This paper considers the modelling of scalar fields exhibiting non-stationary noise in the context of Gaussian Process (GP) regression. We show how a Heteroscedastic GP produces more accurate predictions of the variance of a process of this type compared to the standard Homoscedastic model. We present a parametric model for the noise process and derive analytical solutions to the Log Marginal Likelihood of the data and its gradients with respect to Hyper Parameters of the kernel and the noise process. We compare our parametric model to one which estimates a full GP for the noise and show analogous predictive performance with a model which has greater computational efficiency and is less complex to implement. We also discuss active sample selection in this framework and show through the numerical simulation of an arrested bathymetric front in an estuary, the superiority of using Mutual Information to Fisher Information, Entropy or Random sampling in terms of errors in the first two moments of the predicted distributions.