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.

翻译：地震断层作为弹性连续体中的位移不连续性，主要通过地震运动观测得以显现。这种理想化震源的位移不连续性与表面法向量（n向量）可通过势张量进行测量，该张量是经刚度归一化的地震矩。本研究构建了一个反演问题，旨在从势张量的面密度场重建光滑的三维断层面。此处，曲面通过高程场表示，而势密度节点的平面代表表面法向（n向量）场，从而将问题简化为n向量到高程的变换。尽管该变换在二维空间中是一对一映射，但在三维空间中却成为超定问题，因为n向量具有两个自由度，而标量高程仅有一个自由度，通常无解。这种超定性源于将具有六个自由度的非弹性应变（势密度）建模为仅五个自由度的位移不连续性。虽然该超定性在点势源中表现为违反无行列式约束，但在面势密度中则体现为与n向量场全局一致性的冲突。认识到势密度描述与位移不连续性不相容的非弹性应变的能力，我们引入先验约束，将断层定义为最能将非弹性应变近似为位移不连续性的光滑曲面。我们推导了该公式的解析解，并展示了其从含噪声合成n向量重建三维曲面的能力。我们将此公式集成到势密度张量反演中，并将其应用于2013年俾路支地震。估算的三维几何形态与观测断层迹线的吻合度优于以往的准二维方法，验证了本研究的有效性。