Because seismic data are almost always sampled uniformly in time, conventional methods perform the spatial resampling along each constant time slice or each constant frequency slice. The problem of two dimensional resampling is thus reduced to one dimension. In the 1950s, Yen (1956 and 1957) developed an algorithm for resampling irregularly sampled, one-dimensional data. In his algorithm, the irregularly sampled data are modeled as the interpolation of unknown uniformly spaced data. By solving a linear system analytically, Yen derived an exact, explicit formula for evaluating data at arbitrary points. Hale (1980) tested Yen's algorithm and pointed out that, although the method is exact for noise-free data, it may give erroneous results for noisy data. With Yen's method, a single noisy sample, even one far away from the correct sampling positions, may have an undesirable influence on the values of the resampled data. Moreover, Yen's algorithm is computationally expensive. To overcome these two problems, Hale uses a truncated, tapered sinc function as the basis function for interpolation. Although this procedure sacrifices accuracy at high frequencies, it performs better and at much less computational cost than Yen's method for data contaminated with truncations, aliasing data or noisy data. In the previous SEP report, Zhang (1991) described another one dimensional resampling scheme in which unevenly spaced data samples are modeled by a Fourier series. The coefficients of this Fourier series are found by minimizing the errors between the given data and the modeled data. Once these coefficients are determined, the data can be resampled arbitrarily.
One of the important characteristics of seismic data is the spatial coherence along the events corresponding to reflection or diffraction. These events may have varying slopes and curvatures. All the one-dimensional resampling methods mentioned above ignore this dominant feature of seismic data, and are therefore not optimal for resampling seismic data.
In studying the finite-aperture slant-stack transform, Kostov (1990) developed a two-dimensional trace-resampling method, in which unevenly spaced seismic traces are modeled as the superposition of many plane waves of various dips. The weighting coefficients of these plane waves are determined by minimizing the errors between the given unevenly spaced traces and the modeled traces. Kostov demonstrated his algorithm by filling in the missing traces in a common shot gather.
Two problems may potentially affect the results of resampling data by plane wave decomposition. First, the finite-aperture slant-stack transform is not a local operator. The aperture of this operator is usually equal to the spatial dimensions of input data. For filling large gaps of missing traces, a large aperture operator is needed. However, if the variations of sampling intervals are small perturbations, the large aperture operator becomes unnecessary. The large aperture also implies that a bad trace in the original data may have an undesirable influence on resampled traces far away from it. Second, for data containing many diffracted events or other curved events, plane wave decomposition is not the best way to model the data.
In this note, we describe a wave equation method for resampling unevenly spaced traces. We model the seismic traces recorded by an array of unevenly spaced receivers at the surface as the upward continuation of the wave field evenly sampled at a selected depth level. By solving a least squares inverse problem, we find the evenly sampled data at the selected depth level. Upward continuing the evenly sampled data allows us to resample the data at the surface. This algorithm is generally applicable to both 2-D and 3-D data, and to both poststack and prestack data. Because the wave equation modeling is used, the algorithm is robust and stable. The aperture of the upward continuation operator is determined by the distance from the selected depth level to the surface and the maximum time-dip of input data. Therefore, we can make the algorithm more efficient by reducing this distance or accept large perturbations of sampling spaces by increasing the distance. Furthermore, because the upward continuation operator is time-invariant, the computation can be done for each frequency independently. Therefore, the algorithm can be well implemented on a parallel machine. The algorithm can also be used to downward continue the wave field recorded by a finite aperture receiver array. An example with synthetic data shows that the algorithm produces a promising result.
In the following sections, we first briefly describe the theory of the new trace resampling method. We then give a synthetic example of resampling 2-D unevenly space traces.