VALIDITY OF THE FIRST ORDER APPROXIMATION

To examine the validity of the approximation I numerically calculated the exact operators for an orthorhombic medium. The elastic constants for this medium were generated by calculating the elastic constants for a layered medium with plane-vertical cracks at $20^\circ$ to the x-axis. Each graph in the grid of figure  is a plot of the amplitude of one element of the $3\times 3$ separation operator as a function of slowness. If the first order approximation is exactly valid the symmetric elements would be horizontal lines and the anti-symmetric elements would be lines with constant slope. Clearly, the first order approximation is not exact. However it is a reasonable approximation for a range of slownesses around zero.

Figure  shows the exact operator after a coordinate frame rotation of $20^\circ$ has been applied. This operator should be the combination of the second and third stages of the separation process. Note that the scaling by 0.5 has not been accounted for in this transformation. If the first order approximation and the separation of the rotation operator were exact we would expect to see a flat line at 0.5 for each of the symmetric elements and a straight line passing through (0.,0.) for the anti-symmetric elements. Again the first order approximation is good for a reasonable range of slownesses. The slownesses are expressed in sec./km. The first order approximation appears to be reasonable for slowness values up to 0.15, which for this medium corresponds to a P-wave propagating at 45 degrees to the vertical and an S-wave propagating at 25 degrees to the vertical. This are both large angles for typical surface seismic experiments. I consider the first order approximation to be a reasonable one for this set of elastic constants. Whether it is always a good approximation is a topic that needs further study.

 

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Figure 1 Exact separation operator for a rotated orthorhombic medium. Each element of the operator is plotted as a function of slowness.

 

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Figure 2 Exact separation operator with the rotation component removed. Each element of the operator is plotted as a function of slowness.