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

Reviews of local cosine basis theory

Super-wide angle one-way scheme

Numerical tests Conclusions

2

Outline

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 )

3

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

6

7

B

2. Super-wide angle Scheme

A

c

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

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”

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

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

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

1 2 3 4

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9

17

Regular LCB Method

Iterative Super-wide LCB Method

RTM Method Iterative Super-wide LCB Method

RTM Method Iterative Super-wide LCB Method

Adding upward wavefields

D+H D+H+U

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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