Ghil, Michael. 2015. “A mathematical theory of climate sensitivity or, How to deal with both anthropogenic forcing and natural variability?” Climate Change: Multidecadal and Beyond, edited by C. P. Chang, Michael Ghil, Mojib Latif, and J. M. Wallace, 31–51. World Scientific Publ. Co./Imperial College Press. Abstract

Recent estimates of climate evolution over the coming century still differ by several degrees. This uncertainty motivates the work presented here. There are two basic approaches to apprehend the complexity of climate change: deterministically nonlinear and stochastically linear, i.e., the Lorenz and the Hasselmann approach. The grand unification of these two approaches relies on the theory of random dynamical systems. We apply this theory to study the random attractors of nonlinear, stochastically perturbed climate models. Doing so allows one to examine the interaction of internal climate variability with the forcing, whether natural or anthropogenic, and to take into account the climate system's non-equilibrium behavior in determining climate sensitivity. This non-equilibrium behavior is due to a combination of nonlinear and random effects. We give here a unified treatment of such effects from the point of view of the theory of dynamical systems and of their bifurcations. Energy balance models are used to illustrate multiple equilibria, while multi-decadal oscillations in the thermohaline circulation illustrate the transition from steady states to periodic behavior. Random effects are introduced in the setting of random dynamical systems, which permit a unified treatment of both nonlinearity and stochasticity. The combined treatment of nonlinear and random effects is applied to a stochastically perturbed version of the classical Lorenz convection model. Climate sensitivity is then defined mathematically as the derivative of an appropriate functional or other function of the system’s state with respect to the bifurcation parameter. This definition is illustrated by using numerical results for a model of the El Niño–Southern Oscillation. The concept of a hierarchy of models is the thread that runs across this chapter, and the robustness of elementary bifurcations across such a hierarchy is emphasized.

Groth, Andreas, and Michael Ghil. 2015. “Monte Carlo Singular Spectrum Analysis (SSA) revisited: Detecting oscillator clusters in multivariate datasets.” Journal of Climate 28 (19): 7873–7893. Abstract

Singular spectrum analysis (SSA) along with its multivariate extension (M-SSA) provides an efficient way to identify weak oscillatory behavior in high-dimensional data. To prevent the misinterpretation of stochastic fluctuations in short time series as oscillations, Monte Carlo (MC)–type hypothesis tests provide objective criteria for the statistical significance of the oscillatory behavior. Procrustes target rotation is introduced here as a key method for refining previously available MC tests. The proposed modification helps reduce the risk of type-I errors, and it is shown to improve the test’s discriminating power. The reliability of the proposed methodology is examined in an idealized setting for a cluster of harmonic oscillators immersed in red noise. Furthermore, the common method of data compression into a few leading principal components, prior to M-SSA, is reexamined, and its possibly negative effects are discussed. Finally, the generalized Procrustes test is applied to the analysis of interannual variability in the North Atlantic’s sea surface temperature and sea level pressure fields. The results of this analysis provide further evidence for shared mechanisms of variability between the Gulf Stream and the North Atlantic Oscillation in the interannual frequency band.

Groth, Andreas, Michael Ghil, Stéphane Hallegatte, and Patrice Dumas. 2015. “The Role of Oscillatory Modes in U.S. Business Cycles.” OECD Journal: Journal of Business Cycle Measurement and Analysis, no. 2015/1: 63–81. Abstract

We apply multivariate singular spectrum analysis to the study of U.S. business cycle dynamics. This method provides a robust way to identify and reconstruct oscillations, whether intermittent or modulated. We show such oscillations to be associated with comovements across the entire economy. The problem of spurious cycles generated by the use of detrending filters is addressed and we present a Monte Carlo test to extract significant oscillations. The behavior of the U.S. economy is shown to change significantly from one phase of the business cycle to another: the recession phase is dominated by a five-year mode, while the expansion phase exhibits more complex dynamics, with higher-frequency modes coming into play. We show that the variations so identified cannot be generated by random shocks alone, as assumed in ‘real’ business-cycle models, and that endogenous, deterministically generated variability has to be involved.

Kondrashov, Dmitri, and Pavel S. Berloff. 2015. “Stochastic modeling of decadal variability in ocean gyres.” Geophysical Research Letters 42: 1543–1553.
Chekroun, Mickaël D., Honghu Liu, and S. Wang. 2015. Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II. New York: Springer Briefs in Mathematics, Springer. Publisher's Version Abstract

In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.

