SHEAR WAVE BIREFRINGENCE IN REVERSE VSP:
AN APPROACH TO 3-D SURFACE P TO S
CONVERTED WAVES
Jesus Sierra
Earth Resources Laboratory
Department of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
John H. Queen
Conoco, Inc.
100 South Pine
P.O. Box 1267
Ponca City, OK 74603
ABSTRACT
We present an original method to estimate local shear wave birefringence properties for
3-D surface P to S converted waves. To accomplish this we approach the problem in
reverse VSP (RVSP), and we show that they are equivalent. The method works in the
pre-stack domain and uses the converted P to S waves as hypothetical sources to study
the problem under the propagator matrix method in transmission. The importance of
this method is that no information is required about layering above the zone of interest
to obtain an accurate estimation of the anisotropy parameters. The method involves
solving a nonlinear problem in the frequency domain where a global minimization technique called Simulation Annealing is used. The procedure also allows us to estimate the
axis of anisotropy independently of the offset angle in the range of angles considered in
VSPs and in surface profiles. The proposed method is validated with synthetic RVSP
data for two models with different densities of vertical fractures. Results show good
accuracy in the estimation of the angle of the fractures for the whole range of offsets.
Also, results show a dependence on the frequency and offset ranges considered in the
analysis.
I
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Sierra and Queen
INTRODUCTION
Shear wave birefringence is an important research topic in reservoir geophysics. This
phenomenon, which relates the vertical or near vertical shear wave propagation with the
anisotropy of rocks, has shown its potential in the characterization and development of
fractured oil reservoirs (Ata et al., 1995; Li, 1997). Birefringence is characterized by
shear-wave splitting which can be described by the main direction of particle motion
polarization and by the difference in velocities (time delay) and attenuation of the split
waves (Crampin, 1985). It is also well known that the near-surface has a degenerative
effect on shear wave signals and polarization properties (Crampin, 1985). As a result,
studies of shear wave birefringence using P to S converted waves play an important role
due to the one way shear-wave propagation path. Ata et al. (1995) show a successful
application of P to S converted waves over a fractured carbonate reservoir. They use
three 2-D lines to generate maps of fracture direction and intensity (time delay) over
the reservoir using the layer stripping method (Garotta and Granger, 1988; Winterstein
and Meadows, 1991a,b). The layer stripping method in this case showed its sensitivity
to the layering above the zone of interest, and so errors in estimations from the upper
layers propagated down toward the estimation at the target zone.
In this paper, we will address the problem of shear wave birefringence for the case
of offset RVSP (reverse VSP) to estimate the principal axis of anisotropy. We will also
show that this problem is similar to the 3-D P to S converted waves for surface seismic
profiles. To accomplish this we follow the work of Lefeuvre et al. (1992), who use the
propagator matrix method to describe the change in the state of polarization of the
split shear waves between two different intervals with VSP data. The importance of
this method, compared with others like layer stripping (Lefeuvre et aI., 1991), is that
no information is required about layering above the zone of interest and an accurate
estimation of the anisotropy parameters can be obtained.
PROPAGATOR MATRIX METHOD
Lefeuvre et al. (1992) define the propagator mat.rix as t.he t.ransfer funct.ion between two
steps of polarizat.ion. In this part.icular casl' t hey use YSP dat.a, and the propagator
matrix describes the propagation of downgoing shear waves between t.wo different depths
z] and Z2(Z] :S Z2). This analysis and processing of shear waves assumes the following:
The medium is horizontally strat.ified. t he well is vert.ical where the YSP is recorded,
and the surface seismic source is zero offset to t.he well. Under t.hese considerations the
problem is solved in the frequency domain as a least squares linear parameter estimation
mInImIZIng
(1)
where P is the 2 by 2 propagat.or mat.rix. E and 5 are t.he polarization vectors at Z1
and Z2, respectively; and B t.he error in t.he predict.ion. J{ represents the number of
experiences or polarizations of source which, in t.his case, is limited to 2.
6-2
(
Shear Wave Birefringence in Reverse VSP
As a result of solving (1), Lefeuvre et ai. (1992) are able to find expressions.for
each element of P as function of frequency. This solution together with the linearity
relationship between frequency and phase allow them to estimate eigen directions and
time delay of the split shear waves.
