Earthquake Likelihood Models Research Articles

Spatial Distribution of Earthquake Occurrence for the New Zealand National Seismic Hazard Model 2022

ABSTRACT We develop candidate hybrid models representing the spatial distribution of earthquake occurrence in New Zealand over the next 100 yr. These models are used within the onshore/near-shore, shallow component of the distributed seismicity model within the New Zealand National Seismic Hazard Model 2022. They combine a variety of spatially gridded covariates based on smoothed seismicity, strain rates, and proximity to mapped faults and plate boundaries in both multiplicative and additive hybrids. They were optimized against a standardized catalog of New Zealand earthquakes with magnitude M ≥ 4.95 and hypocentral depth ≤40 km from 1951 to 2020. We extract smoothed seismicity covariates using three different methods. The additive models are linear combinations of earthquake likelihood models derived from individual covariates. We choose three preferred hybrid models based on the information gain statistics, consideration of the ongoing Canterbury sequence and regions of low seismicity, and inclusion of the most informative covariates. Since the hazard model is designed for the next 100 yr, the preferred hybrid models are also combined with 20-year earthquake forecasts from the “Every Earthquake a Precursor According to Scale” model. Thus, in total, six hybrid spatial distribution candidates are advanced for sensitivity analyses and expert elicitation for inclusion in the final logic tree for the New Zealand National Seismic Hazard Model.

Time-Independent Grid-Based Forecast Model for M ≥6.0 Earthquakes in Southeastern Tibetan Plateau Using GNSS Strain Rates and Seismicity

ABSTRACT Earthquake forecasting models play a vital role in earthquake occurrence assessment. Despite improved availability of seismic and geodetic data and processing techniques to produce high-resolution catalogs and deformation history, the implementation of earthquake forecasting models with seismic and geodetic data remains a challenge. In this study, we utilize seismicity and Global Navigation Satellite Systems (GNSS) data to propose time-independent grid-based regional earthquake likelihood models for southeastern Tibetan plateau (RELM-TibetSE). First, we solve the GNSS velocity field and strain rates from 1999 to 2017, deriving geodetic moment rates and introducing empirical correction coefficients to balance them with historical seismic moment rate. Subsequently, we employ a truncated Gutenberg–Richter law and Poisson process to calculate time-independent probabilities for M ≥ 6 earthquakes in 0.2° × 0.2° cells. The grid-based forecasting models indicate that the 30-year probability for M ≥ 6 earthquakes exceeds 1% in more than one-third of the entire study area, highlighting prominently higher earthquake occurrence in these regions. Probability distribution exhibits significant spatial variations. Finally, the predictive performance of the forecasting models is validated based on historical seismicity. The validation indicates that all RELM-TibetSE exhibit good predictive capability relative to a spatially uniform model. The RELM-TibetSE incorporating principal strain rates outperforms those involving maximum shear strain rate in forecasting seismicity. And the differences in forecasting performance between the RELM-TibetSE accounting for spatially varied seismogenic thickness and rigidity and those with uniform thickness and rigidity are not significant. The forecasting models also exhibit better predictive performance for seismic source areas than for epicenters. Moreover, the optimal model highlights zones with higher earthquake occurrence, including the zones about 50 km wide across the Ninglang fault, the zones across the southwestern segment of the Lijiang–Xiaojinhe fault, the China–Myanmar borderland north of the Nantinghe fault, and so on. Therefore, it is justified to conduct multidisciplinary rigorous observations to capture the potential nucleation process of future large earthquakes in these zones.

Prospective evaluation of multiplicative hybrid earthquake forecasting models in California

