Fast automatic wave-equation migration velocity analysis using encoded simultaneous sources

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Appendix A

Equivalence of image-stack-power maximization and data-domain Born wavefield inversion

This appendix shows that maximizing the image stack power (or minimizing its negative) is equivalent to Born wavefield inversion, which minimizes the difference between the modeled and observed primaries. The difference-based objective function for data-domain Born wavefield inversion can be defined as follows:

$$J = \frac{1}{2}({\bf L}{\bf m}-{\bf d}_{\rm obs})^{*}({\bf L}{\bf m}-{\bf d}_{\rm obs}),$$ (19)

where ${\bf d}_{\rm obs}$ is the observed data vector, ${\bf m}$ is the reflectivity vector; ${\bf L}$ is the Born modeling operator that only modeled the angle stacked reflectivity (zero-subsurface-offset reflectivity), which is a function of the velocity vector ${\bf v}$ . Objective function A-1 is minimized by optimizing both ${\bf v}$ and ${\bf m}$ . Expanding equation A-1 yields

$$J = \frac{1}{2}\left({\bf m}^{*}{\bf L}^{*}{\bf L}{\bf m}-{\bf m}... ...}^{*}_{\rm obs}{\bf L}{\bf m} + {\bf d}^{*}_{\rm obs}{\bf d}_{\rm obs} \right).$$ (20)

In the least-squares sense, the reflectivity model ${\bf m}$ can be formally obtained as follows, assuming the Hessian ${\bf H}$ is invertible:

$${\bf m} = ({\bf L}^{*}{\bf L})^{-1}{\bf L}^{*}{\bf d}_{\rm obs} = {\bf H}^{-1}{\bf L}^{*}{\bf d}_{\rm obs}.$$ (21)

Substituting equations A-3 into A-2 and simplifying yield

$$J = \frac{1}{2}\left( -{\bf d}^{*}_{\rm obs}{\bf L}{\bf H}^{-1}{\bf L}^{*}{\bf d}_{\rm obs} + {\bf d}^{*}_{\rm obs}{\bf d}_{\rm obs}\right)$$ (22)

Since ${\bf d}^{*}_{\rm obs}{\bf d}_{\rm obs}$ is a constant, it can be ignored in the above equation, therefore

$$J \approx -\frac{1}{2}{\bf d}^{*}_{\rm obs}{\bf L}{\bf H}^{-1}{\bf L}^{*}{\bf d}_{\rm obs}.$$ (23)

Note that the migration image ${\bf m}_{\rm mig}$ is defined as follows:

$${\bf m}_{\rm mig} = {\bf L}^{*}{\bf d}_{\rm obs}.$$ (24)

Substituting equations A-6 into A-5 yields

$$J \approx -\frac{1}{2} {\bf m}^{*}_{\rm mig}{\bf H}^{-1}{\bf m}_{\rm mig}.$$ (25)

To simplify the problem, I ignore the Hessian ${\bf H}$ in equation A-7 and assume it to be an identity matrix. Therefore, equation A-7 becomes

$$J \approx -\frac{1}{2} {\bf m}^{*}_{\rm mig}{\bf m}_{\rm mig},$$ (26)

which is the same as equation 1 defined in the body of the paper. However, the Hessian ${\bf H}$ in equation A-7 might be important, especially in complex geologies, where the illumination is distorted by complex overburdens. The importance of the Hessian in equation A-7 remains an area for further investigation.

Fast automatic wave-equation migration velocity analysis using encoded simultaneous sources