Evaluation of geophone ground coupling using geophone/hydrophone comparison
Ground coupling defines the transfer function from ground motion to geophone motion. Ground coupling can be described using a variety of models, which must all be adequately parametrized before being used on data. The ocean bottom seismometers and ocean bottom cables are the devices confronted to gr...
Gespeichert in:
| Datum: | 2015 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут гідромеханіки НАН України
2015
|
| Schriftenreihe: | Акустичний вісник |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/116243 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Evaluation of geophone ground coupling using geophone/hydrophone comparison / R. Estival, D. Léandri, Y. Lacroix // Акустичний вісник — 2015. —Т. 17, № 2. — С. 51-57. — Бібліогр.: 7 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-116243 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1162432017-04-23T03:02:49Z Evaluation of geophone ground coupling using geophone/hydrophone comparison Estival, R. Léandri, D. Lacroix, Y. Ground coupling defines the transfer function from ground motion to geophone motion. Ground coupling can be described using a variety of models, which must all be adequately parametrized before being used on data. The ocean bottom seismometers and ocean bottom cables are the devices confronted to ground coupling problems. Besides the velocity measurements made via geophones, the mentioned two types of devices allow pressure measurements due to presence of hydrophones. In this paper, we introduce a method that searches iteratively for the velocity boundary conditions which, when applied to a finite element model, allow the simulation results to fit with pressure data measured by the hydrophones. The iterative process is supported by a genetic algorithm. The obtained velocity values provide the evaluation of ground coupling of the geophones. В этой статье предложен метод итерационного поиска граничных условий для скоростей, которые, будучи примененными в конечно-элементной модели, позволяют согласовать результаты моделирования с данными о давлении, измеренном с помощью гидрофонов. Итерационный процесс реализован в виде генетического алгоритма. Полученные значения скорости обеспечивают оценку сцепления сейсмоприемников с грунтом. У цій статті запропоновано метод ітераційного пошуку граничних умов для швидкостей, які, будучи застосованими в скінченно-елементної моделі, дозволяють узгодити результати моделювання з даними про тиск, виміряний за допомогою гідрофонів. Ітераційний процес реалізовано у вигляді генетичного алгоритму. Отримані значення швидкості забезпечують оцінку зчеплення сейсмоприймачів із грунтом. 2015 Article Evaluation of geophone ground coupling using geophone/hydrophone comparison / R. Estival, D. Léandri, Y. Lacroix // Акустичний вісник — 2015. —Т. 17, № 2. — С. 51-57. — Бібліогр.: 7 назв. — англ. 1028-7507 http://dspace.nbuv.gov.ua/handle/123456789/116243 534.1 en Акустичний вісник Інститут гідромеханіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Ground coupling defines the transfer function from ground motion to geophone motion. Ground coupling can be described using a variety of models, which must all be adequately parametrized before being used on data. The ocean bottom seismometers and ocean bottom cables are the devices confronted to ground coupling problems. Besides the velocity measurements made via geophones, the mentioned two types of devices allow pressure measurements due to presence of hydrophones. In this paper, we introduce a method that searches iteratively for the velocity boundary conditions which, when applied to a finite element model, allow the simulation results to fit with pressure data measured by the hydrophones. The iterative process is supported by a genetic algorithm. The obtained velocity values provide the evaluation of ground coupling of the geophones. |
| format |
Article |
| author |
Estival, R. Léandri, D. Lacroix, Y. |
| spellingShingle |
Estival, R. Léandri, D. Lacroix, Y. Evaluation of geophone ground coupling using geophone/hydrophone comparison Акустичний вісник |
| author_facet |
Estival, R. Léandri, D. Lacroix, Y. |
| author_sort |
