Sensitivity kernels for finite-frequency
signals: Applications in migration
velocity updating and tomography
Xiao-Bi Xie
University of California at Santa Cruz
Sanya, China July 24-28, 2011
Outline
 A brief introduction
 Data domain vs. depth domain
 Sensitivity Kernel for Migration Velocity Analysis
 The Inversion System
 Velocity model partitioning and sensitivity kernel storage
 Numerical Result
 Conclusions
Outline
 A brief introduction
 Data domain vs. depth domain
 Sensitivity Kernel for Migration Velocity Analysis
 The Inversion System
 Velocity model partitioning and sensitivity kernel storage
 Numerical Result
 Conclusions
Imag
df =arg(u/u0)
arg(u/u0) = imag(U/u0)
U
u
U / u0
u0
Real
u / u0
u0/ u0
u 
U 

imag

 
 u0 
 u0 
d  arg 


G  r; rs ,   G r; rg , 
2
K r, rs , rg ,   imag  2k0

G rg ; rs , 





 


In applied seismology
Huge data size. Efficiency is crucial. Suggested methods
could be one-way propagator or Gaussian beam method.
Complex background models. The velocity perturbations
overlapped on the initial model are large (some times are
more than 100%).
Including not only transmitted observations, where the
information is from the surface data, but also reflection type
observation, where the information is collected in image
domain.
Outline
 A brief introduction
 Data domain vs. depth domain
 Sensitivity Kernel for Migration Velocity Analysis
 The Inversion System
 Velocity model partitioning and sensitivity kernel storage
 Numerical Result
 Conclusions
Complexity in
data domain
▼
Simplicity in
depth domain
A synthetic shot record. The shot is located above
relatively complicated structures. There are many
complicated features in this synthetic section.
Outline
 A brief introduction
 Data domain vs. depth domain
 Sensitivity Kernel for Migration Velocity Analysis
 The Inversion System
 Velocity model partitioning and sensitivity kernel storage
 Numerical Result
 Conclusions
Migration velocity updating
Data
Source
Target
Back project to
modify the velocity
Measuring
incoherence in image
The incoherence information are
RMOs from different common image
gathers.
Offset index CIG
Shot index CIG
Angle index CIG
The methods that converting the RMO
into velocity corrections.
Parameterized semblance
Ray-based tomography
Wave-equation based inversion
How the RMO sense the velocity perturbation:
--- Direct measurements
The actual sensitivity map for a shot image (how the depth image senses the V-model error).
To generate this map, we use an velocity error patch to scan the model. At each location, we
conduct a migration and measure the RMO from the depth image. The RMOs are then
presented in the model to show the sensitivity of the depth image to the velocity error. The
sensitivity map is very complex. A positive error can generate either positive or negative
RMOs; the sensitivity area is much broader than the ray based theory predicted. Our goal is to
derive theoretical equations to express this sensitivity map and use it for velocity updating.
Direct measured sensitivity maps for shots at different locations
in 2D SEG/EAGE salt model
Source side kernel
K DF
 2 GD  r; rS  G  r; rI  
r
,
r
,
r

imag
 S I
2k0

GD  rI ; rS 


Receiver side kernel
KUF
 2 GU  r; rS  G  r; rI  

r, rS , rI   imag 2k0

G
r
;
r


U  I S
Source
Image point
Imaginary source
The GB Green’s functions used to
construct the sensitivity kernel
for migration velocity analysis.
(a) Down-going Green’s function
, (b) up-going Green’s function ,
and (c) Green’s function .
Comparison of different
kernels for a shot gather
The sketch of a ray-based
kernel
The sensitivity kernel
calculated using the finitefrequency theory
The actual sensitivity map
directly measured from
migration
Outline
 A brief introduction
 Data domain vs. depth domain
 Sensitivity Kernel for Migration Velocity Analysis
 The Inversion System
 Velocity model partitioning and sensitivity kernel storage
 Numerical Result
 Conclusions
R  rI , rS   
v0  rI 
m  r   K D  r, rS , rI   KU  r, rS , rI  dv

V
2 cos   rI , rS  
=
RMO
∫
B
δv/v
Velocity model
error
B
dV
Sensitivity
kernel
Comparison between inversions
using a finite-frequency sensitivity
kernel and a ray kernel
Sketch illustrating the relative residual
moveout measurement from a pair of
shots
The differential sensitivity kernel for a
pair of shots. Note the complexity and
volumetric distribution of a finitefrequency kernel
The ray based kernel for a pair of shots.
Note the sensitivity distribution is
unrealistic and the uneven ray
distribution can cause singularities in
inversions.
Differential RMO
Differential kernel
d R rS1, rS 2 , rI   V m r K B r, rS1, rS 2 , rI dv
K B  r, rS1 , rS 2 , rI  
v0  rI 
 K DB  r, rS 2 , rI   KUB  r, rS 2 , rI  