SUMMARYThe Regional Earthquake Likelihood Models (RELM) experiment, conducted within the Collaboratory for the Study of Earthquake Predictability (CSEP), showed that the smoothed seismicity (HKJ) model by Helmstetter et al. was the most informative time-independent earthquake model in California during the 2006–2010 evaluation period. The diversity of competing forecast hypotheses and geophysical data sets used in RELM was suitable for combining multiple models that could provide more informative earthquake forecasts than HKJ. Thus, Rhoades et al. created multiplicative hybrid models that involve the HKJ model as a baseline and one or more conjugate models. In retrospective evaluations, some hybrid models showed significant information gains over the HKJ forecast. Here, we prospectively assess the predictive skills of 16 hybrids and 6 original RELM forecasts at a 0.05 significance level, using a suite of traditional and new CSEP tests that rely on a Poisson and a binary likelihood function. In addition, we include consistency test results at a Bonferroni-adjusted significance level of 0.025 to address the problem of multiple tests. Furthermore, we compare the performance of each forecast to that of HKJ. The evaluation data set contains 40 target events recorded within the CSEP California testing region from 2011 January 1 to 2020 December 31, including the 2016 Hawthorne earthquake swarm in southwestern Nevada and the 2019 Ridgecrest sequence. Consistency test results show that most forecasting models overestimate the number of earthquakes and struggle to explain the spatial distribution of epicenters, especially in the case of seismicity clusters. The binary likelihood function significantly reduces the sensitivity of spatial log-likelihood scores to clustering, however; most models still fail to adequately describe spatial earthquake patterns. Contrary to retrospective analyses, our prospective test results show that none of the models are significantly more informative than the HKJ benchmark forecast, which we interpret to be due to temporal instabilities in the fit that forms hybrids. These results suggest that smoothing high-resolution, small earthquake data remains a robust method for forecasting moderate-to-large earthquakes over a period of 5–15 yr in California.

A Geodesy‐ and Seismicity‐Based Local Earthquake Likelihood Model for Central Los Angeles

AbstractWe estimate time‐independent earthquake likelihoods in central Los Angeles using a model of interseismic strain accumulation and the 1932–2017 seismic catalog. We assume that on the long‐term average, earthquakes and aseismic deformation collectively release seismic moment at a rate balancing interseismic loading, mainshocks obey the Gutenberg‐Richter law (a log linear magnitude‐frequency distribution [MFD]) up to a maximum magnitude and a Poisson process, and aftershock sequences obey the Gutenberg‐Richter and “Båth” laws. We model a comprehensive suite of these long‐term systems, assess how likely each system would be to have produced the MFD of the instrumental catalog, and use these likelihoods to probabilistically estimate the long‐term MFD. We estimate Mmax = 6.8 + 1.05/−0.4 (every ~300 years) or Mmax = 7.05 + 0.95/−0.4 assuming a truncated or tapered Gutenberg‐Richter MFD, respectively. Our results imply that, for example, the (median) likelihood of one or more Mw ≥ 6.5 mainshocks is 0.2% in 1 year, 2% in 10 years, and 18–21% in 100 years.

Multiplicative earthquake likelihood models incorporating strain rates

We examine the potential for strain-rate variables to improve long-term earthquake likelihood models. We derive a set of multiplicative hybrid earthquake likelihood models in which cell rates in a spatially uniform baseline model are scaled using combinations of covariates derived from earthquake catalogue data, fault data and strain rates for the New Zealand region. Three components of the strain rate estimated from Global Positioning System data over the period 1991–2011 are considered: the shear (SSR), rotational (RSR) and dilatational strain rates (DSR). The hybrid model parameters are optimized for earthquakes of M5 and greater over the period 1987–2006 and tested on earthquakes from the period 2012–2015, which is independent of the strain-rate estimates. The SSR is overall the most informative individual covariate, as indicated by Molchan error diagrams as well as multiplicative modelling. Most models including strain rates are significantly more informative than the best models excluding strain rates in both the fitting and testing period. A hybrid that combines the SSR and the DSR with a smoothed seismicity covariate is the most informative model in the fitting period, and a simpler model without the DSR is the most informative in the testing period. These results have implications for probabilistic seismic hazard analysis and can be used to improve the background model component of medium-term and short-term earthquake forecasting models.

Improved Seismicity Forecast with Spatially Varying Magnitude Distribution

ABSTRACT In this article, we present a time‐independent earthquake rate forecast for California. Our model features spatial variations of the Gutenberg–Richter b ‐value using the method of Kamer and Hiemer (2015). We account for lessons learned from the outcome of the Regional Earthquake Likelihood Models (RELM) experiment and use RELM’s framework to investigate different concepts for modeling the spatial distribution of seismicity. Our resulting forecast shows a significant information gain with respect to all first‐generation RELM mainshock forecasts. Our findings indicate that large‐scale b ‐value variations are a considerable feature for increasing the skill of Californian seismicity forecasts. We underline the importance of statistical rigor when implementing earthquake occurrence hypotheses. Our results have implications for seismic hazard studies, in which the b ‐value is either chosen as a regional constant or varies spatially between local zones. Future improvements of our model may serve as a basis for choosing either of these approaches.

