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As mentioned above, expression (5) will be used to estimate the 2N-dimensional slowness vector $\mbox{\boldmath$S$}$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 Si obtained from the individual rays:
S \ =\ \frac{1}{M} \sum_{i=1}^{M} S_i \ =\ \frac{1}{M} \sum_{i=1}^{M} 
\frac{t_i}{l_i},\end{displaymath} (7)
where li is the source-receiver distance and M the total number of traveltimes.

When the model is anisotropic, the 2-D vector $\mbox{\boldmath$S$}$ 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 Sx2 and Sz2. 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 $\mbox{\boldmath$S$}$ of the homogeneous medium. Defining $W_x \ =\ S_x^2$ and $W_z \ =\ S_z^2$,the least-squares problem is
\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}\pmatrix{ W_x \cr
 =\ \mbox{\boldmath$d$},\end{displaymath} (8)

\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}\ =\ \pma...
 ...}^2 \cr
 \vdots & \vdots \cr
 {\Delta x_M}^2 & {\Delta z_M}^2},\end{displaymath}


\mbox{\boldmath$d$} \ =\ \pmatrix{ t_1^2 \cr
 t_2^2 \cr
 \vdots \cr
 t_M^2}. \end{displaymath}

Equation (8) can be solved in different ways. The most popular approach is by using the normal equations, resulting
\pmatrix{ W_x \cr
 W_z } \ =\ ({\displaystyle \mathop{\mbox{...
 ...e \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}}^T \mbox{\boldmath$d$}.\end{displaymath} (9)
However, the normal equations may have undesirable features with respect to numerical stability because the condition number of ${\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}}^T {\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}}$ is the square of the condition number of $\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}$. If $\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}$ is only moderately ill-conditioned, ${\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}}^T {\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}}$ is severely ill-conditioned. For this reason, methods that do not amplify the condition number of $\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}$ should be used to solve systems like (8) (for example QR factorization, Gill at al., 1990).

For estimating Wx and Wz simultaneously and accurately, $\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}$ has to be well conditioned. Note that this is not the case when most of the elements of the matrix satisfy either ${\Delta x_i}^2 \gg {\Delta z_i}^2$or ${\Delta z_i}^2 \gg {\Delta x_i}^2$. 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 $\mbox{\boldmath$S$}$ because of the limited view of the measurements translates immediately into severe ill-conditioning. This can be understood by trying to estimate Wx and Wz from the simple cross-well experiment shown in Figure [*], where ${\Delta x}^2 \gg {\Delta z_i}^2$.In this case

\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}\ =\ \pma...
 ...ta x}^2 & {\Delta z_1}^2 \cr
 {\Delta x}^2 & {\Delta z_2}^2 }. \end{displaymath}

The eigenvalues of this matrix are

\lambda_{\pm} \ =\ \frac{{\Delta x}^2 + {\Delta z_2}^2 \pm \...
 ...ta x}^2 -
{\Delta z_2}^2)^2 + 4{\Delta x}^2{\Delta z_1}^2}}{2}.\end{displaymath}

Because ${\Delta x}^2 \gg {\Delta z_i}^2$, the eigenvalues are approximately
\lambda_{+} \ =\ {\Delta x}^2 \nonumber \\ \lambda_{-} \ =\ 0. \nonumber\end{eqnarray}

Figure 2
Cross-well experiment with two rays.

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 Sz and the smallest one is related to Sx (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 $({\displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}}}^T \displaystyle \mathop{\mbox{\bf M}}_{\mbox{$\sim$}})$ tends to $\sum_{i=1}^{M} {\Delta x_i}^4$ and the smallest one tends to zero again.

The previous inversion scheme for homogeneous models can be generalized for estimating $\mbox{\boldmath$S$}$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 $\mbox{\boldmath$S$}$ 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 Sx is related to the larger singular value and Sz 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 Sx and the smallest ones will be related to lateral variations in Sz. We will demonstrate in the field data examples that estimating horizontal variations in Sz is indeed a difficult problem whereas it is always easier to estimate vertical variations in Sx.

Equation (5) can be used to estimate $\mbox{\boldmath$S$}$ 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 Sj and Sj+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 $\mbox{\boldmath$S_0$}$:
t_i( \mbox{\boldmath$S$}) & \approx & t_i( \mbox{\boldmath$S_0$...
 ...aystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}_{ij}
(S_j - S_{0j})\end{eqnarray}
where the elements of the Jacobian ${\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}_{ij}$ are

{\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}_{ij} \...
 ...0j}}{t_{ij}} & \mbox{if $N+1 \leq j \leq 2N$}\end{array}\right.\end{displaymath}

and tij is the traveltime of the ith ray in the jth cell of the model $\mbox{\boldmath$S_0$}$ (equation (2)). If we assume that $t_i ( \mbox {\boldmath$S$})$ represents one component of the vector of measured traveltimes, we can compute the perturbations $\Delta \mbox{\boldmath$S$}_j = (S_j - S_{oj})$ once the traveltimes in the reference model $\mbox{\boldmath$S_0$}$ has been calculated. The perturbation $\Delta \mbox{\boldmath$S$} = ( \mbox {\boldmath$S$} - 
\mbox{\boldmath$S_0$})$ is the solution of the following system of equations
\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}\Delta \mbox{\boldmath$S$} \ =\ \Delta \mbox{\boldmath$t$}\end{displaymath} (11)
where $ \Delta \mbox{\boldmath$t$}_i = 
t_i ( \mbox{\boldmath$S$}) -
t_i ( \mbox{\boldmath$S_0$})$.

Note that the matrix $\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$ depends explicitly on the slowness of the reference model $\mbox{\boldmath$S_0$}$ 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 $\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$ 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 $\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$ caused by to the limited view of the measurements is better handled than by solving the normal equations (in the overdetermined case).

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