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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:
where L is an ideal seismic-wave propagator, and is an ideal underground medium model.
From the irregular observed seismic-data, an underground medium model can be estimated:
where is the practical seismic wave propagator, which can be written as a complex matrix. is a conjugate transpose matrix of the matrix .Substituting the estimated model into equation (), the estimated and regular data set can be found:
In equation (), the ideal wave propagator L is unknown, but it can be replaced with the practical wave propagator .Therefore, equation () can be rewritten as
where is a Hessian matrix which acts as a filter. If we choose the filter as an ideal full-pass one, that is, , equation () can be rewritten as
which is a general seismic data mapping frame. Up to now, the seismic data mapping can be implemented with or (, , , ).
Redatuming also can be carried out with equation (). The formula for redatuming should be modified as follows:
where is a new and regular data set extrapolated from a topographic surface to another surface which may be a horizontal or non-horizontal datum. is the propagator corresponding to the topographic surface, and is the propagator to a horizontal datum. Now the Hessian matrix 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
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.