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## Kinematic relations

Let us suppose that a data gather P(t,xr) is recorded in an area that has a subsurface structure Q(x,z) and slowness model s(x,z). If this gather is migrated with a different slowness model , the resulting image will be a distortion of the true image of the subsurfaces.

Residual depth migration is the transformation from image to image Q(x,z). The kinematic operator that effects this transformation is defined by the mapping functions between the image locations (x,z) and . In my last report, I showed that these mapping functions are expressed implicitly in the following pair of equations:
 (1)
where xs is the surface location of shot and xr is the surface location of receiver. The terms and are traveltimes associated with slowness models s(x,z) and , respectively. The terms p and represent surface horizontal slownesses defined as follows
 (2)
where xc=xs or xr. The angles and are between ray directions and the vertical at the surface.

The partial derivative of xs with respect to xr depends on the type of a data gather. For post-stack data,

For pre-stack data with common shot geometry,

with constant offset geometry,

Therefore, equation (1) is generally applicable.

For each image location (x,z) on Q(x,z), equation (1) implicitly tells us the corresponding image location on . We want to solve in this equation as functions of (x,z). For general slowness models, this nonlinear equation set has to be solved using a numerical method. Searching is a straightforward method, in which, for each xr and (x,z), all points around (x,z) are checked to find the that satisfies equation (1). However, this algorithm is time-consuming, especially when the image dimensions are large.

I have developed an algorithm that solves equation (1) using finite-difference techniques. The ideas are similar to those used in the finite-difference calculation of traveltimes (Van Trier, 1990). The solution of equation (1) is known on the surface. We can extrapolate this solution in depth once we know the derivative of the solution with respect to the depth.

Next: Partial differential equations Up: REVIEW Previous: REVIEW
Stanford Exploration Project
12/18/1997