  Spatial prediction in an arbitrary location, based on a spatial set of
observations, is usually performed by Kriging, being the best linear unbiased
predictor (BLUP) in a least-square sense. In order to predict a continuous
surface over a spatial domain a grid representation is most often used. Kriging
predictions and prediction variances are computed in the nodes of a grid
covering the spatial domain, and the continuous surface is assessed from this
grid representation. A precise representation usually requires the number of
grid nodes to be considerably larger than the number of observations. For a
Gaussian random field model the Kriging predictor coinsides with the
conditional expectation of the spatial variable given the observation set. An
alternative expression for this conditional expectation provides a spatial
predictor on functional form which does not rely on a spatial grid
discretization. This functional predictor, called the Kernel predictor, is
identical to the asymptotic grid infill limit of the Kriging-based grid
representation, and the computational demand is primarily dependent on the
number of observations - not the dimension of the spatial reference domain nor
any grid discretization. We explore the potential of this Kernel predictor with
associated prediction variances. The predictor is valid for Gaussian random
fields with any eligible spatial correlation function, and large computational
savings can be obtained by using a finite-range spatial correlation function.
For studies with a huge set of observations, localized predictors must be used,
and the computational advantage relative to Kriging predictors can be very
large. Moreover, model parameter inference based on a huge observation set can
be efficiently made. The methodology is demonstrated in a couple of examples.
