Boris Gershgorin

Quantifying uncertainty in climate change science through empirical information theory

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 coarse-grained 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 CO2. 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.

Low-Frequency Climate Response and Fluctuation–Dissipation Theorems: Theory and Practice

Abstract The low-frequency response to changes in external forcing or other parameters for various components of the climate system is a central problem of contemporary climate change science. The fluctuation–dissipation theorem (FDT) is an attractive way to assess climate change by utilizing statistics of the present climate; with systematic approximations, it has been shown recently to have high skill for suitable regimes of an atmospheric general circulation model (GCM). Further applications of FDT to low-frequency climate response require improved approximations for FDT on a reduced subspace of resolved variables. Here, systematic mathematical principles are utilized to develop new FDT approximations on reduced subspaces and to assess the small yet significant departures from Gaussianity in low-frequency variables on the FDT response. Simplified test models mimicking crucial features in GCMs are utilized here to elucidate these issues and various FDT approximations in an unambiguous fashion. Also, the...

Improving model fidelity and sensitivity for complex systems through empirical information theory

In many situations in contemporary science and engineering, the analysis and prediction of crucial phenomena occur often through complex dynamical equations that have significant model errors compared with the true signal in nature. Here, a systematic information theoretic framework is developed to improve model fidelity and sensitivity for complex systems including perturbation formulas and multimodel ensembles that can be utilized to improve both aspects of model error simultaneously. A suite of unambiguous test models is utilized to demonstrate facets of the proposed framework. These results include simple examples of imperfect models with perfect equilibrium statistical fidelity where there are intrinsic natural barriers to improving imperfect model sensitivity. Linear stochastic models with multiple spatiotemporal scales are utilized to demonstrate this information theoretic approach to equilibrium sensitivity, the role of increasing spatial resolution in the information metric for model error, and the ability of imperfect models to capture the true sensitivity. Finally, an instructive statistically nonlinear model with many degrees of freedom, mimicking the observed non-Gaussian statistical behavior of tracers in the atmosphere, with corresponding imperfect eddy-diffusivity parameterization models are utilized here. They demonstrate the important role of additional stochastic forcing of imperfect models in order to systematically improve the information theoretic measures of fidelity and sensitivity developed here.

Test models for improving filtering with model errors through stochastic parameter estimation

The filtering skill for turbulent signals from nature is often limited by model errors created by utilizing an imperfect model for filtering. Updating the parameters in the imperfect model through stochastic parameter estimation is one way to increase filtering skill and model performance. Here a suite of stringent test models for filtering with stochastic parameter estimation is developed based on the Stochastic Parameterization Extended Kalman Filter (SPEKF). These new SPEKF-algorithms systematically correct both multiplicative and additive biases and involve exact formulas for propagating the mean and covariance including the parameters in the test model. A comprehensive study is presented of robust parameter regimes for increasing filtering skill through stochastic parameter estimation for turbulent signals as the observation time and observation noise are varied and even when the forcing is incorrectly specified. The results here provide useful guidelines for filtering turbulent signals in more complex systems with significant model errors.

Improving filtering and prediction of spatially extended turbulent systems with model errors through stochastic parameter estimation

The filtering and predictive skill for turbulent signals is often limited by the lack of information about the true dynamics of the system and by our inability to resolve the assumed dynamics with sufficiently high resolution using the current computing power. The standard approach is to use a simple yet rich family of constant parameters to account for model errors through parameterization. This approach can have significant skill by fitting the parameters to some statistical feature of the true signal; however in the context of real-time prediction, such a strategy performs poorly when intermittent transitions to instability occur. Alternatively, we need a set of dynamic parameters. One strategy for estimating parameters on the fly is a stochastic parameter estimation through partial observations of the true signal. In this paper, we extend our newly developed stochastic parameter estimation strategy, the Stochastic Parameterization Extended Kalman Filter (SPEKF), to filtering sparsely observed spatially extended turbulent systems which exhibit abrupt stability transition from time to time despite a stable average behavior. For our primary numerical example, we consider a turbulent system of externally forced barotropic Rossby waves with instability introduced through intermittent negative damping. We find high filtering skill of SPEKF applied to this toy model even in the case of very sparse observations (with only 15 out of the 105 grid points observed) and with unspecified external forcing and damping. Additive and multiplicative bias corrections are used to learn the unknown features of the true dynamics from observations. We also present a comprehensive study of predictive skill in the one-mode context including the robustness toward variation of stochastic parameters, imperfect initial conditions and finite ensemble effect. Furthermore, the proposed stochastic parameter estimation scheme applied to the same spatially extended Rossby wave system demonstrates high predictive skill, comparable with the skill of the perfect model for a duration of many eddy turnover times especially in the unstable regime.

