Robust probabilistic estimation of statistical variations in earthquake records: application to induced seismicity in western Canada

Summary Accurate characterization of the magnitude-frequency distribution of seismicity, and its associated uncertainties, is essential for seismic hazard assessment. This distribution is commonly described by the Gutenberg–Richter (GR) relation, parameterized by the b-value, which has been identified as a potential proxy for investigating many spatiotemporally varying Earth phenomena. Estimating the spatiotemporal variability of b-values often requires windowing, forcing a trade-off between resolution and statistical reliability. New probabilistic methods circumvent this by inferring both the number and locations of change points directly from earthquake catalogs. Nevertheless, accurately determining the b-value remains difficult because the GR relation only holds over a limited range of magnitudes. This research develops a general statistical model to address several methodological challenges in estimating the magnitude-frequency distribution of observed seismicity, including variations in space or time. The approach simultaneously solves for the b-value and magnitude-range limits. This avoids potential bias due to inaccurate manual truncation of earthquake catalogs. The model considers the entire observed catalog and parameterizes the decay of the distribution at both low and high magnitudes. Consequently, robust uncertainties in estimated b-values reflect uncertainty in the range of magnitudes over which the GR relation is observed to be valid. Importantly, spatiotemporal variations in the parameters that define the magnitude range are considered to be independent from the b-value, as we assume the physical factors that influence the GR relation are independent of the factors that limit the observed earthquake catalog. We demonstrate this methodology through application to simulated and observed earthquake catalogs. In particular, the value of our approach is highlighted through application to observed records of induced seismicity associated with fluid-injection operations in western Canada. Our results demonstrate accurate b-value estimates and associated uncertainties. Furthermore, the additional parameters that define the magnitude range serve as proxies for other factors including seismic network performance, recording duration, potential geometric limitations on earthquake size, and potential injection characteristics (in induced seismicity cases). Our approach also allows for the investigation of how these other factors may vary in space/time. Results from this work contribute to rigorous propagation of accurate b-value estimates, including uncertainties, into subsequent analyses such as seismic hazard models and regulatory protocols that are applied to industrial activity.