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Deconvolution in the time domain can be implemented in terms of the following fitting goal for each (*x*,*z*) location:
| |
(4) |

where is a convolution matrix whose columns are downshifted versions of the source wavefield .
The least-squares solution of this problem is

where is the adjoint of .A damped solution may be used to guarantee to be invertible as in
where is a small positive number.
Equation (6) can be written in terms of the fitting goals
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(5) |

| |

where is the identity matrix.
This approach can be computational efficient if the time window is not too large and we use a Conjugate Gradient as optimization engine. However, it has the disadvantage of relying on a linear inversion process that may or may not converge to the global minimum.
A way to overcome this problem, obtaining an analytical solution, is to implement equation (6) in the Fourier domain, as we do in the next section.

** Next:** Deconvolution in the Fourier
** Up:** Valenciano and Biondi: Deconvolution
** Previous:** Reflector mapping imaging condition
Stanford Exploration Project

11/11/2002