Multiplicative Earthquake Likelihood Models Based on Fault and Earthquake Data

Abstract Potential earthquake sources are revealed by the locations of earthquakes in historical and instrumental catalogs and of geologically mapped faults, including plate boundaries. We derive a set of multiplicative hybrid earthquake likelihood models that combine earthquake and fault data for the New Zealand region. In these models, the cell rates in a spatially uniform baseline model are scaled using selected subsets of five covariates derived from the magnitudes and locations of past earthquakes, the location of the boundary between the Australian and Pacific plates, and the location and slip rate of mapped faults. The hybrid model parameters are optimized for earthquakes of M 5 and greater over the period 1987–2006 and tested on earthquakes from the period 2007–2014. No updating of models is undertaken during the fitting or testing period, but we consider two cases of the earthquake-based covariates in the tests: (1) all data prior to 1987 and (2) all data prior to 2007, respectively. Hybrids containing the earthquake-based covariates perform better in the latter case. The most informative hybrid models in the fitting and testing period are composed of three and four covariates, respectively, including both earthquake- and fault-based variables. Proximity to mapped faults is overall the most informative individual covariate. These results can be used to inform better modeling of long-term earthquake occurrence rates for probabilistic seismic-hazard analysis.

Regional Earthquake Likelihood Models II: Information Gains of Multiplicative Hybrids

The Regional Earthquake Likelihood Models experiment in California tested the performance of earthquake likelihood models over a five‐year period. First‐order analysis showed a smoothed‐seismicity model by Helmstetter et al. (2007) to be the best model. We construct optimal multiplicative hybrids involving the best individual model as a baseline and one or more conjugate models. Conjugate models are transformed using an order‐preserving function. Two parameters for each conjugate model and an overall normalizing constant are fitted to optimize the hybrid model. Many two‐model hybrids have an appreciable information gain (log probability gain) per earthquake relative to the best individual model. For the whole of California, the Bird and Liu (2007) Neokinema and Holliday et al. (2007) pattern informatics (PI) models both give gains close to 0.25. For southern California, the Shen et al. (2007) geodetic model gives a gain of more than 0.5, and several others give gains of about 0.2. The best three‐model hybrid for the whole region has the Neokinema and PI models as conjugates. The best three‐model hybrid for southern California has the Shen et al. (2007) and PI models as conjugates. The information gains of the best multiplicative hybrids are greater than those of additive hybrids constructed from the same set of models. The gains tend to be larger when the contributing models involve markedly different concepts or data. These results need to be confirmed by further prospective tests. Multiplicative hybrids will be useful for assimilating other earthquake‐related observations into forecasting models and for combining forecasting models at all timescales.

A parimutuel gambling perspective to compare probabilistic seismicity forecasts

SUMMARY Using analogies to gaming, we consider the problem of comparing multiple probabilistic seismicity forecasts. To measure relative model performance, we suggest a parimutuel gambling perspective which addresses shortcomings of other methods such as likelihood ratio, information gain and Molchan diagrams. We describe two variants of the parimutuel approach for a set of forecasts: head-to-head, in which forecasts are compared in pairs, and round table, in which all forecasts are compared simultaneously. For illustration, we compare the 5-yr forecasts of the Regional Earthquake Likelihood Models experiment for M4.95+ seismicity in California.

