Steven M. Day

Dynamics of fault interaction: parallel strike‐slip faults

We use a two-dimensional finite difference computer program to study the effect of fault steps on dynamic ruptures. Our results indicate that a strike-slip earthquake is unlikely to jump a fault step wider than 5 km, in correlation with field observations of moderate to great-sized earthquakes. We also find that dynamically propagating ruptures can jump both compressional and dilational fault steps, although wider dilational fault steps can be jumped. Dilational steps tend to delay the rupture for a longer time than compressional steps do. This delay leads to a slower apparent rupture velocity in the vicinity of dilational steps. These “dry” cases assumed hydrostatic or greater pore-pressures but did not include the effects of changing pore pressures. In an additional study, we simulated the dynamic effects of a fault rupture on ‘undrained’ pore fluids to test Sibsons (1985, 1986) suggestion that “wet” dilational steps are a barrier to rupture propagation. Our numerical results validate Sibsons hypothesis by demonstrating that the effect of the rupture on the ‘undrained’ pore fluids is to inhibit the rupture from jumping dilational stepovers. The basis of our result differs from Sibsons hypothesis in that our model is purely elastic and does not necessitate the opening of extension fractures between the fault segments.

The SCEC Southern California Reference Three-Dimensional Seismic Velocity Model Version 2

We describe Version 2 of the three-dimensional (3D) seismic velocity model of southern California developed by the Southern California Earthquake Center and designed to serve as a reference model for multidisciplinary research activities in the area. The model consists of detailed, rule-based representations of the major southern California basins (Los Angeles basin, Ventura basin, San Gabriel Valley, San Fernando Valley, Chino basin, San Bernardino Valley, and the Salton Trough), embedded in a 3D crust over a variable depth Moho. Outside of the basins, the model crust is based on regional tomographic results. The model Moho is represented by a surface with the depths determined by the receiver function technique. Shallow basin sediment velocities are constrained by geotechnical data. The model is implemented in a computer code that generates any specified 3D mesh of seismic velocity and density values. This parameterization is convenient to store, transfer, and update as new information and verification results become available.

Dynamic 3D simulations of earthquakes on En Echelon Faults

One of the mysteries of earthquake mechanics is why earthquakes stop. This process determines the difference between small and devastating ruptures. One possibility is that fault geometry controls earthquake size. We test this hypothesis using a numerical algorithm that simulates spontaneous rupture propagation in a three-dimensional medium and apply our knowledge to two California fault zones. We find that the size difference between the 1934 and 1966 Parkfield, California, earthquakes may be the product of a stepover at the southern end of the 1934 earthquake and show how the 1992 Landers, California, earthquake followed physically reasonable expectations when it jumped across en echelon faults to become a large event. If there are no linking structures, such as transfer faults, then strike-slip earthquakes are unlikely to propagate through stepover s >5 km wide.

Fault steps and the dynamic rupture process: 2‐D numerical simulations of a spontaneously propagating shear fracture

Fault steps may have controlled the sizes of the 1966 Parkfield, 1968 Borrego Mountain, 1979 Imperial Valley, 1979 Coyote Lake and the 1987 Superstition Hills earthquakes. This project investigates the effect of fault steps of various geometries on the dynamic rupture process. We have used a finite difference code to simulate spontaneous rupture propagation in two dimensions. We employ a slip-weakening fracture criterion as the condition for rupture propagation and examine how rupture on one plane initiates rupture on parallel fault planes. The geometry of the two parallel fault planes allows for stepover widths of 0.5 to 10.0 km and overlaps of −5 to 5 km. Our results demonstrate that the spontaneous rupture on the first fault segment continues to propagate onto the second fault segment for a range of geometries for both compressional and dilational fault steps. A major difference between the compressional and dilational cases is, that a dilational step requires a longer time delay between the rupture front reaching the end of the first fault segment and initiating rupture on the second segment. Therefore our dynamic study implies that a compressional step will be jumped quickly, whereas a dilational step will cause a time delay leading to a lower apparent rupture velocity. We also find that the rupture is capable of jumping a wider dilational step than compressional step.

