Regularization and Interpolation
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Regularization is the procedure by which the recorded data are reconfigured into a regularly sampled data set suitable for subsequent processing. This refers to the preparation of data to generate 3D CMP gathers with uniform fold prior to pre-stack time and depth migration.
FLOOD
In regularization, (FLOOD) data are binned and missing traces interpolated in common offsets to normalize the fold of coverage of the survey. Optionally traces within each CMP bin can be interpolated to bin centres. Typically FLOOD is applied in the crossline direction, but can be applied in the inline direction also.
A number of interpolation algorithms are available: linear for non-aliased dips, t-p and FX algorithms for steep dip, aliased data, and an advanced anti-leakage Fourier transform method (FROID). The t-p and FX methods can either just interpolate missing traces or optionally interpolate all live traces to bin centres. FROID automatically interpolates missing traces and interpolates live traces to bin centres.
FROID estimates the Fourier coefficients for an input data set and then can reconstruct the data on any desired grid. For irregularly sampled data, difficulties arise from the non-orthogonality of the global basis functions on an irregular grid. As a consequence, energy from one Fourier coefficient leaks onto other coefficients (spectral leakage). The anti-leakage Fourier transform overcomes this by “re-orthogonalizing” the global Fourier basis functions on irregularly sampled grids; this leads to an accurate estimation of the spectrum of the Fourier coefficients.
All the algorithms in FLOOD are designed for irregular input trace spacing and use the live trace actual mid points for the interpolation. A maximum distance is set for valid interpolation, so if ‘holes’ are too large, interpolation of missing data will not occur. The process is run within common-offset bins and is normally run following offset split. All trace headers are interpolated as part of FLOOD. Offset and azimuth are interpolated along with (x,y) mid-point positions and these are used to construct appropriate shot and receiver locations for migration.
FX Interpolation
Traces are interpolated using linear prediction filters. Prediction filters are computed within user specified time and space gates and the prediction coefficients used in a least squares system to compute the interpolated amplitudes. If N traces are being interpolated between each pair of input traces, then the prediction coefficients for the interpolated traces at a frequency f are mapped from those of the input data at frequency f/(1+N). Consequently, amplitudes at low frequencies and for high orders of interpolation may be potentially determined from below the seismic bandwidth of the input data. To avoid this, un-aliased low frequencies are interpolated using sinc functions and the FX algorithm applied only above a cut-off frequency determined by the maximum dip in the data (user specified). Higher orders of interpolation may be computed either in a single pass or by using a cascaded method.