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

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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.