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Earthquake nowcasting has been proposed as a means of tracking the change in large earthquake potential in a seismically active area. The method was developed using observable seismic data, in which probabilities of future large earthquakes can be computed using Receiver Operating Characteristic methods. Furthermore, analysis of the Shannon information content of the earthquake catalogs has been used to show that there is information contained in the catalogs, and that it can vary in time. So an important question remains, where does the information originate? In this paper, we examine this question using stochastic simulations of earthquake catalogs. Our catalog simulations are computed using an Earthquake Rescaled Aftershock Seismicity (“ERAS”) stochastic model. This model is similar in many ways to other stochastic seismicity simulations, but has the advantage that the model has only 2 free parameters to be set, one for the aftershock (Omori-Utsu) time decay, and one for the aftershock spatial migration away from the epicenter. Generating a simulation catalog and fitting the two parameters to the observed catalog such as California takes only a few minutes of wall clock time. While clustering can arise from random, Poisson statistics, we show that significant information in the simulation catalogs arises from the “non-Poisson” power-law aftershock clustering, implying that the practice of de-clustering observed catalogs may remove information that would otherwise be useful in forecasting and nowcasting. We also show that the nowcasting method provides similar results with the ERAS model as it does with observed seismicity.

Nowcasting is a term originating from economics, finance, and meteorology. It refers to the process of determining the uncertain state of the economy, markets or the weather at the current time by indirect means. In this paper, we describe a simple two-parameter data analysis that reveals hidden order in otherwise seemingly chaotic earthquake seismicity. One of these parameters relates to a mechanism of seismic quiescence arising from the physics of strain-hardening of the crust prior to major events. We observe an earthquake cycle associated with major earthquakes in California, similar to what has long been postulated. An estimate of the earthquake hazard revealed by this state variable time series can be optimized by the use of machine learning in the form of the Receiver Operating Characteristic skill score. The ROC skill is used here as a loss function in a supervised learning mode. Our analysis is conducted in the region of 5° × 5° in latitude-longitude centered on Los Angeles, a region which we used in previous papers to build similar time series using more involved methods (Rundle & Donnellan, 2020, https://doi.org/10.1029/2020EA001097; Rundle, Donnellan et al., 2021, https://doi.org/10.1029/2021EA001757; Rundle, Stein et al., 2021, https://doi.org/10.1088/1361-6633/abf893). Here we show that not only does the state variable time series have forecast skill, the associated spatial probability densities have skill as well. In addition, use of the standard ROC and Precision (PPV) metrics allow probabilities of current earthquake hazard to be defined in a simple, straightforward, and rigorous way.

Abstract: The earthquake cycle of stress accumulation and release is associated with the elastic rebound hypothesis proposed by H.F. Reid following the M7.9 San Francisco earthquake of 1906. However, observing details of the actual values of time- and space-dependent tectonic stress is not possible at the present time. In two previous papers, we have proposed methods to image the earthquake cycle in California by means of proxy variables. These variables are based on correlations in patterns of small earthquakes that occur nearly continuously in time. The purpose of the present paper is to compare these two methods by evaluating their information content using decision thresholds and Receiver Operating Characteristic methods together with Shannon information entropy. Using seismic data from 1940 to present in California, we find that both methods provide nearly equivalent information on the rise and fall of earthquake correlations associated with major earthquakes in the region. We conclude that the resulting timeseries can be viewed as proxies for the cycle of stress accumulation and release associated with major tectonic activity. Article Highlights: The current state of the earthquake cycle of tectonic stress accumulation and release is unobservableWe review two methods for visualizing the current state of the earthquake cycle from correlation in small earthquake patternsMachine learning techniques indicate that signals in a correlation time series corresponding to future large earthquakes can be detected

