Next: The kinematics of offset Up: Three-dimensional seismic data regularization Previous: Second-order reflection traveltime derivatives

Solving the Cauchy problem

To obtain an explicit solution of the Cauchy problem (-) for equation (), it is convenient to apply the following simple transform of the wavefield P:
 (287)
Here the Heavyside function H is included to take into account the causality of the reflection seismic gathers (note that the time tn=0 corresponds to the direct wave arrival). We can extrapolate Q as an even function to negative times, writing the reverse of () as follows:
 (288)
With the change of function (), equation () transforms to
 (289)
Applying the change of variables
 (290)
and Fourier transform in the midpoint coordinate y
 (291)
I further transform equation () to the canonical form of a hyperbolic-type partial differential equation with two variables:
 (292)

 rim Figure 1 Domain of dependence of a point in the transformed coordinate system.

The initial value conditions () and () in the space are defined on a hyperbola of the form . Now the solution of the Cauchy problem follows directly from Riemann's method Courant (1962). According to this method, the domain of dependence of each point is a part of the hyperbola between the points and (Figure ). If we let denote this curve, the solution takes an explicit integral form:
 (293)
Here R is the Riemann's function of equation (), which has the known explicit analytical expression
 (294)
where J0 is the Bessel function of zeroth order. Integrating by parts and taking into account the connection of the variables on the curve , we can simplify equation () to the form
 (295)
where
 (296) (297)

Applying the explicit expression for the Riemann function R () and performing the inverse transform of both the function and the variables allows us to rewrite equations (), (), and () in the original coordinate system. This yields the integral offset continuation operators in the domain
 (298)
where
 (299) (300)
 (301) (302)

The inverse Fourier transforms of equations () and () are reduced to analytically evaluated integrals Gradshtein and Ryzhik (1994) to produce explicit integral operators in the time-and-space domain
 (303)
where
 (304) (305)
The range of integration in () and () is defined by the inequality
 (306)

Equations (), (), and () coincide with (), (), and () in the main text.

Next: The kinematics of offset Up: Three-dimensional seismic data regularization Previous: Second-order reflection traveltime derivatives
Stanford Exploration Project
12/28/2000