Estival, R. |
| title |
Evaluation of geophone ground coupling using geophone/hydrophone comparison |
| title_short |
Evaluation of geophone ground coupling using geophone/hydrophone comparison |
| title_full |
Evaluation of geophone ground coupling using geophone/hydrophone comparison |
| title_fullStr |
Evaluation of geophone ground coupling using geophone/hydrophone comparison |
| title_full_unstemmed |
Evaluation of geophone ground coupling using geophone/hydrophone comparison |
| title_sort |
evaluation of geophone ground coupling using geophone/hydrophone comparison |
| publisher |
Інститут гідромеханіки НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/116243 |
| citation_txt |
Evaluation of geophone ground coupling using geophone/hydrophone comparison / R. Estival, D. Léandri, Y. Lacroix // Акустичний вісник — 2015. —Т. 17, № 2. — С. 51-57. — Бібліогр.: 7 назв. — англ. |
| series |
Акустичний вісник |
| work_keys_str_mv |
AT estivalr evaluationofgeophonegroundcouplingusinggeophonehydrophonecomparison AT leandrid evaluationofgeophonegroundcouplingusinggeophonehydrophonecomparison AT lacroixy evaluationofgeophonegroundcouplingusinggeophonehydrophonecomparison |
| first_indexed |
2025-07-08T10:04:58Z |
| last_indexed |
2025-07-08T10:04:58Z |
| _version_ |
1837072750613626880 |
| fulltext |
ISSN 1028 -7507 Акустичний вiсник. 2015. Том 17, N 2. С. 51 – 57
UDC 534.1
EVALUATION OF GEOPHONE GROUND COUPLING
USING GEOPHONE/HYDROPHONE COMPARISON
R. E S T IVA L1, D. L ÉA N DR I2,4, Y. LA CR O IX3,4
1Total, Pau, France
E-mail: remi.estival@gmail.com
2IGS, UMR 7256, Luminy, Marseille, France
E-mail: joba@club-internet.fr
3MEMOCS, Università Dell’Aquila, Cisterna di Latina, Italy
E-mail: yves.lacroix@univ-tln.fr
4SEATECH, Avenue Georges Pompidou,
83162 La Valette du Var Cedex, France
Received 03.08.2015
Ground coupling defines the transfer function from ground motion to geophone motion. Ground coupling can be described
using a variety of models, which must all be adequately parametrized before being used on data. The ocean bottom
seismometers and ocean bottom cables are the devices confronted to ground coupling problems. Besides the velocity
measurements made via geophones, the mentioned two types of devices allow pressure measurements due to presence
of hydrophones. In this paper, we introduce a method that searches iteratively for the velocity boundary conditions
which, when applied to a finite element model, allow the simulation results to fit with pressure data measured by
the hydrophones. The iterative process is supported by a genetic algorithm. The obtained velocity values provide the
evaluation of ground coupling of the geophones.
KEY WORDS: geophone, ground coupling, hydrophone, velocity boundary conditions, acoustic pressure, iterative process,
genetic algorithm
Ступiнь зчеплення з грунтом визначає передавальну функцiю, яка пов’язує змiщення грунту iз змiщеннями
сейсмоприймача (геофони). Зчеплення з грунтом може бути описане з використанням рiзних моделей, якi повиннi
бути належним чином параметризованi перш, нiж використовувати їх з даними вимiрювань. Океанськi доннi
сейсмометри й океанськi доннi кабелi є пристроями, для яких гостро стоїть проблема оцiнки зчеплення з грунтом.
Окрiм вимiрювань швидкостi, якi проводяться за допомогою сейсмоприймачiв, зазначенi два типи пристроїв
дозволяють вимiрювати тиск завдяки наявностi гiдрофонiв. У цiй статтi запропоновано метод iтерацiйного пошуку
граничних умов для швидкостей, якi, будучи застосованими в скiнченно-елементної моделi, дозволяють узгодити
результати моделювання з даними про тиск, вимiряний за допомогою гiдрофонiв. Iтерацiйний процес реалiзовано
у виглядi генетичного алгоритму. Отриманi значення швидкостi забезпечують оцiнку зчеплення сейсмоприймачiв
iз грунтом.
КЛЮЧОВI СЛОВА: сейсмоприймач, зчеплення з грунтом, гiдрофон, граничнi умови для швидкостей, акустичний
тиск, iтерацiйний процес, генетичний алгоритм
Степень сцепления с грунтом определяет передаточную функцию, связывающую смещения грунта со смещениями
сейсмоприемника (геофона). Сцепление с грунтом может быть описано с использованием различных моделей,
которые должны быть надлежащим образом параметризованы прежде, чем использовать их с данными измерений.
