Keppenne, Christian L., and Michael Ghil. “Adaptive filtering and prediction of noisy multivariate signals: an application to subannual variability in atmospheric angular momentum.” International Journal of Bifurcation and Chaos 3 (1993): 625–634. Abstract

Principal component analysis (PCA) in the space and time domains is applied to filter adaptively the dominant modes of subannual (SA) variability of a 12-year long multivariate time series of Northern Hemisphere atmospheric angular momentum (AAM); AAM is computed in 23 latitude bands of equal area from operational analyses of the U.S. National Meteorological Center. PCA isolates the leading empirical orthogonal functions (EOFs) of spatial dependence, while multivariate singular spectrum analysis (M-SSA) yields filtered time series that capture the dominant low-frequency modes of SA variability. The time series prefiltered by M-SSA lend themselves to prediction by the maximum entropy method (MEM). Whole-field predictions are made by combining the forecasts so obtained with the leading spatial EOFs obtained by PCA. The combination of M-SSA and MEM has predictive ability up to about a month. These methods are essentially linear but data-adaptive. They seem to perform well for short, noisy, multivariate time series, to which purely nonlinear, deterministically based methods are difficult to apply.

Keppenne, Christian L., and Michael Ghil. “Adaptive filtering and prediction of the Southern Oscillation index.” Journal of Geophysical Research: Atmospheres 97, no. D18 (1992): 20449–20454.
Vautard, Robert, Pascal Yiou, and Michael Ghil. “Singular-spectrum analysis: A toolkit for short, noisy chaotic signals.” Physica D 58, no. 1–4 (1992): 95–126. Abstract
Singular-spectrum analysis (SSA) is developed further, based on experience with applications to geophysical time series. It is shown that SSA provides a crude but robust approximation of strange attractors by tori, in the presence of noise. The method works well for short, noisy time series. The lagged-covariance matrix of the processes studied is the basis of SSA. We select subsets of eigenelements and associated principal components (PCs) in order to provide (i) a noise-reduction algorithm, (ii) a detrending algorithm, and (iii) an algorithm for the identification of oscillatory components. Reconstructed components (RCs) are developed to provide optimal reconstruction of a dynamic process at precise epochs, rather than averaged over the window length of the analysis. SSA is combined with advanced spectral-analysis methods - the maximum entropy method (MEM) and the multi-taper method (MTM) - to refine the interpretation of oscillatory behavior. A combined SSA-MEM method is also used for the prediction of selected subsets of RCs. The entire toolkit is validated against a set of four prescribed time series generated by known processes, quasi-periodic or chaotic. It is also applied to a time series of global surface air temperatures, 130 years long, which has attracted considerable attention in the context of the global warming issue and provides a severe test for noise reduction and prediction.
Penland, Cecile, Michael Ghil, and Klaus M. Weickmann. “Adaptive filtering and maximum entropy spectra with application to changes in atmospheric angular momentum.” Journal of geophysical research 96, no. D12 (1991): 22659–22671. Abstract
The spectral resolution and statistical significance of a harmonic analysis obtained by low-order maximum entropy methods (MEM) can be improved by subjecting the data to an adaptive filter. This adaptive filter consists of projecting the data onto the leading temporal empirical orthogonal functions obtained from singular spectrum analysis (SSA). The combined SSA-MEM method is applied both to a synthetic time series and a time series of atmospheric angular momentum (AAM) data. The procedure is very effective when the background noise is white and less so when the background noise is red. The latter case obtains in the AAM data. Nevertheless, we detect reliable evidence for intraseasonal and interannual oscillations in AAM. The interannual periods include a quasi-biennial one and low-frequency one of 5 years, both related to the El Niño/Southern Oscillation. In the intraseaonal band, separate oscillations of about 48..5 and 51 days are ascertained.
Ghil, Michael, and Kingtse Mo. “Intraseasonal Oscillations in the Global Atmosphere. Part I: Northern Hemisphere and Tropics.” Journal of the Atmospheric Sciences 48, no. 5 (1991): 752–779. Abstract
We have examined systematically oscillatory modes in the Northern Hemisphere and in the tropics. The 700 mb heights were used to analyze extratropical oscillations, and the outgoing longwave radiation to study tropical oscillations in convection. All datasets were band-pass filtered to focus on the intraseasonal (IS) band of 10-120 days. Leading spatial patterns of variability were obtained by applying EOF analysis to these IS data. The leading principal components (PCs) were subjected to singular spectrum analysis (SSA). SSA is a statistical technique related to EOF analysis, but in the time domain, rather than the spatial domain. It helps identify nonlinear oscillations in short and noisy time series.In the Northern Hemisphere, there are two important modes of oscillation with periods near 48 and 23 days, respectively. The 48-day mode is the most important of the two. It has both traveling and standing components, and is dominated by a zonal wavenumber two. The 23-day mode has the spatial structure and propagation properties described by Branstator and by Kushnir.In the tropics, the 40-50 day oscillation documented by Madden and Julian, Weickmann, Lau, their colleagues, and many other authors dominates the Indian and Pacific oceans from 60°E to the date line. From 170°W to 90°W, however, a 24-28 day oscillation is equally strong. The extratropical modes are often independent of, and sometimes lead, the tropical modes.
Ghil, Michael, and Kingtse Mo. “Intraseasonal Oscillations in the Global Atmosphere. Part II: Southern Hemisphere.Journal of the Atmospheric Sciences 48 (1991): 780–792. Abstract
In Part II of this two-part article, we complete the systematic examination of oscillatory modes in the global atmosphere by studying 12 years of 500 mb geopotential heights in the Southern Hemisphere. As in Part I, for the tropics and Northern Hemisphere extratropics, the data were band-pass filtered to focus on intraseasonal (IS) phenomena, and spatial EOFs were obtained. The leading principal components were subjected to singular spectrum analysis (SSA), in order to identify nonlinear IS oscillations with high statistical confidence.In the Southern Hemisphere, the dominant mode has a period of 23 days, with spatial patterns carried by the second and third winter EOF of the IS band. It has a zonal wavenumber-four structure. The 40-day mode is second, and dominated by wavenumbers three and four, while a 16-day mode is too weak to separate its spatial behavior from the previous two. The IS dynamics in the Southern Hemisphere is more complex and dominated by shorter wavenumbers than the Northern Hemisphere. No statistically significant correlations between the Southern Hemisphere and the tropics or the Northern Hemisphere are apparent in the IS band.
Ghil, Michael, and Paola Malanotte-Rizzoli. “Data assimilation in meteorology and oceanography.” Advances in Geophysics 33 (1991): 141–266.
Ghil, Michael, and Robert Vautard. “Interdecadal oscillations and the warming trend in global temperature time series.” Nature 350, no. 6316 (1991): 324–327. Abstract