Ghil, Michael, and I. Zaliapin. 2015. “Understanding ENSO variability and its extrema: A delay differential equation approach.” Extreme Events: Observations, Modeling and Economics, edited by M. Chavez, Michael Ghil, and J. Urrutia-Fucugauchi, 63–78. American Geophysical Union & Wiley. Abstract

The El-Nino/Southern-Oscillation (ENSO) phenomenon is the most prominent signal of seasonal-to-interannual climate variability. The past 30 years of research have shown that ENSO dynamics is governed, by and large, by the interplay of the nonlinear mechanisms, and that their simplest version can be studied in autonomous or forced delay differential equation (DDE) models. This chapter briefly reviews the results of Ghil et al., Zaliapin and Ghil, and Ghil and Zaliapin and pursues their DDE model analysis by focusing on multiple model solutions for the same parameter values and the dynamics of local extrema. It first introduces the DDE model of ENSO variability, reviews the main theoretical results concerning its solutions, and comments on the appropriate numerical integration methods. Novel results on multiple solutions and their extrema are reported and illustrated. After discussing the model's pullback attractor, the chapter explores parameter dependence in the model over its entire 3D parameter space.

Podladchikova, T. V., Y. Y. Shprits, Dmitri Kondrashov, and A. C. Kellerman. 2014. “Noise statistics identification for Kalman filtering of the electron radiation belt observations I: Model errors.” Journal of Geophysical Research: Space Physics 119 (7): 5700–5724. Publisher's Version
Podladchikova, T. V., Y. Y. Shprits, A. C. Kellerman, and Dmitri Kondrashov. 2014. “Noise statistics identification for Kalman filtering of the electron radiation belt observations: 2. Filtration and smoothing.” Journal of Geophysical Research: Space Physics 119 (7): 5725–5743. Publisher's Version
Kondrashov, Dmitri, R. Denton, Y. Y. Shprits, and H. J. Singer. 2014. “Reconstruction of gaps in the past history of solar wind parameters.” Geophysical Research Letters 41 (8): 2702–2707. Publisher's Version
Kellerman, A. C., Y. Y. Shprits, Dmitri Kondrashov, D. Subbotin, R. A. Makarevich, E. Donovan, and T. Nagai. 2014. “Three-dimensional data assimilation and reanalysis of radiation belt electrons: Observations of a four-zone structure using five spacecraft and the VERB code.” Journal of Geophysical Research: Space Physics 119 (11): 8764–8783. Publisher's Version
Roques, Lionel, Mickaël D. Chekroun, Michel Cristofol, Samuel Soubeyrand, and Michael Ghil. 2014. “Parameter estimation for energy balance models with memory.” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 470 (2169). The Royal Society. Publisher's Version Abstract
We study parameter estimation for one-dimensional energy balance models with memory (EBMMs) given localized and noisy temperature measurements. Our results apply to a wide range of nonlinear, parabolic partial differential equations with integral memory terms. First, we show that a space-dependent parameter can be determined uniquely everywhere in the PDE’s domain of definition D, using only temperature information in a small subdomain E⊂D. This result is valid only when the data correspond to exact measurements of the temperature. We propose a method for estimating a model parameter of the EBMM using more realistic, error-contaminated temperature data derived, for example, from ice cores or marine-sediment cores. Our approach is based on a so-called mechanistic-statistical model that combines a deterministic EBMM with a statistical model of the observation process. Estimating a parameter in this setting is especially challenging, because the observation process induces a strong loss of information. Aside from the noise contained in past temperature measurements, an additional error is induced by the age-dating method, whose accuracy tends to decrease with a sample’s remoteness in time. Using a Bayesian approach, we show that obtaining an accurate parameter estimate is still possible in certain cases.
Groth, Andreas. 2014. “Interannual variability in the North Atlantic SST and wind forcing.” Seminar at International Research Institute for Climate and Society, Columbia. Abstract

Groth, Andreas. 2014. “Oscillatory behavior and oscillatory modes.” SSA workshop Bournemouth, September 2014. Abstract

Chekroun, Mickaël D., J. David Neelin, Dmitri Kondrashov, James C. McWilliams, and Michael Ghil. 2014. “Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances.” Proceedings of the National Academy of Sciences 111 (5): 1684-1690. Abstract

Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them—as filtered through an observable of the system—is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap—defined as the distance between the subdominant RP resonance and the unit circle—plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño–Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.

