Computing the Distribution of Pareto Sums Using Laplace Transformation and Stehfest Inversion

摘要

In statistical seismology, the properties of distributions of total seismic moment are important for constraining seismological models, such as the strain partitioning model (Bourne et al. J Geophys Res Solid Earth 119(12): 8991-9015, 2014). This work was motivated by the need to develop appropriate seismological models for the Groningen gas field in the northeastern Netherlands, in order to address the issue of production-induced seismicity. The total seismic moment is the sum of the moments of individual seismic events, which in common with many other natural processes, are governed by Pareto or "power law" distributions. The maximum possible moment for an induced seismic event can be constrained by geomechanical considerations, but rather poorly, and for Groningen it cannot be reliably inferred from the frequency distribution of moment magnitude pertaining to the catalogue of observed events. In such cases it is usual to work with the simplest form of the Pareto distribution without an upper bound, and we follow the same approach here. In the case of seismicity, the exponent beta appearing in the power-law relation is small enough for the variance of the unbounded Pareto distribution to be infinite, which renders standard statistical methods concerning sums of statistical variables, based on the central limit theorem, inapplicable. Determinations of the properties of sums of moderate to large numbers of Pareto-distributed variables with infinite variance have traditionally been addressed using intensive Monte Carlo simulations. This paper presents a novel method for accurate determination of the properties of such sums that is accurate, fast and easily implemented, and is applicable to Pareto-distributed variables for which the power-law exponent beta lies within the interval 0, 1. It is based on shifting the original variables so that a non-zero density is obtained exclusively for non-negative values of the parameter and is identically zero elsewhere, a property that is shared by the sum of an arbitrary number of such variables. The technique involves applying the Laplace transform to the normalized sum (which is simply the product of the Laplace transforms of the densities of the individual variables, with a suitable scaling of the Laplace variable), and then inverting it numerically using the Gaver-Stehfest algorithm. After validating the method using a number of test cases, it was applied to address the distribution of total seismic moment, and the quantiles computed for various numbers of seismic events were compared with those obtained in the literature using Monte Carlo simulation. Excellent agreement was obtained. As an application, the method was applied to the evolution of total seismic moment released by tremors due to gas production in the Groningen gas field in the northeastern Netherlands. The speed, accuracy and ease of implementation of the method allows the development of accurate correlations for constraining statistical seismological models using, for example, the maximum-likelihood method. It should also be of value in other natural processes governed by Pareto distributions with exponent less than unity.