Surface Displacements Due to Underground Faults

The mathematical formulas used in this Demonstration are taken from [1], which treats the surface displacements due to planar dislocations at arbitrary depth in a fully elastic half-space. The formulas have been simplified for a Poisson solid (shear modulus equals Lame's modulus ) and for an elemental potency (dislocation times fault area) of 1 . Even so, the expressions are fairly complex; however, Mathematica is very efficient in evaluating them for specific parameters. The tensile fault involves a dislocation perpendicular to the fault plane while the strike-slip and normal faults involve dislocations parallel to the fault plane (horizontal and vertical, respectively). The fault plane is oriented along the axis, and is rotated around the axis by the dip, starting at 0° dip for a horizontal fault plane and going to 90° dip for a vertical fault plane. The formulation for the strike-slip fault is for what is called right-lateral motion. This means that, for instance, in the case of a 90° dip fault, an observer in the negative half-plane would see the positive half-plane move to the right. Results for left-lateral motion are simply the mirror reflection around the axis of the right-lateral results. A similar geometric reflection leads to a reverse fault from the normal fault, with the dislocation of the positive side now going up relative to the negative side rather than down. Results for the reverse fault are therefore obtained by a reflection across the axis of the normal fault results.

Although the results here are for an elemental fault size of 1 , [1] also gives results for fault rectangles of arbitrary size. Those results are not a simple scaling of the elemental fault results. Nonetheless, the results given here for the elemental fault approximate the finite-fault results fairly well when the extent of the finite-fault plane is small relative to the depth of the fault plane. In all cases, for elemental or finite faults, the directions of surface motion are unchanged in the four quadrants of the surface plane.

Reference

[1] Y. Okada, "Surface Deformation Due to Shear and Tensile Faults in a Half-Space," Bulletin of the Seismological Society of America, 75(4), 1985 pp. 1135–1154.