ANADEF: A Nested-Permutation Alarm for Dual-Parameter Earthquake Forecasting
Spatially resolved stress proxies and rate-based seismicity models are increasingly combined for regional earthquake forecasting, yet formally testing their non-redundancy remains largely unaddressed. We present the Nested-Permutation Alarm for Dual-Parameter Earthquake Forecasting (ANADEF) pipeline, integrating a stress-sensitive Gutenberg--Richter $b$-value field, estimated via a penalized 2D B-spline inversion, with a stationary background rate ($\mu$) from space--time ETAS stochastic declustering. Applied to the Zagros Fold--Thrust Belt using an 18-year catalog ($n=40{,}731$, $M_{\mathrm{N}}\geq1.5$, 2006--2024) under a two-stage protocol with non-overlapping training (2006--2014) and target (2015--2024) windows, the model achieves, for $M_w\geq5.0$ ($N=55$), a retrospective Area Skill Score $S=0.69$ (95\% CI: 0.64--0.73), reducing alarmed area from $\tau\approx0.38$ ($\mu$-only) to $\tau\approx0.28$ while retaining hit rate $\nu=92.7\%$; since thresholds are calibrated on the evaluation catalog, these are in-sample, not out-of-sample, estimates. A nested permutation procedure re-optimizing thresholds within each null realization tests whether $b$-value adds information beyond $\mu$, absorbing the optimization bias. Significance is null-model dependent: cell-wise randomization yields $p=0.012$, while a conservative spatial-structure-preserving null gives weaker, non-significant evidence ($p=0.057$)---incremental stress information is suggestive but not unambiguously established. Calibrated thresholds were frozen and applied to updated 2015--2024 fields, generating an unvalidated, forward-looking spatial alarm template for 2025--2029. These results establish a reproducible, statistically transparent methodology for testing, rather than assuming, complementarity between stress-sensitive and rate-based predictors, offering a candidate operational template pending prospective validation.