INTRODUCTION

Geophysicists have known for many years how to compute the prediction-error filter (PEF) for a time series. Given a prediction filter, it is fairly obvious how to extrapolate the signal beyond its ends or how to fill in gaps. (Classical theory focuses on instabilities that can arise, but these instabilities do not arise when using the methods found in my recent book Claerbout (1992b), chapter 8. In essence, instability arises from using a prediction filter recursively. Instability is easily avoided by using a second stage of linear least squares to solve directly for the missing data.)

My book also shows examples of how to find missing portions of signals, both in their interior and exterior. It also shows some applications of multidimensional prediction-error filters. Additionally, the book illustrates the idea of fitting in small windows and choosing edge conditions so as to be able to merge windows fairly seamlessly. In SEP-73 I formalized and extended the ideas behind windowing as a means of coping with nonstationarity and I integrated it with prediction-error filtering in 2-D, and have since extended it to 3-D and prepared more examples Claerbout (1993b), 1993a, 1992c, 1992d, and 1992a.

Here I develop code to restore and extend a 3-D data cube by a two-stage linear least squares process. In the first stage I fit a 3-D prediction-error filter (PEF) to the given cube. Regression equations that involve unknown missing data elements are weighted to zero before solving the least-squares problem by conjugate gradients. In a second stage, I take the PEF as known and I find missing elements in (or beyond) the cube to minimize the power out of the prediction-error filter.