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The U.S. Geological Survey (USGS) has developed a national capability for aftershock forecasting after significant earthquakes. Use of this capability began in August 2018, and the 30 November 2018 Mw 7.1 Anchorage, Alaska, earthquake provided the first opportunity to apply this capability to a damaging earthquake in an urban area of the United States of America and observe how the forecast was discussed in the media. During this sequence, the forecasts were issued by a seismologist using interactive software that implements the Reasenberg and Jones (1989) model as updated in Page et al. (2016). The forecasts are communicated with a tiered template that provides basic information first before providing a more detailed numerical forecast and are posted on the mainshock’s event page on the USGS earthquake program. Experience from the Anchorage sequence showed that the process worked well, and the first forecast was issued only 54 min after the mainshock occurred. Updates over the coming days, weeks, and months adapted the forecast model from the initial generic parameters for the seismotectonic region to Bayesian and sequence‐specific models. Media reports accurately reported the forecast, demonstrating that the forecast template was successful except for a few reports that incorrectly merged the probability of one or more events in a given time–magnitude window with the likely range of the number of events. Changes to the template have been made to prevent that confusion in the future. We also released a special report on the possible duration of the sequence to assist in the federal disaster declaration and assistance process. Both our standard forecasts and this special report would benefit from more rapid determination of a sequence‐specific decay rate.

In this paper we investigate the dynamic behavior of a system of interconnected faults in the Brawley Seismic Zone (BSZ) in southern California. The system of faults includes the southern San Andreas Fault (SSAF), the Imperial Fault (IF), and a set of cross faults in the BSZ that may serve as connecting structures between the two larger faults. Geological and seismic evidence imply that the SSAF and IF may have buried extensions that link them together in a large‐scale step over, with the cross faults in the BSZ cutting between them. Such a configuration poses the question of whether through‐going rupture across the step over is possible in this region, leading to large, plate‐boundary scale earthquakes. We investigate potential earthquakes in this region through 3‐D dynamic finite element spontaneous rupture modeling. We find that under multiple assumptions about fault stress and fault geometry, through‐going rupture is possible, both from north to south and south to north. Participation of the cross faults is facilitated by two factors: absence of rupture on one of the main two faults and a contrast in prestress between the main faults and the cross faults, leading to slow propagation speed on the main faults while maintaining ease of failure on the cross faults. The pattern of rupture propagation and slip is strongly affected by fault‐to‐fault dynamic stress interactions during the rupture process. The results may have implications for both potential earthquakes in this region, as well as for understanding the dynamics of geometrically complex/branched faults in general. Published version

Author(s): Harris, Ruth; Barall, Michael; Ponce, David; Moore, Diane; Graymer, Russell; Lockner, David; Morrow, Carolyn; Funning, Gareth; Kyriakopoulos, Christos; Eberhart-Phillips, Donna | Abstract: The Rodgers Creek-Hayward-Calaveras-Northern Calaveras fault system in California dominates the hazard posed by active faults in the San Francisco Bay Area. Given that this fault system runs through a densely populated area, a large earthquake in this region is likely to affect millions of people. This study produced scenarios of large earthquakes in this fault system, using spontaneous (dynamic) rupture simulations. These types of physics-based computational simulations require information about the 3D fault geometry, physical rock properties, fault friction, and initial stress conditions. In terms of fault geometry, the well-connected multi-fault system includes the Hayward fault, at its southern end the Central and Northern Calaveras faults, and at its northern end the Rodgers Creek fault. Geodetic investigations of the fault system’s slip-rate pattern provide images of where the fault surfaces at depth are creeping or locked interseismically, and this helped us choose appropriate initial stress conditions for our simulations. A 3D geologic model of the fault system provides the 3D rock units and fault structure at depth, while field samples from rocks collected at Earth’s surface provide frictional parameters. We used this suite of information to investigate the behavior of large earthquake ruptures nucleating at various positions along this partially creeping fault system. We found that large earthquakes starting on the Hayward fault or on the Rodgers Creek fault may be slowed, stopped, or unaffected in their progress, depending on how much energy is released by the creeping regions of the Hayward and Central Calaveras faults during the time between large earthquakes. Large earthquakes starting on either the Hayward fault or the Rodgers Creek faults will likely not rupture the Northern Calaveras fault, and large earthquakes starting on either the Northern Calaveras fault or the Central Calaveras fault will likely remain confined to those fault segments.

