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D.J. Andrews

Publications and source records attributed to D.J. Andrews.

14 recordsLinked to original sources
USGS2012

A stochastic fault model. 2. Time-dependent case

A random model of fault motion in an earthquake is formulated by assuming that the slip velocity is a random function of position and time truncated at zero, so that it does not have negative values. This random function is chosen to be self-affine; that is, on change of length scale, the function is multiplied by a scale factor but is otherwise unchanged statistically. A snapshot of slip velocity at a given time resembles a cluster of islands with rough topography; the final slip function is a smoother island or cluster of islands. In the Fourier transform domain, shear traction on the fault equals the slip velocity times an impedance function. The fact that this impedance function has a pole at zero frequency implies that traction and slip velocity cannot have the same spectral dependence in space and time. To describe stress fluctuations of the order of 100 bars when smoothed over a length of kilometers and of the order of kilobars at the grain size, shear traction must have a one-dimensional power spectrum is space proportional to the reciprocal wave number. Then the one-dimensional power spectrum for the slip velocity is proportional to the reciprocal wave number squared and for slip to its cube. If slip velocity has the same power law spectrum in time as in space, then the spectrum of ground acceleration will be flat (white noise) both on the fault and in the far field.

Journal of Geophysical Research Solid Earth↗
USGS2012

Scarp degraded by linear diffusion: Inverse solution for age

Under the assumption that landforms unaffected by drainage channels are degraded according to the linear diffusion equation, a procedure is developed to invert a scarp profile to find its “diffusion age.” Diffusion age, having dimension [length] 2 , is the product of diffusivity times chronological age. The second moment of scarp slope grows linearly with age. This fact, together with an assumption about initial scarp shape, allows the inverse determination of diffusion age. Age found assuming a vertical initial scarp is termed “apparent age”; any nonvertical initial scarp profile has a nonzero initial apparent age. True diffusion age differs from apparent age by a fraction of scarp offset squared. The inverse procedure applied to synthetic data yields the following rules of thumb. Evidence of initial scarp shape has been lost when apparent age reaches twice its initial value. If a scarp is formed by two events, the inversion gives their offset-weighted-mean age with an error that is a fraction of offset squared. A scarp that appears to have been formed by one event may have been formed by two with an interval between them as large as apparent age. After scarps of two fault traces have diffused to appear as one, the error in inferred age may be as large as half the apparent age. Variation of apparent age along strike would indicate multiple fault traces. The simplicity of scarp profile measurement and this inversion makes profile analysis attractive. If linearity of the flow law, time for a free face to be reduced to the angle of repose, and variation of diffusivity with climate and material could be established, profile analysis would become a reliable dating technique.

Journal of Geophysical Research Solid Earth↗
USGS2012

Fitting degradation of shoreline scarps by a nonlinear diffusion model

The diffusion model of degradation of topographic features is a promising means by which vertical offsets on Holocene faults might be dated. In order to calibrate the method, we have examined present-day profiles of wave-cut shoreline scarps of late Pleistocene lakes Bonneville and Lahontan. It may be assumed that these scarps were initially at least as steep as the angle of repose. Offsets range from 1 to 12 m, and present slope angles range from 9° to 29°. A parameter called apparent diffusion age, defined as half the mean square horizontal extent of the slope function of each profile, is plotted as a function of scarp offset. The points show a clear trend of apparent age increasing nearly linearly with offset. If linear diffusion held and scarps were initially vertical, apparent diffusion age would be the same for all the shoreline profiles. The increasing trend can only partly be explained by nonvertical initial scarp slope, and therefore the rate of transport of material downslope must increase significantly faster than a linear law in the range of slopes spanned by the data. The transport law must become linear at small slope to reduce scatter between profiles with varying ambient slopes. The transport law adopted for the purpose of dating is k 0 (1 + 5 s 2 ), where s is local slope. The transport coefficient k 0 is correlated inversely with fan slope, suggesting that there is a dependence on the particle size distribution. A table is included that allows easy application of the model to scarps with simple initial shape.

Journal of Geophysical Research Solid Earth↗
USGS2012

Effect of far-field slope on morphologic dating of scarplike landforms

The principal finding of this paper is that the far-field slope has a first-order effect on model age determinations of scarplike landforms in weakly consolidated terrains. Observationally, this can be demonstrated in two ways using the Lake Bonneville and Lahontan shoreline scarps as separate and combined data sets. Use of the reduced scarp slope, tan θ s - b (where θ s is the maximum scarp angle and b is the far-field or fan slope), instead of tan θ s alone as the measure of scarp slope measurably reduces separation between the two data sets induced by different average fan slopes for the two data sets and significantly reduces scatter in the slope-offset plot for both the separate and combined data sets. Theoretically, the argument can be put even more strongly, at least within the range of linear and nonlinear diffusion models that we consider here together with a mathematical transformation of the empirical approach of R. C. Bucknam and R. E. Anderson: When one correctly takes into account the far-field slope, one will basically get the same age determination no matter which of these models one uses; conversely, without accounting properly for the effect of far-field slope, one is virtually guaranteed to get an erroneous age determination, no matter which model is used.