Following this approach, we study the RVSP problem with offset receiver positions
(Figure 1). We consider two different locations for shear sources with the same polarization in ZI and 22, respectively. The propagator representation of the change in the
state of polarization of the shear waves are:
S Ik
--
PI . E Ik
(2)
and
(3)
where PI and P2 are the propagator matrices and E I .2 , SI,2 represent the source and
receiver polarization vectors for layer 1 and 2, respectively. J( as in (1) is the number
of experiences.
Combining (2) and (3), we can solve this inversion problem in terms of a minimization scheme setting up the nonlinear objective function as:
nl
2:)sf - PI' Ef)+(s[
ij)
- PI' Ef)
+
k
n2
l)S~ - P I P 2 . E~)+(Sf - P 1P 2 . E~)
(4)
,.
where + denotes complex the conjugate. and nl and n2 are the number of experiences
at 21 and Z2. respectively We can see from (4) that to solve for PI and P2 we have to
know the shape and polarization of the sources. Therefore, if 21 and Z2 are close enough
and E I and E 2 have the same polarization. we can approximate
where h is a complex constant.
Solving for E 1 in (2) and usinl'; (5)
the sources do not appear
Il1
(4) we can write an expression for
ij)
where
n
ij)
= 2:)S~ - P 1P 2 P 1- 1 . S[)+(S~ - PIP2PI-1. Sf)
(6)
k
where nl and n2 are equal to n and we invoke the reciprocity source-receiver property
of the propagator matrix (Gilbert and Backus, 1966 ) to assure the existence of the
inverse of PI' Also, P2 has been redefined to include h.
Minimizing this new objective function allows us to estimate the complex elements of
PI and P 2 (a total of 8). Due to the nonlinearity of the problem, the simulated annealing
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Sierra and Queen
method (see the appendix) is used. This method is a computational technique derived
from statistical mechanics for finding near globally-minimum-cost solutions to large
optimization problems.
The simulated annealing method requires a certain amount of redundancy over the
data to assure good statistics of the solution space. To accomplish this we evaluate q;,
not over a single frequency and offset (trace), but over ranges
n
q; =
tT!
Vf
"L"L "L(S~ k
tT;
p j p 2 p j-
j
•
Sf)+(S~ - p j p 2 p j-
j
•
Sf)
(7)
Vi
where IIi and vf are the initial and final frequency and tT; and tTf are the initial and
final offset (trace) of the range considered for each one.
We evaluate small frequency ranges and groups of offsets (traces). We assume that
the response of the medium is approximately the same for the frequency range considered
and that the ray paths are similar for that group of offsets.
INTERPRETATION OF THE PROPAGATOR MATRIX
The propagator matrix is an operator that relates two different states of polarization.
If these polarizations correspond to the upgoing wave in transmission at two different
depths, and if the interval between these two depths corresponds to a series of layers with
the same birefringence properties, then the propagator can be interpreted in terms of
eigen directions (natural directions of polarization), delays and attenuations (Lefeuvre
et ai., 1992).
The propagator matrix is estimated in the frequency domain whose initial coordinate
system (xo, yo, 0) is assumed known. We define P o2 (vrange) as the initial estimate for a
particular frequency (vrange) and offset range. P 04>2(Vrange) is the transfer matrix after
the transmission which refers to another coordinate system (xe¢, ze¢, ze¢) making
an angle ¢ with respect to (xo, yo) and an angle e with respect to the zo direction
(Figure 2):
(8)
P o.p2(Vrange) = R(e)R(¢)P02 R'(¢)R'(e)
where R(¢) and R(e) denote the matrix rotation for ¢ and e, respectively:
R(¢)
=
(cos(¢)
- sm(¢)
Sin(¢)) andR(e)
cos(¢)
=
(cose
0
0)
1
(9)
where superscript t denotes the matrix transpose. R( ¢) represents a rotation in the x-y
plane, while R(e) is the rotation in the x-z plane (Alford, 1986), as shown in Figure 2.
