Calculation of Uncertainty in the Variogram

Abstract

There are often limited data available in early stages of geostatistical modeling. This leads to considerable uncertainty in statistical parameters including the variogram. This article presents an approach to calculate the uncertainty in the variogram. A methodology to transfer this uncertainty through geostatistical simulation and decision making is also presented.

The experimental variogram value $2\widehat\gamma \left( {\text{h}} \right)$ for a separation lag vector h is a mean of squared differences. The variance of a mean can be calculated with a model of the correlation between the pairs of data used in the calculation. The “data” here are squared differences; therefore, we need a measure of a 4-point correlation. A theoretical multi-Gaussian approach is presented for this uncertainty assessment together with a number of examples. The theoretical results are validated by numerical simulation. The simulation approach permits generalization to non-Gaussian situations.

Multiple plausible variograms may be fit knowing the uncertainty at each variogram point, $2\gamma \left( {\text{h}} \right)$. Multiple geostatistical realizations may then be constructed and subjected to process assessment to measure the impact of this uncertainty.