At Issue: The Gaussian Autocorrelation Function

Abstract

This paper focuses on the use of Gaussian autocorrelation functions (ACF) in civil engineering applications involving random processes and random fields. It aims at debunking misgivings, verifying facts and figures, and formulating practical conclusions. A large majority of civil engineers active in random field modelling and reliability analysis is quite content to point out that the routine use of Gaussian autocorrelation functions is part of standard practice and perfectly harmless. A common approach in 2D random field problems, for instance, is to estimate an appropriate correlation length on some physical or empirical basis, and then plug it into a multivariate ACF that is both isotropic, and separable into a product of univariate ACFs: if both of these objectives are to be met, the Gaussian ACF naturally stands out as it is in fact the only real function to possess both of these properties. But as early as the nineteen-sixties, a substantive piece of electrical engineering literature pointed to “issues” and “red flags”. The claim was that the Gaussian ACF produces unrealistic results, violates certain principles concerning both the modelling and the estimation of random properties, and runs into results that possibly defy common sense. Similarly, geostatisticians have been issuing warnings of hyper-predictability, super-smoothness, wildly underestimated estimation errors, and artificial results in applications such as spatial kriging using Gaussian ACFs, leading to the recommendation that the Gaussian model should never be used in practice. This paper revisits the use of the Gaussian ACF and presents a sober but principled look at the entire issue. Importantly, it also considers the pros and cons of replacement ACF models and adjusted ACF models. The paper includes examples and measurable outcomes with the aim of providing a fair assessment and justifiable recommendations.