Revisiting the Relationship between Scale of Fluctuation and Mean Cross Distance

Abstract

Estimating scale of fluctuation is an intriguing issue, for which several methods have been developed, such as simple estimators (e.g., 0.8d¯-estimator) based on the mean cross distance d¯ of a soil property profile, sample autocorrelation function method, maximum likelihood method, Bayesian method, etc. Among these methods, the 0.8d¯-estimator is the simplest one and can be readily used by geotechnical practitioners whose training in probability theory and statistics is usually limited. It, however, shall be noted that the 0.8d¯-estimator was derived from the normal random field with squared exponential correlation function, which is largely ignored in its practical applications. Effects of the distribution type (e.g., normal or lognormal) and correlation function on the performance of the 0.8d¯-estimator remain unexplored and, hence, unknown to geotechnical practitioners, which potentially leads to misuse of the simple relationship. This paper aims to highlight the theoretical assumptions underlying the 0.8d¯-estimator and to, systematically, explore the effects of these theoretical assumptions on its performance (i.e., unbiasedness and variability). It is found that the 0.8d¯-estimator provides reasonably unbiased estimation of scale of fluctuation for the normal random field with squared exponential correlation function when there are, at least, two sampling data within a distance of scale of fluctuation. Whereas, results from the 0.8d¯-estimator for other cases violating the assumptions are biased, and may lead to a significant underestimation of scale of fluctuation. It is also found that the variability of the 0.8d¯-estimator increases as the sampling length decreases.