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When a solution, , to the Helmholtz equation is known, obtaining
a ray trajectory using equation (5) is relatively trivial.
The expression for the wavefield gradient, , divided by the wavefield, , is,
| |
(7) |

An expression for the wavefield gradient vector, , is
obtained by retaining the imaginary component of equation
(7) and using the expression for in equation
(3),
| |
(8) |

Equation (8) may be rewritten explicitly as a system of two
decoupled ordinary differential equations,
| |
(9) |

The solution for ray-path, , is computed through an initial
evaluation of the right hand side of equations (9), and an
iterative forward step by a constant interval using the precomputed
quantity to determine the proper apportioning of the step
along each coordinate.
Note that because these differential equations are first-order, only
one initial condition (position) is required for ray computation, and
rays may be started from any location in the wavefield
solution.
Finally, although the two-dimensional formulation is presented here, the
extension to three dimensions is trivial.

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Stanford Exploration Project

10/14/2003