References

• Claerbout, J. C., 1985, Imaging the Earth's Interior: Blackwell Scientific Publications.

• Hale, I. D., 1983, Dip-moveout by Fourier transform: Ph.D. Thesis, Stanford University.

• Hale, I. D., 1988, Dip-moveout processing: Course notes from SEG continuing education course, SEG.

• Levin, F. K., 1971, Apparent velocity from dipping interface reflections: Geophysics, 36, 510-516.

• Popovici, A. M., and Biondi, B., 1989, Kinematics of prestack partial migration in a variable velocity medium: SEP - 61, 133-147.

• Yilmaz, O., and Claerbout, J. F., 1980, Prestack partial migration: Geophysics, 45, 1753-1779.

• Zhang, L., 1988, A new Jacobian for dip-moveout: SEP - 59, 201-208.

APPENDIX A

The Fourier transform of a finite segment differs only in the amplitude term from the Fourier transform of an infinitely long segment. This is to be expected since we actually multiply the infinite segment by a boxcar filter which in Fourier domain means convolution with a sync function.

A two dimensional function representing a segment of constant amplitude in (t,y) space can be described by

$$H(y) \delta ({{t-t_0} \over p} - y)$$

where

$$\left \{ \begin{array} {ll} H(y)=1; & y \in [a,b] \\ \\ H(y)=0; & y \in (-\infty,a) \cup (b,\infty)\end{array} \right.$$

where t and y are the coordinates, t₀ is the intersection point on the t axis and the value p is the tangent of the slope. The function $\delta ({{t-t_0} \over p}-y)$ has unitary amplitude when the argument is zero or t=t₀+py. The 2-D Fourier transform is

$$\begin{array} {lcl} S(\omega,k_y) & = & \displaystyle{ \int ... ...omega p-k_y)}+i \int_a^b {\sin y(\omega p-k_y)} ] }\end{array}$$

and using the well known trigonometric transformations

$$\begin{array} {ccc} {\sin \alpha - \sin \beta} & = & {2 \sin... ...pha+ \beta} \over 2 } \sin {{\alpha-\beta} \over 2}}\end{array}$$

we obtain

$$\begin{array} {lcl} S(\omega,k_y) & = & \displaystyle{ {{e^{... ...(\omega p -k_y) e^{{{b+a} \over 2}(\omega p -k_y)}}.\end{array}$$ (29)