super-wide angle beamlet propagator based on iterative wavefront reconstruction
DESCRIPTION
Super-wide angle beamlet propagator based on iterative wavefront reconstruction. Zhongmou Xia , Ru -Shan Wu, Hong Liu. Modeling and Imaging Laboratory, IGPP, University of California, Santa Cruz. Institute of Geology and Geophysics, Chinese Academy of Sciences. 1. - PowerPoint PPT PresentationTRANSCRIPT
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Super-wide angle beamlet propagator based on iterative
wavefront reconstructionZhongmou Xia , Ru-Shan Wu, Hong Liu
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Modeling and Imaging Laboratory, IGPP, University of California, Santa Cruz
Institute of Geology and Geophysics, Chinese Academy of Sciences
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Reviews of local cosine basis theory
Super-wide angle one-way scheme
Numerical tests Conclusions
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Outline
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1. Reviews of Local Cosine Basis Theory
For 2D scalar acoustic equation:2 2
22 2x
Pk Pz
Corresponding one-way wave equation:
zP ik Pz
( , , ) ( , , ) exp zik dzx xP z dz k P z k
Expression of analytic solution:Starting Point!!!
(1 )
(2 )
(3 )
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22 approz xk ximatk ion
00
0
( )( ) 1,n
Global reference velocityLocal reference velo
xx n N city
(Wu et. al.,2008, Geophysics)
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Local Cosine Basis
2 1( ) ( ) cos( ( ) )2mn
nn
n n
x xb x B x m
L L
The basis element
( , ) ( , ), ( ) ( )
( , ; ) ( )
mn mnn m
n m mnn m
u x z u x z b x b x
u x z b x
The wavefield at depth Z can be decomposed into local cosine beamlets with windows along the horizontal x-axis
( , ; )n mu x z coefficients of the decomposition beamlets
(4)
(5 )
1( ) /2m nm Lx p
æ ö÷ç= + ÷ç ÷çè ø
(Coifman and Meryer, 1991)
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2 2 2 2/ ( , ) ( , , ) 0x z v x z u x z (6)
( , ; , )l nj mP x x : propagator
(For details see Wu et al., 2008),
Wave equation in frequency and space domain:
,
( , ; )
( , ; )
=
( )
,
,
;
;
( )
,
l j
nl mn m
n m
nj m
mj
nl mn
u x z z
P x
P
u z
x z
x x
u
(7)
Wavefield extrapolation expression:
OldNew
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B
2. Super-wide angle Scheme
A
c
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Weight function
(Jia and Wu, 2009, Geophysics,)
c : Cut angle
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Weight function field in homogeneous medium
9C 2 C [ , ]
2C C
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Two schemes to implement super-wide angle method
Downward
Horizongtal Weighted
summation
(Jia & Wu, 2009, Gepphysics)
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Two schemes 1) “Interpolation method”
2) Iterative reconstruction method1) “Interpolation
method”
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2) Iterative reconstruction method
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X0 + X
Z0 + Z
Z0X0 - X
X0 X0 + X
Z0 + Z
Z0X0 - X
X0
0( , )DP X Z Z 0( , )HP X X Z
Combining Superposition Wavefield (5 points):
0 0 0 0
0 0 0 0
0 0 0 0
( , ) ( , )( , ) ( , )( , ) ( , )
D H
D H
D H
P X Z Z P X Z ZP X X Z P X X ZP X X Z Z P X X Z Z
X0 + X
X0 - X
X0
Z0 + Z
Z0
The First Reconstructed Wavefront
To reconstruct the first wavefront
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Z0 + Z
Z0
X0 - 2 X
X0
0( , 2 )DP X Z Z 0( 2 , )HP X X Z
Combining Superposition Wavefield( 9 points ):
0 0 0 0
0 0 0 0
0 0 0 0
( , 2 ) ( , 2 )( , ) ( , )( 2 , 2 ) ( 2 , 2 )
D H
D H
D H
P X Z Z P X Z ZP X X Z Z P X X Z ZP X X Z Z P X X Z Z
The Second Reconstructed Wavefront
To reconstruct the second wavefront
Z0 + 2 Z
X0 - 2 X
X0 - 2 X
X0 X0 - 2 X
Z0 + Z
Z0
Z0 + 2 Z
Z0 + Z
Z0
Z0 + 2 Z
X0 - 2 X
X0 X0 - 2 X
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Z0 + Z
Z0
Z0 + 2 Z
X0 - 2 X
X0 X0 + 2 X
X0 - m X
X0 + m X
。。。
Z0 + m Z
All the reconstructed wavefronts
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3. Numerical tests
Model1: 2D Salt Model
(layered + salt model)
(Made by Ruiyan)
Model2: Bp Model
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dx=dz=24m fd=15 HZ Nx=1001 Nz=150 dt=0.04s
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Salt model at time 1.6s
Snapshots for 2D Salt Model
Regular one-way method Super-wide one-way LCB method
RTM
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Regular
16Interpolation method
Iterative reconstruction method
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1 2 3 4
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9
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Regular LCB Method
Iterative Super-wide LCB Method
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RTM Method Iterative Super-wide LCB Method
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RTM Method Iterative Super-wide LCB Method
Adding upward wavefields
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D+H D+H+U
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Conclusions
1. Super-wide angle beamlet propagator based on iterative wavefront reconstruction can handle large-angle and super-angle (e.g. turning waves) imaging problem. It can overcome the angle limitation but keep the merits of one-way method2. Cost of super-wide angle beamlet (iterative reconstruction) one-way method is close 1.5-2 to regular one-way method.
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Thanks!
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