Analytic Properties and Covariance Functions of a New Class of Generalized Gibbs Random Fields

Spartan Spatial Random Fields (SSRFs) are generalized Gibbs random fields, equipped with a coarse-graining kernel that acts as a low-pass filter for the fluctuations. SSRFs are defined by means of physically motivated spatial interactions and a small set of free parameters (interaction couplings). This paper focuses on the FGC-SSRF model, which is defined on the Euclidean space $\mathbb{R}^{d}$ by means of interactions proportional to the squares of the field realizations, as well as their gradient and curvature. The permissibility criteria of FGC-SSRFs are extended by considering the impact of a finite-bandwidth kernel. It is proved that the FGC-SSRFs are almost surely differentiable in the case of finite bandwidth. Asymptotic explicit expressions for the Spartan covariance function are derived for $d=1$ and $d=3$; both known and new covariance functions are obtained depending on the value of the FGC-SSRF shape parameter. Nonlinear dependence of the covariance integral scale on the FGC-SSRF characteristic length is established, and it is shown that the relation becomes linear asymptotically. The results presented in this paper are useful in random field parameter inference, as well as in spatial interpolation of irregularly-spaced samples.Comment: 24 pages; 4 figures Submitted for publication to IEEE Transactions on Information Theor

Stochastic Stick - Slip Model Linking Crustal Shear Strength and Earthquake Interevent Times

The current understanding of the earthquake interevent times distribution (ITD) is incomplete. The Weibull distribution is often used to model the earthquake ITD. We link the earthquake ITD on single faults with the Earth's crustal shear strength distribution by means of a phenomenological stick - slip model. We obtain Weibull ITD for power-law stress accumulation, i.e., $\sigma(t) = \alpha t^{\beta}$, where $\beta >0$ for single faults or systems with homogeneous strength statistics. We show that logarithmic stress accumulation leads to the log-Weibull ITD. For the Weibull ITD, we prove that (i) $m= \beta m_s$, where $m$ and $m_s$ are, respectively, the ITD and crustal shear strength Weibull moduli and (ii) the time scale $\tau_s = (S_s/\alpha)^{1/\beta}$ where $S_s$ is the scale of crustal shear strength. We generalize the ITD model for fault systems. We investigate deviations of the ITD tails from the Weibull due to sampling bias, magnitude selection, and non-homogeneous strength parameters. Assuming the Gutenberg - Richter law and independence of $m$ on the magnitude threshold, $M_{L,c},$ we deduce that $\tau_s \propto e^{- \rho_{M} M_{L,c}},$ where $\rho_M \in [1.15, 3.45]$ for seismically active regions. We demonstrate that a microearthquake sequence conforms reasonably well to the Weibull model. The stochastic stick - slip model justifies the Weibull ITD for single faults and homogeneous fault systems, while it suggests mixtures of Weibull distributions for heterogeneous fault systems. Non-universal deviations from Weibull statistics are possible, even for single faults, due to magnitude thresholds and non-uniform parameter values.Comment: 32 pages, 11 figures Version 2; minor correction

Finite-Size Effects on Return Interval Distributions for Weakest-Link-Scaling Systems

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and in the fracture strength of quasi-brittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. We use the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the $\kappa$-Weibull distribution. We conduct statistical analysis of experimental data and simulations that show good agreement with the $\kappa$-Weibull distribution. We show the following: (1) The weakest-link theory for finite-size systems involves the $\kappa$-Weibull distribution. (2) The power-law decline of the $\kappa$-Weibull upper tail can explain deviations from the Weibull scaling observed in return interval data. (3) The hazard rate function of the $\kappa$-Weibull distribution decreases linearly after a waiting time $\tau_c \propto n^{1/m}$, where $m$ is the Weibull modulus and $n$ is the system size in terms of representative volume elements. (4) The $\kappa$-Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we show that the $\kappa$-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.Comment: 33 pages, 11 figure