John B. Rundle

Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems

[1] Earthquakes and the faults upon which they occur interact over a wide range of spatial and temporal scales. In addition, many aspects of regional seismicity appear to be stochastic both in space and time. However, within this complexity, there is considerable self-organization. We argue that the occurrence of earthquakes is a problem that can be attacked using the fundamentals of statistical physics. Concepts of statistical physics associated with phase changes and critical points have been successfully applied to a variety of cellular automata models. Examples include sandpile models, forest fire models, and, particularly, slider block models. These models exhibit avalanche behavior very similar to observed seismicity. A fundamental question is whether variations in seismicity can be used to successfully forecast the occurrence of earthquakes. Several attempts have been made to utilize precursory seismic activation and quiescence to make earthquake forecasts, some of which show promise.

A simplified spring‐block model of earthquakes

The time interval between earthquakes is much larger than the actual time involved during slip in an individual event. The authors have used this fact to construct a cellular automaton model of earthquakes. This model describes the time evolution of a 2-D system of coupled masses and springs sliding on a frictional surface. The model exhibits power law frequency-size relations and can exhibit large earthquakes with the same scatter in the recurrence time observed for actual earthquakes.

Shallow and peripheral volcanic sources of inflation revealed by modeling two-color geodimeter and leveling data from Long Valley caldera, California, 1988-1992

We refined the model for inflation of the Long Valley caldera near Mammoth Lakes, California, by combining both geodetic measurements of baseline length and elevation changes. Baseline length changes measured using a two-color geodimeter with submillimeter precision revealed that the resurgent dome started to reinflate in late 1989. Measurements between late 1989 and mid-1992 revealed nearly 13 cm of extension across the resurgent dome. Geodetic leveling surveys with approximately 2-mm precision made in late 1988 and in mid-1992 revealed a maximum of about 8 cm of uplift of the resurgent dome. Two ellipsoidal sources satisfy both the leveling and two-color measurements, whereas spherical point sources could not. The models primary inflation source is located 5.5 km beneath the resurgent dome with the two horizontal axes being nearly equal in size and the vertical axis being 4 times the length of the horizontal axes. A second ellipsoidal source was added to improve the fit to the two-color measurements. This secondary source is located at a depth between 10 and 20 km beneath the south moat of the caldera and has the geometry of an elongated ellipsoid or pipe that dips down to the northeast. In addition, the leveling data suggest dike intrusion beneath Mammoth Mountain during the 1988-1992 interval, which is likely associated with an intense swarm of small earthquakes during the summer of 1989 at that location. Our analysis shows the dike intrusion to be the shallowest of the three sources with a depth range of 1-3 km below the surface to the top of the intrusion.

Model for the Distribution of Aftershock Interoccurrence Times

In this work the distribution of interoccurrence times between earthquakes in aftershock sequences is analyzed and a model based on a nonhomogeneous Poisson (NHP) process is proposed to quantify the observed scaling. In this model the generalized Omoris law for the decay of aftershocks is used as a time-dependent rate in the NHP process. The analytically derived distribution of interoccurrence times is applied to several major aftershock sequences in California to confirm the validity of the proposed hypothesis.

Simulation-Based Distributions of Earthquake Recurrence Times on the San Andreas Fault System

Earthquakes on a specified fault (or fault segment) with magnitudes greater than a specified value have a statistical distribution of recurrence times. The mean recurrence time can be related to the rate of strain accumulation and the strength of the fault. Very few faults have a recorded history of earthquakes that define a distribution well. For hazard assessment, in general, a statistical distribution of recurrence times is assumed along with parameter values. Assumed distributions include the Weibull (stretched exponential) distribution, the lognormal distribution, and the Brownian passage-time (inverse Gaussian) distribution. The distribution of earthquake waiting times is the conditional probability that an earthquake will occur at a time in the future if it has not occurred for a specified time in the past. The distribution of waiting times is very sensitive to the distribution of recurrence times. An exponential distribution of recurrence times is Poissonian, so there is no memory of the last event. The distribution of recurrence times must be thinner than the exponential if the mean waiting time is to decrease as the time since the last earthquake increases. Neither the lognormal or the Brownian passage time distribution satisfies this requirement. We use the “Virtual California” model for earthquake occurrence on the San Andreas fault system to produce a synthetic distribution of earthquake recurrence times on various faults in the San Andreas system. We find that the synthetic data are well represented by Weibull distributions. We also show that the Weibull distribution follows from both damage mechanics and statistical physics.

