[1] We model ruptures on faults that weaken in response to flash heating of microscopic asperity contacts (within a rate-and-state framework) and thermal pressurization of pore fluid. These are arguably the primary weakening mechanisms on mature faults at coseismic slip rates, at least prior to large slip accumulation. Ruptures on strongly rate-weakening faults take the form of slip pulses or cracks, depending on the background stress. Self-sustaining slip pulses exist within a narrow range of stresses: below this range, artificially nucleated ruptures arrest; above this range, ruptures are crack-like. Natural earthquakes will occur as slip pulses if faults operate at the minimum stress required for propagation. Using laboratory-based flash heating parameters, propagation is permitted when the ratio of shear to effective normal stress on the fault is 0.2-0.3; this is mildly influenced by reasonable choices of hydrothermal properties. The San Andreas and other major faults are thought to operate at such stress levels. While the overall stress level is quite small, the peak stress at the rupture front is consistent with static friction coefficients of 0.6-0.9. Growing slip pulses have stress drops of $3 MPa; slip and the length of the slip pulse increase linearly with propagation distance at $0.14 and $30 m/km, respectively. These values are consistent with seismic and geologic observations. In contrast, cracks on faults of the same rheology have stress drops exceeding 20 MPa, and slip at the hypocenter increases with distance at $1 m/km. Citation: Noda, H., E. M. Dunham, and J. R. Rice (2009), Earthquake ruptures with thermal weakening and the operation of major faults at low overall stress levels,

1 Supplementary Information S1. Model parameters Figure S1 and Table S1 show the model geometry and parameters used in the simulation presented in the main article. S2. Model response that reproduces a range of observationsThe rich model response described in the main text is animated by the two supplementary movies which show the time evolution of slip rate distribution V z (x, z, t) on the fault from 4300 to 4800 years (movie I) and from 3500 to 4000 years (movie II) after the beginning of the numerical simulation. Coseismic slip rates of the order of 0.1 m/s and higher are indicated by white to blue colors. The slower slip rates are shown in progressively darker yellow and orange, with the long-term plate rate of 10 -9 m/s indicated in red. Locked regions which have, by definition, much smaller slip rates than the plate rate are indicated by black patches.The simulated time and the maximum slip rate over the fault at that time are given at upper-right and upper-left corners of the movies, respectively. The movie frames are taken every 30 computational time steps, which vary in the computation, and hence the time intervals between the frames are quite heterogeneous. During coseismic periods, the typical time step is 2/150 s and the typical interframe rate is 0.4 s. In the interseismic periods, the interframe rates reach values of about 1 year.Movie I illustrates the behavior of the model that combines, in the same patch, creep in the interseismic periods and large seismic slip. It starts at a time when the interseismic fault slip rate distribution is as expected for the slow-rate friction properties adopted in the model: patch A is mostly locked and patch B is creeping with the long-term plate rate, like the creeping areas that surround the two patches. During the following interseismic period, the locked region of patch A shrinks by penetration of creep from the surrounding areas, which occurs due to stress concentration at the boundary between creeping and locked regions (e.g., Tse and Rice, 1986;Lapusta et al., 2000). Earthquake rupture nucleates when the creeping region within patch A becomes comparable to the nucleation size estimates (Rice and Ruina, 1983;Rice et al., 2001;Rubin and

Stress drop, a measure of static stress change in earthquakes, is the subject of numerous investigations. Stress drop in an earthquake is likely to be spatially varying over the fault, creating a stress drop distribution. Representing this spatial distribution by a single number, as commonly done, implies averaging in space. In this study, we investigate similarities and differences between three different averages of the stress drop distribution used in earthquake studies. The first one, σ M , is the commonly estimated stress drop based on the seismic moment and fault geometry/dimensions. It is known that σ M corresponds to averaging the stress drop distribution with the slip distribution due to uniform stress drop as the weighting function. The second one, σ A , is the simplest (unweighted) average of the stress drop distribution over the fault, equal to the difference between the average stress levels on the fault before and after an earthquake. The third one, σ E , enters discussions of energy partitioning and radiation efficiency; we show that it corresponds to averaging the stress drop distribution with the actual final slip at each point as the weighting function. The three averages, σ M , σ A , and σ E , are often used interchangeably in earthquake studies and simply called 'stress drop'. Yet they are equal to each other only for ruptures with spatially uniform stress drop, which results in an elliptical slip distribution for a circular rupture. Indeed, we find that other relatively simple slip shapes-such as triangular, trapezoidal or sinusoidal-already result in stress drop distributions with notable differences between σ M , σ A , and σ E . Introduction of spatial slip heterogeneity results in further systematic differences between them, with σ E always being larger than σ M , a fact that we have proven theoretically, and σ A almost always being the smallest. In particular, the value of the energy-related σ E significantly increases in comparison to the moment-based σ M with increasing roughness of the slip distribution over the fault. Previous studies used σ M in place of σ E in computing the radiation ratio η R that compares the radiated energy in earthquakes to a characteristic part of their strain energy change. Typical values of η R for large earthquakes were found to be from 0.25 to 1. Our finding that σ E ≥ σ M allows us to interpret the values of η R as the upper bound. We determine the restrictions placed by such estimates on the evolution of stress with slip at the earthquake source. We also find that σ E can be approximated by σ M if the latter is computed based on a reduced rupture area.

[1] A new methodology for three-dimensional (3-D) simulations of earthquake sequences is presented that accounts not only for inertial effects during seismic events but also for shear-induced temperature variations on the fault and the associated evolution of pore fluid pressure. In particular, the methodology allows to capture thermal pressurization (TP) due to frictional heating in a shear zone. One-dimensional (1-D) diffusion of heat and pore fluids in the fault-normal direction is incorporated using a spectral method, which is unconditionally stable, accurate with affordable computational resources, and highly suitable to earthquake sequence calculations that use variable time steps. The approach is used to investigate the effect of heterogeneous hydraulic properties by considering a fault model with two regions of different hydraulic diffusivities and hence different potential for TP. We find that the region of more efficient TP produces larger slip in model-spanning events. The slip deficit in the other region is filled with more frequent smaller events, creating spatiotemporal complexity of large events on the fault. Interestingly, the area of maximum slip in model-spanning events is not associated with the maximum temperature increase because of stronger dynamic weakening in that area. The region of more efficient TP has lower interseismic shear stress, which discourages rupture nucleation there, contrary to what was concluded in prior studies. Seismic events nucleate in the region of less efficient TP where interseismic shear stress is higher. In our model, hypocenters of large events do not occur in areas of large slip or large stress drop.Citation: Noda, H., and N. Lapusta (2010), Three-dimensional earthquake sequence simulations with evolving temperature and pore pressure due to shear heating: Effect of heterogeneous hydraulic diffusivity,

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