We develop simple relations to estimate dynamic displacement gradients (and hence the strains and rotations) during earthquakes in the lake-bed zone of the Valley of Mexico, where the presence of low-velocity, high-water content clays in the uppermost layers cause dramatic amplification of seismic waves and large strains. The study uses results from a companion article (Bodin et al., 1997) in which the data from an array at Roma, a lake-bed site, were analyzed to obtain displacement gradients. In this article, we find that the deformations at other lake-bed sites may differ from those at Roma by a factor of 2 to 3. More accurate estimates of the dominant components of the deformation at an individual instrumented lake-bed site may be obtained from the maximum horizontal velocity and displacement, νmax and umax, at the surface. The maximum surface strain ɛmax is related to νmax by ɛmax = νmax/C, with C ∼ 0.6 km/sec. From the analysis of data from sites equipped with surface and borehole sensors, we find that the vertical gradient of peak horizontal displacement (Δumax/Δz) computed from sensors at 0 and 30 m equals (umax)z=0/Δz, Δz = 30 m, within a factor of 1.5. This is the largest gradient component, and the latter simple relation permits its estimation from surface records alone. The observed profiles of umax versus depth suggest a larger gradient in some depth range of 10 to 20 m, in agreement with synthetic calculations presented in Bodin et al. (1997).
From the free-field recordings of the 19 September 1985 Michoacan earthquake, we estimate a maximum surface strain, ɛmax, between 0.05% and 0.11%, and a lower bound for the peak vertical gradient (Δumax/Δz) between 0.3% and 1.3%. This implies that (1) the extensive failure of water pipe joints during the Michoacan earthquake in the valley occurred at axial strains of about 0.1%, not 0.38% as previously reported, and (2) the clays of the valley behave almost linearly even at shear strain of about 1%, in agreement with laboratory tests. The available data in the valley can be used to predict deformations during future earthquakes using self-similar earthquake scaling.