Fast, accurate solutions for curvilinear earthquake faults and anelastic strain

Imaging the anelastic deformation within the crust and lithosphere using surface geophysical data remains a significant challenge in part due to the wide range of physical processes operating at different depths and to various levels of localization that they embody. Models of Earth's elastic properties from seismological imaging combined with geodetic modeling may form the basis of comprehensive rheological models of Earth's interior. However, representing the structural complexity of faults and shear zones in numerical models of deformation still constitutes a major difficulty. Here, we present numerical techniques for high-precision models of deformation and stress around both curvilinear faults and volumes undergoing anelastic (irreversible) strain in a heterogenous elastic half-space. To that end, we enhance the software Gamra to model triangular and rectangular fault patches and tetrahedral and cuboidal strain volumes. This affords a means of rapid and accurate calculations of elasto-static Green's functions for localized (e.g., faulting) and distributed (e.g., viscoelastic) deformation in Earth's crust and lithosphere. We demonstrate the correctness of the method with analytic tests, and we illustrate its practical performance by solving for coseismic and postseismic deformation following the 2015 Mw 7.8 Gorkha, Nepal earthquake to extremely high precision

Deformation of a Half-Space from Anelastic Strain Confined in a Tetrahedral Volume

Deformation in the lithosphere-asthenosphere system can be accommodated by faulting and plastic flow. However, incorporating structural data in models of distributed deformation still represents a challenge. Here, I present solutions for the displacements and stress in a half-space caused by distributed anelastic strain confined in a tetrahedral volume. These solutions form the basis of curvilinear meshes that can adapt to realistic structural settings, such as a mantle wedge corner, a spherical shell around a magma chamber, or an aquifer. I provide computer programs to evaluate them in the cases of anti-plane strain, in-plane strain, and three-dimensional deformation. These tools may prove useful in the modeling of deformation data in tectonics, volcanology, and hydrology.Comment: 54 pages, 14 figure

Evidence for postseismic deformation of the lower crust following the 2004 Mw6.0 Parkfield earthquake

Previous studies have shown that postseismic relaxation following the 2004 Mw6.0 Parkfield, CA, earthquake is dominated by afterslip. However, we show that some fraction of the afterslip inferred from kinematic inversion to have occurred immediately below the seismically ruptured area may in fact be a substitute for viscous postseismic deformation of the lower crust. Using continuous GPS and synthetic aperture radar interferometry, we estimate the relative contribution of shallow afterslip (at depth less than 20km) and deeper seated deformation required to account for observed postseismic surface displacements. Exploiting the possible separation in space and time of the time series of displacements predicted from viscoelastic relaxation, we devise a linear inversion scheme that allows inverting jointly for the contribution of afterslip and viscoelastic flow as a function of time. We find that a wide range of models involving variable amounts of viscoelastic deformation can fit the observations equally well provided that they allow some fraction of deep-seated deformation (at depth larger than ∼20 km). These models require that the moment released by postseismic relaxation over 5 years following the earthquake reached nearly as much as 200% of the coseismic moment. All the models show a remarkable complementarity of coseismic and shallow afterslip distributions. Some significant deformation at lower crustal depth (20–26 km) is required to fit the geodetic data. The condition that postseismic deformation cannot exceed complete relaxation places a constraint on the amount of deep seated deformation. The analysis requires an effective viscosity of at least ~10^(18) Pa s of the lower crust (assuming a semi-infinite homogeneous viscous domain). This deep-seated deformation is consistent with the depth range of tremors which also show a transient postseismic response and could explain as much as 50% of the total postseismic geodetic moment (the remaining fraction being due to afterslip at depth shallower than 20 km). Lower crustal postseismic deformation could reflect a combination of localized ductile deformation and aseismic frictional sliding

Gamra: Simple meshing for complex earthquakes

The static offsets caused by earthquakes are well described by elastostatic models with a discontinuity in the displacement along the fault. A traditional approach to model this discontinuity is to align the numerical mesh with the fault and solve the equations using finite elements. However, this distorted mesh can be difficult to generate and update. We present a new numerical method, inspired by the Immersed Interface Method (Leveque and Li, 1994), for solving the elastostatic equations with embedded discontinuities. This method has been carefully designed so that it can be used on parallel machines on an adapted finite difference grid. We have implemented this method in Gamra, a new code for earth modeling. We demonstrate the correctness of the method with analytic tests, and we demonstrate its practical performance by solving a realistic earthquake model to extremely high precision