2 cos   rS 2 , rI   
v0  rI 
 K DB  r, rS1 , rI   KUB  r, rS1 , rI  


2 cos   rS1 , rI   
Outline
 A brief introduction
 Data domain vs. depth domain
 Migration Velocity Analysis
 The Inversion System
 Velocity model partitioning and sensitivity kernel storage
 Numerical Result
 Conclusions
A 5-layer velocity model used to demonstrate the migration velocity analysis.
How to partition the model?
B

d R rS1, rS 2 , rI   V m r  K r, rS1, rS 2 , rI dv
How to store huge amount of kernels?
d R  rS1 , rS 2 , rI    V  r  m  r  K B  r, rS1 , rS 2 , rI dv
k
k
B

m
r
K
 r, rS1 , rS 2 , rI dv 


V  r 
i
 V  r   a1 a2
i
a3


a4  



f1  r   

f 2  r  B
K  r  dv

f3  r 


f4 r  
 a1   1 0 0 0   m  r1  
  
 m r 
a

1

1
0
0
 2
   2 
 a3   1 0 1 0   m  r3  

  

 a4   1 1 1 1  m  r4 
m r 
 
 V  r  Pij m r j fi  r   K B  r , rS1 , rS 2 , rI dv
k
 
  Pij m  r j  FKi 

k
 Pij m r j V  r  fi  r   K B  r , rS1 , rS 2 , rI dv
k
  FK1 FK 2
FK3
Unknown perturbation at
cell corners
 1 0 0 0   m  r1  
 1 1 0 0  m r 
   2 
FK 4  
 1 0 1 0   m  r3  



m
r

1

1

1

1



 
4 

Actual output and stored kernels
FK1  V  r  f1  r  K B  r  dv
i
FK 2  V  r  f 2  r  K B  r  dv
i
FK3  V  r  f3  r  K B  r  dv
i
FK 4  V  r  f 4  r  K B  r  dv
i
Parameter matrix
Stored parameter a1. The 4
groups of kernels are for 4
reflectors; the horizontal
coordinate is for different
image points and the vertical
coordinate is for different
sources.
Model grid size 10m x 10m
Cell size 500m x 500m
31shot x 31 imaging point x 4
reflectors, 32x10 cells spend
286Mb.
Outline
 A brief introduction
 Data domain vs. depth domain
 Sensitivity Kernel for Migration Velocity Analysis
 The Inversion System
 Velocity model partitioning and sensitivity kernel storage
 Numerical Result
 Conclusions
migration velocity updating process
(1) Conduct the migration using an initial model.
(2) Calculate the RMOs from the shot-index CIGs.
(3) Pick the reflector position from the initial depth image.
(4) Use the initial model and reflector locations to calculate
sensitivity kernels.
(5) Input the RMOs and the sensitivity kernels to the inversion
system to do the tomography.
(6) Use the inverted errors to update the initial model and use it for
the next iteration.
A 5-layer velocity model used to demonstrate the migration velocity analysis.
Comparison between the
theoretically calculated
kernels (left column) and
actually measured
sensitivity maps (right
column). From top to
bottom are for different
reflectors.
Coverage of sensitivity kernels in the model. Panels (a) to (d) are kernel coverage for
image points on the 4 reflectors. Panel (e) is the coverage from all kernels. Shown
here is the summed positive parameter FK1.
Velocity models in updating process, with (a) initial
model and (b) model after two iterations.
Depth image improved in the velocity updating process. (a) Image calculated using
the initial model and (b) image calculated using the updated velocity model.
CIGs before and after the velocity updating, with (a) CIGs in the
initial model and (b) CIGs in the updated velocity model.
Outline
 A brief introduction
 Data domain vs. depth domain
 Sensitivity Kernel for Migration Velocity Analysis
 The Inversion System
 Velocity model partitioning and sensitivity kernel storage
 Numerical Result
 Conclusions
Summary
(1) Based on the finite-frequency sensitivity theory, we present a
migration velocity analysis method. The new approach is a
wave-equation based method which naturally incorporates the
wave phenomena and is best teamed with the wave-equation
based migration for velocity analysis.
(2) The finite-frequency sensitivity kernel is used to link the
observed shot gather RMO with the errors in the migration
velocity model. Angle domain decomposition is not required.
(3) We developed method to calculate the broadband sensitivity
kernel in complex velocity models and for irregular reflectors.
Summary (continues)
(4) A new velocity model partitioning approach is tested. This
method partitions the model into small cells and uses interpolation
function to represent the velocity model within cells.
(5) To store the sensitivity kernels, we use interpolation functions
as basis and expanded kernels to these basis. Thus we only need to
store the expansion coefficients. The accuracy of the kernel is
adaptive to the required accuracy of the velocity model. In this way,
we significantly reduce the storage space of sensitivity kernels
while without losing the required accuracy.
Summary (continues)
(6) Using this approach, we demonstrate the velocity model
updating. The updated velocity model improves the depth image
by both flattened the common image gather and bring the image to
the original location of reflectors.
EOF
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