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Detection thresholds in earthquake catalogs frequently change in time due to station coverage improvements and network saturation effects during active periods such as mainshock‐aftershock cascades. This presents a challenge to seismicity‐rate estimation; there is a tradeoff between using as low a minimum magnitude as possible to maximize data while not undercounting the rate due to catalog incompleteness. Here we present a simple method, "a‐positive," which makes use of differential statistics to robustly estimate the seismicity rate in catalogs with time‐varying detection thresholds. We demonstrate the effectiveness of this method for a centuries‐long, hybrid earthquake catalog with both historical and instrumentally‐detected earthquakes in the Central and Eastern U.S., as well as for the 2019 Ridgecrest aftershock sequence in California, which has rapid changes in completeness due to network saturation. We find that the a‐positive method leads to more precise and less biased estimates of seismicity rate than traditional methods. In addition, with our improved estimate of earthquake rate early in the aftershock cascade, we find no evidence of rate‐saturation at short times from the mainshock; that is, the Omori c‐value is not distinguishable from zero. Plain Language Summary: The seismicity rate is a fundamental aspect of statistical seismology, and it is used to create long‐ term models that inform building codes and seismic hazard assessments. However, earthquake catalogs are imperfect records of earthquakes, due to both changing network coverage and short‐ term aftershock incompleteness, where large earthquakes obscure smaller ones in times of high activity. We here present a new method "a‐positive" to estimate the seismicity rate in catalogs with time‐varying detection thresholds. This method uses differential statistics to estimate the seismicity rate from interevent times, where interevent times are measured to the next larger earthquake, with no reference to a completeness magnitude. The "a‐positive" method provides more precise and less biased estimates of seismicity rate than traditional methods. The method has been tested on a centuries‐long, hybrid earthquake catalog with both historical and instrumentally‐detected earthquakes in the Central and Eastern U.S. and on the 2019 Ridgecrest aftershock sequence in California, which has rapid changes in completeness due to short‐term aftershock incompleteness. The results show that the "a‐positive" method is minimally sensitive to catalog incompleteness and provides an unbiased estimate of the earthquake rate even in the first moments of an aftershock sequence. Key Points: Earthquake catalogs are imperfect records of the earthquake rate, suffering from both network sparseness and short‐term rate saturation"a‐positive" estimates seismicity rate based on the time to the next larger earthquake, with no reference to completeness magnitudeUnbiased measurement of the aftershock rate shows no evidence for rate saturation at early times [ABSTRACT FROM AUTHOR]

The 2014 Working Group on California Earthquake Probabilities (WGCEP14) present the time-independent component of the Uniform California Earthquake Rupture Forecast, Version 3 (UCERF3), which provides authoritative estimates of the magnitude, location, and time-averaged frequency of potentially damaging earthquakes in California. The primary achievements have been to relax fault segmentation and include multifault ruptures, both limitations of UCERF2. The rates of all earthquakes are solved for simultaneously and from a broader range of data, using a system-level inversion that is both conceptually simple and extensible. The inverse problem is large and underdetermined, so a range of models is sampled using an efficient simulated annealing algorithm. The approach is more derivative than prescriptive (e.g., magnitude-frequency distributions are no longer assumed), so new analysis tools were developed for exploring solutions. Epistemic uncertainties were also accounted for using 1440 alternative logic-tree branches, necessitating access to supercomputers. The most influential uncertainties include alternative deformation models (fault slip rates), a new smoothed seismicity algo-rithm, alternative values for the total rate of Mw >5 events, and different scaling relationships, virtually all of which are new. As a notable first, three deformation models are based on kinematically consistent inversions of geodetic and geologic data, also providing slip-rate constraints on faults previously excluded due to lack of geologic data. The grand inversion constitutes a system-level framework for testing hypotheses and balancing the influence of different experts. For example, we demonstrate serious challenges with the Gutenberg-Richter hypothesis for individual faults. UCERF3 is still an approximation of the system, however, and the range of models is limited (e.g., constrained to stay close to UCERF2). Never-theless, UCERF3 removes the apparent UCERF2 overprediction of M 6.5-7 earth-quake rates and also includes types of multifault ruptures seen in nature. Although UCERF3 fits the data better than UCERF2 overall, there may be areas that warrant further site-specific investigation. Supporting products may be of general interest, and we list key assumptions and avenues for future model improvements. [ABSTRACT FROM AUTHOR]

