Interpolation with Fourier-radial adaptive thresholding

Many interpolation methods are effective with regularly sampled or randomly sampled data, whereas the spatial sampling of seismic reflectivity data is typically neither regular nor random. Fourier-radial adaptive thresholding (FRAT) is a sparsity-promoting method in which the interpolated result is sparse in the frequency-wavenumber domain and is coherent in a manner consistent with that of a collection of unaliased plane waves. The sparsity and the desired pattern in the f-k domain are promoted by iterative soft thresholding and adaptive weighting; data in the f-k domain are transformed to polar coordinates and then low-pass filtered along the radial axis to generate the nonlinear weight. FRAT interpolates data that are randomly sampled and aliased; i.e., where the minimum distance between adjacent traces is greater than the Nyquist sampling interval. A conventional approach to solving this problem is to apply a cascade of two procedures: first a sparsity-based method, such as projection onto convex sets (POCS) to interpolate the data onto a regularly sampled but aliased grid, followed by a beyond aliasing approach such as Gulunay f-k interpolation to further interpolate the regularly sampled POCS result. In a simple synthetic example of two dipping plane waves with irregular, aliased sampling, FRAT outperformed this cascaded approach. In another experiment, the Sigsbee2A prestack synthetic data set was sampled using the source geometry from a 3D offshore survey where POCS will have difficulty with the semiregularity of this sampling pattern. FRAT produced results superior to those of POCS before and after the data were migrated.