Gravity changes and deformation due to a magmatic intrusion in a two‐layered crustal model

We develop and extend theoretical and computational methods for the calculation of the deformation, gravity and potential change due to a point source of magma injection into a multilayered, elastic-gravitational earth model. In our calculations, which are based upon the method outlined by Rundle, two distinct layers overlying a half-space may be incorporated. The source can be located in either of the layers or the half-space. The method is quite general, and can be readily adapted to calculations in which stresses in either the layers or the half-space relax by viscoelastic flow. The results obtained indicate that the use of homogeneous half-space to represent the Earth may in some cases be too simple a model and that variations in elastic moduli have a more significant effect than variation in reference density on both the surface displacements and gravity changes. As an example, we calculate the displacement and gravity changes due to a subsurface mass injection in a crust-mantle model appropriate to the volcanic island of Lanzarote, which is presently the subject of numerous geophysical experiments. Both historical and recent data indicate that Lanzarote may be subject to some risk of volcanic eruption in the future, thus our calculations may be useful in interpreting observations of preemption phenomena. The results are discussed in terms of prediction versus measurement capabilities.

Scaling and critical phenomena in a cellular automaton slider-block model for earthquakes

The dynamics of a general class of two-dimensional cellular automaton slider-block models of earthquake faults is studied as a function of the failure rules that determine slip and the nature of the failure threshold. Scaling properties of clusters of failed sites imply the existence of a mean-field spinodal line in systems with spatially random failure thresholds, whereas spatially uniform failure thresholds produce behavior reminiscent of self-organized critical behavior. This model can describe several classes of faults, ranging from those that only exhibit creep to those that produce large events.

Application of DInSAR-GPS Optimization for Derivation of Fine-Scale Surface Motion Maps of Southern California

A method based on random field theory and Gibbs-Markov random fields equivalency within Bayesian statistical framework is used to derive 3-D surface motion maps from sparse global positioning system (GPS) measurements and differential interferometric synthetic aperture radar (DInSAR) interferogram in the southern California region. The minimization of the Gibbs energy function is performed analytically, which is possible in the case when neighboring pixels are considered independent. The problem is well posed and the solution is unique and stable and not biased by the continuity condition. The technique produces a 3-D field containing estimates of surface motion on the spatial scale of the DInSAR image, over a given time period, complete with error estimates. Significant improvement in the accuracy of the vertical component and moderate improvement in the accuracy of the horizontal components of velocity are achieved in comparison with the GPS data alone. The method can be expanded to account for other available data sets, such as additional interferograms, lidar, or leveling data, in order to achieve even higher accuracy

The statistical mechanics of earthquakes

Abstract We review recent theoretical developments on the physics of earthquakes. In particular, we focus on the rise of the statistical mechanical view of earthquakes as a kind of ‘phase transition’. This view is appealing in light of the well known scaling relations such as the Gutenberg-Richter magnitude frequency and Omoris law of aftershock decay. Scaling relations such as these, which are in reality power laws, are known to be associated with dynamical systems residing near a critical point in the state space of the system. These second-order critical points are associated with second-order transitions, which are a result of gradual changes of the controlling parameters. At the same time, characteristic earthquakes, which involve the entire fault segment sliding nearly at once, are more reminiscent of a first-order transition, which is characterized by sudden widespread changes in the physical state of the system. In this paper, we review these ideas and show how recent developments are leading to a view of earthquake fault systems based on modern statistical mechanics.

Scaling Properties of the Parkfield Aftershock Sequence

Aftershock sequences present a unique opportunity to study the physics of earthquakes. Important questions concern the fundamental origin of three widely applicable scaling laws: (1) Gutenberg–Richter frequency–magnitude scaling, (2) Omori’s law for aftershock decay rates, and (3) Bath’s law for the difference between the magnitude of the largest aftershock and a mainshock. The high-resolution Parkfield seismic network provided the opportunity for detailed studies of the aftershock sequence following the 28 September 2004, M 6.0 Parkfield earthquake. In this article it is shown that aftershocks satisfy the Gutenberg–Richter scaling relation only for relatively large times after the mainshock. There is a systematic time delay for the establishment of this scaling law. The temporal evolution of the rates of occurrence of aftershocks is quantified using the generalized Omori’s law. This scaling law contains two characteristic times c and τ . The analysis suggests that the parameter c plays the role of a characteristic time for the establishment of Gutenberg– Richter scaling. This time increases systematically with a decreasing lower magnitude cutoff. The systematic time delay is attributed to a cascade of energy from long wavelengths to short wavelengths. The parameter τ is a measure of the average time until the first aftershock occurs. We find that τ slightly varies with the lower magnitude cutoff of the sequence. We also note that the largest aftershock inferred from an extrapolation of Gutenberg–Richter scaling, M 5.0, is equal to the largest observed aftershock. This scaling associated with the universal applicability of Bath’s law is attributed to a constant partitioning of energy between a mainshock and its associated aftershock sequence. We also give in this article the distribution of interoccurrence times between successive aftershocks. We show that this distribution is well approximated by a nonhomogeneous Poissons process driven by the modified Omori’s law. The self-consistency between interoccurrence statistics and decay rates is taken as further evidence for the applicability of our studies.