Seismic data interpolation beyond aliasing using regularized nonstationary autoregression |

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:

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):

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

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

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

where

and the elements of matrix are

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

where

the elements of matrix are

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 |

2013-07-26