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
Received October 20, 2011; revised November 22, 2011; accepted December 14, 2011
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
We proposed a mechanism of elastic waves propagation
in rocks  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  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-
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:
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
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-
opyright © 2012 SciRes. IJG
V. SIDOROV ET AL.
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 :
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
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-
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
Simple transformations lead us to:
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
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
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.
Copyright © 2012 SciRes. IJG
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-
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
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:
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.
Copyright © 2012 SciRes. IJG
V. SIDOROV ET AL.
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-
 V. K. Sidorov Possible Mecha-
fective boundaries. The reflected signal could be gained
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 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
quence of phonon-phonon mechanism of elastic wave
propagation in rocks. Its development gave a distinctly
different from the others model for geologic stratum ap-
proximation—the model of medium with effective
boundaries. Its features are clearly seen when compared
with the layered medium. First of all, in the layered me-
dium all layer borders are caused by change in velocity
(or impedances) while effective boundaries have differ-
ent physical nature and are caused by heterogeneity. It is
heterogeneity of rocks that makes elastic wave to loose
its energy by way of scattered phonons. Abrupt change
of the heterogeneity parameter, or porosity factor in our
case, causes the appearance of effective boundary. Sec-
ond, in layered medium the value of reflection coefficient
for every boundary does not depend on frequency. De-
pendence on frequency appears only for a system of such
boundaries. Effective boundaries (its depths and coeffi-
cients) in general do depend on frequency a priori be-
cause the algorithm for their construction involves the
wavelengths at the averaging stage. Third, the layered
medium being sequentially applied to the system of lay-
ers gives a resonant system [9,10]. This fact complicates
modeling so some methods of regularization are involved.
For example, only primary reflections could be used and
system . The model with effective boundaries in this
sense is more stable from the start.
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Copyright © 2012 SciRes. IJG