An equivalent medium toolkit

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 $6\times6$ (compressed subscript notation) stiffness tensor. The symmetric $6\times6$ matrix is rearranged to give three $3\times3$ submatrices. We call the $6\times6$form the ``parameter'' form and the three $3\times3$ 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 $3\times3$ matrices. The mapping from the layer to group is computed using only $3\times3$ 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 $6\times6$ stiffness matrix to its full $3\times 3\times 3\times 3$ tensor form and then multiplying by a $3\times3$ 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.

 

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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.

 

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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!)