References: Ceverny, 2001 : Seismic Ray Theory, Cambridge University Press. Gazdag, 1984 : Migration of seismic data by phase shift plus interpolation, Geophysics V.49, 124-131. Kosloff and Kessler, 1983 : Migration with the full acoustic wave equation, Geophysics V.48, 667-687.

Kosloff and Kessler, 1987 : Accurate depth migration by a generalized phase-shift method, Geophysics, V.52, 1074-1084. Pai, 1988 : Generalized f-k migration in arbitrarily varying media, Geophysics, V.53, 1547-1555.

Sharma and Agrawal, 2003 : Wide angle and bi-directional beam propagation using the collocation method for the non-paraxial wave equation, Opt. Commun. 216, 41-45. Wenzel, 1991 : Frequency-wave number migration in laterally heterogeneous media, Geophysics, V.56, 1671-1673. G. Zhang, 1993 : System of coupled equations for up-going and down-going waves, Acta Math. Appl, Sinica, V.16,

251-263. Y. Zhang, G. Zhang and N. Bleistein, 2005 : Theory of true-amplitude one way equations and true amplitude com-mon-shot migration, Geophysics, V.70, E1-E10.

W. Zhang, H. Chen and J. Tian, 2009 : Novel three dimension wide-angle beam propagation method based on split-step fast Fourier transform, Chin. Phys. Lett., V.26, 1-4.

A Practical One Way Scalar Wavefield Extrapolation: A Step Forward in True Amplitude Processing

Presented by: Zhengsheng Yao

yao@absoluteimaging.ca

www.absoluteimaging.ca

Summary A new Phase-shift method for wavefield downward continuation is presented.

Unlike conventional Phase-shift which is based on the one way scalar wave

equation that is factorized from homogenous media, this method is derived

by directly solving the full scalar wave equation to obtain one way wave

propagation in an inhomogeneous media.

Introduction Propagation of wavefield in an inhomogeneous media causes reflection and

transmission in energy. A two layer example is shown in figure 1a. Methods

employed for wave equation migration such as Phase-shift (Gazdag, 1978),

are based on wavefield propagation in homogenous media, therefore true

amplitude during wavefield extrapolation cannot be obtained as shown in fig-

ure 1b. In this poster we present a method for true amplitude extrapolation

as shown in figure 1c.

D

A

B

C

E

D

A

B

C

E

D

A

B

C

E

Figure 1. A two layer model for wavefield propagation shown in 1a; wavefield forward and backward propagation in constant velocity shown in 1b; and wavefield forward

and backward propagation in true velocity shown in 1c.

where:

, and

by defining:

we have:

and:

Until now no approximation has been made. However, the coupled terms still

make the practice difficult. By ignoring multi-scattering we have:

where:

It is of interest to get an insight on the reflection term R in the equation

above. If can be replaced by via a Fourier Transform as in the case

of a laterally homogeneous media then:

and it is equivalent to the WKBJ approximation result (Zhang, 2005). Further-

more, in the case of a layered media, the interval between layers k and k+1

can be defined as:

which is the reflection coefficient formula for an acoustic wave (Ceverny,

2001).

Practical Implementation for Migration

With the assumption that the vertical velocity does not change within each

extrapolation step then:

With this equation, the reflected energy for the backward wavefield does not

participate in further forward propagation but will be included in the wavefield

for further backward propagation.

iSiSV

11

2

1

iS

iS

0

0

Si

SiV

/1

/1

2

11

U

UzxV ),(1

U

UR

U

U

U

U

z

SSSS

SSSSzVVR

zz

zz

//

//

2

1/1

URiSUz

U

S

SR z

2

1

222

2

2

1

x

z

kvv

vR

kkkk

kkkk

vv

vvR

coscos

coscos

11

11

)( izzURisUz

U

Synthetic Example Figure 2 shows a shot gather with a simple velocity model shown in figure 1.

Receiver stations are 30 meters apart and the two reflectors are located at

depths of 1000 and 2000 meters respectively.

Imaging condition used for migration is

(reflection coefficients)

Results of migration using conventional Phase-shift and this proposed method

are shown in figures 3a and 3b. Migrated results for the upper reflector are

the same for both methods but different for the lower reflector. Figures 4a

and 4b show the plot for peak amplitudes corresponding to figures 3a and 3b.

The red line is the theoretical reflection coefficient calculated using model

parameters.

dzxU

zxUzxA

),,(

),,(),(

Discussion A new formula for one way wave equation extrapolation is pre-

sented. Comparing this formula with the conventional Phase-shift method shows that they are very similar except for an extra term

related to the reflection. Numerical results for a 2 layer model

shows that applying this method to migration can produce a more

correct image coincident with the reflection coefficient.

Figure 2. Synthetic data of wavefield reflected from two reflectors.

Figure 3. Migrated results 3a with conventional Phase-shift and 3b with equation:

)( izzURisU

z

U

Figure 4. Peak values from figure 3 compared to the reflection coefficient.

Methods The scalar wave propagation problem (constant density) is defined with the

wave equation (2D case)

where:

wavefield

x, z spatial coordinates

temporal frequency

Displacement and stress representation of equation above (Kosloff and Bysal

1983, 1987) is:

where:

and , where

Eigen value decomposition of matrix H:

0),,(),( 2

2

2

2

2

2

zxzxvxz

z/ 0

10

SH 22222 // vxS

1a 1b 1c

),( zx

),(),( zxzxHz

1VVH

2

xk22 / x 3a 3b

Absolute Imaging Inc., L120 Mission Centre, 2303 - 4th Street SW, Calgary, AB T2S 2S7