Nonparametric kernel density estimation of magnitude distribution for the analysis of seismic hazard posed by anthropogenic seismicity

Abstract Frequent significant deviations of the observed magnitude distribution of anthropogenic seismicity from the Gutenberg–Richter relation require alternative magnitude–frequency models for probabilistic seismic hazard assessments. Five nonparametric kernel density estimation (KDE) methods are evaluated on simulated samples drawn from four magnitude distribution models: the exponential, concave and convex bi-exponential, and exponential-Gaussian distributions. The studied KDE methods include Silverman’s and Scott’s rules with Abramson’s bandwidth adaptation, two diffusion-based methods (ISJ and diffKDE), and adaptiveKDE, which formulates the bandwidth estimation as an optimization problem. Their performance is assessed for magnitudes from 2 to 6 with sample sizes of 400 to 5000, using the mean integrated square error of cumulative distribution (MISE F ) over 100,000 simulations. Their suitability in hazard assessments is illustrated by the mean of the mean return period (MRP) for a sample size of 1000. Among the tested methods, diffKDE provides the most accurate cumulative distribution function estimates for larger magnitudes. Even when the data are drawn from an exponential distribution, diffKDE performs comparably to maximum likelihood estimation when the sample size is at least 1000. Given that anthropogenic seismicity often deviates from the exponential model, using diffKDE for probabilistic seismic hazard assessments is recommended whenever a sufficient sample size is available.