We propose a new machine learning-based method for nowcasting earthquakes to image the time-dependent earthquake cycle. The result is a timeseries that may correspond to the process of stress accumulation and release. The timeseries are constructed by using principal component analysis of regional seismicity. The patterns are found as eigenvectors of the cross-correlation matrix of a collection of seismicity timeseries in a coarse grained regional spatial grid (pattern recognition via unsupervised machine learning). The eigenvalues of this matrix represent the relative importance of the various eigenpatterns. Using the eigenvectors and eigenvalues, we compute the weighted correlation timeseries of the regional seismicity. This timeseries has the property that the weighted correlation generally decreases prior to major earthquakes in the region, and increases suddenly just after a major earthquake occurs. As in a previous paper (Rundle & Donnellan, 2020, https://doi.org/10.1029/2020ea001097), we find that this method produces a nowcasting timeseries that resembles the hypothesized regional stress accumulation and release process characterizing the earthquake cycle. We then address the problem of whether the timeseries contain information regarding future large earthquakes. For this, we compute a receiver operating characteristic and determine the decision thresholds for several future time periods of interest (optimization via supervised machine learning). We find that signals can be detected that can be used to characterize the information content of the timeseries. These signals may be useful in assessing present and near-future seismic hazards.

Seismic nowcasting uses counts of small earthquakes as proxy data to estimate the current dynamical state of an earthquake fault system. The result is an earthquake potential score that characterizes the current state of progress of a defined geographic region through its nominal earthquake "cycle." The count of small earthquakes since the last large earthquake is the natural time that has elapsed since the last large earthquake (Varotsos et al., 2006, https://doi.org/10.1103/PhysRevE.74.021123). In addition to natural time, earthquake sequences can also be analyzed using Shannon information entropy ("information"), an idea that was pioneered by Shannon (1948, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x). As a first step to add seismic information entropy into the nowcasting method, we incorporate magnitude information into the natural time counts by using event self-information. We find in this first application of seismic information entropy that the earthquake potential score values are similar to the values using only natural time. However, other characteristics of earthquake sequences, including the interevent time intervals, or the departure of higher magnitude events from the magnitude-frequency scaling line, may contain additional information.

In weather forecasting, current and past observational data are routinely assimilated into numerical simulations to produce ensemble forecasts of future events in a process termed “model steering”. Here we describe a similar approach that is motivated by analyses of previous forecasts of the Working Group on California Earthquake Probabilities (WGCEP). Our approach is adapted to the problem of earthquake forecasting using topologically realistic numerical simulations for the strike-slip fault system in California. By systematically comparing simulation data to observed paleoseismic data, a series of spatial probability density functions (PDFs) can be computed that describe the probable locations of future large earthquakes. We develop this approach and show examples of PDFs associated with magnitude M > 6.5 and M > 7.0 earthquakes in California.

Earthquakes on a specified fault (or fault segment) with magnitudes greater than a specified value have a statistical distribution of recurrence times. The mean recurrence time can be related to the rate of strain accumulation and the strength of the fault. Very few faults have a recorded history of earthquakes that define a distribution well. For hazard assessment, in general, a statistical distribution of recurrence times is assumed along with parameter values. Assumed distributions include the Weibull (stretched exponential) distribution, the lognormal distribution, and the Brownian passage-time (inverse Gaussian) distribution. The distribution of earthquake waiting times is the conditional probability that an earthquake will occur at a time in the future if it has not occurred for a specified time in the past. The distribution of waiting times is very sensitive to the distribution of recurrence times. An exponential distribution of recurrence times is Poissonian, so there is no memory of the last event. The distribution of recurrence times must be thinner than the exponential if the mean waiting time is to decrease as the time since the last earthquake increases. Neither the lognormal or the Brownian passage time distribution satisfies this requirement. We use the "Virtual California" model for earthquake occurrence on the San Andreas fault system to produce a synthetic distribution of earthquake recurrence times on various faults in the San Andreas system. We find that the synthetic data are well represented by Weibull distributions. We also show that the Weibull distribution follows from both damage mechanics and statistical physics. [ABSTRACT FROM AUTHOR]