Journal of Geophysical Research Solid Earth↗
USGS2011

Specifying initial stress for dynamic heterogeneous earthquake source models

Dynamic rupture calculations using heterogeneous stress drop that is random and self-similar with a power-law spatial spectrum have great promise of producing realistic ground-motion predictions. We present procedures to specify initial stress for random events with a target rupture length and target magnitude. The stress function is modified in the depth dimension to account for the brittle-ductile transition at the base of the seismogenic zone. Self-similar fluctuations in stress drop are tied in this work to the long-wavelength stress variation that determines rupture length. Heterogeneous stress is related to friction levels in order to relate the model to physical concepts. In a variant of the model, there are high-stress asperities with low background stress. This procedure has a number of advantages: (1) rupture stops naturally, not at artificial barriers; (2) the amplitude of short-wavelength fluctuations of stress drop is not arbitrary: the spectrum is fixed to the long-wavelength fluctuation that determines rupture length; and (3) large stress drop can be confined to asperities occupying a small fraction of the total rupture area, producing slip distributions with enhanced peaks.

Bulletin of the Seismological Society of America↗
USGS2011

Verifying a computational method for predicting extreme ground motion

In situations where seismological data is rare or nonexistent, computer simulations may be used to predict ground motions caused by future earthquakes. This is particularly practical in the case of extreme ground motions, where engineers of special buildings may need to design for an event that has not been historically observed but which may occur in the far-distant future. Once the simulations have been performed, however, they still need to be tested. The SCEC-USGS dynamic rupture code verification exercise provides a testing mechanism for simulations that involve spontaneous earthquake rupture. We have performed this examination for the specific computer code that was used to predict maximum possible ground motion near Yucca Mountain. Our SCEC-USGS group exercises have demonstrated that the specific computer code that was used for the Yucca Mountain simulations produces similar results to those produced by other computer codes when tackling the same science problem. We also found that the 3D ground motion simulations produced smaller ground motions than the 2D simulations.

Seismological Research Letters↗
USGS2010

Inelastic off-fault response and three-dimensional dynamics of earthquake rupture on a strike-slip fault

Large dynamic stress off the fault incurs an inelastic response and energy loss, which contributes to the fracture energy, limiting the rupture and slip velocity. Using an explicit finite element method, we model three-dimensional dynamic ruptures on a vertical strike-slip fault in a homogeneous half-space. The material is subjected to a pressure-dependent Drucker-Prager yield criterion. Initial stresses in the medium increase linearly with depth. Our simulations show that the inelastic response is confined narrowly to the fault at depth. There the inelastic strain is induced by large dynamic stresses associated with the rupture front that overcome the effect of the high confining pressure. The inelastic zone increases in size as it nears the surface. For material with low cohesion (~5 MPa) the inelastic zone broadens dramatically near the surface, forming a "flowerlike" structure. The near-surface inelastic strain occurs in both the extensional and the compressional regimes of the fault, induced by seismic waves ahead of the rupture front under a low confining pressure. When cohesion is large (~10 MPa), the inelastic strain is significantly reduced near the surface and confined mostly to depth. Cohesion, however, affects the inelastic zone at depth less significantly. The induced shear microcracks show diverse orientations near the surface, owing to the low confining pressure, but exhibit mostly horizontal slip at depth. The inferred rupture-induced anisotropy at depth has the fast wave direction along the direction of the maximum compressive stress.

Journal of Geophysical Research B: Solid Earth↗
USGS2005

Rupture dynamics with energy loss outside the slip zone

Energy loss in a fault damage zone, outside the slip zone, contributes to the fracture energy that determines rupture velocity of an earthquake. A nonelastic two-dimensional dynamic calculation is done in which the slip zone is modeled as a fault plane and material off the fault is subject to a Coulomb yield condition. In a mode 2 crack-like solution in which an abrupt uniform drop of shear traction on the fault spreads from a point, Coulomb yielding occurs on the extensional side of the fault. Plastic strain is distributed with uniform magnitude along the fault, and it has a thickness normal to the fault proportional to propagation distance. Energy loss off the fault is also proportional to propagation distance, and it can become much larger than energy loss on the fault specified by the fault constitutive relation. The slip velocity function could be produced in an equivalent elastic problem by a slip-weakening friction law with breakdown slip Dc increasing with distance. Fracture energy G and equivalent Dc will be different in ruptures with different initiation points and stress drops, so they are not constitutive properties; they are determined by the dynamic solution that arrives at a particular point. Peak slip velocity is, however, a property of a fault location. Nonelastic response can be mimicked by imposing a limit on slip velocity on a fault in an elastic medium.