Океанские донные сейсмометры и океанские донные кабели являются устройствами, для которых остро стоит
проблема оценки сцепления с грунтом. Помимо измерений скорости, проводимых с помощью сейсмоприемников,
указанные два типа устройств позволяют измерять давление благодаря наличию гидрофонов. В этой статье
предложен метод итерационного поиска граничных условий для скоростей, которые, будучи примененными в
конечно-элементной модели, позволяют согласовать результаты моделирования с данными о давлении, измеренном
с помощью гидрофонов. Итерационный процесс реализован в виде генетического алгоритма. Полученные значения
скорости обеспечивают оценку сцепления сейсмоприемников с грунтом.
КЛЮЧЕВЫЕ СЛОВА: сейсмоприемник, сцепление с грунтом, гидрофон, граничные условия для скоростей, аку-
стическое давление, итерационный процесс, генетический алгоритм
INTRODUCTION
Seismic and seismology need accurate
measurement of soil motion. Geophones and
hydrophones are designed to produce a linear
response to velocity or to pressure in accordance to
the ground motion.
In practice, the transfer function of the geophone
is never perfectly linear. Firstly, because of the
geophone itself which response depends on its
resonance frequency. Secondly, because of the so-
called geophone ground coupling term that descri-
bes the transfer function from ground motion
to the output voltage of the geophone. Ground-
coupling affects seismic exploration and monitoring.
Papers [1,2] and others have shown that the effect of
c© R. Estival, D. Léandri, Y. Lacroix, 2015 51
ISSN 1028 -7507 Акустичний вiсник. 2015. Том 17, N 2. С. 51 – 57
geophone coupling may play a prominent role even
for frequencies below 100 Hz.
The two main causes responsible for ground
coupling include the disturbances in the wave-field
influencing ground motions due to the presence of
the device and disturbances resulting from contact
conditions between the device and the ground, also
referred to as weight coupling.
A very general definition is used for ground coupli-
ng:
C(w) =
Vgeophone(w)
Vground(w)
. (1)
Different models of ground coupling have been
developed. Some resolve the equation of motion by
modeling the geophone and the ground as a spring
and mass system like in [3]. Other resolve a weak form
equation of motion as in [4]. These models explain
ground coupling mechanism but depend on tests to
be used on exploration data, which is usually the only
available information.
In marine acquisition, geophones are mainly
used in Ocean Bottom Seismometers and Ocean
bottom cables. These devices often comprise three
geophones and one hydrophone. Our method
compares hydrophone and geophone data to esti-
mate the amplitude spectra of vertical and horizontal
coupling coefficients.
We will start our discussions with the assumpti-
ons required by our method. We will then explain
its principles, describe the finite element model and
the iterative process on which it rests, and we
will conclude this paper with results obtained on
synthetic data.
1. HYPOTHESIS
Geophone ground coupling has various dependenci-
es to seabed parameters as buried depth, contact
surface, seabed density and type, etc. Measuring
those parameters is often complicated and expensi-
ve. To avoid these issues, we have developed a
method that relies on hydrophone and geophone data
comparisons.
To make this comparison possible, we first assume
that the device’s transfer function can be neglected
and that hydrophones, due to their two opposite
ceramics that attenuate acceleration, are not affected
by ground coupling.
We also assume that seabed can be modeled by
an elastic solid with a perfect fluid interface, and we
consider that seismic waves are plane waves. These
assumptions are often used in seismic like in [5].
In these conditions, vertical stress and pressure are
opposite at fluid/solid interfaces and the velocity and
the pressure have the same time dependency in eiwt.
2. GENERAL APPROACH
In “perfect” coupling conditions, when ground
velocity is equal to geophone velocity, the OBS
should behave as a volume of sediment. If we
model this “perfectly” coupled OBS and apply
geophone data as velocity boundary conditions, si-
mulation pressure should be equal to hydrophone
measurements. Otherwise, coupling is involved.
By modifying iteratively geophone data using wei-
ghting on selected frequency bands, we can produce
new boundary conditions and significantly reduce
differences between the simulated pressure and the
hydrophone data.
Comparison between these modified geophone data
and geophone measurements allows us to evaluate
ground coupling.
