We will use a notation consistent with earlier work of Berryman 1994 on crosswell seismic tomography in which the linear inversion problem to be solved takes the form

**M****s** = **t**,
where we assume that the data vector and the linear
forward modeling operator are given and that the model
vector is being sought.

In crosswell tomography example, is
an *n*-vector of wave slownesses associated in either two- or
three-dimensions with cells of constant slowness, is a
matrix of ray-path lengths such that *M*_{ij} is the length of the
*i*-th ray path through the *j*-th cell, and
is an *m*-vector of the traveltimes associated
with the ray paths between specified and numbered pairs of sources and
receivers. The assumption that the ray-path matrix **M** is known
corresponds to assuming that the full inverse problem is being solved
in an iterative fashion -- in which case the ray-path matrix in
question is just the one in use in the latest iteration. We generally
assume in addition that the problem is overdetermined so that *m* > *n*,
*i.e.*, the number of data exceed the dimension of the model
space.

11/11/1997