Kriging Derivatives

Some properties of conditioning a Gaussian random field on point value data and derivative data
is investigated. Linear predictors with or without a linear trend are considered. Derivative data
are seen to improve predictions and the corresponding prediction errors are reduced. The predictors
are extensions to the well known simple and universal kriging predictors. The covariance
functions between different components of the derivative fields are the key component entering
the predictors. They are given by partial derivatives of the covariance function of the Gaussian
random field. Spatial symmetries such as stationarity and isotropy are used to restrict the number
of covariance functions. In particular isotropy reduces complexity and a general framework
for utilising this spatial symmetry is established. Properties of the predictors and the associated
prediction error are studied in detail for some particular isotropic covariance functions. The
importance of the smoothness of the random field on the predictor and the prediction error is
illustrated. This text was first published as part of Abrahamsen (1997).