A moving window semivariance estimator

Abstract

nstract. Use of geostatistics in hydrology, reservoir engineering, soil physics, and environmental science has expanded vastly in the last few years. Since the semivariance is the. backbone of geostatistics, it is very important to estimate it accurately. Building on Matheron's classical formula, there are several improved semivariance estimators, but they are only reliable at small lag distances. For large lag distances all estimat0rs can produce erratic results that may lead to a misinterpretation of the spatial structure of a given data set. The lack of precision is primarily caused by a shrinking number of observation pairs as lag distance increases. This paper presents a new semivariance estimator that uses all given data at all lag distances. In addition, the new e•timator is not restricted to samples taken along lines, and it easily accounts for missing data. We compare the new estimator to existing ones on simulated data sets as well as three outcrop data sets. The results show that the new estimator is unbiased, robust, and resistant to contamination. IntroductionMost spatial data are so complicated that describing them forces us to rely on geostatistics, a hybrid discipline of engineering, geology, mathematics, and statistics. In recent years geostatistics has become an extremely powerful tool to study spatial data. In the sense of statistics the distribution of spatial dat•. is described with statistical moments. This kind of spatial behavior (autocorrelation or structure) is measured by a statistic, designated as the semivariogram, or simply the variogram. The variogram is a plot of the semivariance versus separation or lag distance.The original and most popular semivariance estimator (or "experimental" semivariance) in present use was given by Matheron [1962]. UmOortunately, the estimator is imprecise at moderate to large (compared to the extent of correlation in the data) lag distances. Early on, Matheron [1965] warned of the variability of his estimator for large lag distances. Yang and Lake [1989], analyzing the precision of Matheron's estimator using both analytical and numerical methods, also concluded that the precision of this estimator decreases as the lag distance and/or autocorrelation length increase (Figure 1). Therefore only the first few points of Matheron's estimator can be considered significant [Armstrong, 1984;Dowd, 1984]. Beyond these points the shape of the variogram is distorted, and the spatial structure obscured. The principal reason for the distortion is that the semivariance estimate is based on fewer and fewer data pairs as lag distance increases. Indeed, at the largest lag distance the semivariance is estimated on only one data pair for a one-dimensional problem. Matheron's estimator is unbiased, however. There are two other properties, robustness and resistance, that a good estimator should possess. An estimator is "robust" if its results are insensitive to a lack of conformance to underlying assumptions, usually in the form of the probability density function (PDF). An es...

Deformation bands and their impact on fluid flow in sandstone reservoirs: the role of natural thickness variations

“…The covariance can be estimated from measurements by a moving window estimator (Li & Lake 1994). Also we consider three analytical covariance functions (Monin & Yaglom 1975):…”

Influence of soil inhomogeneity on GPR for landmine detection

“…Here a more robust estimator which implements a moving window is employed. The estimator for regularly discretized 1D data is given by the following equations [6]:…”

Characterizing heterogeneous permeable media with spatial statistics and tracer data using sequential simulated annealing

“…It is crucial for an inversion scheme to capture such "scale effects" exhibited by naturally occurring permeable media. We have examined this aspect by comparing the horizontal variograms for the actual data and the inverted permeability field using a moving window semivariance estimator (Li and Lake, 1994). The moving window estimator describes the sample variance of a moving window of size h as a function of h (scale) and thus reflects changes in heterogeneity at different scales more accurately compared to classical estimators.…”