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Environmental datasets such as those from remote-sensing platforms and sensor net-works are often spatial, temporal, and very large or even massive. Analyzing large spatial or spatio-temporal datasets can be challenging and dimension reduction is usually neces-sary. In this work, we exploit the Spatial Random Effects (SRE) model with a fixed number of known but not necessarily orthogonal (multi-resolutional) spatial basis functions. The SRE model allows a flexible family of nonstationary covariance functions and the fixed number of basis functions results in dimension reduction and thus efficient computation. We propose priors on the parameters of the SRE model in a fully Bayesian framework. These priors are based on the covariance matrix parameterized in terms of Givens angles and eigenvalues, and they recognize the multi-resolutional nature of the basis functions. We compare this Givens-angle prior to other methods in a simulation study, to show its advantages and apply it to a large remote-sensing spatial dataset. We also apply the SRE model with the Givens-angle prior in a Bayesian meta analysis, where outputs from six different regional climate outputs (RCMs) are combined to construct a consensus climat