Comparison between spatio-temporal random processes and application to climate model data

Comparing two spatio-temporal processes are often a desirable exercise. For example, assessments of the difference between various climate models may involve the comparisons of the synthetic climate random fields generated as simulations from each model. We develop rigorous methods to compare two spatio-temporal random processes both in terms of moments and in terms of temporal trend, using the functional data analysis approach. A highlight of our method is that we can compare the trend surfaces between two random processes, which are motivated by evaluating the skill of synthetic climate from climate models in terms of capturing the pronounced upward trend of real-observational data. We perform simulations to evaluate our methods and then apply the methods to compare different climate models as well as to evaluate the synthetic temperature fields from model simulations, with respect to observed temperature fields.