Mechanism of Elastic Waves Reflection in Geological Media

doi:10.4236/ijg.2012.31019

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International Journal of Geosciences, 2012, 3, 175-178

http://dx.doi.org/10.4236/ijg.2012.31019 Published Online February 2012 (http://www.SciRP.org/journal/ijg)

Mechanism of Elastic Waves Reflection in Geological Media

Viktor Sidorov, Michail Tarantin

Organization of Russian Academy of Sciences, Mining Institute Ural Branch RAS, Perm, Russia

Email: [email protected]

Received October 20, 2011; revised November 22, 2011; accepted December 14, 2011

ABSTRACT

Reflecting properties of layered geological media are substantiated in the framework of phonon-phonon mechanism of

elastic wave propagation in porous media. In this scope the reflectio n coefficient is calculated using not impedances bu t

impulses of phonons in adjoining porous media. Assuming for the first approximation that rocks do fulfill an average

time equation we got an expression for the reflection coefficient via porosity factors of that geological medium. For

calculation of reflection coefficient the wavelength is chosen as averaging line scale. These coefficients are calculated at

every depth point for a set of frequencies in seismic range. Resulting curves have special depth points. Being cross-

plotted in time-frequency space such points do form coherent units. These units we call effective boundaries, because

they cause all reflections for the given media in the framework of considered model. Effective boundaries are not

wide-band as for two half spaces but have a cutoff at some low frequency. Geological medium at a whole is character-

ized by the system of such effective boundaries that are capable to form a reflection waves field. To construct this field

an algorithm is developed that solves the direct problem of seismic in the framework of effective boundaries theory.

This algorithm is illustrated with vibroseis survey modeling for a specific geological section.

Keywords: Elastic Waves; Phonons; Heterogeneous Media; Reflection Coefficient; Direct Seismic Problem;

Effective Seismic Boundaries

1. Introduction

We proposed a mechanism of elastic waves propagation

in rocks [1] that ju stified an independence of logarithmic

decrement on frequency or, equivalent, justified an ab-

sence of velocity dispersion when an attenuation does

take place. The problem of decrement arose long time

ago [2] but is still not solved and the proposed mecha-

nism as far as we know is unique. That is why we do not

consider many models of elastic waves propagation in

heterogeneous media among which the Biot model [3,4]

is the most popular. This model describes the elastic

waves propagation in the media with the so lid matrix and

many recent works attend to it [5-7]. But this model does

not comply with the frequency independence of decre-

ment.

Proposed mechanism supposes the elastic wave propa-

gation in heterogeneous media to be the process that is

attended and supported by phonon-phonon interactions

of a special sort. At that the correspondence is proved of

some phonon impulses and their combinations with elas-

tic wave parameters. Justification is based on a specific

type of heterogeneous medium-porous fluid-saturated

rocks with an average ti me equation fulfilled . Th ese were

the media where an experimental testing of a proposed

mechanism was allowed. For that purposes laboratory

data and acoustic logging material in carbonate forma-

tions were analyzed [1,8].

The aim of the paper is to substantiate the reflection

properties of layered geological media in the framework

of phonon-phon on mechanism and to create on that basis

an algorithm for solving the direct seismic problem.

2. The Reflection Coefficient Derivation

For a special type of porous fluid-saturated media the

longitudinal wave propagation is attended and supported

by phonon interactions that are described by an impulse

conservation law expression:

mPmfl P

hmhmhh

VV VV







(1)

In the expression above m denotes the porosity coeffi-

cient, VPm—velocity of longitudinal waves in solid ma-

trix, Vfl—velocity in the fluid, Vp—velocity in th e porous

medium, h is the Plank constant and v is the phonon fre-

quency. It is stated in mechanism that phonon’s veloci-

ties are equal to that of waves in the co rresponding com-

ponents of the porous medium .

Phonons with impulses

mh V



and fl

mh V



are

responsible for wave energy dissipation because they are

directly proportional to porosity m and frequency v and

correspond to attenuation coefficient in porous fluid-sa-

V. SIDOROV ET AL.

176

turated media with the average time equation fulfilled.

Such kind of media in turn has an attenuation coefficient

direct proportional to wave porosity and frequency f [1]:

m



We supposed that reflection properties of a layered

geological medium could be gained in the framework of

phonon-phonon mechanism. Let’s consider from this

point of view the reflection coefficient for two-media

boundary. In the common form it could be written as

Rpp







pV



(2)

where p is the impulse, that turns a production of density

and velocity for elastic waves and is well-

known as impedance. But this time p is the phonon im-

pulse: phV



Let’s consider two adjoined media with porosities m1

and m2. Impulses in the reflection coefficient (2) we ex-

press from left part o f (1) with poros ities m1 and m2. And

so we got

Pm PmflPm

mh mhm

VV VV

Rmh mhm

VV VV



 







 





2 2

Pm fl

h mh

V V

h mh

V V

 



 

 

 



 

 













(3)

Simple transformations lead us to:

()

fl Pm

mV V







mmm 

pV

R (4)

where and V–1 is the average interval time

for two media.

