S U M M A R YDespite the widely recognized importance of the spatio-temporal clustering of earthquakes, there are few robust methods for identifying clusters of causally related earthquakes. Recently, it has been proposed that earthquakes can be linked to their nearest neighbour events using a rescaled distance that depends on space, time and magnitude. These nearest neighbour links may correspond either to causally related event pairs within a clustered sequence or a non-causal relationship between independent events in different sequences. The frequency distribution of these rescaled nearest neighbour distances is consistent with a two-component mixture model where one component models random background events and the other models causally related clusters of events. To distinguish between these populations, a binary threshold has commonly been used to separate the clustered and background events. This has an obvious weakness in that it ignores the overlap of the two distributions and therefore all uncertainty in the event pair classification. It is also restricted so far to treating the two modes as normal distributions. Here we develop a new probabilistic clustering framework using a Markov Chain Monte Carlo mixture modelling approach which allows overlap and enables us to quantify uncertainty in event linkage. We test three hypotheses for the underlying component distributions. The normal and gamma distributions fail to fit the tails of the observed mixture distribution in a well-behaved way. In contrast, the Weibull mixture model is well-behaved in the tail, and provides a better fit to the data. We demonstrate this using catalogues from Southern California, Japan, Italy and New Zealand. We also demonstrate how this new approach can be used to create probabilistic cluster networks allowing investigation of cluster structure and the spatial, temporal and magnitude distributions of different types of clustering and highlight difficulties in applying simple metrics for cluster discrimination.

Recent developments in earthquake forecasting models have demonstrated the need for a robust method for identifying which model components are most beneficial to understanding spatial patterns of seismicity. Borrowing from ecology, we use Log-Gaussian Cox process models to describe the spatially varying intensity of earthquake locations. These models are constructed using elements which may influence earthquake locations, including the underlying fault map and past seismicity models, and a random field to account for any excess spatial variation that cannot be explained by deterministic model components. Comparing the alternative models allows the assessment of the performance of models of varying complexity composed of different components and therefore identifies which elements are most useful for describing the distribution of earthquake locations. We demonstrate the effectiveness of this approach using synthetic data and by making use of the earthquake and fault information available for California, including an application to the 2019 Ridgecrest sequence. We show the flexibility of this modeling approach and how it might be applied in areas where we do not have the same abundance of detailed information. We find results consistent with existing literature on the performance of past seismicity models that slip rates are beneficial for describing the spatial locations of larger magnitude events and that strain rate maps can constrain the spatial limits of seismicity in California. We also demonstrate that maps of distance to the nearest fault can benefit spatial models of seismicity, even those that also include the primary fault geometry used to construct them. Plain Language Summary Recently, many statistical models for earthquake occurrence have been developed with the aim of improving earthquake forecasting. Several different underlying factors might control the location of earthquakes, but testing the significance of each of these factors with traditional approaches has not been straightforward and has restricted how well we can combine different successful model elements. We present a new approach using a point process model to map the spatial intensity of events. This method allows us to combine maps of factors which might affect the location of earthquakes with a random element that accounts for other spatial variation. This allows us to rapidly compare models with different components to see which are most helpful for describing the observed locations. We demonstrate this approach using synthetic data and real data from California as a whole and the 2019 Ridgecrest sequence in particular. Slip rates are found to be beneficial for explaining the spatial distribution of large magnitude events, and strain rates are found useful for constraining spatial limits of observed seismicity. Constructing a fault distance map can also improve models where many events cannot be directly linked to an individual fault.

Abstract. Probabilistic earthquake forecasts estimate the likelihood of future earthquakes within a specified time-space-magnitude window and are important because they inform planning of hazard mitigation activities on different time scales. The spatial component of such forecasts, expressed as seismicity models, generally relies upon some combination of past event locations and underlying factors which might affect spatial intensity, such as strain rate, fault location and slip rate or past seismicity. For the first time, we extend previously reported spatial seismicity models, generated using the open source inlabru package, to time-independent earthquake forecasts using California as a case study. The inlabru approach allows the rapid evaluation of point process models which integrate different spatial datasets. We explore how well various candidate forecasts perform compared to observed activity over three contiguous 5-year time periods using the same training window for the input seismicity data. In each case we compare models constructed from both full and declustered earthquake catalogues. In doing this, we compare the use of synthetic catalogue forecasts to the more widely used grid-based approach of previous forecast testing experiments. The simulated catalogue approach uses the full model posteriors to create Bayesian earthquake forecasts, not just the mean. We show that simulated catalogue based forecasts perform better than the grid-based equivalents due to (a) their ability to capture more uncertainty in the model components and (b) the associated relaxation of the Poisson assumption in testing. We demonstrate that the inlabru models perform well overall over various time periods: The full catalogue models perform favourably in the first testing period (2006–2011) while the declustered catalogue models perform better in the 2011–2016 testing period, with both sets of models performing less well in the most recent (2016–2021) testing period. Together, these findings demonstrate a significant improvement in earthquake forecasting is possible although this has yet to be tested and proven in true prospective mode.

Abstract. Probabilistic earthquake forecasts estimate the likelihood of future earthquakes within a specified time-space-magnitude window and are important because they inform planning of hazard mitigation activities on different timescales. The spatial component of such forecasts, expressed as seismicity models, generally rely upon some combination of past event locations and underlying factors which might affect spatial intensity, such as strain rate, fault location and slip rate or past seismicity. For the first time, we extend previously reported spatial seismicity models, generated using the open source inlabru package, to time-independent earthquake forecasts using California as a case study. The inlabru approach allows the rapid evaluation of point process models which integrate different spatial datasets. We explore how well various candidate forecasts perform compared to observed activity over three contiguous five year time periods using the same training window for the seismicity data. In each case we compare models constructed from both full and declustered earthquake catalogues. In doing this, we compare the use of synthetic catalogue forecasts to the more widely-used grid-based approach of previous forecast testing experiments. The simulated-catalogue approach uses the full model posteriors to create Bayesian earthquake forecasts. We show that simulated-catalogue based forecasts perform better than the grid-based equivalents due to (a) their ability to capture more uncertainty in the model components and (b) the associated relaxation of the Poisson assumption in testing. We demonstrate that the inlabru models perform well overall over various time periods, and hence that independent data such as fault slip rates can improve forecasting power on the time scales examined. Together, these findings represent a significant improvement in earthquake forecasting is possible, though this has yet to be tested and proven in true prospective mode.

The Collaboratory for the Study of Earthquake Predictability (CSEP) is an open and global community whose mission is to accelerate earthquake predictability research through rigorous testing of probabilistic earthquake forecast models and prediction algorithms. pyCSEP supports this mission by providing open-source implementations of useful tools for evaluating earthquake forecasts. pyCSEP is a Python package that contains the following modules: (1) earthquake catalog access and processing, (2) representations of probabilistic earthquake forecasts, (3) statistical tests for evaluating earthquake forecasts, and (4) visualization routines and various other utilities. Most significantly, pyCSEP contains several statistical tests needed to evaluate earthquake forecasts, which can be forecasts expressed as expected earthquake rates in space–magnitude bins or specified as large sets of simulated catalogs (which includes candidate models for governmental operational earthquake forecasting). To showcase how pyCSEP can be used to evaluate earthquake forecasts, we have provided a reproducibility package that contains all the components required to re-create the figures published in this article. We recommend that interested readers work through the reproducibility package alongside this article. By providing useful tools to earthquake forecast modelers and facilitating an open-source software community, we hope to broaden the impact of the CSEP and further promote earthquake forecasting research.