A new hierarchical Bayesian framework (HBM) is proposed for identification of Gaussian processes or fields, which are usually used for simulating uncertainty in temporal variability of loads or spatial variability of material properties. An improved orthogonal series expansion (iOSE) is embedded into the proposed framework by simulating the Gaussian process or field through correlated Gaussian variables, and then HBM is applied to quantify their uncertainty. Hyper parameters to be identified are set to be the mean value and standard deviation vectors of these Gaussian variables, as well as the parameters in autocorrelation function (ACF) of the Gaussian process or field which are used to replace correlation coefficients of correlated Gaussian variables for reducing the number of hyper parameters. With the identified hyper parameters, a simulation model of the Gaussian process or field can be obtained based on the iOSE expression. In addition, model class selection is introduced to select the optimal number of orthogonal functions and integral points involved in iOSE as well as select the category of ACF among several alternative models, known to influence the simulated expression and accuracy. Studies conducted on two dynamic examples verify the effectiveness of proposed framework.