So we have an expression for reflection coefficient for

a boundary of two porous media based on the expression

for phonon interaction. It should be marked that exactly

the same result could be gained if the reflection coeffi-

cient (2) would be expressed via impulses



when

neglecting density changes an d assuming that in adjo ined

media velocities are calculated using the average time

equation. But such a definition would be tautological

because for porous media the average interval time ex-

pression used in this approach is the consequence of the

phonon impulse conservation law.

The next step is to use the coefficient R to determine

the reflection properties of a layered geological media.

We suppose from the beginning that rocks could be clas-

sified as reference medium-porous fluid-saturated with

average time equation fulfilled. Let’s calculate the ex-

pression (4) for a moving window along the medium

cross-section. At the boundary of two spaces this func-

tion will have an extremum corresponded to the real re-

flection coefficient for that media boundary (Figure

1(a)). For a single layer calculated function has a shape

similar to that shown in Figure 1(b). Two kink points

mark the layer boundaries and the corresponding reflec-

tion coefficients. As a size of moving window it is natu-

rally to take a wavelength because the phonon-phonon

interaction is corresponded to the specific elastic wave

frequency. And then an algorithm of calculations would

be as follows.

The log-curve of an interval time (reversal velocity) is

transformed from depth-scale into time-scale. The value

of frequency fn is set. Along the geological section with a

special step moves the interval of a wavelength size with

the reference point at the center. Above and below the

reference point the function (4) is calculated with m be-

ing equal to difference of porosity factors for upper and

(a) (b)

Figure 1. The view of re flection co efficient cur ves near two half-space boundary (a) and near thin layer (b); dT: interval time

of wave, R: reflection coefficient.

V. SIDOROV ET AL. 177

lower half-interval and V–1—an average interval time for

all wavelength interval. Such a way the reflection func-

tion R(fn,t) is calculated. For a general randomly layered

geological medium the corresponding curve have ex-

treme-points and kink-points. Such curves are calculated

for a count of frequencies within a selected seismic range.

For every curve all special points are registered: frequen-

cies, times and coefficient values. Being plotted in the

(fn;t) plane these points make separate boundaries. Let’s

call these boundaries effective ones. These boundaries

accumulate seismic reflections of a layered stratum of

rocks for a selected frequency band and this fact clears a

distinctly new approach to reflected waves field calcula-

tion.

3. Direct Seismic Problem Solving Using

Effective Boundaries Approach

Let’s demonstrate the process of signal calculation using

effective boundaries approach with the specific example

of vibroseis method. Geological section is presented by

carbonate rocks with an embedded unit of terrigenous

sediments (Figure 2). Vertical scale is time in ms and the

start time is taken by convention. Information about ve-

locity is given as log curve of longitudinal waves inverse

velocities. Using this curve and assuming that consider-

ing rocks do fulfill an average time equation parameters

for R(fn,t) calculation are determined. These functions are

calculated for series fn of frequencies within seismic band,

and one for f = 50 Hz is presented as an example in the

corresponding c ol umn (see Figure 2).

All curves R(fn,t) are examined for extremes and kinks.

These points being plotted in frequency-time coordinates

with mark-size depending on the reflection coefficient

value make coherent units as can be seen in the next co-

lumn in Figure 2. These coherent units were defined above

as effective boundaries. Such kind of boundaries in gen-

eral are not wide-band like in the case of two half spaces

but are limited in frequency especially in low-frequency

range. Every boundary has its own frequency dependence

of reflection.

Representation of cross-section model as a chain of

effective boundaries allows the calculation of reflections

from the given geological section to be simply enough:

the result would be the composition of reflections from

every one of effective boundaries. Due to the special

properties of these new objects—variations of reflection

coefficient with depth (or time) and frequency—it is

convenient to calculate the reflection function not for

every boundary but for the section at once. For that pur-

pose for every frequency fn within the selected seismic

band the geometrical sum of all reflected signals allowed

for that frequency is calculated. The differences in the

registration times of reflected wavelets are accounted in

the phase part of the complex coefficient while the real

part is responsible for the amplitude of signal regarding

its sign. As a result, the medium under consideration is

characterized by the spectral function:

( )exp()

nii

Rfa j (5)



where i accounts for all effective boundaries of the stra-

tum at the frequency fn.

Figure 2. Calculation of cross-correlation function (CCF) of reflected signal for vibroseis method; Lith: lithological section,

dT_p: inverse velocity of longitudinal waves, R50: reflection coefficient for a frequency 50 Hz, Ef_Brd: selected extremes of

reflection function, CCF: cross-correlation function of source sweep and reflected signal, CCF_F: pro ce s sed f iel d seismic trace.

V. SIDOROV ET AL.

178

This function accumulates reflection properties of all this gives us not resonant but more stable interference

ple in the Figure 2 the source signal is

4. Conclusions and Discussion

s the conse-

REFERENCES

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by convolution of this function and a signal accepted for

a radiated one. If the radiated signal is sweep then the

result of convolution should be correlated to that sweep

in addition.

In the exam

eep with frequencies 14 - 100 Hz. The result of calcu-

lations using presented algorithm is shown in the 5-th

column; the most right column demonstrate fragment of

seismic trace gained in the standard processing of field

data in the framework of vibroseis method. It is seen that

modeled signal does not conflict with the real one and

has its main features.

Described mechanism of reflection arises a

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