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The Engquist-Osher scheme used by Van Trier and Symes is an
interesting balance between computational speed and
precision. Although the scheme
introduces approximations in the calculation
of the travel time field that can be adverse for accurate
calculations of differential operators (which involve the
local derivative of the traveltime field), the scheme is more
stable than a straightforward finite-difference scheme.
In spite of all the instability problems one may
encounter using this algorithm, all of the
simpler alternatives I have tried are stable only for much smaller
grid sizes.
The fundamental algorithm described by Van Trier and Symes (1989)
is based on the eikonal equation

where
and where *s*(*x*,*z*) is the 2-dimensional slowness model and *t*(*x*,*z*) is the
travel time field.
The second equation used is the equality of the
partial derivatives of the fields *u*(*x*,*z*) and *v*(*x*,*z*),
| |
(2) |

In cylindrical coordinates the eikonal equation becomes

| |
(3) |

where
and the mixed partial derivatives are
| |
(4) |

The finite difference implementation of equations (1),
(2) and in cylindrical coordinates (3), (4)
is based on advancing the computational
front for the functions *u*(*x*,*z*) and *v*(*x*,*z*).
The traveltime field *t*(*x*,*z*) is found subsequently by integrating
the function with respect to *r*.
In cylindrical coordinates the scheme is based on using equation (4)
to compute the values of on a new computational
front of constant radius, by using the values of
and from
the previous computational front. Equation (4) becomes

| |
(5) |

Starting with a constant velocity condition in the immediate vicinity
of the source location ,we can advance the computational front using the finite-difference
equation (5).
Once the values of are known, the values of
can be computed using the eikonal equation:
| |
(6) |

** Next:** An insight into the
** Up:** Popovici : FD Traveltime
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Stanford Exploration Project

12/18/1997