3. THE FINITE-ELEMENT MODEL
The finite element method has been used for years
within the industry and has been described extensi-
vely. For a good review of this method, the reader is
referred to the work of [6]. This section focuses on
describing our modeling choices.
OBS measures its own motion which is supposed to
be similar to ground motion. In optimal conditions,
the OBS should behave like a volume of sediment
; a volume whose size can be considered negligible
compared to the wave length of the measured signal,
to acknowledge the measure as a point.
The OBS is therefore modeled as an elastic solid
moving at ground motion velocity with its upper
surface behaving as fluid/solid interface. Our model
uses velocity as boundary conditions with pressure
calculated from its upper surface.
The finite element method requires discretization
in both space and time as well as the characterization
of the material behavior:
• To reduce computational time, the OBS is
represented as a cube formed of 32 linear
tetrahedrons. Time discretization rests on a
backward Euler method which discrete time usi-
ng consideration on the future step
∆t = tn+1 − tn, (2)
where ∆t is the time step that must be suffi-
ciently small to assume the system response
approximatively constant during [tn, tn+1]. This
method gives an approximation f(x(t) at each
52 R. Estival, D. Léandri, Y. Lacroix
ISSN 1028 -7507 Акустичний вiсник. 2015. Том 17, N 2. С. 51 – 57
time step using its value at the interval endpoint
f(xn+1):
xn+1 = xn + ∆tf(xn+1). (3)
• As an elastic and homogeneous solid, the
mechanical behavior of the OBS is described as
follows:
σ=2µε+λ tr(ε) I , ε=
∇v(x)+∇T v(x)
2
. (4)
• Our model uses a weak form based on a di-
splacement formulation:
u(x, t) ∈ S ∀v(x) ∈ Ω
∫
Ω
ρü(x, t)v(x)dΩ +
∫
Ω
σε∗dΩ =
=
∫
∂Ωf
f(x, t)v(x)dS.
(5)
We then apply boundary conditions to our model:
• On the vertical and the inferior boundaries of the
model, we apply a velocity which, in our method,
either is the velocity measured by the geophone,
or is a modification of this velocity.
• As a fluid/solid interface, the upper surface
∂Ωupp assumes no shear:
σxz = σyz = 0. (6)
These choices allow the construction of a linear
system that we solve using the Newton method.
The simulation pressure can then be obtained at
the upper surface by relating the vertical stress and
the pressure at a fluid/solid interface using:
P = −σzz. (7)
4. VELOCITY BOUNDARY CONDITIONS
As a test, a first simulation is done using geophone
data for the velocity boundary conditions.
We evaluate the results using a method of the least
squares (MLS):
C(ajpj, P ) =
n
∑
i=0
N
∑
j=0
ajpj(i)
P (i)
2
n
∑
i=0
N
∑
j=0
ajpj(i)
2
n
∑
i=0
P 2(i)
. (8)
Values of C(ajpj , P ) closest to 1 indicate close
approximations to hydrophone data.
As pressure results usually diverge from the
hydrophone, we search for new velocities to apply to
the model.
Modification of boundary conditions
According to our hypothesis, with perfect coupli-
ng, the calculated pressure should fit the hydrophone
data. When it doesn’t, coupling is involved. We
must then find better velocity boundary conditi-
ons to obtain a new pressure that assumes better
resemblance.
Coupling has often been described as a resonance
phenomena lightly influenced by phasing [1]. We
decide not to modify the phasing to operate on
the geophone amplitude spectra only. This operati-
on divides each geophone signal in 10 frequency
bands using a forward and backward discrete Fourier
transform:
S(x) =
N−1
∑
n=0
s(n)e−2πkn/N for 0 ≤ k < N, (9)
where S(x) is the geophone signal in the frequency
domain and s(n) in the time domain.
Frequencies out of the band of interest are di-
scarded and the selected 10 frequency bands are wei-
ghted. An inverse Fourier transform brings the signal
back in the time domain:
s(n) =
1
N
N−1
∑
n=0
10
∑
i=1
aiSi(k)e2πkn/N (10)
with ai =0 for if <k<(i−1)f ; f is the size of the
frequency band and ai is the weighted coefficient of
the i-th frequency band.
