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Similar to the conversion of the full wave equation into a one-way wave
equation, we can rewrite the eikonal equation as a one-way equation
that can be solved with standard techniques.
First, use to rewrite equation (1) as

| |
(2) |

By choosing a positive sign in front of the square root,
and by using instead of as the substitution
variable, we limit ourselves to downward-traveling rays.
I will come back to these choices in one of the later sections.
Second, take the derivative of this equation with respect to *x*:

| |
(3) |

where the function is defined as
| |
(4) |

is called the conserved flux; if , the rays do not
``flow'' downward anymore, but travel horizontally.
The (one-way) eikonal equation now has the form of a flux-conservative equation
This is a well-known equation in computational fluid dynamics, and it can
be solved in many ways (for example, see Roache, 1976).
The next section discusses one particular solution.

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Stanford Exploration Project

1/13/1998