This article is concerned with predicting for Gaussian random fields i
n a way that appropriately deals with uncertainty in the covariance fu
nction. To this end, we analyze the best linear unbiased prediction pr
ocedure within a Bayesian framework. Particular attention is paid to t
he treatment of parameters in the covariance structure and their effec
t on the quality, both real and perceived, of the prediction. These id
eas are implemented using topographical data from Davis.