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The finitedifference and finiteelement modeling of seismic wave propagation and earthquake motion
 ACTA PHYS SLOVACA
, 2007
"... Numerical modeling of seismic wave propagation and earthquake motion is an irreplaceable tool in investigation of the Earth’s structure, processes in the Earth, and particularly earthquake phenomena. Among various numerical methods, the finitedifference method is the dominant method in the modelin ..."
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Cited by 19 (2 self)
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Numerical modeling of seismic wave propagation and earthquake motion is an irreplaceable tool in investigation of the Earth’s structure, processes in the Earth, and particularly earthquake phenomena. Among various numerical methods, the finitedifference method is the dominant method in the modeling of earthquake motion. Moreover, it is becoming more important in the seismic exploration and structural modeling. At the same time we are convinced that the best time of the finitedifference method in seismology is in the future. This monograph provides tutorial and detailed introduction to the application of the finitedifference (FD), finiteelement (FE), and hybrid FDFE methods to the modeling of seismic wave propagation and earthquake motion. The text does not cover all topics and aspects of the methods. We focus on those to which we have contributed. We present alternative formulations of equation of motion for a smooth elastic continuum. We then develop alternative formulations for a canonical problem with a welded material interface and free surface. We continue with a model of an earthquake source. We complete the general theoretical introduction by a chapter on the constitutive laws for elastic and viscoelastic media, and brief review of strong formulations of the equation of motion. What follows is
Staggeredgrid splitnode method for spontaneous rupture simulation
 J. Geophys. Res
, 2007
"... [1] We adapt the tractionatsplitnode method for spontaneous rupture simulations to the velocitystress staggeredgrid finite difference scheme. The staggeredgrid implementation introduces both velocity and stress discontinuities via split nodes. The staggered traction components on the fault pla ..."
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[1] We adapt the tractionatsplitnode method for spontaneous rupture simulations to the velocitystress staggeredgrid finite difference scheme. The staggeredgrid implementation introduces both velocity and stress discontinuities via split nodes. The staggered traction components on the fault plane are interpolated to form the traction vector at split nodes, facilitating alignment of the vectors of sliding friction and slip velocity. To simplify the splitnode partitioning of the equations of motion, spatial differencing is reduced from fourth to second order along the fault plane, but in the remainder of the grid the spatial differencing scheme remains identical to conventional spatially fourthorder threedimensional staggeredgrid schemes. The resulting staggeredgrid split node (SGSN) method has convergence rates relative to rupturetime, finalslip, and peakslipvelocity metrics that are very similar to the corresponding rates for both a partly staggered splitnode code (DFM) and the boundary integral method. The SGSN method gives very accurate solutions (in the sense that errors are comparable to the uncertainties in the reference solution) when the median resolution of the cohesive zone is 4.4 grid points. Combined with previous results for other grid types and other faultdiscontinuity approximations, the SGSN results demonstrate that accuracy in finite difference solutions to the spontaneous rupture problem is controlled principally by the scheme used to represent the fault discontinuity, and is relatively insensitive to the grid geometry used to represent the continuum. The method provides an efficient and accurate means of adding spontaneous rupture capability to velocitystress staggeredgrid finite difference codes, while retaining the computational advantages of those codes for problems of wave propagation in complex media.
Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using HighOrder Finite Difference Methods.
, 2011
"... Abstract We develop a stable and highorder accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by preexisting fault interfaces that accommodate relative motion of the material ..."
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Abstract We develop a stable and highorder accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by preexisting fault interfaces that accommodate relative motion of the material on the two sides. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields. The method is based on summationbyparts finite difference operators with irregular geometries handled through coordinate transforms and multiblock meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization. The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex freesurface topography, nonplanar faults, and varying material properties.
Comparison of Fault Representation Methods in Finite Difference Simulations of Dynamic Rupture
"... Abstract Assessing accuracy of numerical methods for spontaneous rupture simulation is challenging because we lack analytical solutions for reference. Previous comparison of a boundary integral method (BI) and finitedifference method (called DFM) that explicitly incorporates the fault discontinuity ..."