In equation (8) the angle of incidence q produces an apparent nonorthogonality of
the split shear waves for orthogonal eigen directions (Li et ai., 1998). To determine P 02
it is necessary to know e. One way is to use ray tracing if there is a velocity model
available. Another way is to work with near offsets « 15°) where the effect of the angle
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Shear Wave Birefringence in Reverse VSP
of incidence is negligible (Li et al., 1998) and() ~ 0° can be assumed. However, if· we
minimize the off diagonal energy of the rotated transfer matrix Po¢z (llrange), we will
be able to estimate <P without knowing ().
Solving (8) for Poz
(10)
and defining the angle <Po that minimizes the quantity IPozyxl2 + lPo2xyl2, <PI and <P2
are the angles that minimize separately the quantities IPozyxlZ, IP02xyIZ, respectively. It
can be shown that it is possible to find <Po, <PI or <pz independently of () in the range of
angles considered in VSPs. This is true because both of the off diagonal elements are
affected in the same way by a factor of cos(()). In particular we can choose () = 0° and
solve. Now (10) reduces to
(11)
where we use the "prime" over PO¢2 to emphasize that it is not the same as in (10).
We perform this rotation for <Po between 0 and 90 degrees because of the symmetrical
properties of the matrix. For the case of <PI and <pz we rotate between 0 and 180°. In
cases where directions of polarization are not orthogonal, <PI and <pz should be used. To
differentiate between fast and slow polarization, an estimation of the time delay between
components is needed. For the purpose of this work, this will not be addressed, and we
will limit ourselves to the estimation of <Po ± 90°.
SYNTHETIC DATA AND APPLICATION
The method is tested with synthetic data. The data was generated using the discrete
wave number method to solve the 3-D full wave equation for general anisotropy (MandaI,
1988; Mandai and Toksiiz, 1989). We analyze two similar models (see Table 1). Both
models deal with vertical fractures where the natural directions of polarization are
orthogonal. In Modell, the second layer has a fracture density of 0.2 (cracks/volume),
while the fracture density in Model 2 is 0.1 (cracks/volume).
To generate the synthetic data, we use 40 receivers at the surface spaced 22.5 m
apart with a near offset of 100 m and a far offset of 1000 m. Four shots with azimuths
(E to N) of 0, 30, 60 and 90 degrees for each depth, zl and z2 were simulated (Figures
3 and 4). The signal spectrum ranges between 0 and 100 Hz, with a center frequency
of 30 Hz. Figures 5 and 6 show the results of applying the method to the synthetics
generated with ModelL The analysis is performed at the same time over the four shots
and over the top and bottom of the fractured zone events.
We use a 200 ms (2 ms sample rate) window around each event, a frequency range
from 20 to 35 Hz, and an offset range of 4 traces (90 m). We can see from Figure 6 that
the estimation of <Po is accurate and close to the true solution out to an offset of 640 m.
However, if we decrease the upper limit of the frequency range, and use 20 to 28 Hz, we
obtain a new estimation of 62.33°for offsets 662.5 m to 752.5 m. Also, we test a larger
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Sierra and Queen
Layer
Bottom depth (m)
Density (kg/m 3 )
Background VP (m/s)
Background VS (m/s)
Filling material VP (m/s)
Filling material VS (m/s)
Crack density
Aspect ratio
VP (m/s)
VS1 (m/s)
VS2 (m/s)
Percentage of VS1
Fast direction (E to N)
1
950
2200
-
-
3800
2500
2500
0.0
-
2
1400
2600
5800
2900
340
0
0.2
0.01
4703
2900
2352
0.811
60°
Table 1: Parameters used to generate the synthetics. VS1 and VS2 are the fast and
slow velocities, respectively. Crack density is 0.1 for Model 2.
offset range of 180 m for offsets starting at 572.5 m and 775.0 m and we obtain values
for !Po of 60.660 and 59.220, respectively. These new estimates show that the frequency
range and the offset range play important roles in the estimation. This dependence
of the offset range is related to the need for redundancy in the simulated annealing
method. The same can be obtained by incorporating more source polarizations. The
frequency range has to be moved to lower values as the time delay gets closer to zero.
For Modell this apparent diminution in the time delay becomes more obvious in the
far offsets.
Figure 7 shows the result of applying the method to the synthetic data generated
using Model 2. This model is characterized by having less birefringence effect compared
with Modell (we decreased the fracture density of layer 2). As we approach far offsets
and the time delay between fast and slow shear waves becomes smaller, they are not
distinguishable in frequency and the estimation is less accurate. In this case, even by
changing the offset range, it is not possible to have a good estimation of !Po .
