Gaussian Processes are a very convenient approach for representing functional uncertainty such as space-time fields, as would appear, for example in the forecast of spatial distribution of atmospheric temperature. In this talk, we discuss the computational challenges of operating with Gaussian Process models of uncertainty, with an emphasis on space-time uncertainty.  In particular, we discuss a new scalable approach for Gaussian process calibration. The approach is based on a recently introduced unbiased stochastic approximation to the score equations for maximum likelihood calculation for Gaussian processes. Under certain conditions, including bounded condition number of the covariance matrix, the approach achieves O(n) storage and nearly O(n) computational effort per optimization step, where n is the number of data sites. As a comparison, the standard approach of using Cholesky factorization for the computation of the log-det term would require O(n^2) storage and O(n^3) computational effort per optimization step. We present several numerical results that validate our findings.