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Problem formulation

If $\bold{D}$ is a regularization operator, and $\bold{m}$ is the estimated model, then Claerbout's interpolation method amounts to minimizing the power of $\bold{D m}$ ($\bold{m}^T \bold{D}^T \bold{D
 m}$) under the constraint  
 \bold{K m = m_k}\;,\end{displaymath} (1)
where $\bold{m_k}$ stands for the known data values, and $\bold{K}$ is a diagonal matrix with 1s at the known data locations and zeros elsewhere. It is easy to implement a constraint of the form (1) in an iterative conjugate-gradient scheme by simply disallowing the iterative process to update model parameters at the known data locations Claerbout (1999).

The operator $\bold{D}$ can be considered as a differential equation that we assume the model to satisfy. If $\bold{D}$ is able to remove all correlated components from the model and produce white Gaussian noise in the output, then $\bold{D}^T \bold{D}$ is essentially equivalent to the inverse covariance matrix of the model, which appears in the statistical formulation of least-squares estimation Tarantola (1987).

In this paper, I propose to use the offset continuation equation Fomel (1995a) for the operator $\bold{D}$. Under certain assumptions, this equation is indeed the one that prestack seismic reflection data can be presumed to satisfy. The equation has the following form:  
h \, \left( {\partial^2 P \over \partial y^2} - {\partial^2 ...
 ...\, t_n \, {\partial^2 P \over {\partial t_n \,
\partial h}} \;,\end{displaymath} (2)
where P(tn,h,x) is the prestack seismic data after the normal moveout correction (NMO), tn stands for the time coordinate after NMO, h is the half-offset, and y is the midpoint. Offset continuation has the following properties:

I propose to adopt a finite-difference form of the differential operator (3) for the regularization operator $\bold{D}$. A simple analysis of equation (3) shows that at small frequencies, the operator is dominated by the first term. The form ${\partial^2 P \over \partial y^2} - {\partial^2 P \over \partial
 h^2}$ is equivalent to the second mixed derivative in the source and receiver coordinates. Therefore, at low frequencies, the offset waves propagate in the source and receiver directions. At high frequencies, the second term in (3) becomes dominating, and the entire method becomes equivalent to the trivial linear interpolation in offset. The interpolation pattern is more complicated at intermediate frequencies.

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Next: Tests Up: Fomel: Offset continuation Previous: Introduction
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