Author(s): Harris, RA; Barall, M; Lockner, DA; Moore, DE; Ponce, DA; Graymer, RW; Funning, G; Morrow, CA; Kyriakopoulos, C; Eberhart-Phillips, D | Abstract: The Hayward fault in California's San Francisco Bay area produces large earthquakes, with the last occurring in 1868. We examine how physics-based dynamic rupture modeling can be used to numerically simulate large earthquakes on not only the Hayward fault, but also its connected companions to the north and south, the Rodgers Creek and Calaveras faults. Equipped with a wealth of images of this fault system, including those of its 3D geology and 3D geometry, in addition to inferences about its interseismic creep-rate pattern and rock-friction behavior, we use a finite-element computer code to perform 3D dynamic earthquake rupture simulations. We find that the rock properties affect the locations and amount of slip produced in our simulated large earthquakes. Crucial factors that control rupture behavior in our modeling are the earthquake nucleation locations, the fault geometry, and the data that reveal where the fault system is creeping or locked. Our findings suggest that large Rodgers Creek-Hayward-Calaveras-Northern Calaveras (RC-H-C-NC) fault-system earthquakes may result from dynamic rupture that starts in a locked part of the fault system, but is then stopped by the creeping parts, leading to high-magnitude-6 earthquakes; or, from dynamic rupture that starts in a locked part of the fault system, then cascades through some of the creeping parts, leading to magnitude-7 earthquakes.

Numerical simulations of Sequences of Earthquakes and Aseismic Slip (SEAS) have made great progress over the past decades to address important questions in earthquake physics and fault mechanics. However, significant challenges in SEAS modeling remain in resolving multiscale interactions between aseismic fault slip, earthquake nucleation, and dynamic rupture; and understanding physical factors controlling observables such as seismicity and ground deformation. The increasing capability and complexity of SEAS modeling calls for extensive efforts to verify codes and advance these simulations with rigor, reproducibility, and broadened impact. In 2018, we initiated a community code-verification exercise for SEAS simulations, supported by the Southern California Earthquake Center (SCEC). Here we report the findings from our first two benchmark problems (BP1 and BP2), designed to test the capabilities of different computational methods in correctly solving a mathematically well-defined, basic problem in crustal faulting. These benchmarks are for a 2D antiplane problem, with a 1D planar vertical strike-slip fault obeying rate-and-state friction, embedded in a 2D homogeneous, linear elastic half-space. Sequences of quasi-dynamic earthquakes with periodic occurrences (BP1) or bimodal sizes (BP2) and their interactions with aseismic slip are simulated. The comparison of >70 simulation results from 11 groups using different numerical methods, uploaded to our online platform, show excellent agreements in long-term and coseismic evolution of fault properties. In BP1, we found that the truncated domain boundaries influence interseismic fault stressing, earthquake recurrence, and coseismic rupture process, and that agreement between models is only achieved with sufficiently large domain sizes. In BP2, we found that complexity of long-term fault behavior depends on how well important physical length scales related to spontaneous nucleation and rupture propagation are resolved. Poor numerical resolution can result in the generation of artificial complexity, impacting simulation results that are of potential interest for characterizing seismic hazard, such as earthquake size distributions, moment release, and earthquake recurrence times. These results inform the development of more advanced SEAS models, contributing to our further understanding of earthquake system dynamics.

It has recently become practical to use triangular fault elements in earthquake simulators instead of the usual rectangular fault elements. A simulator spends most of its time computing how slip on some fault elements affects the stresses on other fault elements. We explore whether rectangles or triangles yield more accurate stress values on curved fault surfaces. One might expect triangles to perform better, because triangles can follow the shape of a curved surface, whereas rectangles leave gaps and overlaps between adjacent fault elements. Our test results show that, contrary to expectations, rectangles overall perform as well as or better than triangles when computing stresses on curved fault surfaces. We also find that one triangulation may perform significantly better than another triangulation.

The Ventura basin in Southern California includes coastal dip-slip faults that can likely produce earthquakes of magnitude 7 or greater and significant local tsunamis. We construct a 3-D dynamic rupture model of an earthquake on the Pitas Point and Lower Red Mountain faults to model low-frequency ground motion and the resulting tsunami, with a goal of elucidating the seismic and tsunami hazard in this area. Our model results in an average stress drop of 6 MPa, an average fault slip of 7.4 m, and a moment magnitude of 7.7, consistent with regional paleoseismic data. Our corresponding tsunami model uses final seafloor displacement from the rupture model as initial conditions to compute local propagation and inundation, resulting in large peak tsunami amplitudes northward and eastward due to site and path effects. Modeled inundation in the Ventura area is significantly greater than that indicated by state of California's current reference inundation line.

