3-D depth migration via McClellan transformations

Three-dimensional seismic wavefields may be extrapolated in depth, one frequency at a time by two-dimensional convolution with a circularly symmetric, frequencyand velocity-dependent filter. This depth extrapolation, performed for each frequency independently, lies at the heart of 3-D finite-difference depth migration. The computational efficiency of 3-D depth migration depends directly on the efficiency of this depth extrapolation. McClellan transformations provide an efficient method for both designing and implementing twodimensional digital filters that have a particular form of symmetry, such as the circularly symmetric depth extrapolation filters used in 3-D depth migration. Given the coefficients of one-dimensional, frequencyand velocity-dependent filters used to accomplish 2-D depth migration, McClellan transformations lead to a simple and efficient algorithm for 3-D depth migration. 3-D depth migration via McClellan transformations is simple because the coefficients of two-dimensional depth extrapolation filters are never explicitly computed or stored: only the coefficients of the corresponding one-dimensional filter are’required. The algorithm is computationally efficient because the cost of applying the two-dimensional extrapolation filter via McClellan transformations increases only lin~nrly with the number of coefficients N in the corresponding one-dimensional filter. This efficiency is not intuitively obvious. because the cost of convolution with a twodimensional filter is generally proportional to N’. Computational efficiency is particularly important for 3-D depth migration, for which long extrapolation filters (large N) may be required for accurate imaging of steep reflectors. gration has motivated many papers describing different INTRODUCTION methods for performing this important step in seismic data processing. This paper describes another method with significant advantages in computational simplicity and effiThe high computational cost of 3-D poststack depth miciency. The development of accurate 3-D poststack depth migration has been hindered by the fact that commonly used finite-difference migration methods cannot be extended easily from 2-D to 3-D. In particular, 2-D depth migration methods that are based on implicit depth extrapolation of the seismic wavefield, such as the common “45-degree” finitedifference method (e.g., Claerbout, 1985), are difficult to extend to 3-D processing. but as these authors have shown, the errors in spllttlng extrapolate alternately along the inline (x) and crossline (.v) depend significantly on reflector dip and azimuth. Specifidirections, independently. Splitting is the most practical way cally, for reflector dips greater than about 20 degrees and reflector azimuths near 45 degrees, measured with respect to to extend implicit extrapolation methods from 2-D to 3-D. either the x or y directions, splitting yields unacceptable errors in reflector positioning. Although techniques for re. . ducing the error in splitting have been demonstrated (RisI. tow, 1980; Graves and Clayton. 1990), these techniques significantly increase both the computational cost and complexity of implicit depth extrapolation methods. Many authors, including Brown (1983), Yilmaz (1987, p. 404-405), and Kitchenside and Jakubowicz (1987). have noted the error in splitting the depth extrapolation process to Some migration methods developed more recently, such as reverse-time migration, may be extended from 2-D to 3-D relatively easily, as demonstrated by Chang and McMechan (1989). Reverse-time migration, because it is based on an explicit backward extrapolation in time requires no splitting. However, several authors (Reshef and Kessler, 1989: Manuscript received by the Editor August 16, 1990; revised manuscript received February 25. 1991 *Department of Geophysics, Colorado School of Mines, Golden, CO 80401.