P TO S CONVERTED WAVES
We have studied the birefringence of shear waves in the RVSP case as an approach to
a more complex problem, the conversion of P to S waves in 3-D.
Consider an incident P wave converting to SV with amplitude Ap-s\f which is a
function of the angle of incidence e. The change in the state of polarization for the
shear waves after propagating through two different intervals containing an anisotropic
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Shear Wave Birefringence in Reverse VSP
medium is found (adapting MacBeth et al., 1998):
S
= Ap_sv(e)R(e)R(¢)AsR'(¢)R'(e)
(
~
)
(12)
where R(e) and R(¢) are the same as in (9), As is the diagonal matrix representing the
S wave propagation through the medium. Notice the similitude between (12) and (2),
(3) and (7) where
f;
= Ap-sv (
~
(13)
)
as a hypothetical shear source. Note that the angle of incidence of the P wave differs
from the angle of the emerging SV converted wave.
In a 3-D surface profile we have P waves coming from different azimuths with approximately the same common conversion point (ccp), providing SV waves polarized
with different azimuths (Figure 8) at each ccp bin. Notice also that the conversion at
ZI has the same polarization as in Z2 because they are in the same plane.
CONCLUSIONS
The propagator matrix can be used to describe the phenomenon of shear wave birefringence in the case of offset RVSP. This case is also equivalent to 3-D P to S converted
wave surface profiles. This method assumes homogeneous flat layers and orthorhombic
or monoclinic anisotropy with a vertical axis of symmetry.
We have shown the following points concerning the method:
1. It estimates the birefringence properties of the medium without any information
about layering above the zone of interest.
2. It uses a global minimization method called Simulated Annealing to solve the
nonlinear objective function. The use of frequency and offset ranges as well as
different source polarizations is needed to accomplish for redundancy over the
data.
3. The estimation of the eigen directions is accurate and depends on the frequency
range we analyze.
4. The estimation of the eigen directions is independent of the angle of incidence in
the range considered in VSP or in surface profiles.
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ACKNOWLEDGMENTS
The authors thank Dr. Reinaldo Michelena (INTEVEP, S.A.) for giving the original idea
for this work, Dr. Rama Rao and Prof. Dale Morgan (M.LT.) for fruitful discussions on
the simulated annealing method, and Ms. Feng Shen and Dr. Dan Burns for reviewing
this paper.
This work was supported by the Borehole Acoustics and Logging/Reservoir Delineation Consortia at the Massachusetts Institute of Technology.
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Shear Wave Birefringence in Reverse VSP
REFERENCES
Alford, R.M., 1986, Shear data in the presence of azimuthal anisotropy: Dilley, Texas,
56th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 476-479.
Ata, E., Michelena, R.J., Gonzales, M., Cerquone, H., and Carry, M., 1995, Exploiting
P-S converted waves: Part 2, application to a fractured reservoir, 64th Ann. Internat.
Mtg., Soc. Expl. Geophys., Expanded Abstracts, g4, 240-243.
Crampin, S., 1985, Evaluation of anisotropy by shear-wave splitting, Geophysics, 50,
142-152.
Garotta, R. and Granger, P., 1988, Acquisition and processing of 3cX3c-D data using
converted waves, 58th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts,
995-997.
Gilbert, F. and Backus, G., 1966, Propagator matrices in elastic wave and vibration
problems, Geophysics, XXXI, 326-332.
Laarhoven, V., 1988, Theoretical and computational aspects of simulated annealing,
Amsterdam, Center for Mathematics and Computer Science.
Laarhoven, V. and Aarts, E., 1987, Simulated annealing: Theory and applications, Amsterdam, Center for Mathematics and Computer Science.
Lefeuvre, F., Nicoletis, L., Ansel, V., and Cliet, C., 1992, Detection and measure of
the shear-wave birefringence from vertical seismic data: Theory and applications,
Geophysics, 57, 1463-1481.