The article of record as published may be found at http://dx.doi.org/10.1785/0220170222 We describe a set of benchmark exercises that are designed to test if computer codes that simulate dynamic earthquake rupture are working as intended. These types of computer codes are often used to understand how earthquakes operate, and they produce simulation results that include earthquake size, amounts of fault slip, and the patterns of ground shaking and crustal deformation. The benchmark exercises examine a range of features that scientists incorporate in their dynamic earthquake rupture simulations. These include implementations of simple or complex fault geometry, off-fault rock response to an earthquake, stress conditions, and a variety of formulations for fault friction. Many of the benchmarks were designed to investigate scientific problems at the forefronts of earthquake physics and strong ground motions research. The exercises are freely available on our website for use by the scientific community. Southern California Earthquake Center (SCEC)

Online Material: Explanation of the concept of a dynamic rupture simulation. Earthquakes are complex events that involve a myriad of interactions among multiple geologic features and processes. One of the tools that is available to assist with their study is computer simulation, particularly dynamic rupture simulation. A dynamic rupture simulation is a numerical model of the physical processes that occur during an earthquake. Starting with the fault geometry, friction constitutive law, initial stress conditions, and assumptions about the condition and response of the near‐fault rocks, a dynamic earthquake rupture simulation calculates the evolution of fault slip and stress over time as part of the elastodynamic numerical solution (![Graphic][1] see the simulation description in the electronic supplement to this article). The complexity of the computations in a dynamic rupture simulation make it challenging to verify that the computer code is operating as intended, because there are no exact analytic solutions against which these codes’ results can be directly compared. One approach for checking if dynamic rupture computer codes are working satisfactorily is to compare each code’s results with the results of other dynamic rupture codes running the same earthquake simulation benchmark. To perform such a comparison consistently, it is necessary to have quantitative metrics. In this paper, we present a new method for quantitatively comparing the results of dynamic earthquake rupture computer simulation codes. Over the past decade, the Southern California Earthquake Center–U.S. Geological Survey (SCEC‐USGS) Spontaneous Rupture Code Verification Project has published a series of computational dynamic rupture benchmark exercises (Harris and Archuleta, 2004; Harris et al. , 2009, 2011). The benchmarks, a subset of which of are shown in Figure 1, are listed on the project’s website (http://scecdata.usc.edu/cvws, last accessed November 2014) and cover many earthquake settings, including a range of fault geometries (strike‐slip faults, normal faults, … [1]: /embed/inline-graphic-1.gif

Integral equation‐based methods in elasticity require the calculation of integrals that involve a singular (or weakly singular) matrix Green’s function and a traction or displacement vector field defined on a surface. A variety of numerical methods based on analytic quadratures have been developed for such calculations, using either rectangular or triangular surface patches. Unfortunately, without correction, these analytic rules are subject to numerical instabilities in certain parameter regimes. In this paper, we present a stable, semi‐analytic collection of quadrature rules that can be applied to both infinite medium and half‐space simulations. We describe our underlying approach and illustrate its performance with numerical examples. Online Material: Fortran routines.

Online Material: Supplemental figures of space‐time and frequency‐magnitude relations, scaling plots, mean and covariance plots of interevent times, probability distribution functions of recurrence intervals, and earthquake density plots. In order to understand earthquake hazards we would ideally have a statistical description of earthquakes for tens of thousands of years. Unfortunately the ∼100‐year instrumental, several 100‐year historical, and few 1000‐year paleoseismological records are woefully inadequate to provide a statistically significant record. Physics‐based earthquake simulators can generate arbitrarily long histories of earthquakes; thus they can provide a statistically meaningful history of simulated earthquakes. The question is, how realistic are these simulated histories? This purpose of this paper is to begin to answer that question. We compare the results between different simulators and with information that is known from the limited instrumental, historic, and paleoseismological data. As expected, the results from all the simulators show that the observational record is too short to properly represent the system behavior; therefore, although tests of the simulators against the limited observations are necessary, they are not a sufficient test of the simulators’ realism. The simulators appear to pass this necessary test. In addition, the physics‐based simulators show similar behavior even though there are large differences in the methodology. This suggests that they represent realistic behavior. Different assumptions concerning the constitutive properties of the faults do result in enhanced capabilities of some simulators. However, it appears that the similar behavior of the different simulators may result from the fault‐system geometry, slip rates, and assumed strength drops, along with the shared physics of stress transfer. This paper describes the results of running four earthquake simulators that are described elsewhere in …

