From Slip Deficit to Local Failure Margin: A Spatially Resolved Framework for Earthquake Recurrence and Pseudo-Prospective Testing at Parkfield, California

Earthquake recurrence is commonly estimated from historical intervals, geodetic loading rates, or renewal models. Such approaches are useful but can obscure the strong spatial heterogeneity of fault locking, effective normal stress, frictional strength, and aseismic slip. This paper introduces a state-of-stress margin recurrence formulation in which a fault is discretized into mechanically meaningful patches and the remaining distance to a Coulomb-type failure threshold is evaluated for each patch. The central quantity is the stress margin Mi = τf,i − τi, where τf,i = ci + μf,i(σn,i − Pf,i). When Mi > 0 and ˙Mi < 0, a local remaining-time diagnostic is defined as Trem,i = Mi/(− ˙M 18 i). A second, independent spatial condition uses a rate-and-state critical nucleation length, Rconn,i = Lconn,i/Lc,i, with Rconn ≥ 1 treated only as a necessary reduced-model condition, not as a timing law. The method is tested pseudo-prospectively at Parkfield using an information set restricted to observations and analyses available before the 28 September 2004 earthquake. The primary geodetic input is the 1991–1998 GPS inversion of Murray et al., which resolves a low interseismic slip-rate zone and a deep loading rate of about 33 mm/yr. The numerical implementation uses an explicitly identified digitization of the published slip-rate field and a calibrated stress-loading proxy; it is therefore a proof-of-concept rather than a full three-dimensional elastic stress inversion. The calculation identifies a strongly locked, high-deficit region before the earthquake. It does not, however, establish an absolute 90% failure state or a ±6 month timing accuracy because the pre-event observations do not uniquely determine the local cohesion, friction coefficient, pore pressure, and absolute stress tensor. The result is consequently a falsifiable framework for improving earthquake-cycle diagnostics rather than a claim of deterministic earthquake prediction. This revision (v4) adds a new Section 4, proposing a thick curved-beam correction to the effective normal stress for fault patches whose trace is curvilinear rather than straight, together with a dimensionless curvature-slenderness criterion for deciding when the correction matters.