P- and S-wave separated elastic wave equation numerical modeling using 2D staggered-grid

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P- and S-wave separated elastic wave equation numerical modeling using 2D staggered-grid

Zhang Jianlei*, Tian Zhenping, Wang Chengxiang; BGP, CNPC

Summary

Traditional numerical modeling method using second-order elastic wave equation can only generate the synthetic seismograms of the Z and X component in isotropic medium, in which the P- and S-wavefields are coupled. To obtain wavefield of the pure P- and S-waves, the general method is wavefield separation processing of the coupled wavefield of each component, but it is difficult to get completely separated seismograms. If we use P- and S-wave equation respectively to generate P- and S-wave, the converted P- and S-wave will not appear in wavefield, thus it is not equal to full wavefield modeling. Carrying out full separation of wavefield modeling of pure P- and S-wave (In reconstructed wavefield, X and Z component comprise P- and S-wave that is fully separated.) makes no need of wavefield separation in following multi-wave processing, what’s more, it is of great practical importance for us to study seismic wave propagation mechanism and structure of geology as well as oil reservoir characterization.

Based on P- and S-wave separation of second-order elastic wave equation and first-order staggered-grid method, a P- and S-wave separated modeling equation in first-order staggered-grid is presented. In this method, P- and S-wave are fully separated instead of coupled together to produce synthetic seismograms of pure P- and S-wave. The feasibility and accuracy are shown by a couple of examples of numerical modeling.

Introduction

Ma Detang (2003) presented an elastic wave modeling method using second-order elastic wave equations to separate P- and S-waves. This leads to a new direction for elastic wave modeling study and inversion method. But this method doesn’t adapt to widely extend used because of its low efficient modeling and serious numerical dispersion. Then, we are facing on a problem how to use highly efficient new arithmetic to realize the thinking.

At present, the most prevailing and appropriate modeling method is using velocity-stress elastic wave equation. This is solved using staggered-grid (half–grid in wave field numerical modeling). The advantage is that it doesn’t need the derivatives to the elastic constant media (the known velocity and density). Thus, numerical modeling using velocity-stress elastic wave equation is more efficient and accurate than that of traditional second-order elastic wave equation. Virieux (1984) proposed finite-difference staggered-grid technique to the velocity-

stress elastic wave equation (Virieux, 1986). Its finite-difference accuracy is onlyO( t2+ x2). Levander (1988) presented the staggered-grid with higher order finite-difference schemes, the accuracy isO( t2+ x4). One of the advantage of staggered-grid method is that, with little increasing of the computation and memory space, the local accuracy improved four times and the performance is better (Igel, 1992) than that of conventional grid. To get high accuracy and resolution for seismic wave modeling of the complex structure and media, finite-difference accuracy and grid dispersion have to be improved.

The paper solves the second-order P- and S-wave separated elastic wave equation with higher order staggered-grid finite-difference method. Synthetic modeling result show the validity and effectiveness of our methods.

Basic Principle

Traditional 3D second-order elastic wave equations can be represented as:

2u2

2 u t2=Vp + 2v+ 2w x2

x y x z + V2 2u 2u 2v 2

w s y+ z x y x z

2v222

（1）

t=V2 up + v+ w x y y y z +

2v222

V2 + v u w

s

x2

z2 x y y z 2w2 2v 2=V2 u+w +

t2p x z y z+ z2 2 2w 2u 2 V2 wv

s + x2

y2 x z y z By introducing new variables:

Sp={up,vp,wp} （2） Ss={us,vs,ws}

where Sp is non-rotational P-wave field, and Ss is non-dispersed pure S-wave field, equation (1) will become (Ma Detang, 2003):

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P- and S-wave separate Elastic wave equation numerical modeling using 2D staggered-grid

u=up+usv=vp+vsw=wp+ws 2up t=V2 2u 2v 2w p x+ x y+ x z

2vp2

2u 2 t2Vp

x y+v 2=w

y2

+ y z

2wp

2

t2=V2 u 2v 2w p （3）

x z+ y z

+ z2

2u2

2s u 2v 2w t2=V2 u s y2+ z2 x y x z 2v22s v 2u 2w t=V2 vs

x

+ z x y y z 2w2

s 2w 2u 2v t2=V2 ws x2+ y2 x z y z

For 2D case, equation (3) will be simplified to the

following equations:

u

=up+us w=wp+ws 2up2 2

u 2w

t=Vp x+ x z 2

2

wp=V2 u 2w t

p x z+ z 2u22

s（4）

t2=V2 us 2

w

z x z 2

w2s=V2 w

2

u

t2s x2 x z

Here,

up+wp

is non-rotational pure P-wave field,

wp+ws is non-dispersed pure S-wave field.

Unlike traditional regular grid, Madariaga (1976)

presented a more advanced staggered-grid finite-difference method which was used by Virieux to modeling SH and P-SV wave in isotropy medium. The accuracy of the traditional finite-difference schemes isO( t2+ x2). Without little loss of computation speed and memory space, the staggered-grid improved the local accuracy to be the fourth order at O( t2+ x4) (Levander, 1988). This makes the performance much better than that of conventional grid. Equation (5) shows 2D isotropic elastic equation, where B is the reciprocal of density.

