Assessment of Experimental Semivariogram Uncertainty in the Presence of a Polynomial Drift

Abstract

The semivariogram, which measures the spatial variability between experimental data, is generally used as a structural input in all two-point geostatistical procedures. However, in most geoscience applications, experimental semivariograms are usually computed from a limited number of sparsely spaced measurements, which results in uncertainty associated with the semivariance values estimated for a specified number of lags. More importantly, considering a spatial variable modelled by a nonstationary random field, uncertainty is not only in the experimental semivariogram of the residuals, but also in the coefficients of the drift model estimated from the available experimental data. Therefore, when assessing the reliability of an experimental semivariogram (or estimated semivariances) in the nonstationary case, both aforementioned uncertainties should be taken into account. The aim of this paper is to extend the “Generalised Bootstrap” procedure to the nonstationary model by propagating the uncertainty associated with the estimated drift coefficients into the uncertainty in the experimental semivariogram of the residuals. The proposed methodology is demonstrated in a case study using abundant geophysical measurements characterised by a nonstationary random function. Two scenarios are evaluated in the case study: (1) it is assumed that the drift coefficients can be estimated without any uncertainty, and (2) uncertainty of the drift coefficients is taken into account. We have explained the methodology that allows to assess the uncertainty of the semivariogram lag estimates in the presence of the drift in the mean. Considering the second scenario, uncertainty is obviously larger than the case where uncertainty of the drift in the mean is ignored. This evaluation should be considered in applications where the data is often rather limited, such as subsurface hydrology (i.e. porosity, transmissivity), soil science (i.e. heavy metal content, soil moisture) and mining (i.e. scoping or pre-feasibility stage of the project). In fact, modern geostatistics should provide not only the semivariogram estimates but also estimation of its uncertainty.