Modal Filtering of Depth-Dependent Stress Drop in a Finite Seismogenic Layer: An Analytical Source Model for Effective Rupture Thickness

Abstract

The finite thickness of the seismogenic layer is a fundamental constraint on earthquake rupture and seismic moment release. In many source models, however, this constraint is incorporated in a largely geometric manner, implicitly assuming a direct mapping between depth-dependent stress drop and fault slip. Here we develop an analytical earthquake source model that explicitly accounts for the elastic boundary conditions governing deformation within a finite-thickness seismogenic layer. The problem is formulated as a two-dimensional antiplane elastic boundary-value problem with a traction-free surface and an elastically coupled basal boundary. Depth-dependent stress-drop distributions are prescribed along the fault plane, while the resulting slip field is obtained through modal decomposition of the vertical elastic problem. This formulation reveals that imposed stress-drop profiles are mechanically filtered by a discrete set of vertical eigenmodes whose structure is controlled by the boundary conditions. We show that the fundamental vertical mode dominates depth-integrated quantities such as slip and seismic moment, whereas higher-order modes contribute oscillatory components that are progressively suppressed by elastic stiffness. Consequently, different depth-dependent stress-drop tapers, despite sharing the same geometric thickness and stress-drop scale, produce systematically different slip distributions and seismic moments. To quantify this effect, we introduce the concept of an effective rupture thickness, which represents the fraction of the seismogenic layer that efficiently contributes to coseismic deformation. The resulting framework generalizes classical seismic moment scaling relations by demonstrating that the thickness term entering the scaling law is an emergent mechanical property rather than a purely geometric parameter.