Comparing climate time series – Part 4: Annual cycles

Abstract. This paper derives a test for deciding whether two time series come from the same stochastic model, where the time series contains periodic and serially correlated components. This test is useful for comparing dynamical model simulations to observations. The framework for deriving this test is the same as in the previous three parts: the time series are first fit to separate autoregressive models, and then the hypothesis that their parameters are equal is tested. This paper generalizes the previous tests to a limited class of nonstationary processes, namely, those represented by an autoregressive model with deterministic forcing terms. The statistic for testing differences in parameters can be decomposed into independent terms that quantify differences in noise variance, differences in autoregression parameters, and differences in forcing parameters (e.g., differences in annual cycle forcing). A hierarchical procedure for testing individual terms and quantifying the overall significance level is derived from standard methods. The test is applied to compare observations of the meridional overturning circulation from the RAPID array to Coupled Model Intercomparison Project Phase 5 (CMIP5) models. Most CMIP5 models are inconsistent with observations, with the strongest differences arising from having too little noise variance, though differences in annual cycle forcing also contribute significantly to discrepancies from observations. This appears to be the first use of a rigorous criterion to decide “equality of annual cycles” in regards to all their attributes (e.g., phases, amplitudes, frequencies) while accounting for serial correlations.