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Abstract Assessing accuracy of numerical methods for spontaneous rupture simulation is challenging because we lack analytical solutions for reference. Previous comparison of a boundary integral method (BI) and finitedifference method (called DFM) that explicitly incorporates the fault discontinuity at velocity nodes (tractionatsplitnode scheme) shows that both converge to a common, gridindependent solution and exhibit nearly identical powerlaw convergence rates with respect to grid spacing Dx. We use this solution as a reference for assessing two other proposed finitedifference methods, the thick fault (TF) and stress glut (SG) methods, both of which approximate the faultjump conditions through inelastic increments to the stress components (inelasticzone schemes). The TF solution fails to match the qualitative rupture behavior of the reference solution and has quantitative misfits in rootmeansquare rupture time of �30 % for the smallest computationally feasible Dx (with �9 gridpoint resolution of cohesive zone, denoted N ¢ c � 9). For sufficiently small values of Dx, the SG method reproduces the qualitative features of the reference solution, but rupture velocity remains systematically low for SG relative to the reference solution, and SG lacks the welldefined powerlaw convergence seen for BI and DFM. The rupturetime error for SG, with N ¢ c � 9, remains well above uncertainty in the reference solution, and the splitnode method attains comparable accuracy with N ¢ c 1/4 as large (and computation timescales as (N ¢ c) 4). Thus, accuracy is highly sensitive to the formulation of the faultjump conditions: The splitnode method attains powerlaw convergence. The SG inelasticzone method achieves solutions that are qualitatively meaningful and quantitatively reliable to within a few percent, but convergence is uncertain, and SG is computationally inefficient relative to the splitnode approach. The TF inelasticzone method does not achieve qualitatively meaningful solutions to the 3D test problem and is sufficiently computationally inefficient that it is not feasible to explore convergence quantitatively.
Scalemodel and numerical simulations of nearfault seismic directivity
, 2008
"... Abstract Foam rubber experiments simulating unilaterally propagating strikeslip earthquakes provide a means to explore the sensitivity of nearfault ground motions to rupture geometry. Subsurface accelerometers on the model fault plane show rupture propagation that approaches a limiting velocity cl ..."
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Abstract Foam rubber experiments simulating unilaterally propagating strikeslip earthquakes provide a means to explore the sensitivity of nearfault ground motions to rupture geometry. Subsurface accelerometers on the model fault plane show rupture propagation that approaches a limiting velocity close to the Rayleigh velocity. The slipvelocity waveform at depth is cracklike (slip duration of the order of narrower fault dimension W divided by Swave speed β). Surface accelerometers record nearfault ground motion enhanced along strike by ruptureinduced directivity. Most experimental features (initiation time, shape, duration and absolute amplitude of acceleration pulses) are successfully reproduced by a 3D spontaneousrupture numerical model of the experiments. Numerical and experimentalmodel acceleration pulses show similar decay with distance away from the fault, and faultnormal components in both models show similar, large amplitude growth with distance along fault strike. This forward directivity effect is also evident in response spectra: the faultnormal spectral response peak (at period ∼W=3β) increases approximately sixfold along strike, on average, in the experiments, with similar increase (about fivefold) in the corresponding numerical simulation. The experimental and numericalmodel response spectra agree with an empirical directivity model for natural earthquakes at long periods (near ∼W=β), and both overpredict shorterperiod empirical directivity effects, with the amount of overprediction increasing systematically with diminishing period. We attribute this difference to rupture and wavefront incoherence in natural earthquakes, due to faultzone heterogeneities in stress, frictional resistance, and elastic properties present in the Earth but absent or minimal in the experimental and numerical models. Rupturefront incoherence is an important component of source models for groundmotion prediction, but finding an effective kinematic parameterization may be challenging.
A SupportOperator Method for Viscoelastic Wave Modeling in 3D Heterogeneous Media
, 2007
"... We apply the method of Support Operators (SOM) to solve the three dimensional, viscoelastic equations of motion for use in earthquake simulations. SOM is a generalized finitedifference method that can utilize meshes of arbitrary structure and incorporate irregular geometry. Our implementation uses ..."
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We apply the method of Support Operators (SOM) to solve the three dimensional, viscoelastic equations of motion for use in earthquake simulations. SOM is a generalized finitedifference method that can utilize meshes of arbitrary structure and incorporate irregular geometry. Our implementation uses a 3D, logicallyrectangular, hexahedral mesh. Calculations are secondorder in space and time. A correction term is employed for suppression of spurious zeroenergy modes (hourglass oscillations). We develop a free surface boundary condition, and an absorbing boundary condition using the method of Perfectly Matched Layers (PML). Numerical tests using a layered material model in a highly deformed mesh show good agreement with the frequencywavenumber method, for resolutions greater than 10 nodes per wavelength. We also test a vertically incident P wave on a semicircular canyon, for which results match boundary integral solutions at resolutions greater that 20 nodes per wavelength. We also demonstrate excellent parallel scalability of our code. Key words: seismic wave propagation, numerical methods, mimetic operators, perfectly matched layer 1
Finite Element Modeling of Branched Ruptures Including OffFault Plasticity (Submitted 8 May ’11 to the Bulletin of the Seismological Society of America,
"... Fault intersections are a geometric complexity that frequently occurs in nature. Here we focus on earthquake rupture behavior when a continuous, planar main fault has a second fault branching off of it. We use the finite element method to examine which faults are activated and how the surrounding ma ..."
