CONJUGATE DIRECTIONS AND CONJUGATE GRADIENTS

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

Ms = t,   where we assume that the data vector ${\bf t}$ and the linear forward modeling operator ${\bf M}$ are given and that the model vector ${\bf s}$ is being sought.

In crosswell tomography example, ${\bf s}^T = (s_1,s_2,\ldots,s_n)$ is an n-vector of wave slownesses associated in either two- or three-dimensions with cells of constant slowness, ${\bf M}$ 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 ${\bf t}^T = (t_1,t_2,\ldots,t_m)$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.