b-Bayesian: The Full Probabilistic Estimate of b-Value Temporal Variations for Non-Truncated Catalogs

The frequency/magnitude distribution of earthquakes can be approximated by an exponential law whose exponent (the so-called $b_{\mathrm{value}}$) is routinely used for probabilistic seismic hazard assessment. The $b_{value}$ is commonly measured using Aki's maximum likelihood estimation, although biases can arise from the choice of completeness magnitude (i.e., the magnitude below which the exponential law is no longer valid). In this work, we introduce the b-Bayesian method, where the full frequency-magnitude distribution of earthquakes is modeled by the product of an exponential law and a detection law. The detection law is characterized by two parameters, which we jointly estimate with the $b_{\mathrm{value}}$ within a Bayesian framework. All available data are used to recover the joint probability distribution. The b-Bayesian approach recovers temporal variations of the $b_{\mathrm{value}}$ and the detectability using a transdimensional Markov-chain Monte Carlo algorithm to explore numerous configurations of their time variations. An application to a seismic catalog of far-western Nepal shows that detectability decreases significantly during the monsoon period, while the $b_{value}$ remains stable around 0.8, albeit with larger uncertainties. This $b_{value}$ lower than 1 is expected in such a region with large interseismic strain accumulation. This confirms that the $b_{\mathrm{value}}$ can be estimated independently of variations in detectability (i.e., completeness). Our results are compared with those obtained using the maximum likelihood estimation, and using the b-positive approach, showing that our method avoids dependence on arbitrary choices such as window length or completeness thresholds.