Abstract: A method and apparatus for performing efficient interpolation of data sequences or signals is disclosed. A preferred embodiment of the present invention determines a suitable Gaussian quadrature to match given bandwidth and accuracy requirements. This Gaussian quadrature is then used to construct a suitable family of interpolating functions to represent a physical data sequence or signal (which, in a preferred embodiment, is seismic data). In one embodiment, Gaussian quadratures are constructed using trigonometric moments of exponential functions. In an alternative embodiment, an interpolating function is constructed using prolate spheroidal wave functions (PSWFs) by adopting Gaussian quadrature points corresponding to a family of PSWFs as interpolation points. The particular family of PSWFs utilized is determined in accordance with bandwidth and accuracy requirements.

Abstract: Disclosed are a method and a system for discrete transformation of measurements such as seismic data into and out of tau-p space which is both exact and practical in terms of processing time. The measurements can be filtered or otherwise processed in tau-p space in ways which are not practical or possible in their original space. Since the transformations into and out of tau-p space are exact, the filtered and transformed measurements are free of errors and distortions that perturb known approximate transforms which can be performed in a reasonable time. When the transformation process is carried out in frequency space, it is done frequency-by-frequency, and when it is carried out in the spatial domain, it can utilize a transformation matrix having a block-circulant structure. In each case, the transformation process and matrix have a structure which substantially reduces storage and processing requirements as compared with known prior art.