Dynamic rupture calculations using heterogeneous stress drop that is random and self-similar with a power-law spatial spectrum have great promise of producing realistic ground-motion predictions. We present procedures to specify initial stress for random events with a target rupture length and target magnitude. The stress function is modified in the depth dimension to account for the brittle-ductile transition at the base of the seismogenic zone. Self-similar fluctuations in stress drop are tied in this work to the long-wavelength stress variation that determines rupture length. Heterogeneous stress is related to friction levels in order to relate the model to physical concepts. In a variant of the model, there are high-stress asperities with low background stress. This procedure has a number of advantages: (1) rupture stops naturally, not at artificial barriers; (2) the amplitude of short-wavelength fluctuations of stress drop is not arbitrary: the spectrum is fixed to the long-wavelength fluctuation that determines rupture length; and (3) large stress drop can be confined to asperities occupying a small fraction of the total rupture area, producing slip distributions with enhanced peaks.

SUMMARY We present a new finite-element technique for calculating dynamic 3-D spontaneous rupture on an earthquake fault, which can reduce the required computational resources by a factor of six or more, without loss of accuracy. The grid-doubling technique employs small cells in a thin layer surrounding the fault. The remainder of the modelling volume is filled with larger cells, typically two or four times as large as the small cells. In the resulting non-conforming mesh, an interpolation method is used to join the thin layer of smaller cells to the volume of larger cells. Grid-doubling is effective because spontaneous rupture calculations typically require higher spatial resolution on and near the fault than elsewhere in the model volume. The technique can be applied to non-planar faults by morphing, or smoothly distorting, the entire mesh to produce the desired 3-D fault geometry. Using our FaultMod finite-element software, we have tested grid-doubling with both slip-weakening and rate-and-state friction laws, by running the SCEC/USGS 3-D dynamic rupture benchmark problems. We have also applied it to a model of the Hayward fault, Northern California, which uses realistic fault geometry and rock properties. FaultMod implements fault slip using common nodes, which represent motion common to both sides of the fault, and differential nodes, which represent motion of one side of the fault relative to the other side. We describe how to modify the traction-at-split-nodes method to work with common and differential nodes, using an implicit time stepping algorithm.

Numerical simulations of earthquake rupture dynamics are now common, yet it has been difficult to test the validity of thesesimulations because there have been few field observations and no analytic solutions with which to compare the results. This paper describes the Southern California Earthquake Center/U.S. Geological Surve(SCEC/USGS) Dynamic Earthquake Rupture Code Verification Exercise, where codes that simulate spontaneous rupture dynamics in three dimensions are evaluated and the results produced by these codes are compared using Web-based tools. This is the first time that a broad and rigorous examination of numerous spontaneous rupture codes has been performed—a significant advance in this science. The automated process developed to attain this achievement provides for a future where testing of codes is easily accomplished. Scientists who use computer simulations to understand earthquakes utilize a range of techniques. Most of these assume that earthquakes are caused by slip at depth on faults in the Earth, but hereafter the strategies vary. Among the methods used in earthquake mechanics studies are kinematic approaches and dynamic approaches. The kinematic approach uses a computer code that prescribes the spatial and temporal evolution of slip on the causative fault (or faults). These types of simulations are very helpful, especially since they can be used in seismic data inversions to relate the ground motions recorded in the field to slip on the fault(s) at depth. However, these kinematic solutions generally provide no insight into the physics driving the fault slip or information about why the involved fault(s) slipped that much (or that little). In other words, these kinematic solutions may lack information about the physical dynamics of earthquake rupture that will be most helpful in forecasting future events. To help address this issue, some researchers use computer codes to numerically simulate earthquakes and construct dynamic, spontaneous rupture (hereafter called “spontaneous rupture”) solutions. For these types of numerical simulations, rather than prescribing the slip function at each location on the fault(s), just the friction constitutive properties and initial stress conditions are prescribed. The subsequent stresses and fault slip spontaneously evolve over time as part of the elasto-dynamic solution. Therefore, spontaneous rupture computer simulations of earthquakes allow us to include everything that we know, or think that we know, about earthquake dynamics and to test these ideas against earthquake observations.

