RMS response of a one-dimensional half-space to SH

We examine the extent to which the response of a perfectly elastic half-space to an SH-wave incident from below can be characterized when knowledge about the elastic structure is limited to the near surface. Elastic properties are modeled as piecewise continuous functions of the depth coordinate. It is found that the site amplification function can be determined with a frequency resolution that depends inversely on the depth to which the elastic structure is known. Specifically, certain spectral averages of the site amplification function, concentrated over bandwidth Δƒ, depend only on the elastic structure down to a two-way travel-time depth of 1/Δƒ. These spectral averages are entirely independent of the elastic properties at greater depth. Equivalently, when the incident motion has a bandlimited white power spectrum of bandwidth Δƒ, the site amplification of the root mean square (rms) ground motion depends only on the elastic structure down to a two-way travel-time depth of 1/Δƒ. When the bandwidth is sufficiently large, the following corollary applies: the rms surface ground motion equals the rms incident motion multiplied by 2√I_b/I₀, where I₀ and I_b are shear impedances at the ground surface and basement depth, respectively. This result provides justification for a procedure conventionally used to correct stochastic estimates of earthquake ground motion to account for local site effects. The analysis also clarifies the limitations of that conventional procedure.

The results define specific site-response parameters that can be computed from knowledge of shallow structure alone and may thereby contribute to improved understanding of the physical basis for, and limitations of, site classification schemes that are based on average S-wave velocity at shallow depth. While the analytical results are rigorous only for infinite Q, numerical experiments indicate that similar results apply to models with finite, frequency-independent Q. The practical utility of the results is likely to be limited primarily by the degree of lateral heterogeneity present near sites of interest and the degree to which the sites respond nonlinearly to incident ground motion.