Comparison between sparse Radon transforms towards an MPI implementation.

In exploration geophysics the linear Radon transform projects seismic data from time-offset (t-x) domain into intercept time-ray parameter (τ-p) domain while the parabolic Radon transform maps t-x domain seismic data into intercept time-curvature (τ-q) domain. Seismic reflections can be approximated as hyperbolae when the dips are gentle and the spread is small (Yilmaz, 2001). Ideal linear Radon transform maps a hyperbolic seismic event in an ellipse with the center at (0,0) and the vertical and horizontal radii equal to the intercept time τ and the reciprocal of the root mean square velocity 1/v, respectively. Normal moveout (NMO) correction approximates hyperbolic seismic events as parabolas that become points when an ideal parabolic Radon transform is applied to the seismic gather. In Radon transform domain Random noise is mapped destructively while seismic reflections are projected constructively and separately. This makes Radon transform methods useful in enhancing signal to noise ratio, doing trace interpolation and attenuating multiples in seismic data processing, especially for marine seismic data. In this work we focus our attention mainly on the parabolic Radon transform. Two main problems that arise in the Radon transform computation are i) the artifacts generated in the transform domain and ii) the long computing time. The artifacts originate from two sources. One is related with a too coarse transform domain sampling rate or a too wide sampling range, the other is related with the limited offset range in t-x domain. Different sampling criterions can be used to deal with the limits of the sampling rate and range (Turner, 1990; Schonewille and Duijindam, 2001). A common statement of the sampling criterions is that a higher frequency coincides with a more stringent limit of sampling rate and range. So we can relieve these requirements by doing a filtering before Radon transform. But we should only filter out the frequencies between the maximum useful frequency of the signal and the Nyquist frequency to avoid degrading the resolution of the seismic data. The limited offset range in t-x domain makes NMO corrected seismic events be projected into butterfly structures instead of points in τ-q domain. This fact causes the inverse parabolic Radon transform to be unable to reconstruct the original data faithfully and leads to the difficulty in doing multiple attenuation with the Radon transform method. To deal with this difficulty different authors developed various sparse Radon transform algorithms to compress the butterfly structures. Sacchi and Ulrych (1995) proposed the frequency domain sparse Radon transform (FDS-RT). Trad et al. (2003) presented the time-frequency domain sparse Radon transform (TFDS-RT). Lu (2013) formulated the iterative shrinkage sparse Radon transform (ISS-RT). We developed our code according to different standard and sparse Radon transform algorithms and compared their performances in the reconstruction of the original data and in the demultiple processing. We also compared our results with the outcomes of the ProMAX® software for what concerns the ability of multiple attenuation. To deal with the long computing time problem, we used Open MPI to parallelize the Radon transform algorithms. Open MPI is a Message Passing Interface (MPI) library project. MPI allows users to simultaneously use multiple processors to perform a calculation.