|
|
|
| Target-oriented joint inversion of incomplete time-lapse seismic data sets | |
|
Next: Target-oriented Hessian
Up: Ayeni and Biondi: Incomplete
Previous: APPENDIX A
From the Born approximation of the linearized acoustic wave equation, the synthetic seismic data recorded by a receiver at
due to a shot at
is given by
|
|
|
(A-1) |
where is frequency,
is reflectivity at image points ,
is source waveform,
and
and
are GreenŐ's functions from
to and from to
respectively.
Taking the true recorded data at
to be , the quadratic cost function is given by
|
(A-2) |
As shown by previous authors (Valenciano, 2008; Plessix and Mulder, 2004), the gradient of this cost function (summed over all frequencies, sources and receivers) with respect to reflectivity is the real part of
|
(A-3) |
and the Hessian (second derivatives) is the real part of
|
|
|
(A-4) |
where denotes all image points and is the complex conjugate of .
Plessix and Mulder (2004) and Valenciano (2008) discuss this derivation in detail.
Subsections
|
|
|
| Target-oriented joint inversion of incomplete time-lapse seismic data sets | |
|
Next: Target-oriented Hessian
Up: Ayeni and Biondi: Incomplete
Previous: APPENDIX A
2009-05-05