and singular value de-

composition analysis was shown reducing coupling be-

tween parameters of viscoelastic medium in the case of

increasing of the frequency range. Also was shown reduc-

ing coupling between parameters in the case of increasing

of the condition number of the numerical problem.

It is concluded that the use of high-quality provides

simultaneous reconstruction of impedances and quality

factors. Study of an independent recovery of the parame-

ters of viscoelastic medium with use of poor-quality data

have planned for the future work.

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