Comparison of finite difference and boundary integral solutions to three‐dimensional spontaneous rupture

The spontaneously propagating shear crack on a frictional interface has proven to be a useful idealization of a natural earthquake. The corresponding boundary value problems are nonlinear and usually require computationally intensive numerical methods for their solution. Assessing the convergence and accuracy of the numerical methods is challenging, as we lack appropriate analytical solutions for comparison. As a complement to other methods of assessment, we compare solutions obtained by two independent numerical methods, a finite difference method and a boundary integral (BI) method. The finite difference implementation, called DFM, uses a traction-at-split-node formulation of the fault discontinuity. The BI implementation employs spectral representation of the stress transfer functional. The three-dimensional (3-D) test problem involves spontaneous rupture spreading on a planar interface governed by linear slip-weakening friction that essentially defines a cohesive law. To get a priori understanding of the spatial resolution that would be required in this and similar problems, we review and combine some simple estimates of the cohesive zone sizes which correspond quite well to the sizes observed in simulations. We have assessed agreement between the methods in terms of the RMS differences in rupture time, final slip, and peak slip rate and related these to median and minimum measures of the cohesive zone resolution observed in the numerical solutions. The BI and DFM methods give virtually indistinguishable solutions to the 3-D spontaneous rupture test problem when their grid spacing Δx is small enough so that the solutions adequately resolve the cohesive zone, with at least three points for BI and at least five node points for DFM. Furthermore, grid-dependent differences in the results, for each of the two methods taken separately, decay as a power law in Δx, with the same convergence rate for each method, the calculations apparently converging to a common, grid interval invariant solution. This result provides strong evidence for the accuracy of both methods. In addition, the specific solution presented here, by virtue of being demonstrably grid-independent and consistent between two very different numerical methods, may prove useful for testing new numerical methods for spontaneous rupture problems.

Estimation of Q for Long-Period (>2 sec) Waves in the Los Angeles Basin

We simulate 0- to 0.5-Hz 3D wave propagation through the Southern California Earthquake Center seismic velocity reference model, version 2, for the 1994 Northridge earthquake in order to examine the effects of anelastic attenuation and amplification within the near-surface sediments. We use a fourth-order finite-difference staggered-grid method with the coarse-grained frequency-independent anelastic scheme of Day and Bradley (2001) and a variable slip distribution from kinematic inversion for the Northridge earthquake. We find that the near-surface material with S -wave velocity ( V s) as low as 500 m/sec significantly affects the long-period peak ground velocities, compared with simulations in which the S -wave velocity is constrained to 1 km/sec and greater. Anelastic attenuation also has a strong effect on ground-motion amplitudes, reducing the predicted peak velocity by a factor of up to 2.5, relative to lossless simulations. Our preferred Q model is Q s/ V s = 0.02 ( V s in meters per second) for V s less than 1–2 km/sec, and much larger Q s/ V s (0.1, V s in meters per second) for layers with higher velocities. The simple model reduces the standard deviation of the residuals between synthetic and observed natural log of peak velocity from 1.13 to 0.26, relative to simulations for the lossless case. The anelastic losses have their largest effect on short-period surface waves propagating in the Los Angeles basin, which are principally sensitive to Q s in the low-velocity, near-surface sediments of the basin. The low-frequency ground motion simulated here is relatively insensitive to Q p, as well as to the values of Q s at depths greater than roughly that of the 2-km/sec S -wave velocity isosurface.

Misfit Criteria for Quantitative Comparison of Seismograms

We have developed and numerically tested quantitative misfit criteria for comparison of seismograms. The misfit criteria are based on the time-frequency representation of the seismograms obtained as the continuous wavelet transform with the analyzing Morlet wavelet. The misfit criteria include time-frequency envelope and phase misfits, time-dependent envelope and phase misfits, frequency-dependent envelope and phase misfits, and single-valued envelope and phase misfits. We tested properties of the misfit criteria using canonical signals. The canonical signals, taken as the reference signals, were specifically amplitude, phase shift, time shift, and frequency modified to demonstrate the ability of the misfit criteria to prop- erly quantify the misfits and recognize the character and cause of the misfits between the reference and modified signals. In all cases the misfit criteria properly quantified and characterized the misfits. The misfit criteria were also calculated for four different numerical solutions for a single layer over half-space (the SCEC LOH.3 problem) and the reference FK so- lution. The misfit criteria provided useful insight into the misfits between individual numerical solutions and the reference solution. The standard RMS misfit matches the single-valued envelope misfit only in the case of a pure amplitude modification of the signal. In all other cases RMS consid- erably overestimates the misfits and does not characterize them.