Shprits, Yuri, Adam Kellerman, Dmitri Kondrashov, and Dmitriy Subbotin. 2013. “Application of a new data operator-splitting data assimilation technique to the 3-D VERB diffusion code and CRRES measurements.” Geophysical Research Letters 40 (19): 4998–5002. Publisher's Version
Ghil, Michael. 2013. “Lecture 1: Data Assimilation: How We Got Here and Where To Next?” Workshop on Mathematics of Climate Change, Related Hazards and Risks, CIMAT, Guanajuato, Mexico. Abstract

Lecture 1: Data Assimilation: How We Got Here and Where To Next?
Ghil, Michael. 2013. “Lecture 2: Toward a Mathematical Theory of Climate Sensitivity.” Workshop on Mathematics of Climate Change, Related Hazards and Risks, CIMAT, Guanajuato, Mexico. Abstract

Lecture 2: Toward a Mathematical Theory of Climate Sensitivity
Ghil, Michael. 2013. “Lecture 3 : The Coupled Dynamics of Climate and Economics.” Workshop on Mathematics of Climate Change, Related Hazards and Risks, CIMAT, Guanajuato, Mexico. Abstract

Lecture 3 : The Coupled Dynamics of Climate and Economics
Sella, Lisa, Gianna Vivaldo, Andreas Groth, and Michael Ghil. 2013. “Economic Cycles and their Synchronization: A spectral survey.” Fondazione Eni Enrico Mattei (FEEM) 105 (105). Fondazione Eni Enrico Mattei (FEEM): 1. Publisher's Version Abstract

The present work applies several advanced spectral methods to the analysis of macroeconomic fluctuations in three countries of the European Union: Italy, The Netherlands, and the United Kingdom. We focus here in particular on singular-spectrum analysis (SSA), which provides valuable spatial and frequency information of multivariate data and that goes far beyond a pure analysis in the time domain. The spectral methods discussed here are well established in the geosciences and life sciences, but not yet widespread in quantitative economics. In particular, they enable one to identify and describe nonlinear trends and dominant cycles –- including seasonal and interannual components –- that characterize the deterministic behavior of each time series. These tools have already proven their robustness in the application on short and noisy data, and we demonstrate their usefulness in the analysis of the macroeconomic indicators of these three countries. We explore several fundamental indicators of the countries' real aggregate economy in a univariate, as well as a multivariate setting. Starting with individual single-channel analysis, we are able to identify similar spectral components among the analyzed indicators. Next, we consider combinations of indicators and countries, in order to take different effects of comovements into account. Since business cycles are cross-national phenomena, which show common characteristics across countries, our aim is to uncover hidden global behavior across the European economies. Results are compared with previous findings on the U.S. indicators \citepGroth.ea.FEEM.2012. Finally, the analysis is extended to include several indicators from the U.S. economy, in order to examine its influence on the European market.

de Viron, O., J. O. Dickey, and Michael Ghil. 2013. “Global modes of climate variability.” Geophysical Research Letters 40 (9): 1832-1837. Abstract

The atmosphere, hydrosphere and cryosphere form a fully coupled climate system. This system exhibits a number of large-scale phenomena, such as the El Nino Southern Oscillation (ENSO), the Asian Monsoon, the North Atlantic Oscillation (NAO), and the Madden-Julian Oscillation (MJO). While these modes of variability are not exactly periodic, they are oscillatory in character, and their state is monitored using so-called climate indices. Each of these scalar indices is a combination of several climate variables. Here, we use a comprehensive set of 25 climate indices for time intervals that range between 1948 and 2011, and estimate an optimal set of lags between these indices to maximize their correlation. We show that most of the index pairs drawn from this set present a significant correlation on interannual time scales. It is also shown that, on average, about two-thirds of the total variability in each index can be described by using only the four leading principal components of the entire set of lagged indices. Our index set's leading orthogonal modes exhibit several interannual frequencies and capture separately variability associated with the North Atlantic and the North Pacific. These modes are associated, in turn, with large-scale variations of sea surface temperatures.