Next: Dip picking Up: Ji: Interpolation Previous: Introduction

# THREE STEP INTERPOLATION

Many trace interpolation schemes function in two stages. The first stage finds dips of plane waves or finds filters which kill dips in a given data set. The second stage linearly interpolates missing traces along the dips found or finds missing traces in the least square sense by minimizing the filtered output.

In order to interpolate multi-dips which cross each other, the approach that finds missing traces by minimizing filtered output is preferable. In this approach, the filter's spectrum should be the inverse of the data spectrum desired after interpolation. For the filter which has an inverse spectrum, an approach, such as Claerbout's (1991) and Spitz's (1991) algorithm, uses a prediction-error filter (PEF) obtained from a low frequency portion of data (Ji, 1991). A shortcoming of the approach using the prediction error filter is that we cannot obtain the filter when the missing traces are randomly located among traces. However, an approach that interpolates linearly along the most coherent dip at a point presents no difficulty in such a situation.

In order to overcome the pitfalls of both approaches, I propose a three-step algorithm. The step of finding missing traces uses the approach of minimization of the filtered output in a least square sense because this approach correctly handles the amplitude when dips cross each other. The step of finding a prediction-error filter can be understood as finding zeros in the spectrum to minimize the filtered output. Therefore the spectrum of the prediction error filter resembles the inverse spectrum of a given data set. If we decompose the step of finding the prediction-error filter into two parts as a step for finding dips and a step for finding filters which kill dips found, the following three steps can be used as an interpolation scheme:

1.
Pick several dominant dips from the dip spectrum.
2.
Find a filter which kills the dips found.
3.
Find missing traces that minimize the filtered output in the least square sense.

Next: Dip picking Up: Ji: Interpolation Previous: Introduction
Stanford Exploration Project
11/18/1997