We have pursued two approaches to estimating the time variant stress field within a Rayleigh wave. The first is a simple half-space analytical method. While this approach is oversimplified it has the attraction of a straight forward analytical solution. The second method is a brute force numerical approach which can handle the complexities of velocity structure and higher modes.

Method 1
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We use a simplified expression for Rayleigh particle displacement assuming a Poisson half space. Spatial derivatives in the x and z for each displacement yield four strain fields - xx, zz, xz and zx. Constituitive laws and an assumption about the shear modulus can be used to estimate the stress fields. (see derivation)

The figure at right displays the Rayleigh wave partical motion, strains and stresses as a function of time for a 30 s Rayleigh wave traveling 3.7 km/s assuming a shear moulus of 35 x 10⁹ Pa. Notable features include:

• The horizontal and vertical normal strain and stress fields (xx and zz orientations) occur in phase with one another and in phase with vertical ground displacement. It should be noted that this is also in phase with the radial direction ground velocity, not shown here.
• The shear strains and stresses (xzand zx orientations) occur a half cycle out of phase with this motion. All stresses necessarily vanish at the surface except for the horizontal normal stress.
• The maximum horizontal normal stress (xx) is significantly larger in amplitude than any other stress component.
• It should be noted that the half space approximation ignores the significant perturbations introduced by realistic velocity structure as shown below in method two. This approach is suitable for exploring the phase relationship between different elements of stress and motion and for rough characterization of the stress field amplitudes.

Method 2
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This approach is based on calculating fundamental mode synthetic surface wave seismograms for displacement in the vertical and radial directions. By recalculating the synthetics for slightly different locations in x and z, the four spatial derivatives required for strain can be determined numerically. (We chose to shift the locations of the receivers by 100 m). Synthetic seismograms were calculated using the Computer Programs in Seismology suite by R. Herrmann, though other methods could be used. Once the spatial derivatives have been determined, strain and stress are calculated using the same approach outlined in the analytic method above. In the test shown at right we use the same parameters as in the analytic example except we replace the half-space with a layered velocity model. The gross features are the same. The maximum strains and stresses in the shallow Earth are are similar in amplitude and the phase relationships are unchanged. The primary difference is the stress concentrations that occur near boundaries in the velocity model. This is particularly clear near the bounday at 15 km. These results are entirely model dependent. However they demonstrates the effects that should be anticipated in a realistic Earth.