    Seismic data interpolation beyond aliasing using regularized nonstationary autoregression  Next: Missing data interpolation Up: Step 1: Adaptive PEF Previous: Step 1: Adaptive PEF

### Regular trace interpolation

An important property of PEFs is scale invariance, which allows estimation of PEF coefficients (including the leading '' and prediction coefficients ) for incomplete aliased data that include known traces and unknown or zero traces . For trace decimation, zero traces interlace known traces. To avoid zeroes that influence filter estimation, we interlace the filter coefficients with zeroes. For example, consider a 2-D PEF with seven prediction coefficients: (1)

Here, the horizontal axis is time, the vertical axis is space, and '' denotes zero. Rescaling both time and spatial axes assumes that the dips represented by the original filter in equation 1 are the same as those represented by the scaled filter (Claerbout, 1992): (2)

For nonstationary situations, we can also assume locally stationary spectra of the data because trace decimation makes the space between known traces small enough, thus making adaptive PEFs locally scale-invariant. For estimating adaptive PEF coefficients, nonstationary autoregression allows coefficients to change with both and . The new adaptive filter can look something like (3)

In other words, prediction coefficients are obtained by solving the least-squares problem,    (4)

where = , which represents the causal translation of , with time-shift index and spatial-shift index scaled by decimation interval . Note that predefined constant uses the interlacing value as an interval; i.e., the shift interval equals 2 in equation 3. Subscript is the general shift index for both time and space, and the total number of and is . is the regularization operator, and is a scalar regularization parameter. All coefficients are estimated simultaneously in a time/space variant manner. This approach was described by Fomel (2009) as regularized nonstationary autoregression (RNA). If is a linear operator, least-squares estimation reduces to linear inversion (5)

where (6) (7)

and the elements of matrix are (8)

Shaping regularization (Fomel, 2007) incorporates a shaping (smoothing) operator instead of and provides better numerical properties than Tikhonov's regularization (Tikhonov, 1963) in equation 4 (Fomel, 2009). Inversion using shaping regularization takes the form (9)

where (10)

the elements of matrix are   (11)

and is a scaling coefficient. One advantage of the shaping approach is the relative ease of controlling the selection of and in comparison with and . We define as Gaussian smoothing with an adjustable radius, which is designed by repeated application of triangle smoothing (Fomel, 2007), and choose to be the mean value of .

Coefficients at zero traces get constrained (effectively smoothly interpolated) by regularization. The required parameters are the size and shape of the filter and the smoothing radius in .    Seismic data interpolation beyond aliasing using regularized nonstationary autoregression  Next: Missing data interpolation Up: Step 1: Adaptive PEF Previous: Step 1: Adaptive PEF

2013-07-26