Unifying the Gutenberg-Richter Law with Probabilistic Catalog Completeness

We propose a probabilistic approach to modeling catalog incompleteness through four candidate augmented Gutenberg-Richter (GR) laws, which incorporates incompleteness into the frequency-magnitude distribution (FMD) using two parameters, mc, the transition magnitude, and σc, which defines the transition range from incompleteness to completeness. The four GR models are tested on synthetic and empirical catalogs, using multiple performance evaluation metrics. The GR-AEReLU model, which allows for an asymmetry in the convergence to the pure linear GR law for m > mc relative to the censorship of earthquakes of sizes smaller than mc, is found to consistently outperform, providing more robust estimates of seismological parameters (e.g., b-value) that better reflect realistic physical conditions and observational characteristics. This augmented framework offers three main advantages: (1) unified modeling of incompleteness into the FMD, (2) parameters with clear physical and statistical meaning, and (3) the ability to capture nonlinear and asymmetric detection behaviors. Finally, our analysis reveals systematic regional variations in earthquake b-values that deviate significantly from the assumed universal value of 1.0, challenging a fundamental paradigm in seismology and demonstrating the need for region-specific values that reflect local tectonic conditions in seismic hazard assessments.