Unifying the Gutenberg–Richter Law with Probabilistic Catalog Completeness

Abstract Replacing the traditional sharp completeness threshold with a smooth probabilistic transition allows for a more realistic representation of earthquake detection probabilities, acknowledging both missed small events below the completeness magnitude and occasional omissions above it. We propose a probabilistic approach to modeling catalog incompleteness through four candidate augmented Gutenberg–Richter (GR) laws, which incorporate incompleteness into the complementary cumulative frequency–magnitude distribution (CCFMD) using two parameters, mc the transition magnitude and σc, which defines the transition range from incompleteness to completeness. The four augmented GR models are tested within a dedicated validation framework using both synthetic and empirical catalogs. The synthetic catalogs are generated based on the concept of a smooth probabilistic mc, and the models are rigorously evaluated within a novel framework of correct specification (where each model is fitted to data generated by itself) and misspecification (where a model is fitted to data generated by a different model), using multiple performance evaluation metrics. The GRAE model, which allows for an asymmetry in the convergence to the pure linear GR law for m>mc relative to the censoring of earthquakes of sizes smaller than mc, is found to consistently outperform the others in this framework of synthetic and observational tests, providing more robust estimates of seismological parameters (e.g., b value) than existing parameter estimation algorithms, thereby better reflecting realistic physical conditions and observational characteristics. This augmented framework offers three main advantages: (1) unified modeling of incompleteness into the CCFMD, (2) parameters with clear physical and statistical meaning, and (3) the ability to capture nonlinear and asymmetric detection behaviors. It can enhance seismicity models by incorporating small, previously discarded events with appropriate probabilistic weighting, leading to improved seismic hazard assessments and a deeper understanding of earthquake processes.