Online Material: Specifications for the standard earthquake simulator input and output files. An earthquake simulator is a computer program that simulates a long sequence of earthquakes, which may range from tens of thousands of years up to millions of years. Different simulator codes have different input requirements, but in general the inputs to a simulator program include the fault system geometry, fault slip rates, rake angles, friction parameters, initial stress state, and elastic parameters. The output from a simulator program is a list of events, each of which is characterized by time, location, magnitude, rupture area, mean slip, stress change, and, optionally, a map that shows the distribution of slip over the fault surface. It is challenging to demonstrate that an earthquake simulator program is reliable, in the sense that it produces a simulated history whose statistical properties are similar to the properties of real earthquake sequences. The Southern California Earthquake Center (SCEC) Earthquake Simulator Comparison Project addresses this challenge by comparing the results from five different earthquake simulators with each other and, to the extent possible, with what is known about real earthquakes (Tullis et al. , 2012a). The five simulators are independently developed, so agreement among the simulators can provide some confidence in their reliability. To carry out the comparison, it is necessary for each simulator code to receive precisely the same inputs. It is also necessary for each simulator code to generate output in the same format, so that output from all the codes can be analyzed by the same set of statistical tools. We have achieved this by designing standard file formats for transferring data into and out of simulator codes. Although the standard files are formatted as ASCII text and are human‐readable, they are intended primarily for transferring data from one computer program to another. …

This report presents a new method for generating random stress distributions on an earthquake fault, suitable for use as initial conditions in a dynamic rupture simulation. The method employs concepts from thermodynamics and statistical mechanics. A pattern of fault slip is considered to be analogous to a micro-state of a thermodynamic system. The energy of the micro-state is taken to be the elastic energy stored in the surrounding medium. Then, the Boltzmann distribution gives the probability of a given pattern of fault slip and stress. We show how to decompose the system into independent degrees of freedom, which makes it computationally feasible to select a random state. However, due to the equipartition theorem, straightforward application of the Boltzmann distribution leads to a divergence which predicts infinite stress. To avoid equipartition, we show that the finite strength of the fault acts to restrict the possible states of the system. By analyzing a set of earthquake scaling relations, we derive a new formula for the expected power spectral density of the stress distribution, which allows us to construct a computer algorithm free of infinities. We then present a new technique for controlling the extent of the rupture by generating a random stress distribution thousands of times larger than the fault surface, and selecting a portion which, by chance, has a positive stress perturbation of the desired size. Finally, we present a new two-stage nucleation method that combines a small zone of forced rupture with a larger zone of reduced fracture energy.

Large earthquakes strike infrequently and close-in recordings are uncommon. This situation makes it difficult to predict the ground motion very close to earthquake-generating faults, if the prediction is to be based on readily available observations. A solution might be to cover the Earth with seismic instruments so that one could rely on the data from previous events to predict future shaking. However, even in the case of complete seismic data coverage for hundreds of years, there would still be one type of earthquake that would be difficult to predict: those very rare earthquakes that produce very large ground motion.

The lithosphere crack model, the main alternative to the mantle plume model for age-progressive magma emplacement along the Hawaiian-Emperor volcano chain, requires the maximum horizontal tensile stress to be normal to the volcano chain. However, published stress fields calculated from Pacific lithosphere tractions and body forces (e.g., subduction pull, basal drag, lithosphere density) are not optimal for southeast propagation of a stress-free, vertical, tensile crack coincident with the Hawaiian segment of the Hawaiian-Emperor chain. Here we calculate the thermoelastic stress rate for present-day cooling of the Pacific plate using a spherical shell finite element representation of the plate geometry. We use observed seafloor isochrons and a standard model for lithosphere cooling to specify the time dependence of vertical temperature profiles. The calculated stress rate multiplied by a time increment (e.g., 1 Ma) then gives a thermoelastic stress increment for the evolving Pacific plate. Near the Hawaiian chain position, the calculated stress increment in the lower part of the shell is extensional with maximum extension normal to the chain direction. Near the projection of the chain trend to the southeast beyond Hawaii, the stress increment is compressive. This incremental stress field, if the main contribution to the total Pacific plate stress field, could produce a tensile crack or similar propagating flaw model for the Hawaiian volcano chain.