Viscoelastic‐gravitational deformation by a rectangular thrust fault in a layered Earth

Abstract

Previous papers in this series have been concerned with developing the numerical techniques required for the evaluation of vertical displacements which are the result of thrust faulting in a layered, elastic‐gravitational earth model. This paper extends these methods to the calculation of fully time‐dependent vertical surface deformation from a rectangular, dipping thrust fault in an elastic‐gravitational layer over a viscoelastic‐gravitational half space. The elastic‐gravitational solutions are used together with the correspondence principle of linear viscoelasticity to give the solution in the Laplace transform domain. The technique used here to invert the displacements into the time domain is the Prony series technique, wherein the transformed solution is fit to the transformed representation of a truncated series of decaying exponentials. Purely viscoelastic results obtained are checked against results found previously using a different inverse transform method, and agreement is excellent. The major advantage in using the Prony series technique is that deformations can be computed for arbitrary time intervals. A series of results are obtained for a rectangular, 30° dipping thrust fault in an elastic‐gravitational layer over viscoelastic‐gravitational half space. Time‐dependent displacements are calculated out to 50 half space relaxation times τa, or 100 Maxwell times 2τm = τa. Significant effects due to gravity are shown to exist in the solutions as early as several τa. The difference between the purely viscoelastic solution and the viscoelastic‐gravitational solutions grows as time progresses. Typically, the solutions with gravity reach an equilibrium value after 10–20 relaxation times, when the purely viscoelastic solutions are still changing significantly. Additionally, the length scaling which was apparent in the purely viscoelastic problem breaks down in the viscoelastic‐gravitational problem. Two independent length scales, one of which changes with time, are now seen to characterize the problem.

Effects of Earth's layered structure, gravity and curvature on coseismic deformation

“…After the gravity effect was first investigated by Rundle (1980), it was further studied by some researchers (Rundle 1982;Pollitz 1997;Wang 2005). Rundle (1982) claimed that the self-gravitating effect can be neglected. However, Wang's (2005) results indicate that the gravity effect on the coseismic deformation and post-seismic is different.…”

“…Sato (1971) and Sato & Matsu'ura (1973) investigated deformations in a multilayered medium. Rundle (1982) developed the viscoelastic-gravitational dislocation model. Ma & Kusznir (1994) researched the effects of layering and gravity based on a 3-D subsurface fault displacement fields.…”

Displacement field for an edge dislocation in a layered half‐space

“…The edge dislocation in an elastic half-space also furnishes a representation of the deformation due to the injection of a long (measured along strike) inclined dike or sill. Clearly, the representation could be improved by using a more complicated Earth model than a uniform to the faulting problem already available for a layered half-space [Rundle, 1982;Fernandez ei al., 1996;Kusznir, 1992, 1994]. However, two-dimensional solutions such as employed here are useful because of their comparative simplicity, and one of the principal uses of the two-dimensional solutions is to improve understanding of the deformation [e.g., Savage, 1987].…”

Hydrodynamic response to strike‐ and dip‐slip faulting in a half‐space

“…However, we hope that the results of this study will provide a means with which to evaluate hydrologic conditions after an earthquake on both spatial and temporal scales. It has been suggested that postseismic pore pressure plays a role in aftershocks [Nur and Booker, 1972] [Rundle, 1982] tends to occur over longer timescales or in narrow fault zones. Second, the dislocation method, specifying the fault slip as dislocation along certain boundaries, has been widely employed to obtain stress-strain distributions.…”