Pseudo-dynamic source modelling with 1-point and 2-point statistics of earthquake source parameters

Ground motion prediction is an essential element in seismic hazard and risk analysis. Empirical ground motion prediction approaches have been widely used in the community, but efficient simulation-based ground motion prediction methods are needed to complement empirical approaches, especially in the regions with limited data constraints. Recently, dynamic rupture modelling has been successfully adopted in physics-based source and ground motion modelling, but it is still computationally demanding and many input parameters are not well constrained by observational data. Pseudo-dynamic source modelling keeps the form of kinematic modelling with its computational efficiency, but also tries to emulate the physics of source process. In this paper, we develop a statistical framework that governs the finite-fault rupture process with 1-point and 2-point statistics of source parameters in order to quantify the variability of finite source models for future scenario events. We test this method by extracting 1-point and 2-point statistics from dynamically derived source models and simulating a number of rupture scenarios, given target 1-point and 2-point statistics. We propose a new rupture model generator for stochastic source modelling with the covariance matrix constructed from target 2-point statistics, that is, auto- and cross-correlations. Our sensitivity analysis of near-source ground motions to 1-point and 2-point statistics of source parameters provides insights into relations between statistical rupture properties and ground motions. We observe that larger standard deviation and stronger correlation produce stronger peak ground motions in general. The proposed new source modelling approach will contribute to understanding the effect of earthquake source on near-source ground motion characteristics in a more quantitative and systematic wa

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

Rupture Reactivation during the 2011 M_w 9.0 Tohoku Earthquake: Dynamic Rupture and Ground-Motion Simulations

Near‐source ground‐motion observations and kinematic source inversions suggest that the rupture process of the 2011 M_w 9.0 Tohoku earthquake involved rupture reactivation, that is, repeated rupture nucleation in the same hypocentral area. This unusual phenomenon may have provided a second breath to the rupture that enhanced its final size. Here, we propose that rupture reactivation may have been governed by a slip‐weakening friction model with two sequential strength drops, the second one being activated at large slip. Such frictional behavior has been previously observed in laboratory experiments and attributed to pressurization of fault‐zone fluids by mineral decomposition reactions activated by shear heating, such as dehydration and decarbonation. Further evidence of this double‐slip‐weakening friction model is obtained here from the dynamic stress changes in the hypocentral region derived from a finite source inversion model. We incorporate this friction model in a dynamic rupture simulation comprising two main asperities constrained by source inversion models and several deep small asperities constrained by backprojection source imaging studies. Our simulation produces ground‐motion patterns along the Japanese coast consistent with observations and rupture patterns consistent with a kinematic source model featuring rupture reactivation. The deep small asperities serve as a bridge to connect the two main asperities, and the rupture reactivation mechanism is needed to reproduce the observed ground‐motion pattern. Therefore, we argue that rupture reactivation during the 2011 Tohoku earthquake is consistent with a second strength drop, possibly caused by activation of thermochemical weakening processes at large slip

Evaluating the effect of network density and geometric distribution on kinematic source inversion models

The effect of network density and geometric distribution on kinematic non-linear source inversion is investigated by inverting synthetic ground motions from a buried strike-slip fault (Mw 6.5), that have been generated by dynamic spontaneous rupture modelling. For the inversion, we use a physics-based regularized Yoffe function as slip velocity function. We test three different cases of station network geometry: (i) single station, varying azimuth and epicentral distance; (ii) multistation circular configurations, that is stations at similar distances from the fault, and regularly spaced around the fault; (iii) irregular multistation configurations using different numbers of stations. Our results show: (1) single station tests suggest that it may be possible to obtain a relatively good source model even using a single station. The best source model using a single station is obtained with stations at which amplitude ratios between three components are not large. We infer that both azimuthal angle and source-to-station distance play an important role in the design of optimal seismic network for source inversion. (2) Multistation tests show that the quality of the inverted source systematically correlates neither with the number of stations, nor with waveform misfit. (3) Waveform misfit has a direct correlation with the number of stations, resulting in overfitting the observed data without any systematic improvement of the source. It suggests that the best source model is not necessarily derived from the model with minimum waveform misfit. (4) A seismic network with a small number of well-spaced stations around the fault may be sufficient to obtain acceptable source inversio

