Seismic data interpolation beyond aliasing using regularized nonstationary autoregression

Missing data interpolation

Irregular gaps occur in the recorded data for many different reasons, and prediction-error filters are known to be a powerful method for interpolating missing data. Missing data interpolation is a particular case of data regularization, where the input data are already given on a regular grid, and one needs to reconstruct only the missing values in empty bins (Fomel, 2001). One can use existing traces to directly estimate adaptive PEF coefficients instead of scaling the filter as in regular trace interpolation problem. However, finding the adaptive PEF needs to avoid using any regression equations that involve boundaries or missing data. This can be achieved by creating selection mask operator $K(t,x)$ , a diagonal matrix with ones at the known data locations and zeros elsewhere, for both causal translations and input data (Claerbout, 2010).

Analogously to the stationary prediction-error filter (1), adaptive PEF coefficients $B_n(t,x)$ use the unscaled format and appear as

$$\begin{array}{ccccc} B_3(t,x) &B_4(t,x) &B_5(t,x) &B_6(t,x) &B_7(t,x) \\ \cdot &\cdot &-1 &B_1(t,x) &B_2(t,x) \end{array}$$ (12)

The nonstationary coefficients $B_n(t,x)$ can be obtained by solving the least-squares problem

$$\widehat{B_n}(t,x) = \arg\min_{B_n}\Vert K(t,x)[S(t,x)-\sum_{n=1}^{N} B_n(t,x)S_n(t,x)]\Vert _2^2$$

$$+ \epsilon^2\, \sum_{n=1}^{N} \Vert\mathbf{D}[B_n(t,x)]\Vert _2^2\;.$$ (13)

where $S_n(t,x)=S(t-i,x-j)$ . By using shaping regularization, adaptive PEF coefficients are smoothly filled at missing trace locations.

Seismic data interpolation beyond aliasing using regularized nonstationary autoregression