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Absorption effects on plane waves in layered media

U.S. GEOLOGICAL SURVEY, 345 MIDDLEFIELD ROAD, MENLO PARK, CALIFORNIA 94025

Abstract

We present equations which can be used in conjunction with estimates of material properties to determine whether exact linear viscoelastic modeling will give significantly different results from elastic ray calculations. We apply the viscoelastic version of Snell's law to assess the effects of anelasticity on phase velocities, attenuation coefficients, transmission angles, and travel times for a plane wave propagating through a stack of plane viscoelastic layers described by the velocity and Q for a homogeneous wave in each layer.

We include a survey of the viscoelastic theory of plane waves to provide a context for the results. Our analysis shows that the most general version of Snell's law leads to a complex two-dimensional vector ray parameter in which case plane P-wave motion does not decouple from horizontal transverse S-wave motion. We show that in a layered medium the fields generated by a symmetric point source, a vector point force, a double-couple point force, or a plane of tractions corresponding to a complex vector parameter, as well as other "reasonable" sources, can be calculated by integrating over real ray parameters, but in all of these cases, one must consider the coupling in laterally heterogeneous media.

Differences from elastic ray behavior are found to be well correlated with the behavior of the parameter where is the angle of inhomogeneity of the wave. We present equations for in terms of layer parameters and both the vector ray parameter and incident wave parameters. Examples presented to illustrate the method show significant (5 to 10 per cent) deviations from elastic travel times, and offsets in a given layer may be possible in very lossy layered materials.