Stress interactions and sliding characteristics of faults with random fractal waviness in a purely elastic medium differ both qualitatively and quantitatively from those of faults with planar surfaces. With nonplanar fault models, solutions for slip diverge as resolution of the fractal features increases, and the scaling of fault slip with fault rupture dimension becomes nonlinear. We show that the nonlinear scaling of slip and divergence of solutions arise because stresses from geometric interactions at irregularities along nonplanar faults grow with increasing slip and produce backstresses that progressively impede slip. However, in real materials with finite strength, yielding will halt the growth of the interaction stresses, which will profoundly affect slip of nonplanar faults. We infer that in the brittle seismogenic portion of the Earth's crust, off-fault yielding occurs on pervasive secondary faults. Predicted rates of stress relaxation with distance from major faults with random fractal roughness follow a power-law relationship that is consistent with reported clustering of background seismicity up to 15 kilometers from faults.

S U M M A R YRecent studies suggest that small and large earthquakes nucleate similarly, and that they often have indistinguishable seismic waveform onsets. The characterization of earthquakes in real time, such as for earthquake early warning, therefore requires a flexible modeling approach that allows a small earthquake to become large as fault rupture evolves over time. Here, we present a modeling approach that generates a set of output parameters and uncertainty estimates that are consistent with both small/moderate (≤M6.5) and large earthquakes (>M6.5) as is required for a robust parameter interpretation and shaking forecast. Our approach treats earthquakes over the entire range of magnitudes (>M2) as finite line-source ruptures, with the dimensions of small earthquakes being very small (<100 m) and those of large earthquakes exceeding several tens to hundreds of kilometres in length. The extent of the assumed line source is estimated from the level and distribution of high-frequency peak acceleration amplitudes observed in a local seismic network. High-frequency motions are well suited for this approach, because they are mainly controlled by the distance to the rupturing fault. Observed ground-motion patterns are compared with theoretical templates modeled from empirical ground-motion prediction equations to determine the best line source and uncertainties. Our algorithm extends earlier work by Böse et al. for large finite-fault ruptures. This paper gives a detailed summary of the new algorithm and its offline performance for the 2016 M7.0 Kumamoto, Japan and 2014 M6.0 South Napa, California earthquakes, as well as its performance for about 100 real-time detected local earthquakes (2.2 ≤ M ≤ 5.1) in California. For most events, both the rupture length and the strike are well constrained within a few seconds (<10 s) of the event origin. In large earthquakes, this could allow for providing warnings of up to several tens of seconds. The algorithm could also be useful for resolving fault plane ambiguities of focal mechanisms and identification of rupturing faults for earthquakes as small as M2.5.

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The ShakeAlert earthquake early warning system is designed to automatically identify and characterize the initiation and rupture evolution of large earthquakes, estimate the intensity of ground shaking that will result, and deliver alerts to people and systems that may experience shaking, prior to the occurrence of shaking at their location. It is configured to issue alerts to locations within the West Coast of the United States. In 2018, ShakeAlert 2.0 went live in a regional public test in the first phase of a general public rollout. The ShakeAlert system is now providing alerts to more than 60 institutional partners in the three states of the western United States where most of the nation’s earthquake risk is concentrated: California, Oregon, and Washington. The ShakeAlert 2.0 product for public alerting is a message containing a polygon enclosing a region predicted to experience modified Mercalli intensity (MMI) threshold levels that depend on the delivery method. Wireless Emergency Alerts are delivered for M 5+ earthquakes with expected shaking of MMI≥IV. For cell phone apps, the thresholds are M 4.5+ and MMI≥III. A polygon format alert is the easiest description for selective rebroadcasting mechanisms (e.g., cell towers) and is a requirement for some mass notification systems such as the Federal Emergency Management Agency’s Integrated Public Alert and Warning System. ShakeAlert 2.0 was tested using historic waveform data consisting of 60 M 3.5+ and 25 M 5.0+ earthquakes, in addition to other anomalous waveforms such as calibration signals. For the historic event test, the average M 5+ false alert and missed event rates for ShakeAlert 2.0 are 8% and 16%. The M 3.5+ false alert and missed event rates are 10% and 36.7%. Real-time performance metrics are also presented to assess how the system behaves in regions that are well-instrumented, sparsely instrumented, and for offshore earthquakes.

IntroductionIn her comment (Hardebeck, 2015) on our stress heterogeneity article (Smith and Heaton, 2011), Hardebeck suggests a different focal-mechanism error distribution than what we used in our 2011 article and suggests that this new error distribution will reduce our estimates of stress heterogeneity. In response to this, we have rerun our calculations three ways: (1) with the original mechanism error distribution from Smith and Heaton (2011), (2) with a mechanism error distribution similar to the one presented by Hardebeck (2015), and (3) with a mechanism error distribution derived from repeating earthquake statistics. We find the two new mechanism error models, relative to the original mechanism error distribution, reduce the heterogeneity ratio (HR) estimates by approximately 35%-40% (using Hardebeck's suggested distribution) and by approximately 8%-10% (using the repeating earthquake based error distribution).Applying these two new mechanism error distribution models helps parameterize the estimates of stress heterogeneity amplitude but does not change the main novel points of the Smith and Heaton (2011) article. Namely, we find that focal-mechanism data are still compatible with a heterogeneous stress that is more dissimilar at large interevent distances and more correlated at small interevent distances and that a heterogeneous stress can bias traditional stress inversions toward the stressing rate function.Last, we demonstrate that as the size of the stress inversion region decreases and as the maximum variability of the heterogeneous stress decreases, the normalized stress inversion bias also decreases. This is consistent with taking the model of Smith and Heaton (2011) to the limit where region size decreases to a point source; however, most stress inversions may require dimensions closer to the outer scale of the stress (∼60 km for southern California) and hence experience significant stress inversion biasing toward the stressing rate.Hardebeck (2015) also refers to her 2010 article (Hardebeck, 2010) as a basis for refuting biasing of stress inversions. We disagree with Hardebeck's stated refutation of stress inversion orientations in Smith and Heaton (2011) and Smith and Dieterich (2010); however, addressing the Hardebeck (2010) article is best accomplished via a direct modeling test in a separate comment that takes into account the complexities of the nucleation process, including rapidly evolving slip over the period of days.