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Fault intersections are a geometric complexity that frequently occurs in nature. Here we focus on earthquake rupture behavior when a continuous, planar main fault has a second fault branching off of it. We use the finite element method to examine which faults are activated and how the surrounding material responds for both elastic and elasticplastic offfault descriptions. Compared to an elastic model, a noncohesive, elasticplastic material, intended to account for zones of damaged rock bordering maturely slipped faults, will inhibit rupture on compressional side branches and promote rupture of extensional side branches. Activation of extensional side branches can be delayed and is triggered by continued rupture propagation on the main fault. We examine the deformation near the branching junction and find that fault opening is common for elastic materials, especially for compressional side branches. An elasticplastic material is more realistic since elevated stresses around the propagating rupture tip and at the branching junction should bring the surrounding material to failure. With an elasticplastic material model, fault opening is inhibited for a range of realistic material parameters. For large cohesive strengths opening can occur, but with material softening, a real feature of plastically deforming rocks, opening can be prevented. We also discuss algorithmic artifacts that may arise due to the presence of such a triple junction. When opening does not occur, the behavior at the triple junction is simplified and standard contact routines in finite element programs are able to properly represent the physical situation. 2 2
A domain decomposition approach to implementing fault slip in finiteelement models of quasistatic and dynamic crustal deformation
 Journal of Geophysical Research: Solid Earth
"... This information is distributed solely for the purpose of predissemination peer review and must not be disclosed, released, or published until after approval by the U.S. Geological Survey (USGS). It is deliberative and predecisional information and the findings and conclusions in the document have n ..."
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Cited by 3 (0 self)
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This information is distributed solely for the purpose of predissemination peer review and must not be disclosed, released, or published until after approval by the U.S. Geological Survey (USGS). It is deliberative and predecisional information and the findings and conclusions in the document have not been formally approved for release by the USGS. It does not represent and should not be construed to represent any USGS determination or policy.
Galerkin boundary integral equation method for spontaneous rupture propagation problems: SHcase,Geophys
 J. Int
, 2008
"... We develop a Galerkin finite element boundary integral equation method (GaBIEM) for spontaneous rupture propagation problems for a planar fault embedded in a homogeneous full 2D space. A 2D antiplane rupture propagation problem, with a slipweakening friction law, is simulated by the GaBIEM. This ..."
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We develop a Galerkin finite element boundary integral equation method (GaBIEM) for spontaneous rupture propagation problems for a planar fault embedded in a homogeneous full 2D space. A 2D antiplane rupture propagation problem, with a slipweakening friction law, is simulated by the GaBIEM. This method allows one to eliminate the strong singularities from the integral representation of the traction, and to separate explicitly the expression for the traction into an instantaneous component; static and timedependent components with weakly (logarithmic) singular kernels; and a dynamic component and a quasistatic component, with continuous, bounded, kernels. Simulated results throw light into the performance of the GaBIEM and highlight differences with respect to that of the traditional, collocation, boundary integral equation method (BIEM). Both methods converge with a power law with respect to grid size, with different exponents. There is no restriction on the CFL stability number for the GaBIEM since an implicit, unconditionally stable method is used for the time integration. The error of the approximation increases with the time step, as expected, and it can remain below that of the BIEM. Key words: rupture propagation, elastodynamics
ⒺKinematic Inversion of Physically Plausible Earthquake Source Models Obtained from Dynamic Rupture Simulations
"... Abstract One approach to investigate earthquake source processes is to produce kinematic source models from inversion of seismic records and geodetic data. The setup of the inversion requires a variety of assumptions and constraints to restrict the range of possible models. Here, we evaluate to what ..."
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Abstract One approach to investigate earthquake source processes is to produce kinematic source models from inversion of seismic records and geodetic data. The setup of the inversion requires a variety of assumptions and constraints to restrict the range of possible models. Here, we evaluate to what extent physically plausible earthquake scenarios are reliably restituted in spite of these restrictions. We study which characteristics of ruptures, such as rupture velocity, slip distribution, stress drop, rise time, and slip function, can be reliably determined from the inversion of nearfield seismic and geodetic data. Using spontaneous dynamic rupture simulations, we generate five earthquake scenarios, each of which has different characteristics of the source process. Then we conduct a blind test by modeling the synthetic nearsource data using a standard inversion scheme that optimizes the fit to the observations while searching for solutions with minimum roughness. The inversion procedure assumes a rupture front propagating away from the hypocenter with variable rupture velocity and a simple cosine sliptime function. Our results show that, overall, slip distribution and stress drop are reasonably well determined even for input models with relatively complex histories (such as a subshear rupture transitioning to supershear speeds). Depthaveraged rupture velocities are also reasonably well resolved although their estimate progressively deteriorates away from the hypocenter. The local rise time and slip function are not well resolved, but there is some sensitivity to the rupture pulse width, which can be used to differentiate between pulselike and cracklike ruptures. Our test for understanding the inaccuracies in Green’s functions shows that random 3D perturbations of 5 % standard deviation do not lead to significant degradation of the estimation of earthquake source parameters. As remedies to the current limitations, we propose smoothing slip function parameters and using more complicated inversion schemes only if data necessitates them. Online Material: Figures showing snapshots of forward and inverse modeling of rupture, L curves, slip models, and waveform fits.