As mentioned above, expression (5) will be used to
estimate the 2*N*-dimensional slowness vector given the traveltimes from a cross-well experiment. However,
we can investigate some of the difficulties in estimating such a vector
by first studying the case of a homogeneous medium (*N*=1)

When the model is isotropic, we usually estimate the slowness *S*
of the homogeneous medium that best fits the traveltimes
by simply averaging all
the slownesses *S*_{i} obtained from the
individual rays:

(7) |

When the model is anisotropic, the 2-D vector
that best fit the traveltimes can be obtained
by generalizing
the average (7).
This
generalization is, as expected, in a least-squares sense. Note that
expression (1b) is linear in *S*_{x}^{2} and *S*_{z}^{2}. Therefore, for a
given set of traveltimes and source-receiver
locations, it is possible to set up a least-squares problem to find
the vector of the
homogeneous medium. Defining and ,the least-squares problem is

(8) |

Equation (8) can be solved in different ways. The most popular approach is by using the normal equations, resulting

(9) |

For estimating
*W*_{x} and *W*_{z} *simultaneously* and *accurately*,
has to be well conditioned. Note that this is not the case
when most of the elements of the matrix satisfy
either or . These two
conditions describe cases when rays
are traveling close to the horizontal or the vertical. In such cases, it is
impossible to determine simultaneously
both components of the vector because of
the limited view of the measurements translates
immediately into severe ill-conditioning.
This can be understood by trying to
estimate *W*_{x} and *W*_{z} from the simple cross-well experiment
shown in Figure , where .In this case

Figure 2

In other words, the smallest eigenvalue (zero in this case) is
related to the vertical component of the slowness whereas the largest
one is related to the horizontal component.
On the contrary, for a VSP-like geometry largest eigenvalue
is related to *S*_{z} and the smallest one is related
to *S*_{x} (Dellinger, 1989).
Having more rays (*M*) without increasing the aperture
does not solve the problem. In such a case, the largest
eigenvalue of the matrix tends to
and the smallest one tends to zero again.

The previous inversion scheme for homogeneous models can be generalized for estimating in an heterogeneous medium. All we have to do is to solve systems of equations like (8) at each cell. In other words, the problem in the heterogeneous model is separated into many subproblems in homogeneous models. This approach might be easily implemented when the ray paths are used as basis functions for describing the slowness (Harris et al., 1990a) if, instead of averaging the slownesses of the different rays where they intersect, system of equations like (8) are solved to estimate the two components of the slowness.

Although this idea will not be
exploited
in the present paper, it can help us to understand
*intuitively* which components
of the slowness vector are easier (or more difficult)
to estimate from cross-well traveltimes measurements. In general, vertical
variations in the medium
are easier to estimate than horizontal variations.
Vertical variations correspond to singular vectors associated with
the largest singular values of the problem whereas lateral variations
are associated to the smallest singular values (Pratt and Chapman, 1990).
As explained earlier, in homogeneous models *S*_{x}
is related to the larger singular value and *S*_{z} is related to the
smaller one. Therefore, if the problem in a heterogeneous model
is solved as many separate subproblems in homogeneous models, the
largest singular values will be related to vertical
variations in *S*_{x} and the smallest ones will be related
to lateral variations in *S*_{z}. We will demonstrate in the field data
examples that estimating horizontal variations in *S*_{z} is indeed
a difficult problem whereas it is always easier to estimate
vertical variations in *S*_{x}.

Equation (5) can be used
to estimate for *all* the cells
at the same time (rather than in a cell-by-cell basis,
as explained before).
This equation is obviously non-linear in *S*_{j} and
*S*_{j+N}. One way to solve the problem is by a sequence of
linearized steps. We start by approximating (5) by a first order
Taylor series expansion centered in a given model :

(10) |

(11) |

Note that the matrix depends *explicitly* on the slowness of the
reference model in contrast with the isotropic case
where the matrix depends only on the lengths of the rays in each pixel.
In the isotropic case if the rays are straight, the estimation
of the slowness becomes a linear problem because is a constant.
In the anisotropic case, however, the problem is still non-linear even if the
rays are straight. Ray bending introduces another source of
non-linearity.

In the examples shown later, equation (11) will be solved using the LSQR variant of the conjugate gradients algorithm (Nolet, 1987). We will show that by doing a few iterations with this method at each linearized step, the ill-conditioning of caused by to the limited view of the measurements is better handled than by solving the normal equations (in the overdetermined case).

12/18/1997