The global climate system is dominated by large-scale spatial patterns of atmospheric and oceanic variability that are fundamentally important for regional climate. These patterns are often defined in terms of Empirical Orthogonal Function (EOF) analysis. Two limitations of standard EOF analysis are: 1. The assumption of a stationary process over a long period of time so that the EOFs may be determined; 2. The absence of associated measures of uncertainty or variability. We build a parametric model and develop the necessary computational strategies to address these limitations, using the North Atlantic Oscillation (NAO) as a case study. A parametric representation of the covariance matrix is built to reflect the expected correlation pattern of measurements on grid points in the spatial domain. We adopt a Bayesian approach to estimate the parameters of the resulting model. Prior information is combined with a Gaussian likelihood to obtain a posterior distribution, which provides both estimates for the parameters and associated measures of uncertainty. Posterior computation is handled via Markov chain Monte Carlo. A common approach to the study of non-stationarity of the pattern is to divide the time period into short windows and estimate the pattern in each window. However the information for each time window may not be sufficient to support accurate estimates. The parametric model provides a natural way to explore non-stationary behavior. When the pattern is believed to evolve smoothly over time, it is plausible to develop a state-space model where measurements (e.g., of mean sea level pressure) in each time window are linked to the underlying parameters of the parametric model, and the parameters at successive time windows are linked through system equations. We explore these methods and illustrate with both simulations and real data.