FORWARD MODELING

We start by defining the equations needed to do the forward modeling step in the inversion algorithm. In an homogeneous transversely isotropic media, the traveltime between two different points separated by a distance $l=\sqrt{\Delta x^2 + \Delta z^2}$ can be expressed as

$$t \ =\ \sqrt{ {\Delta x}^2 S_{x}^{2} + {\Delta z}^2 S_{z}^{2}}$$ (1)

or

$$t^2 \ =\ {\Delta x}^2 S_{x}^{2} + {\Delta z}^2 S_{z}^{2},$$ (2)

where S_x and S_z are the horizontal and vertical slownesses respectively. Since the medium is homogeneous, the ray path is straight.

An heterogeneous medium can be approximated as a superposition of non-overlapping homogeneous regions. For this medium, the previous expression for the traveltime between two points can be easily generalized as follows:

$$\begin{eqnarray} t_i & = & \sum_{j=1}^{N} \sqrt{ \Delta x_{ij}^2 S_{x_j}^2 + \De... ...hspace{.5in} i=1,...,M, \nonumber \\ & = & \sum_{j=1}^{N} t_{ij} \end{eqnarray}$$ (2)

where t_ij is the traveltime of the i^th ray in the j^th cell and S_xj and S_zj are the horizontal and vertical slownesses respectively in that cell. $\Delta x_{ij}$ and $\Delta z_{ij}$ are the horizontal and vertical distances traveled by the i^th ray in each cell. If the slowness contrasts among adjacent cells are small, the ray paths can be approximated by straight lines. In equation (2), N is the total number of cells and M is the total number of traveltimes.

The slowness model can be seen as a vector $\mbox{\boldmath$S$}$ whose components contain the horizontal and vertical slownesses of each cell. This vector can be defined as follows:

$$\begin{eqnarray} S_i \ =\ S_{x_i} \\ S_{i+N} \ =\ S_{z_i}.\end{eqnarray}$$ (1) (2)

Then, the slowness vector $\mbox{\boldmath$S$}$ has the following form:

$$\mbox{\boldmath$ S$} \ =\ (S_1,S_2,\ldots,S_N,S_{N+1},S_{N+2},S_{N+3},\ldots,S_{2N})^T$$ (4)

where T means transpose. The first N components correspond to the horizontal slownesses of all the cells and the second N components correspond to the vertical slownesses. When the model is homogeneous, $\mbox{\boldmath$\space S $}$ is 2-dimensional and in general, for an heterogeneous model described by N cells, $\mbox{\boldmath$\space S $}$ is 2N-dimensional. Figure  shows the vector $\mbox{\boldmath$\space S $}$for the particular case of a layered medium.

 

S-in-layers
Figure 1 Slowness vector in a layered medium.

Using the new variables introduced in (3), the traveltime equation (2) can be written as

$$t_i \ =\ t_i ( \mbox{\boldmath$S$}) \ =\ \sum_{j=1}^{N} \sqrt{ {\Delta x_{ij}}^2 S_j^2 + {\Delta z_{ij}}^2 S_{j+N}^2}.$$ (5)

Notice that when the medium is isotropic (S_j = S_j+N), equation (5) reduces to the familiar equation that approximates the traveltimes computed in an isotropic model described by cells (McMechan, 1983):

$$\begin{eqnarray} t_i & = & \sum_{j=1}^{N} S_j \sqrt{{\Delta x_{ij}}^2 + {\Delta z_{ij}}^2 } \nonumber \\ & = & \sum_{j=1}^{N} S_j l_{ij}\end{eqnarray}$$ (6)

where l_ij is the length of the i^th ray in the j^th cell.

In the next section we will see that when expression (5) is linearized, it can be used for estimating the horizontal and vertical slownesses in an heterogeneous anisotropic model given a set of traveltimes measurements from a cross-well configuration. Equation (5) can also be used for surface geometries, as long as the depths of the reflectors are known a priori.