Surface-oriented wavefront synthesis

The simplest case in wavefront synthesis is synthesizing a horizontal plane-wave source at the surface Taner (1976). It can be done by stacking traces along a common receiver gather and is equivalent to the wave stack equation () with $\gamma_j$ equal to 1 for all j. Figure  shows this synthesis schematically.

Schultz and Claerbout 1978 have noted that reflectors of different dips can be shown more clearly by synthesizing various slanted plane waves and used the synthesis to estimate the velocity. Such a slanted plane wave synthesis is obtained by applying the wave stack equation () with

 

$$\gamma_j = \exp [i\omega \tau_j]$$ (5)

where

$$\tau_j = \tau_{j-1} - p \Delta x .$$

In equation (), $\omega$ is the temporal frequency (recall that equation () is formulated for a monochromatic field.), $\tau_j$ is the time shifting of each source with respect to the reference source and depends on p, which is the ray parameter that controls the direction of wavefront propagation, and $\Delta x$ is the interval between sources. Figure  explains schematically a simple example of the slanted wavefront synthesis.

For a more realistic example, the Marmousi data set is used. Figure  is the stacked section obtained by synthesizing a plane wave with p = 0 s/m at the surface (Figure ) and the stacked section shown in Figure  is obtained by synthesizing a plane wave with p = 0.0001 s/m (Figure ). The value of p (= 0.0001 s/m) was chosen so that the plane wavefront would have a 10 degrees incidence angle to a flat reflector whose velocity is 1700 m/s, which is the average velocity of the medium near the bottom of the water. In Figure  we can observe that some of the dipping events are enhanced and the water bottom reflection that is strong in Figure  becomes weak. In these stacked sections, we can see that the reflectors whose dip is normal to the direction of the wavefront propagation are more clearly visible than the others, respectively.

 

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Figure 4 A plane-wave synthesis at the surface.

 

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Figure 5 A slanted plane-wave synthesis at the surface.

 

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Figure 6 Synthesis operator to synthesize a plane-wave with p = 0 (s/m) at surface.

 

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Figure 7 Wave stacked section of Marmousi data set with synthesis operator shown in Figure .

 

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Figure 8 Synthesis operator that synthesizes a plane-wave with p = .0001 (s/m) at surface.

 

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Figure 9 Wave-stacked section of Marmousi data set with synthesis operator shown in Figure .