Steep events on seismic sections are often aliased because of the insufficient spatial sampling of wave fields by receivers. Such aliased events may create many kinds of coherent noises in the output of seismic event separation and seismic imaging processes. The solution to this problem is to interpolate seismic traces before these processes are applied.
A linear method of seismic trace interpolation usually consists of two steps, finding prediction filters from the known traces and then using the prediction filters to generate the missing traces. Two such methods have been developed. Claerbout (1991a) formulated the problem in the t-x domain. With his method, two-dimensional prediction filters are computed from known traces that are subsampled in time. Squeezing these filters both in time and in space gives the filters for generating the missing traces. Spitz (1991) noticed that seismic data are sufficiently sampled in time, and simplified the 2-D interpolation problem into a 1-D interpolation in the -x domain. For each frequency, the filter that predicts data along the x-axis is computed. Stretching the coefficients of the prediction filters along the frequency axis then gives the filters that are used to generate the missing traces in the frequency domain. Spitz applied his algorithm to field data. His results showed that the linear two-step approach works well and that the trace interpolation improves the resolution of migrated images and reduces coherent noises.
However, these two methods have a common weakness. Both methods use the low-frequency components of data to estimate the prediction filters for the high-frequency components of data, which assumes that the dip structure of the seismic events are the same for all frequency components of data. This approach may fail to find the correct filters when data are completely aliased or when the dip structures of the high-frequency components of the seismic events are different from those of the low-frequency components.
Claerbout (1991b) observed that if a seismic section only contains linear events, the aliased components are distinguishable from the unaliased components in the -kx domain. Using this observation, he proposed a nonlinear optimization scheme for interpolating missing data. We have applied Claerbout's principle to the process of finding the correct prediction filter and developed a new algorithm for interpolating seismic traces. With our method, the prediction filter at one frequency is obtained from the component of data at that frequency; no squeezing or stretching of the filter is required. The effects of aliased data on the prediction filters are removed by selecting the zeros of the prediction filters with a neural net. Consequently, this method does not assume a correlation among the dip structures of data within the different frequency bands. It can interpolate the missing traces even when all the data contained in the known traces are completely aliased.
This paper is organized as follows: In the first section, we define linear events and summarize the linear two-step algorithm for seismic trace interpolation. This section also introduces the notation that is required in the subsequent sections. The second section describes our method of finding the prediction filters from the aliased data and clarifies the assumptions that the method involves. The third section first compares the results of Spitz's algorithm and ours with synthetic data. It then shows how the interpolation process improves the results of removing tube waves in cross-well data.