Finite difference modelling of rupture propagation with strong velocity-weakening friction

We incorporate rate- and state-dependent friction in explicit finite difference (FD) simulations of mode II dynamic ruptures in elastic media, using the Mimetic Operators Split-Node (MOSN) method, with adjustable order of spatial accuracy (second-, fourth- or mixed-order accurate), including an option that is fourth-order accurate at the fault discontinuity as well as in the elastic volume. At fault points, the rate and state equations combined with the spatially discretized momentum conservation equations form a coupled system of ordinary differential equations (ODEs) for slip velocity and state variable. As a consequence of the rapid damping of velocity perturbations due to the direct effect, this system exhibits numerical stiffness that is inversely proportional to velocity squared. Approximate solutions to this velocity-state system are achieved by two different implicit schemes: (i) a fourth-order Rosenbrock integration of the full system using multiple substeps and (ii) low order integrations (backward Euler and trapezoidal) of the velocity equation, time-staggered with analytic integration of the state equation under the approximation of constant slip velocity over the time step. In assessing the numerical schemes, we use three test problems: ruptures with frictional resistance controlled by (i) a slip evolution law with strong velocity-weakening behaviour at high slip rates, representing thermal weakening due to flash heating of microscopic asperity contacts, (ii) the classic (low-velocity) slip evolution law and (iii) the classic aging evolution law. A convergence analysis is carried out using reference solutions from a spectral boundary integral equation method (BIEM) (a method restricted to homogeneous media, with nominal spectral accuracy in space and second-order accuracy in time for smooth solutions). Errors are measured by root-mean-square differences of fault-plane time histories (slip, slip rate, traction and state). MOSN shows essentially the same convergence rates as BIEM: second-order convergence for slip and state-variable misfits, with slower (but at least first-order) convergence for slip rates and tractions. For a given grid spacing, fourth-order MOSN is as accurate as BIEM for all variables except slip-rate. MOSN-Rosenbrock nominally has fourth-order temporal accuracy for the fault-plane velocity-state ODE integration (compared to lower-order accuracy for the other two MOSN schemes) and therefore provides an important theoretical benchmark. However, it is sensitive to details of the elastic calculation scheme and occasionally its adaptive substepping performs poorly, leading to large excursions from the reference solution. In contrast, MOSN-trapezoidal is robust and reliable, much easier to implement than MOSN-Rosenbrock, and in all cases achieves precision as good as the latter without recourse to substepping. MOSN-Euler has the same advantages as MOSN-trapezoidal, except that its nominal first-order temporal accuracy ultimately leads to larger errors in slip and state variable compared with the higher-order MOSN schemes at sufficiently small grid spacings and time step

Rupture Reactivation during the 2011 M_w 9.0 Tohoku Earthquake: Dynamic Rupture and Ground-Motion Simulations

Near‐source ground‐motion observations and kinematic source inversions suggest that the rupture process of the 2011 M_w 9.0 Tohoku earthquake involved rupture reactivation, that is, repeated rupture nucleation in the same hypocentral area. This unusual phenomenon may have provided a second breath to the rupture that enhanced its final size. Here, we propose that rupture reactivation may have been governed by a slip‐weakening friction model with two sequential strength drops, the second one being activated at large slip. Such frictional behavior has been previously observed in laboratory experiments and attributed to pressurization of fault‐zone fluids by mineral decomposition reactions activated by shear heating, such as dehydration and decarbonation. Further evidence of this double‐slip‐weakening friction model is obtained here from the dynamic stress changes in the hypocentral region derived from a finite source inversion model. We incorporate this friction model in a dynamic rupture simulation comprising two main asperities constrained by source inversion models and several deep small asperities constrained by backprojection source imaging studies. Our simulation produces ground‐motion patterns along the Japanese coast consistent with observations and rupture patterns consistent with a kinematic source model featuring rupture reactivation. The deep small asperities serve as a bridge to connect the two main asperities, and the rupture reactivation mechanism is needed to reproduce the observed ground‐motion pattern. Therefore, we argue that rupture reactivation during the 2011 Tohoku earthquake is consistent with a second strength drop, possibly caused by activation of thermochemical weakening processes at large slip