Estimation of gradients from sparse data by universal kriging

Abstract

The determination of a gradient, or directional derivative, of a spatial variable is a common problem in many Earth science applications. For hydraulic heads, for example, the gradient defines the direction of the groundwater flow and is used in solving groundwater flow and transport equations. Kriging provides a methodology for estimating gradients directly from experimental data without the need to estimate the variable at the nodes of a regular grid (even when the data are widely and irregularly scattered) and without the need to use finite difference approximations. Kriging is more efficient than other methods of gradient estimation because it uses a probabilistic approach that gives an estimator that is unbiased and with minimum estimation variance. In addition, it provides a standard error of estimation, which is not available from deterministic methods. We extend the kriging formulation of gradient estimation to the universal kriging formulation to account for the frequently observed drift in spatial variables; this is almost always present at the regional scale, for example, for hydraulic head. We present an alternative approach to kriging the gradient that is simpler than previous formulations. This new formulation uses linear systems theory, which facilitates the inference of the required covariances as functions of the covariance of the primary variable whose gradient is to be estimated. An example of estimating the gradient of hydraulic head in an aquifer in southern Spain is presented to illustrate the methodology.