A framework for assessing and understanding sources of error in Earth System Model emulation

Abstract. Full-scale Earth system models are too computationally expensive to keep pace with the growing demand for climate projections across a large range of emissions pathways. Climate emulators, reduced-order models that reproduce the output of full-scale models, are poised to fill this niche. However, the large number of emulation techniques available and lack of a comprehensive theoretical basis to understand their relative strengths and weaknesses compromises fundamental methodological comparisons. Here, we present a theoretical framework that connects disparate emulation techniques, using it to analyze sources of emulator error focusing on memory effects, hidden variables, system noise, and nonlinearities. This framework includes popular emulation techniques such as pattern scaling and response functions, relating them to less commonly used methods, such as Dynamic Mode Decomposition and the Fluctuation Dissipation Theorem (FDT). To support our theoretical contributions, we provide practical implementation details for each technique, evaluating performance across a series of experiments designed to highlight different potential sources of error. We find that response function-based emulators outperform other techniques, particularly pattern scaling, across all scenarios tested. We additionally outline potential advantages of incorporating statistical mechanics into climate emulation through the use of the FDT, though this technique requires greater computational resources and non-standard scenarios for training. Results highlight the relative utility of each technique discussed, along with the importance of designing future scenarios for Earth system models with emulation in mind, suggesting that large-ensemble experiments utilizing the FDT could benefit climate modeling and impacts communities.