Three new velocities are then created (one per
geophone) and are used as new boundary conditions
for a new simulation.
We now need to find the right coefficients for the
simulated pressure to be close enough to the pressure
measured by the hydrophone.
Weighting coefficients
Searching randomly for a solution would mostly be
inefficient and even searching over a space of finite
possibilities would rather be impossible due to the
time needed to cover a space containing a sufficient
amount of possibilities to provide a suitable solution.
We therefore select an iterative process based on a
genetic algorithm [7]. Genetic algorithm have strong
convergence properties and are often used to select a
R. Estival, D. Léandri, Y. Lacroix 53
ISSN 1028 -7507 Акустичний вiсник. 2015. Том 17, N 2. С. 51 – 57
Fig. 1. Outline of our method
solution in important value sets. Genetic algorithms
copy natural selection which can be summarized in
four steps:
• creation of a population: we create six indi-
viduals, each containing 30 weighting coeffici-
ents and three geophone datasets divided in 10
frequency bands.
• selection: a simulation pressure is calculated for
each individual and compared to the hydrophone
signal using MLS. Two individuals are then
chosen randomly with a greater probability for
individuals with MLS values closest to 1 to be
selected.
• reproduction: two new individuals are created
from the two selected individuals sharing half of
the weighting coefficients of each parents.
• mutation: each new individual has a 10 % chance
to undergo random modification for one of his
coefficients.
New individuals replace the two worst individuals in
the population. The process is then repeated with the
new population and ends when the desired MLS or
an iteration number is reached.
Our method is summarized in fig. 1.
5. GEOPHONE GROUND COUPLING
Once the new velocities are found, ground coupling
coefficients C(t) can be calculated using:
C(w) =
Vgeophone(w)
Vmodified(w)
. (11)
54 R. Estival, D. Léandri, Y. Lacroix
ISSN 1028 -7507 Акустичний вiсник. 2015. Том 17, N 2. С. 51 – 57
6. NUMERICAL RESULTS
We present here the results from four different
datasets. These datasets are synthetic data obtai-
ned using a software developed in SEATECH that
calculates the amplitude and the propagation time of
different reflections of a wave propagating in a strati-
fied media. This software does not take into account
multiple reflections.
With this software we have produced a unique
dataset of 2.5 s duration, on which we have applied 4
different coupling coefficients. These coupling coeffi-
cients are calculated using the G. G. Drijkoningen
model [4].
Coupling coefficients are calculated for a cylinder
of 3 cm radius, 60 cm length, an OBS density of
4700 kg/m3 and a soil density of 1700 kg/m3. The
signal created lasts 2.5 s and is sampled at 250 Hz. It
has been shown that horizontal and vertical coupli-
ng often differ. Even if G. G. Drijkoningen’s theory
is only made for vertical motion, we also apply that
coupling on horizontal components. Our purpose is to
show the capacity of our approach to recover coupling
coefficients.
Dataset 1 has a low coupling contribution over
its horizontal component (obtained using G. G. Dri-
jkoningen’s elastic model with vp =800 m/s and
vs =400 m/s) and a strong one applied to the vertical
component (with vp =200 m/s and vs =100 m/s).
Dataset 2 is similar to dataset 1 except that the
same low coupling contribution is applied to the verti-
cal component and the strong contribution to the
horizontal one.
On dataset 3, the cylinder has the same coupling on
both components (vp =200 m/s and vs =100 m/s).
Dataset 4 is obtained using strong coupling
contributions on both components (with hori-
zontal coupling coefficient obtained for vp =200 m/s
and vs =100 m/s, and vertical coupling coeffici-
ent obtained for ρ=2200 kg/m3, vp =800 m/s and
vs =80 m/s).
After applying our method on these datasets, we
study frequencies in the 50 to 150 Hz domain.
Fig. 2 displays the amplitude spectra of the
coupling coefficient applied to the dataset and
those recovered using our method. Results show
erratic variations all over the studied frequencies. On
low coupling contributions, differentiating between
artefacts and coupling effects becomes challenging.
Genetic algorithms are not determinist and using
them on a short 2.5 s signal and a simple run of
our method certainly explains most of these artefacts.
Still, our method seems to enhance strong coupling
contributions (cf. fig. 2, b, c, e, f, g and h).
