REFERENCES

Biondi, B., and Palacharla, G., 1993, 3-D wavefield depth extrapolation by rotated McClellan filters: SEP-77, 27-36.

Hale, D., 1991, 3-D depth migration via McClellan transformations: Geophysics, 56, 1778-1785.

Lu, W.-S., and Antoniou, A., 1992, Two-dimensional digital filters: Marcel Dekker Inc.

McClellan, J., and Chan, D., 1977, A 2-D FIR filter structure derived from the Chebyshev recursion: IEEE Trans. Circuits Syst., CAS-24, 372-384.

Nautiyal, A., Gray, S. H., Whitmore, N. D., and Garing, J. D., 1993, Stability versus accuracy for an explicit wavefield extrapolation operator: Geophysics, 58, 277-283.

Parks, T., and Burrus, C., 1987, Digital filter design: John Wiley and Sons.

Soubaras, R., 1992, Explicit 3-D migration using equiripple polynomial expansion and Laplacian synthesis: 62nd Ann. Internat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, 905-908.

APPENDIX A

In this Appendix, I give the Chebyshev recursion formula and the different approximations to $\cos{\sqrt{k_x^2 + k_y^2}}$. The Chebyshev recursion formula is given by,  

$$\cos{nk} = 2\cos{\left[(n-2)k \right ]}\cos{k} - \cos{\left [(n-1)k \right ]}$$ (9)

The original McClellan transformation is given by

 

$$\cos{\sqrt{k_x^2 + k_y^2}} \approx -\frac{1}{2} + \frac{1}{2}\cos{k_x} + \frac{1}{2}\cos{k_y} + \frac{1}{2}\cos{k_x}\cos{k_y}$$ (10)

The modified McClellan transformation is given by (taken from Hale (1991)),  

$$\cos{\sqrt{k_x^2 + k_y^2}} \approx -\frac{1}{2} + \frac{1}{2... ...(1+\cos{k_y})} - \frac{c}{2}{(1- \cos{(2k_x)})(1-\cos{(2k_y)})}$$ (11)

where c is chosen, by exactly matching a particular value k along the diagonal k_x=k_y. c=0.0255 was used here, which corresponds to k=$\pi/3$.

The 2-D filter in the x-y domain corresponding to the original McClellan transformation is given by,

1/8 1/4 1/8

1/4 -1/2 1/4

1/8 1/4 1/8

The 2-D filter in the x-y domain corresponding to the modified McClellan transformation is given by (taken from Hale (1991)),

-c/8 0 c/4 0 -c/8

1/8 1/4 1/8

c/4 1/4 -(1+c)/2 1/4 c/4

1/8 1/4 1/8

-c/8 c/4 -c/8

The 2-D filters have a quadrantal symmetry, so the number of floating point operations required for the 2-D convolution of the filter with data is reduced.