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

Abstract

Earthquake faults as observed by seismic motions primarily manifest as displacement discontinuities within elastic continua. The displacement discontinuity and the surface normal vector (n-vector) of such an idealized earthquake source are measured by the tensor of potency, which is seismic moment normalized by stiffness. This study formulates an inverse problem to reconstruct a smooth 3D fault surface from an areal density field of the potency tensor. Here, the surface is represented by an elevation field, while nodal planes of the potency density represent the surface normal (n-vector) field, reducing the problem to an n-vector-to-elevation transform. Although this transform is a one-to-one mapping in 2D, it becomes overdetermined in 3D because the n-vector has two degrees of freedom while the scalar elevation has only one, admitting no solution in general. This overdeterminacy originates from modeling the potency density, the inelastic strain with six degrees of freedom, as a displacement discontinuity of five degrees of freedom. Whereas this overdeterminacy is the violation of the determinant-free constraint in point potency sources, it appears as a conflict with the global consistency of the n-vector field in areal potency densities. Recognizing this capacity of the potency density to describe inelastic strain incompatible with displacement discontinuity, we introduce an a priori constraint to define the fault as the smooth surface that best approximates inelastic strain as displacement discontinuity. We derive an analytical solution for this formulation and demonstrate its ability to reproduce 3D surfaces from noisy synthetic n-vectors. We integrate this formula into potency density tensor inversion and apply it to the 2013 Balochistan earthquake. The estimated 3D geometry shows better agreement with observed fault traces than previous quasi-2D methods, validating our proposal.

Figures (13)

• Figure 1: Parametrization of a fault surface $\Gamma$ using a reference surface $\Gamma^\circ$. The normal vector ${\bf n}$ of $\Gamma$ depends on the location $\boldsymbol\xi$ parametrized as $(x,y,z)$. The $x$-$y$ plane describes the coordinate of $\boldsymbol\xi$ along $\Gamma^\circ$, and the $z$ axis represents the vertical elevation of $\Gamma$ relative to $\Gamma^\circ$. In the schematic, $\Gamma^\circ$ is drawn as a flat surface being perpendicular to a reference vector ${\bf n}^\circ$ and passing through a point $\boldsymbol\xi^\circ$ on $\Gamma$.
• Figure 2: Three types of $n$-vector characterizations based on different fault properties, indicated by arrows of different colors. (Red) The normal-vector field of a fault surface $\Gamma$, denoted by $n$. (Black) The reference normal $n^\circ$, which is the $n$-vector of the reference surface $\Gamma^\circ$. (Blue) The normal-vector field $n_*$ expected from the potency $D$, or precisely, from $a$ ($D$ parametrized by a basis function expansion along the reference surface.)
• Figure 3: Geometric relationship between the normal vector ${\bf n}$ and the tangent plane $d\boldsymbol\xi$ of a surface. A Cartesian coordinate is defined such that the reference-surface normal ${\bf n}^\circ$ aligns with the $z$ axis. The $xy$-plane represents the coordinate space spanned along the reference surface. The horizontal projection ${\bf n}_\parallel$ of ${\bf n}$ indicates the surface slope direction, while the ratio of the horizontal component $n_\perp$ to the vertical component $n_z$ of ${\bf n}$ determines the downward surface slope.
• Figure 4: The ground truth of our synthetic test 1: $z=[(y-\bar{y})/W]\sin[(x-\bar{x})/L]$ with $W=2$ and $L=30$, where $(\bar{x},\bar{y})$ denotes the reference plane center.
• Figure 5: Examples of (a) informed noisy $n$-vectors and (b) reconstructed elevation and $n$-vector fields in our synthetic test 1. The $n$-vector randomly varies from the ground truth with a standard deviation of 0.05.