The Matérn variogram model: Implications for uncertainty propagation and sampling in geostatistical surveys

Abstract The Matern variogram model has been advocated because it is flexible and can represent varied behaviour at small lags. We show how the constraints on the spherical and exponential variogram at short lags ignore a possible source of uncertainty in the variogram and so in kriging surveys, that the Matern model can describe. Matern, spherical and exponential variogram models were fitted by maximum likelihood to a set of log 10 (K) observations made on a regular grid at Broom's Barn Farm, Suffolk, England. The likelihood profiles of the Matern parameter estimates were asymmetric. Thus the uncertainty of these estimates could only be adequately assessed by a Bayesian approach. The uncertainty of estimated parameters of the Matern variogram was larger than for the exponential variogram. This is an indication that the assumption of an exponential model limits the behaviour that may be described by the variogram. Thus uncertainty analyses where an exponential variogram is assumed may underestimate the uncertainty of kriged estimates. Bayesian analysis of the kriged estimates of log 10 (K) at Broom's Barn Farm using the Matern variogram revealed an observable component of uncertainty due to variogram uncertainty. When an exponential variogram model was used, the estimate of this component of uncertainty was negligible. The Matern variogram should therefore be used rather than the exponential model when assessing the adequacy of a variogram estimate. A method of designing sample schemes which is suitable for both estimating a Matern variogram and interpolation is suggested.