Mixture Models for Improved Earthquake Forecasting with Short-to-Medium Time Horizons

Abstract Optimal mixtures of three space–time–magnitude earthquake likelihood models are found for forecasting earthquakes with magnitudes of 5.0 and greater in the New Zealand and California catalogs, with forecasting time horizons ranging from 0 to 3000 days. The models are the Epidemic‐Type Aftershock (ETAS) short‐term clustering model, the Every Earthquake a Precursor According to Scale (EEPAS) medium‐term clustering model, and the Proximity to Past Earthquakes (PPE) quasi‐time‐invariant smoothed seismicity model. The ETAS model is by far the most informative of these models for short time horizons of a few days, but even with a zero time horizon, an optimal mixture of the three models, here called the Janus model, outperforms it with an information gain per earthquake (IGPE) of about 0.1. For time horizons of 10–3000 days, the Janus model outperforms the most informative of its component models with IGPEs ranging from 0.2 to 0.5. As the time horizon lengthens beyond six months in New Zealand and two years in California, the EEPAS model becomes the most informative of the individual models and the major component of the optimal mixture. Changes in the Janus model parameters with the forecasting time horizon reveal features of time‐and‐area scaling of precursory seismicity. The results suggest that both cascades of triggering and the precursory scale increase phenomenon contribute to earthquake predictability and that these contributions are largely independent.

Regional Earthquake Likelihood Models I: First-Order Results

Abstract The Regional Earthquake Likelihood Models (RELM) working group designed a 5-year experiment to forecast the number, spatial distribution, and magnitude distribution of subsequent target earthquakes, defined to be those with magnitude ≥4.95 (M 4.95+) in a well-defined California testing region. Included in the experiment specification were the description of the data source, the methods for data processing, and the proposed evaluation metrics. The RELM experiment began on 1 January 2006 and involved 17 time-invariant forecasts constructed by seismicity modelers; by the end of the experiment on 1 January 2011, 31 target earthquakes had occurred. We analyze the experiment outcome by applying the proposed consistency tests based on likelihood measures and additional comparison tests based on a measure of information gain. We find that the smoothed seismicity forecast by Helmstetter et al., 2007 based on M 2+ earthquakes since 1981, is the best forecast, regardless of whether aftershocks are included in the analysis. The RELM experiment has helped to clarify ideas about testing that can be applied to more wide-ranging earthquake forecasting experiments conducted by the Collaboratory for the Study of Earthquake Predictability (CSEP). Online Material: Figures and tables showing the RELM testing region and collection region definitions, numerical results associated with the RELM experiment, and the uncorrected forecast by Ebel et al. (2007).

Bayesian Forecast Evaluation and Ensemble Earthquake Forecasting

The assessment of earthquake forecast models for practical purposes requires more than simply checking model consistency in a statistical framework. One also needs to understand how to construct the best model for specific forecasting applications. We describe a Bayesian approach to evaluating earthquake forecasting models, and we consider related procedures for constructing ensemble forecasts. We show how evaluations based on Bayes factors, which measure the relative skill among forecasts, can be complementary to common goodness‐of‐fit tests used to measure the absolute consistency of forecasts with data. To construct ensemble forecasts, we consider averages across a forecast set, weighted by either posterior probabilities or inverse log‐likelihoods derived during prospective earthquake forecasting experiments. We account for model correlations by conditioning weights using the Garthwaite–Mubwandarikwa capped eigenvalue scheme. We apply these methods to the Regional Earthquake Likelihood Models (RELM) five‐year earthquake forecast experiment in California, and we discuss how this approach can be generalized to other ensemble forecasting applications. Specific applications of seismological importance include experiments being conducted within the Collaboratory for the Study of Earthquake Predictability (CSEP) and ensemble methods for operational earthquake forecasting. Online Material: Tables of likelihoods for each testing phase and code to analyze the RELM experiment.

Implications of the Regional Earthquake Likelihood Models test of earthquake forecasts in California

