Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty
 Eulogio PardoIgúzquiza,
 Peter Dowd
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Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: secondorder stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourthorder moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for Kdimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.
 Title
 Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty
 Journal

Mathematical Geology
Volume 33, Issue 4 , pp 397419
 Cover Date
 20010501
 DOI
 10.1023/A:1011097228254
 Print ISSN
 08828121
 Online ISSN
 15738868
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 confidence interval
 sampling variance
 uncertainty
 variance–covariance matrix
 variogram
 bias
 Industry Sectors
 Authors

 Eulogio PardoIgúzquiza ^{(1)}
 Peter Dowd ^{(1)}
 Author Affiliations

 1. Department of Mining and Mineral Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom