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Next: Common-offset prestack time migration Up: Wang and Shan: Imaging Previous: relationship between data space

Seismic data preprocessing

As discussed above, seismic-wave imaging needs a suitable data set. In general cases, real data sets have some drawbacks. For example, the spatial sampling of land data commonly is too coarse and/or irregular; marine data sets commonly show feathering. Therefore, seismic-data preprocessing is necessary. Seismic-data preprocessing deals with the signal-to-noise enhancement, wavelet correction, seismic-data regularization and interpolation, and redatuming. The latter three terms are closely related to seismic-wave imaging. Seismic data regularization, interpolation and redatuming can be seen as a seismic-data mapping under the least-squares theory. An irregular seismic-data set from on-site field acquisition can be expressed as follows:  
\textbf{d}^{obs}=L\textbf{m},\end{displaymath} (2)
where L is an ideal seismic-wave propagator, and $\textbf{m}$ is an ideal underground medium model. From the irregular observed seismic-data, an underground medium model can be estimated:  
\hat{\textbf{m}}_{inv}=\left[\left( \hat{L}^{*}\right)^{T}\hat{L} \right]^{-1}\left( \hat{L}^{*}\right) ^{T}\textbf{d}^{obs}\end{displaymath} (3)
where $\hat{L}$ is the practical seismic wave propagator, which can be written as a complex matrix. $\left( \hat{L}^{*}\right) ^{T}$ is a conjugate transpose matrix of the matrix $\hat{L}$.Substituting the estimated model into equation ([*]), the estimated and regular data set can be found:  
\hat{\textbf{d}}_{reg}=L\left[\left( \hat{L}^{*}\right)^{T}\hat{L} \right]^{-1}\left( \hat{L}^{*}\right) ^{T}\textbf{d}^{obs}.\end{displaymath} (4)
In equation ([*]), the ideal wave propagator L is unknown, but it can be replaced with the practical wave propagator $\hat{L}$.Therefore, equation ([*]) can be rewritten as  
\hat{\textbf{d}}_{reg}=\hat{L}\left[\left( \hat{L}^{*}\right...}=\hat{L}H^{-1}\left( \hat{L}^{*}\right)^{T}\textbf{d}^{obs},\end{displaymath} (5)
where $\textit{H}=\left( \hat{L}^{*}\right)^{T}\hat{L}$ is a Hessian matrix which acts as a filter. If we choose the filter as an ideal full-pass one, that is, $\textit{H}=\textit{I}$, equation ([*]) can be rewritten as  
\hat{\textbf{d}}_{reg}=\hat{L}\left( \hat{L}^{*}\right)^{T}\textbf{d}^{obs},\end{displaymath} (6)
which is a general seismic data mapping frame. Up to now, the seismic data mapping can be implemented with $\mathbf{DMO}+\mathbf{DMO}^{-1}$ or $\mathbf{PSTM}+\mathbf{PSTM}^{-1}$ (, , , ).

Redatuming also can be carried out with equation ([*]). The formula for redatuming should be modified as follows:  
\hat{\textbf{d}}_{datum}=\hat{L}_{2}\left[\left( \hat{L}^{*}...
 ...{L}_{2}H^{-1}\left( \hat{L}^{*}_{1}\right)^{T}\textbf{d}^{obs},\end{displaymath} (7)
where $\hat{\textbf{d}}_{datum}$ is a new and regular data set extrapolated from a topographic surface to another surface which may be a horizontal or non-horizontal datum. $\hat{L}^{T}_{1}$ is the propagator corresponding to the topographic surface, and $\hat{L}_{2}$ is the propagator to a horizontal datum. Now the Hessian matrix $\textit{H}$ has a relation to the topography and the acquisition configuration. Similarly, if the Hessian matrix is an ideal full-pass filter, equation ([*]) can be rewritten as  
\hat{\textbf{d}}_{datum}=\hat{L}_{2}\left( \hat{L}^{*}_{1}\right)^{T}\textbf{d}^{obs}.\end{displaymath} (8)
However, if a suitable Hessian filter is chosen, the quality of data mapping will be improved further. Next, we discuss data regularization with common-offset prestack time migration and the necessity of anti-aliasing for processing land data sets.