Target-oriented wavefront synthesis

If the reflector we are interested in is not flat or if the velocity of the overburden medium has strong lateral variation, the surface-oriented wavefront synthesis does not produce a stack that has equally strong amplitude along the reflection from the reflector. The irregular amplitude of the reflected wavefield can be explained by the wavefront distortion resulting from the strong lateral velocity change or the variation of the incidence angle of the wavefront to the reflector from place to place.

Therefore, clear illumination of reflectors that are nonflat or located under a complex velocity overburden require target-oriented wavefront synthesis Claerbout (1976); Rietveld et al. (1992); Schultz and Claerbout (1978). In other words, we need a synthesis operator whose wavefront is a plane at a certain depth or along an irregular reflector, not at the surface. Using the property of the wavefield extrapolation operator that is close to unitary, we can easily generate a synthesis operator for a wave stack whose wavefront has a predefined shape at a certain subsurface location.

Suppose that we want to have a downgoing wavefield ${\bf \bar s} (z_n)$ at a depth level z_n. This wavefield can be obtained by propagating a wavefield at the surface, ${\bf \bar s}(z_0)$,to the depth level z_n, as follows:

$${\bf \bar s}(z_n) = W(z_n,z_0) {\bf \bar s}(z_0),$$ (6)

where W(z_n,z₀) represents a wave propagation operator from z₀ to z_n in forward time. If we assume the propagation operator, W(z_n,z₀), is a unitary operator, ${\bf \bar s}(z_0)$ can be obtained by applying the adjoint operator to both sides of the above equation as follows:

$$\begin{eqnarray} W^{\ast}(z_0,z_n) {\bf \bar s}(z_n) & = & W^{\ast}(z_0,z_n) W(z_n,z_0){\bf \bar s}(z_0) \nonumber \\ & = & {\bf \bar s}(z_0)\end{eqnarray}$$ (7)

where $W^{\ast}(z_0,z_n)$ is the adjoint to W(z_n,z₀), which represents a wave propagation operator from z_n to z₀ and backward in time. Even though it is a widely accepted fact that wavefield extrapolation is a pseudounitary operator, the closeness to the unitary property of the extrapolation operator is different depending on the algorithm chosen in implementation. For this thesis, I used the split-step Fourier method Stoffa and Fokkema (1990) for the wavefield extrapolation; the pseudounitary property of the split-step Fourier extrapolation is demonstrated in Appendix .

Assuming that every shot wavefield, ${\bf s}_j(z_0)$,is an impulse source at x_j, the wavefield ${\bf \bar s}(z_0)$ obtained from equation () becomes the synthesis operator

$${\bf \gamma } = {\bf \bar s}(z_0).$$ (8)

Therefore, the target-oriented wavefront synthesis is obtained by applying the wave stack equation () with ${\bf \gamma}$given above.

For synthesizing a slanted wavefront at a certain depth, we only need to start with a slanted plane wave such as ${\bf \bar s} (z_n)$ in equation (); that is

$$\bar s_j(z_n) = \exp [i\omega \tau_j]$$

Figure  is the stacked section obtained by synthesizing a plane wave with p = 0. (s/m) at 1200 m depth (Figure ) and Figure  is obtained by synthesizing a plane wave with p = 0.0001 (s/m) (Figure ). In Figures  and   we can observe that some events that were not visible in Figures  and   are now visible.

 

marm-syn-pln-1
Figure 10 Synthesis operator that synthesizes a plane-wave with p = 0 (s/m) at 1200 m depth.

 

marm-stk-pln-1
Figure 11 Wave-stacked section of Marmousi data set with synthesis operator shown in Figure .

 

marm-syn-pos-1
Figure 12 Synthesis operator that synthesizes a plane-wave with p = .0001 (s/m) at 1200 m depth.

 

marm-stk-pos-1
Figure 13 Wave-stacked section of Marmousi data set with synthesis operator shown in Figure .