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** Up:** NONLINEAR TRAVELTIME TOMOGRAPHY
** Previous:** Traveltime and slowness model

The tomographic inversion is usually done through the optimization process.
Given a slowness model, we can simulate
the wave propagation between two wells and compute the first arrival
traveltime, denoted by .We want to find a set of parameters that minimizes
the differences between the traveltimes picked from the recorded data and
the traveltimes computed from the model, subjected to some linear constraints.
The mathematical description of this problem is as follows:

| |
(2) |

where is the unknown slowness parameter
vector. Matrix and vector are chosen to constrain the
inversion process. Because the traveltimes are generally
nonlinear functions of the
slowness model parameters, equation (2) defines
a nonlinear optimization problem. A nonlinear optimization is usually
accomplished iteratively. In each iteration, we
linearize the traveltime functions as follows:
| |
(3) |

where are
the slowness model parameters obtained
in the previous iteration, and are
the traveltimes calculated from the previous model. We define
and ,
and then form two vectors as follows:
We also define to be an matrix with the elements
and define
Then, equation (2) can be approximated as
| |
(4) |

Now the problem becomes a linear least squares problem. We can
use a gradient method to solve it. If we use the steepest descent
method, we need to calculate the gradient vector defined as follows:
| |
(5) |

This vector has a dimension of *M*.
If we choose the conjugate gradient method, we need to calculate,
in addition to the gradient vector,
the conjugate gradient vector defined as follows:
| |
(6) |

which has a dimension of *N* plus the number of conditions.

** Next:** FINITE DIFFERENCE METHODS
** Up:** NONLINEAR TRAVELTIME TOMOGRAPHY
** Previous:** Traveltime and slowness model
Stanford Exploration Project

12/18/1997