Fractal Rupture Geometry and Nonlinear Stress-Drop Scaling in Earthquake Source Parameters

Abstract. Static stress drop is commonly interpreted within a Euclidean self-similar rupture framework in which rupture area scales as seismic moment scales as A ∝ L², seismici moment scales as M₀ ∝ L³ and stress drop is independent of earthquake size. Here, I develop and test an alternative geometrical scaling framework in which the effective rupture support is non-Euclidean and scales as A_eff ∝ L^D_f, with 2 < D_f < 3. For self-similar slip, this assumption predicts a power-law stress-drop scaling, Δσ ∝ M₀^β with β = (D_f  - 2) / (D_f + 1), thereby linking an observable source-scaling exponent to an equivalent effective rupture-support dimension. Synthetic tests show that weak positive exponents in the theoretically predicted range are recoverable under realistic lognormal scatter. The framework is then applied to an open source-parameter catalog from the Southeastern Alps, using 1521 earthquakes with finite positive seismic moment and static stress drop. The full-catalog log–log regression gives β^{^}_obs=0.223, with bootstrap median 0.222 and 95 % confidence interval [0.180, 0.263]. A binned-median regression gives a consistent exponent of approximately 0.192. Under self-similar slip, the observed exponent maps to an equivalent rupture-support dimension D_f^eq ≃ 2.86 , with an observationally compatible range approximately [2.66, 3^-]. Moment-threshold tests, residual diagnostics, and leave-one-block-out jackknife analyses indicate that the positive scaling is stable within the analyzed catalog. These results do not constitute direct evidence that earthquake ruptures are fractal objects. Rather, they show that a non-Euclidean effective rupture-support framework provides a mathematically derived, observationally compatible, and testable explanation for positive stress-drop scaling in a regional source-parameter catalog.