Lefeuvre, F., Winterstein, D., Meadows, M., and Nicoletis, L., 1991, Propagator matrix
and layer stripping methods: A comparison of shear-wave birefringence detection on
two data sets from Railroad Gap and Lost Hills fields, 618t Ann. Internal. Mig..
Soc. Expl. Geophys., Expanded Abstracts, 91, 55-60.
Li, X.-Y., 1997, Fractured reservoir delineation using multicomponent seismic data.
Geophys. Prosp., 45, 39-64.
Li, X.-Y., MacBeth, C., and Crampin, S., 1998, Interpreting non-orthogonal split shear
waves for seismic anisotropy in multicomponent VSPs, Geophys. Prosp .. 46. 1-27.
MacBeth, C., Boyd, M., Rizer, WI., and Queen. J .. 1998. Estimation of reservoir fracturing from marine VSP using local shear-wave conversion, Geoph:/js. Pm'11 .. 46.
29-50.
MandaI, B., 1988, Computation of complete wave-field synthetic seismograms for layered anisotropic media, 58th Ann. Internat. Mtg., Soc. Expl. Ge01lhys.. Expanded
Abstracts, 88.
MandaI, B. and Toks6z, M.N., 1989, Synthetic seismograms from surface and borehole sources in anisotropic media, 59th Ann. Internal. Mtg .. Soc. Expl. Geophys..
Expanded Abstracts, 89, 769.
Metropolis, N., Rosenbluth A., Rosenbluth, M., Teller, A.. and Teller E., 1953, Equation
of state calculations by fast computing machines, 1. Chern.. Physics, 21. 1087-1092.
Winterstein, D.F. and Meadows, M.A., 1991a, Shear-wave polarizations and subsurface
stress directions at Lost Hills field, Geophysics, 56,1331-1348.
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Sierra and Queen
Winterstein, D.F. and Meadows, M.A., 1991b, Changes in shear-wave polarization azimuth with depth in Cymric and Railroad Gap oil fields, Geophysics, 56, 1349-1364.
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Shear Wave Birefringence in Reverse VSP
APPENDIX
SIMULATED ANNEALING
The simulated annealing algorithm (Laarhoven and Aarts, 1987; Laarhoven, 1988) is
based on the analogy between the simulation of the annealing of solids and the problem
of solving large combinatorial optimization problems. For this reason the algorithm
became known as "simulated annealing." In condensed matter physics, annealing denotes a physical process in which a solid in a heat bath is heated up by increasing the
temperature of the heat bath to a maximum value at which all particles of the solid
randomly arrange themselves in the liquid phase. Then the heat bath is cooled slowly.
In this way, all the particles arrange themselves in the low energy ground state of a
corresponding lattice, provided the maximum temperature is sufficiently high and the
cooling is carried out sufficiently slowly. Starting off at the maximum value of the temperature, the cooling phase of the annealing process can be described as follows. At each
temperature value T, the solid is allowed to reach thermal equilibrium, characterized
by a probability of being in a state with energy E given by the Boltzmann distribution:
prE) =
Z(~) exp ( -
k:T) ,
(A-I)
where Z(T) is a normalization factor, known as the partition function, depending on
the temperature, T, and k is the Boltzmann constant. The factor exp ( is
known as the Boltzman factor. As the temperature decreases, the Boltzmann distribution concentrates on the states with lowest energy, and finally, when the temperature
approaches zero, only the minimum energy states have a non-zero probability of occurrence. However, if the cooling is too rapid, i.e., if the solid is not allowed to reach
thermal equilibrium for each temperature value, defects can be "frozen" into the solid
and metastable amorphous structures can be reached rather than the low energy crystalline lattice structure.
To simulate the evolution to thermal equilibrium of a solid for a fixed value of
the temperature T, Metropolis et al. (1953) proposed a Monte Carlo method, which
generates sequences of states of the solid in the following way. Given the current state
of the solid, characterized by the positions of its particles, a small, randomly generated,
perturbation is applied by a small displacement of a randomly chosen particle. If the
difference in energy, t:>.E, between the current state and the slightly perturbed one is
negative, then the process continues with the new state. If t:>.E 2: 0, then the probability
of acceptance of the perturbed state is given by exp ( This acceptance rule for
new states is referred to as the Metropolis criterion.
We can resume the whole procedure of the simulated annealing method and show
how the algorithm is as follows:
k;T)
B
k;T ).
