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 appears as the violation of the determinant-free constraint in point potency sources, it raises 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：地震断层通过地震运动观测主要表现为弹性连续体中的位移不连续性。此类理想化地震源的位移不连续性与曲面法向量（n向量）由势张量度量，该张量为刚度归一化的地震矩。本研究构建了一个反问题，旨在从势张量的面密度场重建光滑三维断层曲面。其中，曲面由高程场表示，而势密度的节面则代表曲面法向量场，从而将问题简化为n向量至高程的变换。尽管该变换在二维情形下为一一映射，但在三维情形下变为超定问题——这是因为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（即势密度）建模为具有五个自由度的位移不连续性。尽管该超定性在点势源中表现为行列式约束的违反，但在面势密度中则与n向量场的全局一致性相冲突。认识到势密度描述与位移不连续性不兼容的非弹性应变的能力后，我们引入先验约束，将断层定义为最佳近似非弹性应变（作为位移不连续性）的光滑曲面。我们推导了该公式的解析解，并展示其从含噪合成n向量重建三维曲面的能力。将该公式集成到势密度张量反演中，并应用于2013年俾路支地震。与先前准二维方法相比，估计的三维几何形状与观测断层迹线具有更好的一致性，验证了我们的提议。