Memory-Efficient Simulation of Anelastic Wave Propagation

Realistic anelastic attenuation can be incorporated rigorously into finite difference and other numerical wave propagation methods using internal or memory variables. The main impediment to the realistic treatment of anelastic attenuation in 3D is the very large computational storage requirement imposed by the additional variables. We previously proposed an alternative to the conventional memory-variable formulation, the method of coarse-grain memory variables, and demonstrated its effectiveness in acoustic problems. We generalize this memory-efficient formulation to 3D anelasticity and describe a fourth-order, staggered-grid finite-difference implementation. The anelastic coarse-grain method applied to plane wave propagation successfully simulates frequency-independent Q p and Q s . Apparent Q values are constant to within 4% tolerance over approximately two decades in frequency and biased less than 4% from specified target values. This performance is comparable to that achieved previously for acoustic-wave propagation, and accuracy could be further improved by optimizing the memory-variable relaxation times and weights. For a given assignment of relaxation times and weights, the coarse-grain method provides an eight-fold reduction in the storage requirement for memory variables, relative to the conventional approach. The method closely approximates the wavenumber-integration solution for the response of an anelastic half-space to a shallow dislocation source, accurately calculating all phases including the surface-diffracted SP phase and the Rayleigh wave. The half-space test demonstrates that the wave field-averaging concept underlying the coarse-grain method is effective near boundaries and in the presence of evanescent waves. We anticipate that this method will also be applicable to unstructured grid methods, such as the finite-element method and the spectral-element method, although additional numerical testing will be required to establish accuracy in the presence of grid irregularity. The method is not effective at wavelengths equal to and shorter than 4 grid cell dimensions, where it produces anomalous scattering effects. This limitation could be significant for very high-order numerical schemes under some circumstances (i.e., whenever wave-lengths as short as 4 grids are otherwise within the usable bandwidth of the scheme), but it is of no practical importance in our fourth-order finite-difference implementation.

The 1999 Izmit, Turkey, earthquake: A 3D dynamic stress transfer model of intraearthquake triggering

Before the August 1999 Izmit (Kocaeli), Turkey, earthquake, theoretical studies of earthquake ruptures and geological observations had provided estimates of how far an earthquake might jump to get to a neighboring fault. Both numerical simulations and geological observations suggested that 5 km might be the upper limit if there were no transfer faults. The Izmit earthquake appears to have followed these expectations. It did not jump across any step-over wider than 5 km and was instead stopped by a narrower step-over at its eastern end and possibly by a stress shadow caused by a historic large earthquake at its western end. Our 3D spontaneous rupture simulations of the 1999 Izmit earthquake provide two new insights: (1) the west- to east-striking fault segments of this part of the North Anatolian fault are oriented so as to be low-stress faults and (2) the easternmost segment involved in the August 1999 rupture may be dipping. An interesting feature of the Izmit earthquake is that a 5-km-long gap in surface rupture and an adjacent 25° restraining bend in the fault zone did not stop the earthquake. The latter observation is a warning that significant fault bends in strike-slip faults may not arrest future earthquakes. Manuscript received 30 August 2000.

Scalable Earthquake Simulation on Petascale Supercomputers

Petascale simulations are needed to understand the rupture and wave dynamics of the largest earthquakes at shaking frequencies required to engineer safe structures (> 1 Hz). Toward this goal, we have developed a highly scalable, parallel application (AWP-ODC) that has achieved “M8”: a full dynamical simulation of a magnitude-8 earthquake on the southern San Andreas fault up to 2 Hz. M8 was calculated using a uniform mesh of 436 billion 40-m3 cubes to represent the three-dimensional crustal structure of Southern California, in a 800 km by 400 km area, home to over 20 million people. This production run producing 360 sec of wave propagation sustained 220 Tflop/s for 24 hours on NCCS Jaguar using 223,074 cores. As the largest-ever earthquake simulation, M8 opens new territory for earthquake science and engineering—the physics-based modeling of the largest seismic hazards with the goal of reducing their potential for loss of life and property.