Smooth surface reconstruction of earthquake faults from distributed moment-potency-tensor solutions

SUMMARY The earthquake fault as observed by seismic motion primarily manifests as a surface of displacement discontinuity within a linear elastic continuum. The displacement discontinuity and the surface normal vector (n-vector) of this idealized earthquake source are measured by the tensor of potency, which is seismic moment normalized by stiffness. We exploit this theoretical relation to formulate an inverse problem of reconstructing a smooth, 3-D fault surface from an areal density field of the potency tensor. In this problem, the surface is represented by an elevation field that parametrizes the vertical variation of the surface relative to a reference and the nodal planes of a given potency-density-tensor field describe the n-vector field. The remaining subject is the n-vector-to-elevation transform, the operation inverse to defining the n-vector field on a given surface. Whereas this transform is a well-posed one-to-one mapping in two dimensions where the n-vector has one degree of freedom, the transform becomes overdetermined in three dimensions because the n-vector has two degrees of freedom while the scalar elevation has only one, generally admitting no solution. This overdetermination originates from a reduction in degrees of freedom from six to five upon modelling the source as a displacement discontinuity rather than general potency density, namely inelastic strain. The sixth degree of freedom unmodelled by displacement discontinuities and n-vectors is a local violation of the determinant-free constraint in point potency sources; however, in areal sources of potency density, this simple zero-determinant approach results in a conflict with the global consistency of the n-vector field. Recognizing that this conflict derives from the capacity of the potency-density-tensor field to describe an inelastic strain source incompatible with displacement discontinuity on a surface, we explicitly introduce an a priori constraint to define the fault surface as the smooth surface that best approximates the surface distribution of inelastic strain by displacement discontinuity. We derive an analytical solution for the surface reconstruction thus formulated and demonstrate its ability to reproduce smooth 3-D surfaces from synthetic noisy n-vector fields. Lastly, we integrate the derived formula into the potency density tensor inversion and validate it in an application to the 2013 Balochistan earthquake. The estimated fault geometry agrees better with the observed fault trace than that of the previously proposed quasi-2-D surface reconstruction, highlighting the importance of accounting for 3-D fault geometry.