We present implementation details, testing, and results from a new in-version-based methodology, known colloquially as the "grand inversion," developed for the Uniform California Earthquake Rupture Forecast (UCERF3). We employ a parallel simulated annealing algorithm to solve for the long-term rate of all ruptures that extend through the seismogenic thickness on major mapped faults in California while simultaneously satisfying available slip-rate, paleoseismic event-rate, and magnitude-distribution constraints. The inversion methodology enables the relaxation of fault segmentation and allows for the incorporation of multifault ruptures, which are needed to remove magnitude-distribution misfits that were present in the previous model, UCERF2. The grand inversion is more objective than past methodologies, as it eliminates the need to prescriptively assign rupture rates. It also provides a means to easily update the model as new data become available. In addition to UCERF3 model results, we present verification of the grand inversion, including sensitivity tests, tuning of equation set weights, convergence metrics, and a synthetic test. These tests demonstrate that while individual rupture rates are poorly resolved by the data, in-tegrated quantities such as magnitude-frequency distributions and, most importantly, hazard metrics, are much more robust. [ABSTRACT FROM AUTHOR]

Probabilistic seismic hazard assessment is requiring an increasingly broad compilation of earthquake sources. Fault systems are often divided into characteristic ruptures based on geometric features such as bends or steps, though events such as the 2002 M 7.9 Denali, and 2011 M 9.0 Tohoku-Oki earthquakes raise the possibility that earthquakes can involve subsidiary faults and/or rupture through identified geometric barriers. Here we introduce a method to discriminate among a wide range of possible earthquakes within a large fault system and to quantify the probability of a rupture passing through a bend or step. We note that many of the conditions favoring earthquake rupture propagation can be simulated using a static Coulomb stress change approximation. Such an approach lacks inertial effects inherent in a full dynamic simulation but does capture many of the empirical observations drawn from examining past ruptures, such as continuity of rake and strike, as well as distance across gaps or stepovers. We make calculations for a test region in northern California and find that the method provides a quantitative basis for ranking possible ruptures within localized fault systems. [ABSTRACT FROM AUTHOR]

Previous approaches used to determine the rates of different earthquakes on a fault have made assumptions regarding segmentation, have been difficult to document and reproduce, and have lacked the ability to satisfy all available data constraints. We present a relatively objective and reproducible inverse methodology for determining the rate of different ruptures on a fault or fault system. The data used in the inversion include slip rate, event rate, and other constraints such as an optional a priori magnitude-frequency distribution. We demonstrate our methodology by solving for the long-term rate of ruptures on the southern San Andreas fault. Our results imply that a Gutenberg-Richter distribution is consistent with the data available for this fault; however, more work is needed to test the robustness of this assertion. More importantly, the methodology is extensible to an entire fault system (thereby including multifault ruptures) and can be used to quantify the relative benefits of collecting additional paleoseismic data at different sites. [ABSTRACT FROM AUTHOR]

Model uncertainty is prevalent in probabilistic seismic hazard analysis (PSHA) because the underlying statistical signatures for hazard are unknown. Although methods for incorporating parameter uncertainty of a particular model in PSHA are well understood, methods for incorporating model uncertainty are more difficult to implement because of the high degree of dependence between different earthquake-recurrence models. We show that the method used by the 2002 Working Group on California Earthquake Probabilities (WGCEP-2002) to combine the probability distributions given by multiple earthquake-recurrence models has several adverse effects on their results. In particular, WGCEP-2002 uses a linear combination of the models that ignores model dependence and leads to large uncertainty in the final hazard estimate. Furthermore, model weights were chosen based on data, which has the potential to systematically bias the final probability distribution. The weighting scheme used in the Working Group report also produces results that depend on an arbitrary ordering of models. In addition to analyzing current statistical problems, we present alternative methods for rigorously incorporating model uncertainty into PSHA. [ABSTRACT FROM AUTHOR]