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A catalog of low frequency events (LFEs) beneath Vancouver Island is analyzed in the context of a granular flow model. The catalog contains origin-times and magnitudes of 269,423 LFEs grouped within 130 families and recorded between 2003 and 2013. Each family represents a distinct location within the boundary between the subducting Juan de Fuca and overriding North American plates. The LFEs occurred during 10 episodic tremor and slip (ETS) events that recurred at ∼14-month intervals and lasted for about a week. With one exception, each family was active in all 10 ETS episodes. Our analysis suggests that LFEs do not follow Gutenberg-Richter statistics, but are normally distributed with respect to magnitude and, therefore, log-normally distributed with respect to moment. The Kostrov strain associated with the moments in a given family is used to estimate its size as L = 350 ± 15 m. These observations are explained through a model of granular flow in a channel that accommodates displacement along the plate boundary. In this model the LFE families correspond to granular jams in flow that persist over many ETS episodes. Each LFE is a slip between two grains in the jam. The log-normal distribution of LFE moments can be ascribed to a theoretically predicted log-normal distribution of grain sizes in each jam. This model explains the weak dependence of seismic duration on LFE moment because frictional slip between grains in an over-pressured environment need not scale like a growing rupture in an elastic medium.

A recent study of repeating earthquakes on the San Andreas Fault in central California by Nadeau and Johnson [1998] found that the smallest events occurred on patches having a linear dimension of the order of 0.5 m, displacements of about 2 cm, and stress drops of the order of 2000 MPa, roughly 10 times larger than rock strengths measured in the laboratory. The stress drop for larger events was observed to decrease as a power law of the seismic moment reaching the commonly observed value of 10 MPa at about magnitude 6. These large strengths are shown here to be consistent with laboratory data if the preexisting microcracks are all healed. A hierarchical fractal asperity model is presented, which is based on recent laboratory observations of contact distributions in sliding friction experiments. This “Cantor dust” model is shown to be consistent with the observed power law decrease in stress drop and increase in displacement with increasing event size. The spatial distribution of hypocenters in the Parkfield area is shown to be consistent with this simple fractal model and with a hierarchical clustering of asperities having a fractal dimension of D=1 and discrete rescaling factor of about 20.

A composite model is proposed to describe the time-dependent response of the Earth's lower crust. The motivation for such a model is twofold: First, new observations of widespread postseismic deformation indicate that the deep continental crust responds viscoelastically, having both long- and short-term decay times. Second, by any number of observationally based rationales, the lower crust is compositionally and structurally heterogeneous over many length scales. For heterogeneities that have much smaller characteristic lengths than the minimum deformation wavelength of interest, the aggregate rheology can be described by composite media theory. For wavelengths of the order of the thickness of the lower crust (≈25–40 km) and larger, composite theory may be applied to heterogeneities that are smaller than about several hundred meters, or equivalent to the vertical extent of a thick lower crustal mylonitic shear zone. The composite media theory developed here is constructed using both Eshelby-Mori-Tanaka theory for aligned generalized spheroidal inclusions and a generalized self-consistent method. The inclusions and matrix are considered to be Maxwellian viscoelastic: a rheology that is consistent with past homogeneous models of postseismic stress relaxation. The composite theory presented here introduces a transient response to a suddenly imposed stress field which does not appear in homogeneous Maxwell models. Analytic expressions for the amplitude and duration of the transient and for the effective long- and short-term viscosities of the composite are given which describe the sensitivity to inclusion concentration (Φ), to shape, and to ratio of inclusion-to-matrix viscosity (R).

We discuss in detail the concept of discrete scale invariance and show how it leads to complex critical exponents and hence to the log-periodic corrections to scaling exhibited by various measures of seismic activity close to a large earthquake singularity. Discrete scale invariance is first illustrated on a geometrical fractal, the Sierpinsky gasket, which is shown to be fully described by a complex fractal dimension whose imaginary part is a simple function (inverse of the logarithm) of the discrete scaling factor. Then, a set of simple physical systems (spins and percolation) on hierarchical lattices is analyzed to exemplify the origin of the different terms in the discrete renormalization group formalism introduced to tackle this problem. As a more specific example of rupture relevant for earthquakes, we propose a solution of the hierarchical time-dependent fiber bundle of Newman et al. [1994] which exhibits explicitly a discrete renormalization group from which log-periodic corrections follow. We end by pointing out that discrete scale invariance does not necessarily require an underlying geometrical hierarchical structure. A hierarchy may appear “spontaneously” from the physics and/or the dynamics in a Euclidean (nonhierarchical) heterogeneous system. We briefly discuss a simple dynamical model of such mechanism, in terms of a random walk (or diffusion) of the seismic energy in a random heterogeneous system.

