Regularization of multidimensional sparse seismic data using Delaunay tessellation

Abstract

Due to geographical or economic limitations, permanent earthquake seismometers or receiver stations in seismic surveys are often deployed irregularly. The data acquired at such irregularly distributed stations hinders the use of most geophysical data-processing tools, which have been developed based on an array of receiver stations at constant intervals. Therefore, to obtain better results, pre-processing trace regularization is essential before applying geophysical data-processing techniques. In two-dimensional cases, regularization can be performed using a method that finds the proper basis, such as matching pursuit interpolation. Due to computational time costs, however, it is difficult to extend its application to three or more dimensions. Higher-dimensional problems can use linear interpolation at the desired position using several traces from neighbouring stations. However, this requires additional steps for finding the proper neighbouring stations, increasing the time cost. This study proposes an efficient linear interpolation algorithm for higher-dimensional regularization using Delaunay tessellation, which has been used for various purposes, such as surface reconstruction and determining the nearest neighbourhood in computational geometry. Delaunay tessellation is a tiling method that numerically stable creates simplices whose vertices are arbitrary points distributed in multidimensional space. After tessellation, the points inside the simplex are transformed into barycentric coordinates, and the data are made by piece-wise linear combination of traces placed in vertices using the coordinates as weighting. In this way, irregular and sparse geophysical data are efficiently regularized in three-dimensional space while minimizing the additional computational time cost. Numerical results show that the overall artifact was reduced compared to the previous algorithm, and especially the aliasing on the f-k spectrum was clearly reduced.