SUMMARY Finite‐fault source inversions reveal the spatial complexity of earthquake slip or pre‐stress distribution over the fault surface. The basic assumption of this study is that a stochastic model can reproduce the variability in amplitude and the long‐range correlation of the spatial slip distribution. In this paper, we compute the stochastic model for the source models of four earthquakes: the 1979 Imperial Valley, the 1989 Loma Prieta, the 1994 Northridge and 1995 Hyogo‐ken Nanbu (Kobe). For each earthquake (except Imperial Valley), we consider both the dip and strike slip distributions. In each case, we use a 1‐D stochastic model. For the four earthquakes, we show that the average power spectra of the raw, that is, non‐interpolated, data follow a power‐law behaviour with scaling exponents that range from 0.78 to 1.71. For the four earthquakes, we have found that a non‐Gaussian probability law, that is, the Lévy law, is better suited to reproduce the main features of the spatial variability embedded in the slip amplitude distribution, including the presence and frequency of large fluctuations. Since asperities are usually defined as regions with large slip values on the fault, the stochastic model will allow predicting and modelling the spatial distribution of the asperities over the fault surface. The values of the Lévy parameters differ from one earthquake to the other. Assuming an isotropic spatial distribution of heterogeneity for the dip and the strike slip of he Northridge earthquake, we also compute a 2‐D stochastic model. The main conclusions reached in the 1‐D analysis remain appropriate for the 2‐D model. The results obtained for the four earthquakes suggest that some features of the slip spatial complexity are universal and can be modelled accordingly. If this is proven correct, this will imply that the spatial variability and the long‐range correlation of the slip or pre‐stress spatial distribution can be described with the help of five parameters: a scaling exponent controlling the spatial correlation and the four parameters of the Lévy distribution constraining the spatial variability.

Abstract.Conditions under which a single oscillator model coupled with Dieterich-Ruina's rate and state dependent friction exhibits chaotic dynamics is studied. Properties of spring-block models are discussed. The parameter values of the system are explored and the corresponding numerical solutions presented. Bifurcation analysis is performed to determine the bifurcations and stability of stationary solutions and we find that the system undergoes a Hopf bifurcation to a periodic orbit. This periodic orbit then undergoes a period doubling cascade into a strange attractor, recognized as broadband noise in the power spectrum. The implications for earthquakes are discussed.

[1] Finite-fault source inversions reveal the spatial complexity of earthquake slip or prestress distribution over the fault surface. In this paper we discuss a stochastic model that reproduces the spatial variability and the long-range spatial correlation of the slip distribution of the 1979 Imperial Valley earthquake. We have found that stochastic models based on non-Gaussian distributions are better suited to describe the spatial variability of the slip amplitude over the fault. We also show that a stochastic modeling of the slip amplitude based on a Gaussian distribution fails to reproduce the spatial variability observed in the original slip distribution. The stochastic models can be used to deduce ground motion from other earthquakes statistically similar to Imperial Valley.

[1] We analyzed 315 dynamic strike-slip rupture models computed up to 5.0 Hz to get a quantitative understanding of the correlation and amplitude distributions of parameters describing the earthquake source, such as slip and rupture velocity. To account for the epistemic uncertainty of the problem, we constructed a database of dynamic ruptures computed by ourselves and other authors. This database contains ruptures computed using different models of initial stress, peak stress, and critical slip-weakening distance. Using the set of computed ruptures, we constructed probability density functions (pdfs) for the amplitude distributions of the source parameters and for the correlation between the source parameters. We tried to extract parameter pairs that showed a small variability in the spatial correlation given the large epistemic uncertainty in the input. We only analyzed the areas on the fault with subshear propagation speed. The principal findings are as follows: (1) Final slip amplitude does not show correlation with the local rupture velocity. (2) Final slip amplitude correlates well with risetime. (3) Rupture velocity correlates well with peak slip rate and the duration of the impulsive part of the slip rate function. (4) The pdf of rupture velocity, risetime, and peak slip rate depends on the distance from the nucleation zone. (5) Fracture energy is not the single controlling factor for the rupture velocity; the slope of the linear slip-weakening curve has a significant effect on the rupture velocity. (6) The crack length (length that is slipping at a given time) decreases with the distance from the nucleation zone.

Abstract. Existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability K as the variable of concern. We treat K as a multifractal and investigate its scaling and fractality using measured horizontal K data from two locations in the United States. The first data set was from a shoreline sandstone near Coalinga, California, and the second was from an eolian sandstone [Goggin, 1988]. By applying spectral analyses and computing the scaling of moments of various orders (using the double trace moment method [Lavallee, 1991; Lavallee et al., 1992]), we found that K is multiscaling (i.e., scaling and multifractal). We also found that the so-called universal multifractal (UM) [Schertzer and Lovejoy, 1987] model (essentially a log-Levy multifractal), was able to reproduce the multiscaling behavior reasonably well. The UM model has three parameters: a, rr, and H, representing the multifractality index, the codimension of the mean field, and the "distance" to stationary multifractal, respectively. We found (a = 1.7, rr = 0.23, H = 0.22) and (a = 1.6, rr = 0.11, H = 0.075) for the shoreline and eolian data sets, respectively. The fact that a values were less than 2 indicates that the underlying statistics are non-Gaussian. We generated stationary and nonstationary multifractals and illustrated the role of the UM parameters on simulated fields. Studies that treated Log K as the variable of concern have pointed out the necessity for large data records, especially when the underlying distribution is Levy-stable. Our investigation revealed that even larger data records are required when treating K as a multifractal, because Log K is less intermittent (or irregular) than K.•