A record-breaking earthquake has a larger magnitude than any previous earthquake in the study region; a starting date and minimum magnitude must be specified. The first earthquake to satisfy this condition is, by definition, a record-breaking earthquake. The next record-breaking earthquake has a larger magnitude than the first and so forth. In this article we utilize the global Centroid Moment Tensor Project (CMT) catalog for the years 1977 to 2006. We consider earthquakes with moment magnitudes greater than 5.5. We determine the number of record-breaking earthquakes nrb during 15 specified two-year intervals. We then average the nrb at specified subintervals of time. We compare the results with the predictions for a random independent and identically distributed (i.i.d.) process. The expected number of record-breaking earthquakes nrb in a specified period is independent of the statistical distribution of magnitudes. Good statistical agreement between the observations and the predictions is obtained. We carry out the same process for the magnitudes of the record-breaking earthquakes. We again compare the results with the predictions for a random (i.i.d.) process and find good agreement. For our analyses to be valid, it is necessary that the global earthquakes occur randomly, that is, they are not correlated. Thus, aftershocks and foreshocks will be sources of error. Studies of record-breaking temperatures have shown a sensitivity to global warming. A future direction for research is to apply the approach used here to regional earthquake catalogs. Aftershocks are expected to strongly influence the results. [ABSTRACT FROM AUTHOR]

Aftershock sequences present a unique opportunity to study the physics of earthquakes. Important questions concern the fundamental origin of three widely applicable scaling laws: (1) Gutenberg--Richter frequency--magnitude scaling, (2) Omori's law for aftershock decay rates, and (3) Båth's law for the difference between the magnitude of the largest aftershock and a mainshock. The high-resolution Parkfield seismic network provided the opportunity for detailed studies of the aftershock sequence following the 28 September 2004, M 6.0 Parkfield earthquake. In this article it is shown that aftershocks satisfy the Gutenberg--Richter scaling relation only for relatively large times after the mainshock. There is a systematic time delay for the establishment of this scaling law. The temporal evolution of the rates of occurrence of aftershocks is quantified using the generalized Omori's law. This scaling law contains two characteristic times c and τ. The analysis suggests that the parameter c plays the role of a characteristic time for the establishment of Gutenberg--Richter scaling. This time increases systematically with a decreasing lower magnitude cutoff. The systematic time delay is attributed to a cascade of energy from long wavelengths to short wavelengths. The parameter τ is a measure of the average time until the first aftershock occurs. We find that τ slightly varies with the lower magnitude cutoff of the sequence. We also note that the largest aftershock inferred from an extrapolation of Gutenberg--Richter scaling, M 5.0, is equal to the largest observed aftershock. This scaling associated with the universal applicability of Båth's law is attributed to a constant partitioning of energy between a mainshock and its associated aftershock sequence. We also give in this article the distribution of interoccurrence times between successive aftershocks. We show that this distribution is well approximated by a nonhomogeneous Poissons process driven by the modified Omori's law. The self-consistency between interoccurrence statistics and decay rates is taken as further evidence for the applicability of our studies. [ABSTRACT FROM AUTHOR]

In 1906 the great San Francisco earthquake and fire destroyed much of the city. As we approach the 100-year anniversary of that event a critical concern is the hazard posed by another such earthquake. In this article, we examine the assumptions presently used to compute the probability of occurrence of these earthquakes. We also present the results of a numerical simulation of interacting faults on the San Andreas system. Called Virtual California, this simulation can be used to compute the times, locations, and magnitudes of simulated earthquakes on the San Andreas fault in the vicinity of San Francisco. Of particular importance are results for the statistical distribution of recurrence times between great earthquakes, results that are difficult or impossible to obtain from a purely field-based approach. [ABSTRACT FROM AUTHOR]

Corrections to the paper "Scaling Properties of the Parkfield Aftershock Sequence," by Robert Sheherbakov, Donald L. Turcotte and John B. Rundle that was published in the September 2006 issue are presented.