HR: 09:00h
AN: S51A-04    [PDF]
TI: Optimized steep dip Fourier finite-difference migration using Chebyshev polynomials and Simulated Annealing
AU: * Sen, S
EM: ssen@geosc.psu.edu
AF: Pennsylvania State University, Dept. of Geosciences, Deike building, University Park, State College, PA 16802 United States
AU: Anandakrishnan, S
EM: sak@essc.psu.edu
AF: Pennsylvania State University, Dept. of Geosciences, Deike building, University Park, State College, PA 16802 United States
AB: The two most important problems in 2D seismic imaging of complex structures using wave equation migration are (1) steep dips and (2) moderate to severe lateral velocity variations. To tackle this problem in wave-equation based migration schemes we have developed an optimized Fourier finite-difference (FFD) method based on a Taylor's series expansion of the square root operator in the one-way wave equation followed by a two-stage optimization process. Unlike other methods, we expand the square root operator at the reference velocity only, resulting in large errors during the truncation of the infinite series. This degrades the phase approximation severely, even when the lateral velocity variations are moderate To reduce this truncation error we do a first stage of optimization using Chebyshev polynomials. These are a special class of polynomials in which the generating function is a cosine and thus they have a maximum magnitude of 1 on the interval (-1, 1). We use these polynomials to improve the efficiency of the truncated power series (i.e. obtaining higher accuracy with fewer terms as well as a reduction of the truncation error). This requires converting our original power series to a form that can use Chebyshev polynomials. The crucial step of this conversion is a mapping of the interval in which the dependent variable in the original power series is defined onto the interval on which Chebyshev polynomials are defined i.e. (-1, 1). Once this mapping is done we use Chebyshev polynomials to rewrite the power series and then truncate it. After the truncation is done we then invert this truncated Chebyshev polynomial series to recover a truncated and optimized form of our original power series. This gives us the phase approximation after the first stage which is similar in form to Ristow and Ruhl's (1994) FFD approximation. In the second stage of optimization we optimize two coefficients in the phase approximation that we obtained after the first stage with the goal of maximizing the dip angle that is accurately migrated (defined as the dip angle where the relative phase error first exceeds one percent). This optimization is done by using a search algorithm based on simulated annealing (SA). The aim is to provide us with values of the two coefficients that maximize the dip angle under the one percent error constraint. The phase approximation that we finally obtain can migrate dip angles in the range of 65-69 degrees under extremely large velocity contrasts (ratio between the reference and actual velocity being as low as 0.33). This is comparable in accuracy to the Globally optimized Fourier finite-difference method (Huang et al.) and is far more accurate than all other existing migration schemes based on the one-way wave equation (for example 16-20 degrees more accurate than the unoptimized FFD). We demonstrate the accuracy of our method using impulse responses and synthetic examples.
DE: 0900 EXPLORATION GEOPHYSICS
DE: 0902 Computational methods, seismic
DE: 0910 Data processing
DE: 0935 Seismic methods (3025)
SC: Seismology [S]
MN: 2003 Fall Meeting