Although many studies have shown that faults and fractures are self-similar over a large range of scales, none have tested the fault structure for self-similarity in three dimensions. In this study, earthquake hypocentral locations in central and southern California were used to illuminate three-dimensional (3-D) fault structures, for which we measured the fractal capacity dimension, D0(3-D). Hypocentral distributions from the Joshua Tree, Big Bear, and Upland aftershock sequences, as well as background seismicity at Parkfield were found to be fractal, where D0(3-D) increased with increasing event density, asymptotically approaching a stable value. The Joshua Tree data set stabilized at D0(3-D) = 1.92 ± 0.02, the Parkfield data set asymptotically approached D0(3-D) = 1.82, and the Big Bear data set approached D0(3-D) = 2.01. As a test of the effects of location errors upon the measured value of D0(3-D), the Upland aftershock data were located with both the southern California Hadley and Kanamori (1977) (H-K) velocity model, and the more accurate Hauksson and Jones (1991) (H-J) velocity model. Events located with the H-K model asymptotically approached D0(3-D) = 2.07, and events located with the H-J model approached D0(3-D) = 1.79, suggesting that improved hypocentral locations may decrease the measured fractal dimension. One interpretation of our results of D0(3-D) ≤ 2.0 for all of the hypocentral data is that earthquakes only occur on the “percolation backbone” of a fault network, i.e., the active part of the network that accommodates finite strain deformation (Sahimi et al., 1993). We show that a percolation model that allows for healing of previously broken bonds is consistent with this interpretation.

[1] Seismicity clusters within fault zones can be connected to the structure, geometric complexity and size of asperities which perturb and intensify the stress field in their periphery. To gain further insight into fault mechanical processes, we study stick-slip sequences in an analog, laboratory setting. Analysis of small scale fracture processes expressed by acoustic emissions (AEs) provide the possibility to investigate how microseismicity is linked to fault heterogeneities and the occurrence of dynamic slip events. The present work connects X-ray computer tomography (CT) scans of faulted rock samples with spatial maps of b values (slope of the frequency-magnitude distribution), seismic moments and event densities. Our current experimental setup facilitates the creation of a series of stick-slips on one fault plane thus allowing us to document how individual stick-slips can change the characteristics of AE event populations in connection to the evolution of the fault structure. We found that geometric asperities identified in CT scan images were connected to regions of low b values, increased event densities and moment release over multiple stick-slip cycles. Our experiments underline several parallels between laboratory findings and studies of crustal seismicity, for example, that asperity regions in lab and field are connected to spatial b value anomalies. These regions appear to play an important role in controlling the nucleation spots of dynamic slip events and crustal earthquakes.

A radially stratified and incompressible earth model with secular variation of the second degree gravity field J-dot(2) is used here to test sensitivity of data to viscosity increases with depth and convective boundary layer structure. Prior estimates and observed nontidal J-dot(23)(-C-dot(20)) are consistent with a layered lower mantle viscosity. Details of this layering are examined by comparing predicted and observed J-dot(3), J-dot(4). Speculation that a high-velocity layer exists above D-double prime is considered. With a 650-km thick deep high-viscosity layer, the remaining lower mantle is in one of two ranges: 1.5 to 3.5 x 10 exp 20 or 3.5 to about 10 exp 22 Pa s.

