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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
t_{n}=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
hyperbolictype 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 J_{0} 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
timeandspace 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: Threedimensional seismic data regularization
Previous: Secondorder reflection traveltime derivatives
Stanford Exploration Project
12/28/2000