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We did this by developing a set of utilities for calculating the
elastic properties of equivalent media. The toolkit enables the user to
derive the elastic properties of layered and fractured media using S&M
theory.
The calculation of equivalent media is based on the parameterisation of
a layer by its thickness, density, and (compressed subscript
notation) stiffness tensor. The symmetric matrix is
rearranged to give three submatrices. We call the form the ``parameter'' form and the three matrix form the
``model'' form. This change is merely a reordering of elements, no
arrithmetic is performed.
Layers are combined in a ``group domain'' by simple addition of group
elements. In the group domain the parameterisation takes the form of
two scalars and three matrices. The mapping from the layer
to group is computed using only matrix operations as is the
inverse mapping from group to layer.
The other operation we wish to perform is rotation of the coordinate
frame of the elastic constants of a layer. This is done by expanding
the stiffness matrix to its full tensor form and then multiplying by a transformation
operator.
To find the elastic constants for the boundary grid cell, we add the
two bounding media in the appropriate ratio, and then rotate the result
so the layer dips the right amount.
block
Figure 2
A snapshot from a finite-difference model showing the results of a
wavefield hitting a gently sloping reflector. The source is a vertical
point force located at the top center. The top edge of the model is a
free surface, while all the other edges are absorbing boundaries. The
stairsteps in the discretized sloping interface generate noticeable
diffractions. The plot has been clipped at a level suitable for the
P-S conversions, since the higher spatial frequency S waves display the
diffractions better; as a result the primaries are strongly
overclipped.
fuzz
Figure 3
A snapshot from the same finite-difference model used in
Figure 2, but this time Schoenburg-Muir theory has
been used to interpolate the interface. The unwanted diffractions are
almost completely avoided. (As a result, the remaining artifacts are
more noticeable. The direct P wave can be seen to have slightly
reflected off the absorbing boundaries, and to have slightly wrapped
around. The artifact that can be hard to explain is the apparent
semicircular wavefront attached to and just behind the direct P wave.
This ``artifact'' is a laser-printer ghost of the strong direct S wave
at the top of the model!)
Next: EXAMPLE
Up: IMPLEMENTATION
Previous: Smooth gridding
Stanford Exploration Project
1/13/1998