Transcript Mapping the Conversion Point in the VTI Media

Mapping the P-S Conversion
Point in VTI Media
* Jianli Yang
Don C. Lawton
Outline
 Introduction
 Theory
 Numerical modeling methodology and results
 NORSAR2D anisotropy ray mapping
 Discussion and conclusions
 Future work
 Acknowledgement
Source
P-wave
MP
Receiver
S-wave
The geometry of converted wave obeying Snell’s law
Source
MD
Receiver
S-wave
P-wave
P-S trajectory
The conversion point traces a trajectory in the multilayered model
Source
 
Elliptical wavefront
Spherical wavefront
Ray
The definitions of the phase angle and ray angle


v 2P ( )  α 02 1  εsin2θ  D* ( )
2
2


α
α
2
2
2
*
0
0
v SV ( )  β 0 1  2 εsin θ  2 D ( )
β0
 β0



2
v SH
(θ)  β 02 1  2sin2θ
1


2
2
2
*




α
4(1

β
α

ε)ε
1
4δ


2
2
4
0
0
  1 
D* ( )   1 
sin
θcos
θ

sin
θ

1


2
2
2
2

2
β    (1  β 0 α 0 )
(1  β 0 α 0 )




2
0
2
0
1 dv
(tanθ 
)
v dθ
tanφ(θ) 
tanθ dv
(1 
)
v dθ
Thomsen’s exact equations

v p     0 1   sin2  cos2    sin4 

  02

2
2
v sv     0  1  2    sin  cos  
0



v sH     0 1   sin2 θ


1
1 dv 
tan  tan 1 



sin

cos

v

d



Thomsen’s linear approximations

v P  2   0
0
VP ( π 4)
 VP ( π 2)

δ  4
1  
1
VP (0)  
VP (0) 

 
v SH  2   0
0
Thomsen’s definition of the anisotropy parameters
Angles and offsets included in the algorithm
Find the
corresponding  S
Calculate the Pwave ray parameter
for  P
VTI
by Snell’s law
Isotropic
Calculate the V P
and  P
Calculate the V S
and  S
Calculate X P
Calculate
XS
X P + X S = offset
X P (VTI ) -
X P (Isotropic)
= displacement
=0.20
=0.10
=0.05
=0.10, exact equations
=0.20
=0.20
=0.10
=0.10
=0.05
=0.05
=0.10, Thomsen’s linear approximation
0
Source
Receiver
-100
MP
-200
-300
m
-400
Isotropic raypath
-500
-600
-700
VTI raypath
-800
-900
-1000
0
200
400
600
800
m
=0.20, =0.05, offset/depth=1
1000
Source
Receiver
MP
Isotropic raypath
VTI raypath
=0.20, =0.10, offset/depth=1
Source
Receiver
MP
VTI raypath
Isotropic raypath
=0.20, =0.20, offset/depth=1
Source
Receiver
MP
Isotropic raypath
VTI raypath
=0.20, =0.15, offset/depth=1
Source
Receiver
MP
VTI raypath
Isotropic raypath
=0.20, =0.25, offset/depth=1
= 0.25
= 0.50
= 0.75
= 1.0
= 1.25
offset/depth
= 1.5
Isotropic
case
S wave
P wave
Isotropic
Isotropic
VTI
The VTI model designed for NORSAR2D experiment
An example of the synthetic seismogram obtained
from NORSAR2D anisotropy ray tracing on the model
and displayed by PROMAX
For
=0.10
Displacement
from
NORSAR2D (m)
=0.20
Displacement
from linear
equations (m)
Displacement
from exact
equations (m)
236.1
244.18
316.42
=0.15
142.3
139.16
163.58
=0.10
47
41.56
49.53
=0.05
-50
-56.50
-49.39
=0.00
-146
-151.26
-140.15
= -0.05
-244
-237
-218.68
Table 1 , NORSAR 2D experiments in VTI media, with =0.10
Discussion and Conclusions

The location of the conversion point in VTI media is
different to that in the isotropic case.

The displacement of the conversion point is
dependent on the offset/depth, velocity ratio,
anisotropic parameters  and .

When  is greater than , the conversion point is
displaced towards the source relative to its location
in the isotropic case.
Discussion and Conclusions

When  is less than , the conversion point
moves towards the receiver compared to its
location in isotropic case.

Results using linear approximations are similar
to those obtained from NORSAR code.

Accurate placement of the conversion point is
necessary for P-S survey design and data
processing.
Future work

Further investigation of the relation
between the displacement of the
conversion point and Vp/Vs
 Apply results of this work in the 3-C
seismic survey design

Compare results using Thomsen’s effective
Acknowledgements
 We thank Dr. Larry Lines and Dr. Jim Brown for
valuable suggestions
 CREWES Sponsors’ financial support is also greatly
appreciated