VARBOOT: A spatial bootstrap program for semivariogram uncertainty assessment

In applied geostatistics, the semivariogram is commonly estimated from experimental data, producing an empirical semivariogram for a specified number of discrete lags. In a second stage, a model defined by a few parameters is fitted to the empirical semivariogram. As the experimental data are usually few and sparsely located, there is considerable uncertainty about the calculated semivariogram values (uncertainty of the empirical semivariogram) and about the parameters of any model fitted to them (uncertainty of the estimated model parameters). In this paper, the uncertainty in the modeling of the empirical semivariogram is numerically assessed by the generalized bootstrap, which is an extension of the classic bootstrap procedure modified for spatially correlated data. A computer program is described and provided for the assessment of those uncertainties. In particular, the program provides for the empirical semivariogram: the standard errors, the bootstrap percentile confidence intervals, the complete variance-covariance matrix, standard deviation correlation matrix. A public domain, natural dataset is used to illustrate the performance of the program. A promising result is that, for any distance, the median of the bootstrap distribution for the empirical semivariogram approximates more closely the underlying semivariogram than the estimate derived from the empirical sample.