Problem 9.29
Derive the finite-difference migration equation (9.29d).
Background
The wave equation (2.5a) becomes, for two dimensions,
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(9.29a)
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We can simplify equation (9.29a) by replacing
with
(see problem 9.28). To replace derivatives in the
coordinate system with derivatives in the
system, we follow the procedure used in problem 2.6:
Substituting these expressions into equation (9.29a), we obtain the result
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(9.29b)
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However,
varies slowly because the coordinate system rides along with the wavefront and so we omit it. This gives the simplified wave equation,
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(9.29c)
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Using the method of finite differences discussed below, this equation can be changed to
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(9.29d)
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Approximate solutions of differential equations can be found using the method of finite differences. If we denote the value of
at
by the symbol
, an approximate value of the derivative is
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(9.29e)
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the error decreasing as
. The second derivative at a given point can be found by finding the difference between two first derivatives close to the given point and dividing the difference by the distance between the two points. Derivatives with respect to more than one variable can be found using the same principle.
Solution
We use points spaced at intervals
,
,
to evaluate the two derivatives in equation (9.29c):
Writing equation (9.29c) in the form
we substitute the above values of the two derivatives and obtain
Rearranging, we have
Solving for
gives
Multiplying the first bracket by
and the second one by
, we get
which is equation (9.29d).
Continue reading
Also in this chapter
External links
find literature about Finite-difference migration – book
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