Journal of Geophysical Research B: Solid Earth↗
USGS2005

The wrinkle-like slip pulse is not important in earthquake dynamics

A particular solution for slip on an interface between different elastic materials, the wrinkle-like slip pulse, propagates in only one direction with reduced normal compressive stress. More general solutions, and natural earthquakes, need not share those properties. In a 3D dynamic model with a drop in friction and heterogeneous initial stress, the wrinkle-like slip pulse is only a small part of the solution. Rupture propagation is determined primarily by the potential stress drop, not by the wrinkle-like slip pulse. A 2D calculation with much finer resolution shows that energy loss to friction might not be significantly reduced in the wrinkle-like slip pulse. Copyright 2005 by the American Geophysical Union.

Geophysical Research Letters↗
USGS2001

Solving the dynamic rupture problem with different numerical approaches and constitutive laws

We study the dynamic initiation, propagation and arrest of a 2-D in-plane shear rupture by solving the elastodynamic equation by using both a boundary integral equation method and a finite difference approach. For both methods we adopt different constitutive laws: a slip-weakening (SW) law, with constant weakening rate, and rate- and state-dependent friction laws (Dieterich-Ruina). Our numerical procedures allow the use of heterogeneous distributions of constitutive parameters along the fault for both formulations. We first compare the two solution methods with an SW law, emphasizing the required stability conditions to achieve a good resolution of the cohesive zone and to avoid artificial complexity in the solutions. Our modelling results show that the two methods provide very similar time histories of dynamic source parameters. We point out that, if a careful control of resolution and stability is performed, the two methods yield identical solutions. We have also compared the rupture evolution resulting from an SW and a rate- and state-dependent friction law. This comparison shows that despite the different constitutive formulations, a similar behaviour is simulated during the rupture propagation and arrest. We also observe a crack tip bifurcation and a jump in rupture velocity (approaching the P-wave speed) with the Dieterich-Ruina (DR) law. The rupture arrest at a barrier (high strength zone) and the barrier-healing mechanism are also reproduced by this law. However, this constitutive formulation allows the simulation of a more general and complex variety of rupture behaviours. By assuming different heterogeneous distributions of the initial constitutive parameters, we are able to model a barrier-healing as well as a self-healing process. This result suggests that if the heterogeneity of the constitutive parameters is taken into account, the different healing mechanisms can be simulated. We also study the nucleation phase duration Tn, defined as the time necessary for the crack to reach the half-length Ic. We compare the Tn values resulting from distinct simulations calculated using different constitutive laws and different sets of constitutive parameters. Our results confirm that the DR law provides a different description of the nucleation process than the SW law adopted in this study. We emphasize that the DR law yields a complete description of the rupture process, which includes the most prominent features of SW.

Geophysical Journal International↗
USGS1999

Test of two methods for faulting on finite-difference calculations

Tests of two fault boundary conditions show that each converges with second order accuracy as the finite-difference grid is refined. The first method uses split nodes so that there are disjoint grids that interact via surface traction. The 3D version described here is a generalization of a method I have used extensively in 2D; it is as accurate as the 2D version. The second method represents fault slip as inelastic strain in a fault zone. Offset of stress from its elastic value is seismic moment density. Implementation of this method is quite simple in a finite-difference scheme using velocity and stress as dependent variables.

Bulletin of the Seismological Society of America↗
USGS1994

Dynamic growth of mixed-mode shear cracks

A pure mode II (in-plane) shear crack cannot propagate spontaneously at a speed between the Rayleigh and S -wave speeds, but a three-dimensional (3D) or two-dimensional (2D) mixed-mode shear crack can propagate in this range, being driven by the mode III (antiplane) component. Two different analytic solutions have been proposed for the mode II component in this case. The first is the solution valid for crack speed less than the Rayleigh speed. When applied above the Rayleigh speed, it predicts a negative stress intensity factor, which implies that energy is generated at the crack tip. Burridge proposed a second solution, which is continuous at the crack tip, but has a singularity in slip velocity at the Rayleigh wave. Spontaneous propagation of a mixed-mode rupture has been calculated with a slip-weakening friction law, in which the slip velocity vector is colinear with the total traction vector. Spontaneous trans-Rayleigh rupture speed has been found. The solution depends on the absolute stress level. The solution for the in-plane component appears to be a superposition of smeared-out versions of the two analytic solutions. The proportion of the first solution increases with increasing absolute stress. The amplitude of the negative in-plane traction pulse is less than the absolute final sliding traction, so that total in-plane traction does not reverse. The azimuth of the slip velocity vector varies rapidly between the onset of slip and the arrival of the Rayleigh wave. The variation is larger at smaller absolute stress.

Bulletin of the Seismological Society of America↗