Imaging the synthesized stack

After the wavefront synthesis, the synthesized stack is expressed as equation (), and if we write it with a full operator, it becomes

$${\bf \bar g}(z_0) = \sum_{n=1}^{N} W(z_0,z_n) R(z_n) W(z_n,z_0) {\bf \bar s}(z_0).$$ (14)

We can notice that this equation has the same form as the forward model of a shot record (equation ()). Therefore, the imaging is performed in the same manner as profile imaging Claerbout (1985), with ${\bf \bar g}(z_0)$ as the upcoming wave and ${\bf \bar s}(z_0)$ as the downgoing wave. According to the profile imaging concept, the reflectors exist in the earth at places where the downgoing wave is time-coincident with an upcoming wave Claerbout (1985). It can be explained by two procedures: extrapolation and imaging.

By recalling that the extrapolation operator is unitary, after extrapolating the upcoming and downgoing wavefield recursively, we have

$$W^\ast(z_m,z_0) {\bf \bar g}(z_0) = R(z_m) W(z_m,z_0) {\bf \bar s}(z_0)$$ (15)

where $\ast$ denotes the adjoint. For each depth level, extrapolation is performed as follows:  

$${\bf \bar g}(z_n) = W^\ast(z_n,z_m) {\bf \bar g}(z_m)$$ (16)

and  

$${\bf \bar s}(z_n) = W(z_n,z_m) {\bf \bar s}(z_m).$$ (17)

The upcoming waves are extrapolated backward in time, the downgoing waves, forward in time. Keeping in mind that the upcoming wavefield is the convolution of the reflectivity with the downgoing wave field, we can obtain the reflection coefficient from equations  () and  () by deconvolving ${\bf \bar s}(z)$ from ${\bf \bar g}(z)$ as follows:

 

$$R(z_n) = \sum_{\omega} {{ {\bf \bar g}(z_n) {\bf \bar s}^\as... ...over { {\bf \bar s}(z_n){\bf \bar s}^\ast(z_n) + \epsilon^2}}.$$ (18)

In equation (), $\epsilon^2$ represents a small positive value that is introduced for stability because ${\bf \bar s} (z_n)$ may contain zeros, and the summation of all frequency components is equivalent to inverse Fourier transforming and takes the zero-time component.

 

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Figure 17 Image obtained by synthesizing a plane-wave with p = 0 (s/m) at surface.

 

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Figure 18 Image obtained by synthesizing a plane-wave with p = .0001 (s/m) at surface.

 

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Figure 19 Image obtained by synthesizing a plane-wave with p = 0 (s/m) at 1200 m depth.

 

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Figure 20 Image obtained by synthesizing a plane-wave with p = .0001 (s/m) at 1200 m depth.

 

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Figure 21 Image obtained by synthesizing a plane-wave with 10 degrees angle at 1200 m depth.