Quantifying uncertainty in climate change science through empirical information theory

Abstract

Quantifying the uncertainty for the present climate and the predictions of climate change in the suite of imperfect Atmosphere Ocean Science (AOS) computer models is a central issue in climate change science. Here, a systematic approach to these issues with firm mathematical underpinning is developed through empirical information theory. An information metric to quantify AOS model errors in the climate is proposed here which incorporates both coarsegrained mean model errors as well as covariance ratios in a transformation invariant fashion. The subtle behavior of model errors with this information metric is quantified in an instructive statistically exactly solvable test model with direct relevance to climate change science including the prototype behavior of tracer gases such as CO 2 . Formulas for identifying the most sensitive climate change directions using statistics of the present climate or an AOS model approximation are developed here; these formulas just involve finding the eigenvector associated with the largest eigenvalue of a quadratic form computed through suitable unperturbed climate statistics. These climate change concepts are illustrated on a statistically exactly solvable one-dimensional stochastic model with relevance for low frequency variability of the atmosphere. Viable algorithms for implementation of these concepts are discussed throughout the paper. model errors | practical algorithms | unbiased empirical estimates T he climate is an extremely complex coupled system involving significant physical processes for the atmosphere, ocean, and land over a wide range of spatial scales from millimeters to thousands of kilometers and time scales from minutes to decades or centuries (1, 2) Climate change science focuses on predicting the coarse-grained planetary scale long time changes in the climate system due to either changes in external forcing or internal variability such as the impact of increased carbon dioxide (3) For several decades the predictions of climate change science have been carried out with some skill through comprehensive computational atmospheric and oceanic simulation (AOS) models (1-4) which are designed to mimic the complex physical spatio-temporal patterns in nature. Such AOS models either through lack of resolution due to current computing power or through inadequate observation of nature necessarily parameterize the impact of many features of the climate system such as clouds, mesoscale and submesoscale ocean eddies, sea ice cover, etc. Thus, there are intrinsic model errors in the AOS models for the climate system and a central scientific issue is the effect of such model errors on predicting the coarse-grained large scale long time quantities of interest in climate change science.

Link between statistical equilibrium fidelity and forecasting skill for complex systems with model error

“…Statistical equilibrium fidelity (5,6,10) consistent with the L measurements of a coarse-grained variable~u for an imperfect model arises when…”

“…The main goal of the present paper is to provide such a direct link by utilizing fluctuation dissipation theorems (FDTs) for complex dynamical systems (11)(12)(13) together with the framework of empirical information theory for improving imperfect models developed recently (5,6). After a summary of relevant formulas of empirical information theory, the main link utilizing FDT is developed.…”

“…For long-range forecasting, it is natural to guarantee statistical equilibrium fidelity for an imperfect model, and a framework using information theory is a natural way to achieve this in an unbiased fashion (5)(6)(7)(8)10). First, equilibrium statistical fidelity for an imperfect model depends on the choice of coarse-grained variables utilized (5,6); second, equilibrium model fidelity is a necessary but not sufficient condition to guarantee long-range forecasting skill (8). For example, ref.…”

“…Accurate predictions are hampered by the fact that the true dynamics of the system in nature are actually unknown due to inadequate scientific understanding or inadequate spatiotemporal resolution in the imperfect computer models used for these predictions; in other words, there are significant model errors compared to the true signal from nature. Recently, information theory has been utilized in different ways to systematically improve model fidelity and sensitivity (5,6), to quantify the role of coarse-grained initial states in long-range forecasting (7,8), and to make an empirical link between model fidelity and forecasting skill (9,10). Imperfect models for complex systems are constrained by their capability to reproduce certain statistics in a training phase where the natural system has been observed; for example, this training phase in climate science is roughly the 60-y dataset of extensive observations of the Earth's climate system.…”

Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems

“…Moreover, from an information theory perspective 8 , the relative entropy measures loss/change of information. Relative entropy for highdimensional systems was used as measure of loss of information in coarse-graining 2,20,24 , and sensitivity analysis for climate modeling problems 28 .…”