A unified continuum representation of post-seismic relaxation mechanisms: semi-analytic models of afterslip, poroelastic rebound and viscoelastic flow

We present a unified continuum mechanics representation of the mechanisms believed to be commonly involved in post-seismic transients such as viscoelasticity, fault creep and poroelasticity. The time-dependent relaxation that follows an earthquake, or any other static stress perturbation, is considered in a framework of a generalized viscoelastoplastic rheology whereby some inelastic strain relaxes a physical quantity in the material. The relaxed quantity is the deviatoric stress in case of viscoelastic relaxation, the shear stress in case of creep on a fault plane and the trace of the stress tensor in case of poroelastic rebound. In this framework, the instantaneous velocity field satisfies the linear inhomogeneous Navier's equation with sources parametrized as equivalent body forces and surface tractions. We evaluate the velocity field using the Fourier-domain Green's function for an elastic half-space with surface buoyancy boundary condition. The accuracy of the proposed method is demonstrated by comparisons with finite-element simulations of viscoelastic relaxation following strike-slip and dip-slip ruptures for linear and power-law rheologies. We also present comparisons with analytic solutions for afterslip driven by coseismic stress changes. Finally, we demonstrate that the proposed method can be used to model time-dependent poroelastic rebound by adopting a viscoelastic rheology with bulk viscosity and work hardening. The proposed method allows one to model post-seismic transients that involve multiple mechanisms (afterslip, poroelastic rebound, ductile flow) with an account for the effects of gravity, non-linear rheologies and arbitrary spatial variations in inelastic properties of rocks (e.g. the effective viscosity, rate-and-state frictional parameters and poroelastic properties)

Change of apparent segmentation of the San Andreas fault around Parkfield from space geodetic observations across multiple periods

Sequences of earthquakes are commonly represented as a succession of periods of interseismic stress accumulation followed by coseismic and postseismic phases of stress release. Because the recurrence time of large earthquakes is often greater than the available span of space geodetic data, it has been challenging to monitor the evolution of interseismic loading in its entire duration. Here we analyze large data sets of surface deformation at different key episodes around the Cholame, Parkfield and creeping segments of the San Andreas Fault that show evidence of significant deceleration of fault slip during the interseismic period. We compare the average fault slip rates before and after the 2004 Mw6 Parkfield earthquake, in the 1986–2004 and 2006–2012 periods, respectively, avoiding 2 years of postseismic deformation after 2004. Using a combination of GPS data from the Plate Boundary Observatory, the Southern California Earthquake Center Crustal Motion Map and the Bay Area Velocity Unification networks and interferometric synthetic aperture radar from the Advanced Land Observing Satellite (ALOS) and Envisat satellites, we show that the area of coupling at the transition between the Parkfield and Cholame segments appears larger later in the interseismic period than it does earlier on. While strong plate coupling is uniform across the Parkfield and Cholame segments in the 1986–2004 period, creep occurs south of the 2004 epicenter after 2006, making segmentation of the San Andreas Fault south of Parkfield more clearly apparent. These observations indicate that analyses of surface deformation late in the earthquake cycle may overestimate the area of plate coupling. A fault surface creeping much below plate rate may in some case be a region that does not promote earthquake nucleation but rather just be at a slower stage of its evolution. Our analysis also shows signs of large variation of slip velocity above and below plate rate in the creeping segment indicating that cycles of weakening and hardening can also be at play in dominantly aseismic areas

Fast, accurate solutions for curvilinear earthquake faults and anelastic strain

Imaging the anelastic deformation within the crust and lithosphere using surface geophysical data remains a significant challenge in part due to the wide range of physical processes operating at different depths and to various levels of localization that they embody. Models of Earth's elastic properties from seismological imaging combined with geodetic modeling may form the basis of comprehensive rheological models of Earth's interior. However, representing the structural complexity of faults and shear zones in numerical models of deformation still constitutes a major difficulty. Here, we present numerical techniques for high-precision models of deformation and stress around both curvilinear faults and volumes undergoing anelastic (irreversible) strain in a heterogenous elastic half-space. To that end, we enhance the software Gamra to model triangular and rectangular fault patches and tetrahedral and cuboidal strain volumes. This affords a means of rapid and accurate calculations of elasto-static Green's functions for localized (e.g., faulting) and distributed (e.g., viscoelastic) deformation in Earth's crust and lithosphere. We demonstrate the correctness of the method with analytic tests, and we illustrate its practical performance by solving for coseismic and postseismic deformation following the 2015 Mw 7.8 Gorkha, Nepal earthquake to extremely high precision