The glass transition has been experimentally detected in basalt as (1) an increase in the aggregate linear thermal expansion coefficient αL, (2) an abrupt change in the temperature dependence of Young's modulus dE/dT, and (3) a change in stress relaxation behavior that effectively separates the T> TG and T TG. Collectively, the mechanical results suggest that for Hawaiian olivine tholeiite at 1-atm pressure, the principal material responses are (1) elastic (T ≤ 600°C), (2) reduced creep (600 980°C). Fracture surface morphologies developed during solidification suggest that the presence of the supercooled melt grain boundary phase may participate in the regulation of the thermal cracking process. Well-preserved fracture surfaces formed by incremental crack growth are found to be covered with striations that correspond to the inferred sequential stopping positions of the advancing fracture front. These striae may be rationalized in terms of experimental analogues that have been produced in viscoelastic polymers by cyclic tension-tension loading. The fracture surface morphology, mechanical data, and the controlled crack growth analogues suggest that thermal fracture in solidifying basalt is an incremental and cyclic process, involving three steps: (1) the accumulation of elastic strain energy in cooling rock at temperatures below that required for stress relaxation due to viscous flow in the intercrystalline liquid phase, (2) fracture at a ΔT determined primarily by the aggregate thermal expansion coefficient αυ and Young's modulus E, (3) the penetration by the advancing crack tip, of the thermal horizon capable of relaxing stress due to the creep of intercrystalline supercooled melt, producing the rough surface texture associated with the termination of a striation. Further crack growth must now await the migration of the solidus. The cycle then repeats. Striations measured in deep Hawaiian lava lakes have been compared with the crack advance increments expected in the vicinity of the glass transition, based on two tests: (1) thermal gradients measured in Kilauea Iki, combined with the mechanical properties of olivine tholeiite evaluated near TG, and (2) the crack advance required to match the recorded seismic stopping phases for prexisting cracks of the dimensions expected for Kilauea Iki. The observed versus predicted comparisons are (1) 31 versus 36 cm; and (2) 31 versus 30 cm. We envision this incremental crack growth process as contributing to the control on the downward movement of the thermal cracking front—and its associated hydrothermal circulation zone—in the upper portions of solidifying subaerial and submarine ponded basalt.

We have compared estimates of the activation volume V* based on several theoretical models with measured values in metals, alkali halides, and olivine. The theoretical methods tested include one based upon the empirical correlation between activation energy and melting temperature and several which are based upon simple elastic models for the defect structure. For metals and olivine, the melting relation works well, but for alkali halides, the melting model predicts too large a V* by an approximate factor of 1.5. Of the elastic models, the dilatational strain energy model introduced by Zener (1942) provides reasonable estimates of V* for metals and olivine, but it also overestimates V* for alkali halides. Zener's assertion that the experimental value of V* should be bounded by theoretical values calculated from strain energy models which assume pure shear (Keyes, 1963) and pure dilatation is supported by the available data for metals, oxides, and alkali halides. These provide upper and lower estimates for the variation of viscosity with depth in the mantle.

The recent suggestion that earthquakes at Long Valley caldera, California, are caused by rapid tensile failure under high fluid pressure contradicts the assumption that the speed of propagation of a fluid-driven tensile crack is limited by the speed of motion of the fluid within the crack, so that such cracks cannot radiate seismic waves. In this paper we analyze fluid-driven cracks of various geometries and find examples of both stable and catastrophic propagation. For an isolated crack in a homogeneous medium with remotely applied compressive stresses the stress intensity decreases as the crack tip extends beyond the driving fluid (assumed stationary), so that crack propagation is stable and limited by the rate of fluid flow. On the other hand, for two cracks approaching each other, initially stable behavior gives way to unstable propagation, and the cracks join catastrophically even if the fluid is stationary. A crack approaching a free surface also exhibits a transition from stable to unstable propagation, with the final episode occurring catastrophically. A crack emanating from a pressurized reservoir of cylindrical or spherical shape can also undergo an episode of catastrophic propagation. Propagation begins when the reservoir pressure rises to a critical value, which depends on the regional stress field, the fracture toughness of the rock, and the size of the largest flaw at the surface of the reservoir. This flaw propagates catastrophically until it reaches a certain size and thereafter propagates stably. Microearthquakes caused by hydraulic fracturing have dimensions of at most a few tens of borehole radii and therefore probably are difficult to detect. Volcanic earthquakes can have source dimensions comparable to the radius of the associated magma chamber, so earthquakes as large as those at Long Valley caldera (source dimensions ≅ 10 km) could be caused by tensile failure.