The Regional Earthquake Likelihood Models (RELM) test was the first competitive comparison of prospective earthquake forecasts. The test was carried out over 5 years from 1 January 2006 to 31 December 2010 over a region that included all of California. The test area was divided into 7682 0.1°x0.1° spatial cells. Each submitted forecast gave the predicted numbers of earthquakes <em>N<sub>emi</sub></em> larger than <em>M</em>=4.95 in 0.1 magnitude bins for each cell. In this paper we present a method that separates the forecast of the number of test earthquakes from the forecast of their locations. We first obtain the number <em>N<sub>em</sub></em> of forecast earthquakes in magnitude bin <em>m</em>. We then determine the conditional probability <em>λ<sub>emi</sub></em>=<em>N<sub>emi</sub>/</em><em>N<sub>em</sub></em> that an earthquake in magnitude bin <em>m</em> will occur in cell <em>i</em>. The summation of <em>λ<sub>emi</sub></em> over all 7682 cells is unity. A random (no skill) forecast gives equal values of <em>λ<sub>emi</sub></em> for all spatial cells and magnitude bins. The <em>skill</em> of a forecast, in terms of the location of the earthquakes, is measured by the success in assigning large values of <em>λ<sub>emi</sub></em> to the cells in which earthquakes occur and low values of <em>λ<sub>emi</sub></em> to the cells where earthquakes do not occur. Thirty-one test earthquakes occurred in 27 different combinations of spatial cells <em>i</em> and magnitude bins <em>m</em>, we had the highest value of <em>λ<sub>emi</sub></em> for that <em>mi</em> cell. We evaluate the performance of eleven submitted forecasts in two ways. First, we determine the number of <em>mi</em> cells for which the forecast <em>λ<sub>emi</sub></em> was the largest, the best forecast is the one with the highest number. Second, we determine the mean value of <em>λ<sub>emi</sub></em> for the 27 <em>mi</em> cells for each forecast. The best forecast has the highest mean value of <em>λ<sub>emi</sub></em>. The success of a forecast during the test period is dependent on the allocation of the probabilities λemi between the mi cells, since the sum over the mi cells is unity. We illustrate the forecast distributions of <em>λ<sub>emi</sub></em> and discuss their differences. We conclude that the RELM test was successful in illustrating the choices required when a forecast of the location of a future earthquake is made.

Forecasting Earthquakes: The RELM Test

Earthquakes don't have precursors that can be reliably used for earthquake prediction. However, distributions of earthquakes (seismicity) can be used to forecast an earthquake hazard. These distributions in space, time, and magnitude can be studied in a variety of ways. The Regional Earthquake Likelihood Models (RELM) test compares the different approaches.

An Earthquake Likelihood Model Based on Proximity to Mapped Faults and Cataloged Earthquakes

Abstract Known earthquake sources include mapped active faults and locations where historical earthquakes have occurred. We propose a long‐term earthquake likelihood model that makes use of both types of sources but does not invoke the concepts of fault segmentation or characteristic earthquakes. The model has two components, one based on proximity to the locations of past earthquakes, taking into account their magnitudes, and another based on proximity to mapped faults, taking into account their slip rates. The Gutenberg–Richter law is invoked for earthquake magnitudes and an inverse power law for the diminution of earthquake rate density with distance from past earthquakes and mapped faults. The model is applied to the New Zealand region using the National Seismic Hazard Model fault‐source model and the earthquake catalog since 1951. The parameters of each component are optimized for the period 1987–1996, and the rate density of the combined model is estimated as an optimal linear combination of the two components. In a test on the independent period from 1997–2006, the combined model has an information gain (log‐likelihood increase) per earthquake of about 0.1 over each of the component models. The model is also optimized over the longer period 1987–2006 for comparison with other long‐term models in the New Zealand Earthquake Forecast Testing Centre.

Evaluating the RELM Test Results

We consider implications of the Regional Earthquake Likelihood Models (RELM) test results with regard to earthquake forecasting. Prospective forecasts were solicited forM≥4.95earthquakes in California during the period 2006–2010. During this period 31 earthquakes occurred in the test region withM≥4.95. We consider five forecasts that were submitted for the test. We compare the forecasts utilizing forecast verification methodology developed in the atmospheric sciences, specifically for tornadoes. We utilize a “skill score” based on the forecast scoresλfiof occurrence of the test earthquakes. A perfect forecast would haveλfi=1, and a random (no skill) forecast would haveλfi=2.86×10-3. The best forecasts (largest value ofλfi) for the 31 earthquakes had values ofλfi=1.24×10-1toλfi=5.49×10-3. The best mean forecast for all earthquakes wasλ̅f=2.84×10-2. The best forecasts are about an order of magnitude better than random forecasts. We discuss the earthquakes, the forecasts, and alternative methods of evaluation of the performance of RELM forecasts. We also discuss the relative merits of alarm-based versus probability-based forecasts.