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

Understanding and improving the predictive skill of imperfect models for complex systems in their response to external forcing is a crucial issue in diverse applications such as for example climate change science. Equilibrium statistical fidelity of the imperfect model on suitable coarse-grained variables is a necessary but not sufficient condition for this predictive skill, and elementary examples are given here demonstrating this. Here, with equilibrium statistical fidelity of the imperfect model, a direct link is developed between the predictive fidelity of specific test problems in the training phase where the perfect natural system is observed and the predictive skill for the forced response of the imperfect model by combining appropriate concepts from information theory with other concepts based on the fluctuation dissipation theorem. Here a suite of mathematically tractable models with nontrivial eddy diffusivity, variance, and intermittent non-Gaussian statistics mimicking crucial features of atmospheric tracers together with stochastically forced standard eddy diffusivity approximation with model error are utilized to illustrate this link.

Filtering skill for turbulent signals for a suite of nonlinear and linear extended Kalman filters

The filtering skill for turbulent signals from nature is often limited by errors due to utilizing an imperfect forecast model. In particular, real-time filtering and prediction when very limited or no a posteriori analysis is possible (e.g. spread of pollutants, storm surges, tsunami detection, etc.) introduces a number of additional challenges to the problem. Here, a suite of filters implementing stochastic parameter estimation for mitigating model error through additive and multiplicative bias correction is examined on a nonlinear, exactly solvable, stochastic test model mimicking turbulent signals in regimes ranging from configurations with strongly intermittent, transient instabilities associated with positive finite-time Lyapunov exponents to laminar behavior. Stochastic Parameterization Extended Kalman Filter (SPEKF), used as a benchmark here, involves exact formulas for propagating the mean and covariance of the augmented forecast model including the unresolved parameters. The remaining filters use the same nonlinear forecast model but they introduce model error through different moment closure approximations and/or linear tangent approximation used for computing the second-order statistics of the augmented stochastic forecast model. A comprehensive study of filter performance is carried out in the presence of various moment closure errors which are enhanced by additional model errors due to incorrect parameters inducing additive and multiplicative stochastic biases. The estimation skill of the unresolved stochastic parameters is also discussed and it is shown that the linear tangent filter, despite its popularity, is completely unreliable in many turbulent regimes for both parameter estimation and filtering; moreover, regimes of filter divergence for the linear tangent filter are identified. The results presented here provide useful guidelines for filtering turbulent, high-dimensional, spatially extended systems with more general model errors, as well as for designing more skillful methods for superparameterization of unresolved intermittent processes in complex multi-scale models. They also provide unambiguous benchmarks for the capabilities of linear and nonlinear extended Kalman filters using incorrect statistics on an exactly solvable test bed with rich and realistic dynamics.

High skill in low-frequency climate response through fluctuation dissipation theorems despite structural instability

Climate change science focuses on predicting the coarse-grained, planetary-scale, longtime changes in the climate system due to either changes in external forcing or internal variability, such as the impact of increased carbon dioxide. The predictions of climate change science are carried out through comprehensive, computational atmospheric, and oceanic simulation models, which necessarily parameterize physical features such as clouds, sea ice cover, etc. Recently, it has been suggested that there is irreducible imprecision in such climate models that manifests itself as structural instability in climate statistics and which can significantly hamper the skill of computer models for climate change. A systematic approach to deal with this irreducible imprecision is advocated through algorithms based on the Fluctuation Dissipation Theorem (FDT). There are important practical and computational advantages for climate change science when a skillful FDT algorithm is established. The FDT response operator can be utilized directly for multiple climate change scenarios, multiple changes in forcing, and other parameters, such as damping and inverse modelling directly without the need of running the complex climate model in each individual case. The high skill of FDT in predicting climate change, despite structural instability, is developed in an unambiguous fashion using mathematical theory as guidelines in three different test models: a generic class of analytical models mimicking the dynamical core of the computer climate models, reduced stochastic models for low-frequency variability, and models with a significant new type of irreducible imprecision involving many fast, unstable modes.

Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum and intermittency.

This paper motivates, develops and reviews elementary models for turbulent tracers with a background mean gradient which, despite their simplicity, have complex statistical features mimicking crucial aspects of laboratory experiments and atmospheric observations. These statistical features include exact formulas for tracer eddy diffusivity which is non-local in space and time, exact formulas and simple numerics for the tracer variance spectrum in a statistical steady state, and the transition to intermittent scalar probability density functions with fat exponential tails as certain variances of the advecting mean velocity are increased while satisfying important physical constraints. The recent use of such simple models with complex statistics as unambiguous test models for central contemporary issues in both climate change science and the real-time filtering of turbulent tracers from sparse noisy observations is highlighted throughout the paper.

Renormalized waves and discrete breathers in Beta-Fermi-Pasta-Ulam chains.

We demonstrate via numerical simulation that in the strongly nonlinear limit the Beta-Fermi-Pasta-Ulam (Beta-FPU) system in thermal equilibrium behaves surprisingly like weakly nonlinear waves in properly renormalized normal variables. This arises because the collective effect of strongly nonlinear interactions effectively renormalizes linear dispersion frequency and leads to effectively weak interaction among these renormalized waves. Furthermore, we show that the dynamical scenario for thermalized Beta-FPU chains is spatially highly localized discrete breathers riding chaotically on spatially extended, renormalized waves.