7. DISCUSSIONS
Our method suffers from two main limitations:
• the first is the link to the variations that may be
introduced in the results and can complicate the
interpretation of the coupling contribution. A
better accuracy could be obtained by working on
a longer signal and multiplying simulations with
various configurations (changing the number of
coefficients, the size of the frequency bands) to
combine results and find a trend.
• the second limitation is the computational time.
The finite element method is computationally
intensive even with our simple model. With a
1.4 GHz core and 4 Gb RAM, it takes close
to 48 hours to produce the results displayed
in fig. 2. Improving the calculation time should
also improve the method accuracy allowing the
method to search in a bigger space of solution.
This could be done by parallelizing the code and
using analytical or empirical calculation instead
of the finite element model. We have already
made some trials that divide the computational
time by 60, but further tests are still required.
CONCLUSIONS
In this paper, we have presented a new method whi-
ch evaluates geophone ground coupling by compari-
ng geophone and hydrophone data. The method
searches the velocity boundary conditions of a finite
element model required to produce simulation results
close to the pressure measured by the hydrophone.
The calculated velocities allow the estimation of the
geophone ground coupling. Current results produce
artefacts that could conclude to misinterpretation
and the calculation time is too important to apply
this method to a huge amount of data. However
different solutions are to be tested to solve those
problems and results on strongly coupled datasets
are encouraging for both horizontal and vertical
components. As this method does not need any preli-
minary test, it can then be applied to historical
measurements, at first to test the method and, if the
tests confirm its relevance, to estimate OBS ground
coupling.
ACKNOWLEDGEMENT
We would like to thank G. G. Drijkoningen for
providing his Matlab code and CGG-VERITAS for
financing our work.
R. Estival, D. Léandri, Y. Lacroix 55
ISSN 1028 -7507 Акустичний вiсник. 2015. Том 17, N 2. С. 51 – 57
a b
c d
e f
g h
Fig. 2. Comparison between the amplitude spectrum of the coupling coefficients applied to different datasets
and those found using our method. Left figures concern amplitude of horizontal components,
right figures of vertical components. Amplitude spectrum of couplings coefficients
applied to components are represented by solid lines, dash point lines are
amplitude spectrum of coupling coefficients obtained using our method:
a, b – coefficients from dataset 1, c, d – coefficients from dataset 2,
e, f – coefficients from dataset 3, g, h – coefficients from dataset 4
56 R. Estival, D. Léandri, Y. Lacroix
ISSN 1028 -7507 Акустичний вiсник. 2015. Том 17, N 2. С. 51 – 57
1. Krohn C. E. Geophone ground coupling // Geophys.–
1984.– 49, № 6.– P. 722–731.
2. Drijikoningen G. G. The usefulness of geophone
ground-coupling experiments to seismic data //
Geophys.– 1999.– 65, № 6.– P. 1780–1787.
3. Duennebier F. K., Sutton G. H. Fidelity of ocean
bottom seismic observations // Marine Geophys. Res.–
1995.– 17.– P. 535–555.
4. Drijkoningen G. G., Rademakers F., Slob E. C.,
Fokkema J. T. A new elastic model for ground coupli-
ng of geophones with spikes // Geophys.– 2006.– 71,
№ 2.– P. Q9–Q17.
5. Lavergne M. Méthodes Sismiques.– Paris: Technip,
1986.– 212 p.
6. Dhatt G., Touzot G., Lefranзois E. Méthode des
éléments finis.– Paris: Lavoisier, 2005.– 600 p.
7. Fraser A. S. and Burnell D. Computer models in
genetics.– New York: McGraw-Hill, 1970.– 206 p.
NOMENCLATURE
C – geophone ground coupling coefficients;
u, v, ü – displacement, velocity, acceleration;
V̇geophone – Velocity measured by geophone;
V̇ground – Velocity of the ground;
Vpi
– P-wave velocity in the media i;
Vsi
– S-wave velocity in the media i;
w – frequency;
σ – stress;
µ, λ – Lamé coefficients;
ρ – density;
ρw – water density;
ε – strain;
P – pressure.
R. Estival, D. Léandri, Y. Lacroix 57
|