Results of the Regional Earthquake Likelihood Models (RELM) test of earthquake forecasts in California

The Regional Earthquake Likelihood Models (RELM) test of earthquake forecasts in California was the first competitive evaluation of forecasts of future earthquake occurrence. Participants submitted expected probabilities of occurrence of M ≥ 4.95 earthquakes in 0.1° × 0.1° cells for the period 1 January 1, 2006, to December 31, 2010. Probabilities were submitted for 7,682 cells in California and adjacent regions. During this period, 31 M ≥ 4.95 earthquakes occurred in the test region. These earthquakes occurred in 22 test cells. This seismic activity was dominated by earthquakes associated with the M = 7.2, April 4, 2010, El Mayor-Cucapah earthquake in northern Mexico. This earthquake occurred in the test region, and 16 of the other 30 earthquakes in the test region could be associated with it. Nine complete forecasts were submitted by six participants. In this paper, we present the forecasts in a way that allows the reader to evaluate which forecast is the most "successful" in terms of the locations of future earthquakes. We conclude that the RELM test was a success and suggest ways in which the results can be used to improve future forecasts.

Efficient testing of earthquake forecasting models

Computationally efficient alternatives are proposed to the likelihood-based tests employed by the Collaboratory for the Study of Earthquake Predictability for assessing the performance of earthquake likelihood models in the earthquake forecast testing centers. For the conditional L-test, which tests the consistency of the earthquake catalogue with a model, an exact test using convolutions of distributions is available when the number of earthquakes in the test period is small, and the central limit theorem provides an approximate test when the number of earthquakes is large. Similar methods are available for the R-test, which compares the likelihoods of two competing models. However, the R-test, like the N-test and L-test, is fundamentally a test of consistency of data with a model. We propose an alternative test, based on the classical paired t-test, to more directly compare the likelihoods of two models. Although approximate and predicated on a normality assumption, this new T-test is not computer-intensive, is easier to interpret than the R-test, and becomes increasingly dependable as the number of earthquakes increases.

Asperity-based earthquake likelihood models for Italy

The Asperity Likelihood Model (ALM) hypothesizes that small-scale spatial variations in the b-value of the Gutenberg-Richter relationship have a central role in forecasting future seismicity. The physical basis of the ALM is the concept that the local b-value is inversely dependent on the applied shear stress. Thus low b-values (b 1.1), which can be found, for example, in creeping sections of faults, suggest a lower probability of large events. To turn this hypothesis into a forecast model for Italy, we first determined the regional b-value (b = 0.93 ±0.01) and compared it with the locally determined b-values at each node of the forecast grid, based on sampling radii ranging from 6 km to 20 km. We used the local b-values if their Akaike Information Criterion scores were lower than those of the regional b-values. We then explored two modifications to this model: in the ALM.IT, we declustered the input catalog for M ≥2 and smoothed the node-wise rates of the declustered catalog with a Gaussian filter. Completeness values for each node were determined using the probability-based magnitude of completeness method. In the second model, the hybrid ALM (HALM), as a «hybrid» between a grid-based and a zoning model, the Italian territory was divided into eight distinct regions that depended on the main tectonic regimes, and the local b-value variability was thus mapped using the regional b-values for each tectonic zone.

Setting up an earthquake forecast experiment in Italy

We describe here the setting up of the first earthquake forecasting experiment for Italy within the Collaboratory for the Study of Earthquake Predictability (CSEP). The CSEP conducts rigorous and actual prospective forecast experiments for different tectonic environments in several forecast-testing centers around the globe. These forecasts are issued for future periods, and are tested only against future observations, to avoid any possible bias. As such, the experiments need to be completely defined. This includes exact definitions of the testing area, of the learning data for the forecast models, and of the observation data against which the forecasts will be tested to evaluate their performance. We present the rules that were taken from the Regional Earthquake Likelihood Models experiments and extended and modified for the Italian experiment. We also present the characterizations of the learning and observational catalogs that describe the completeness of these catalogs, and reveal inhomogeneities in the magnitudes between these catalogs. A particular focus lies on the stability of the earthquake recordings of the observational network. These catalog investigations provide guidance for CSEP modelers for the development of earthquakes forecasts for submission to the forecast experiments in Italy.