In order to apply staggered-grid technique, we let

vx t=B τxx

x

+ τxz z

vz τzz t=B τxz

x

+ z （5） τxx vx v t=(λ+2µ) x

+λ

z

z τzz

t

=(λ+2µ)

vz z+λ vx x τxz t=µ vx

z+ vz x

where vx and vz are the velocity variables of particles in X- and Z-directions, respectively; τxxandτzz are normal stress;

τxz

is tangential stress；

λ

and

µ

Lames´

coefficient.

vx=vxp+vxs

vz=vzp+vzs

v=V2xpp t2λ+2µ( τxx x+ τ

zz x

vzp=V2

p( τxx τ

t2λ+2µ z+zz z) vxs2 1 τxz

1 τzz τxx t=Vs µ z 2λµ+2µ2 (λ+2µ) x λ

x

vzs=V2 1 τxz t

s µ x 1

2λµ+2µ (λ+2µ) τxx z λ τzz z （6）

τ xx=(λ+2µ) vx+ v τt xλz

zz vzx

=(λ+2µ) vz+λ

t τ z xxz vx

t=µ z+ vz

x

Here,

Vp is the P-wave velocity and Vs is S-wave

velocity. Equation (6) can be solved to get synthetic seismogram of pure P- and S-waves:vp=vxp+vzp

and

vs=vxs+vzs

. These pure seismograms contain

converted waves which will be absent from pure P- and S-wave equation modeling (without converted wave

component). This means that no further separations of P- and S-waves are needed.

Numerical modeling

To demonstrate the correctness of this method, we first test an example of horizontal layer model. The P- and S-

wave velocity model of the horizontal medium is shown in Figure 1. P-wave velocities are 2500m/s, 3000m/s and 4000m/s; S-wave velocities are 1732m/s, 2809m/s and 3309m/s; the densities are 2.0g

/cm3, 3.0g/cm3 and

3.5g/cm3. We use pure P-wave explosive source, which is located at the center of the model in horizontal direction and at the depth of 300m below the surface.

Distance(km)

Depth(k

Figure 1 P-wave velocity field

of a horizontal model. Figure 2 shows the snapshots of each component of the wavefield at the time 700 ms. Figures 2a and 2b are the P- and S-wave field of the Z component, respectively. Correspondingly, Figures 2c and 2d show the P- and S-wave field of the X component. P- and S-wave can be found in the first two snapshots and other waves such as reflection wave, transmitted wave, converted wave, refraction P-wave and refraction S-wave are also displayed in the figures.

Distance (km)

(a)

Depth (km)

Transmitted P-wave

(b)

Depth (km)

(c)

Depth (km)

(d)Depth (km)

Figure 2. Snapshotof each component of the wavefield at the time 700ms, (a) and (b) shows the P- and S-wave field of Z component; (c) and (d)

shows the P- and S-wave field of X component.

The synthetic seismogram is shown in Figure 3, where Figures 3a and 3b are P- and S-wave synthetic seismograms of the Z component, respectively. Figures 3c and 3d are P- and S-wave synthetic seismograms of the X component, respectively.

Trace number Trace number

(a) (b)

Time (s)

(c)

(d)

Time (s)

(e) (f)

时

间(s)

Figure 3. Modeling seismogram, (a) and (b) is P-andS-wave of Z component, (c) and (d) is P- and S-wave of X component, (e) and (f) is Z and X component seismogram that is got by applied elastic wave

equation modeling.

Figures 3e and 3f are the synthetic seismograms for the Z and X component from elastic wave equation modeling. These results just confirm that our 2D P- and S-wave separated elastic wave equation can model both pure P- and S-wave seismograms just like wavefield snapshot shown.

Distance (m)

Depth (m)

Figure 4 The P-wave velocity of complex model

Trace number Trace number(a)

(b)

Time (s)

(c)

(d)

Time (s)

(e)

(f)

Time (s)

Figure 5. The complex model synthetic seismogram, (a) and (b) are P- and S-wave synthetic seismogram of Z component; (c) and (d) are P- and S-wavesynthetic seismogram of X component; (e) and (f) are Z and X

synthetic seismograms from elastic wave equationmodeling.

To further test our modeling method, we design a relatively complex model. Figure 4 is the P-wave velocity for this complex model. The explosive source, which mainly generates a pure P-wave, is located 13000m away from the left boundary at a depth of 300m below the surface. The trace interval is 10m and maximum offset is 5000m. Modeling seismogram is shown in Figure 5, where Figures 5a and 5b are P- and S-wave modeling seismograms of the Z component, Figures 5c and 5d are P- and S-wave modeling seismograms of the X component, Figures 5e and 4f are the Z and X component seismograms from elastic wave equation modeling. This relatively complex model demonstrates that our modeling method works well.

Conclusion

The paper presents 2D P- and S-wave separate staggered-grid finite-difference elastic wave numerical modeling method; this method can generate separated P- and S-wave using elastic wave numerical modeling of complex 2D model. The P- and S-waves are naturally separated in the numerical modeling instead of the separation of wavefield afterwards. It may help understand elastic wave modeling and inversion.

Acknowledgement

The authors would like to thank the BGP of CNPC for encouraging this work and for permission to present this paper.