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Sierra and Queen
begin
Initialize variables and configuration
loop controlling decrease of Temperature (iterations)
loop controlling reach a qnasi eqnilibrinm at a particnlar Temperature
T
Perturbation of the configuration and compntation of t:.E
if t:.E ::: then accept new configuration
else
if exp ( > random number [0,1] then accept new configuration
equilibrium is approached sufficiently closely
decrease Temperature T
°
k;T)
stop criterion
=
true (systen1 is "frozen")
end
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Shear Wave Birefringence in Reverse VSP
.......
near
y
,f
:-.
.
~
offset
~:
zl
sources
/'.---
~
5 0 = azimuth
x
Figure I: Reverse offset VSP geometry.
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Sierra and Queen
-
s
""~~~'="""-~-
-
...,
xo
,,
,,
,
,
,,,
,,,
,,
-
...... ~
-
zo
Figure 2: Geometric definitions for equation (8).
6-14
Shear Wave Birefringence in Reverse VSP
x
100
225
y
4:()
675
UID
225
Offset (m)
4:()
675
HID
400_
600_
700_
(ms)
(a)
Offset (m)
100
225
4:()
c!. ~(,(.f..l(
m ~t«(((((,
700_
225
IT
',,\
',"
~~~
:~)
~)'
I(
800_
(m»
1ffil
tl
600_
900_
675
I II
IIr
)II
r,/rW
([.
l(I
4:()
675
1ffil
II
WI ~<'~, Ij
I)' ~ (
IJ
II
~ ~
II
(b)
Figure 3: X and Y components of synthetics generated using Model 1 with source
azimuth 30 degrees, (a) at Zj. (b) at Zz.
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Sierra and Queen
x
100
450
Z25
y
1~
675
Offset (m)
450
Z25
675
HXXl
IT
300-
,(
400_
./.
...'J I)
,~
lU[
500-
" U
600-
(ms)
(a)
Offset (m)
100
Z25
1WJ
675
500_
600-
. ,
u 1.<'1.«, •{
,","
:I ,(
700-
U{(
225
l,t
{~'IY
, ,
11'1' Yi'i'''' ..
450
','.
HID
675
,
)
,i
800_
900-
(ms)
(b)
Figure 4: X and Y components of synthetics generated using Model 2 with source
azimuth 30 degrees. (aj at Zj. (b) at Z2.
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Shear Wave Birefringence in Reverse VSP
-4.1t
IU
3
,.X.
05
...
.'
.'.
Figure 5: Example of finding <Po. Frequency range 20-28 Hz. Offset range 572.5 m to
775.0 m. Observe the two minimums 90° apart.
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Sierra and Queen
7Or--,----,.--,..---,----,r---,-----r--,-----,
....
,
I
.
_
-
-
.
10
'-;-_-;=_-;:::' __:-:-':
' __:-!:__~
~00
200
roo
400
500
000
.L, _ _-,-_ _-,-_ _.J
700
000
900
1000
Offsets (m)
Figure 6: Estimation of <Po for Modell: (a) (solid) Frequency range 20-35 Hz. Offset.
range 90 m. (b) (dotted) Frequency range 20-28 Hz. Offset. range 180 m. True
solution 60°.
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Shear Wave Birefringence in Reverse VSP
70 r - - - - - , - - - - , - - , - - - - , - - - - , - - - - - - , - - - - - , , - - - - - , - - - - - ,
oor-----.======f'----I----~
101-_ _...L_ _---.L
100
200
300
1 -_ _...L_ _-L_ _-'-'
.wo
000
000
700
"'-_ _..J
800
000
Offsets (m)
Figure 7: Estimation of cPo for Model 2: Frequency range
m. True solution 60°.
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20~28
Hz. Offset range 180
Sierra and Queen
sources
cdp
,, , ,
,, , ,
, ,
'-,, '
,,
,,
,
-- -,
ccp
---
~~!
zl
----
plain view
z2
-
- -
-;.:.--1'---L!III:
I
hypotetical
shear sources
Figure 8: Geometric definition of hypothetical shear sources in 3-D P to S converted
wave profiles. Illustration of two different azimuths and the relative position between
the cdp and the ccp location.
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