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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
can be expressed as
 
(1) 
or
 
(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
nonoverlapping
homogeneous regions. For this medium,
the previous expression for the traveltime
between two points can be easily generalized as follows:
 

 (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. and
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
whose components contain
the horizontal and vertical slownesses of each cell. This vector can
be defined as follows:
 
(1) 
 (2) 
Then, the slowness vector
has the following form:
 
(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, is 2dimensional and in general, for
an heterogeneous model described by N cells, is
2Ndimensional.
Figure shows the vector for the particular case of a layered medium.
Sinlayers
Figure 1 Slowness vector in a layered medium.
Using the new variables introduced in (3), the traveltime equation (2)
can be written as
 
(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):
 

 (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 crosswell configuration.
Equation (5) can also be used for surface geometries, as long
as the depths of the reflectors are known a priori.
Next: INVERSE MODELING
Up: Michelena & Muir: Anisotropic
Previous: Introduction
Stanford Exploration Project
12/18/1997