The ability to distinguish a warming trend from natural variability is critical for an understanding of the climatic response to increasing greenhouse-gas concentrations. Here we use singular spectrum analysis1 to analyse the time series of global surface air tem-peratures for the past 135 years2, allowing a secular warming trend and a small number of oscillatory modes to be separated from the noise. The trend is flat until 1910, with an increase of 0.4 °C since then. The oscillations exhibit interdecadal periods of 21 and 16 years, and interannual periods of 6 and 5 years. The interannual oscillations are probably related to global aspects of the El Niño-Southern Oscillation (ENSO) phenomenon3. The interdecadal oscillations could be associated with changes in the extratropical ocean circulation4. The oscillatory components have combined (peak-to-peak) amplitudes of >0.2 °C, and therefore limit our ability to predict whether the inferred secular warming trend of 0.005 °Cyr-1 will continue. This could postpone incontrovertible detection of the greenhouse warming signal for one or two decades.

Ghil, Michael. “Meteorological data assimilation for oceanographers. Part I: Description and theoretical framework.” Dynamics of Atmospheres and Oceans 13, no. 3-4 (1989): 171–218.
Vautard, Robert, and Michael Ghil. “Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series.” Physica D 35, no. 3 (1989): 395–424. Abstract

We distinguish between two dimensions of a dynamical system given by experimental time series. Statistical dimension gives a theoretical upper bound for the minimal number of degrees of freedom required to describe tje attractor up to the accuracy of the data, taking into account sampling and noise problems. The dynamical dimension is the intrinsic dimension of the attractor and does not depend on the quality of the data. Singular Spectrum Analysis (SSA) provides estimates of the statistical dimension. SSA also describes the main physical phenomena reflected by the data. It gives adaptive spectral filters associated with the dominant oscillations of the system and clarifies the noise characteristics of the data. We apply SSA to four paleoclimatic records. The principal climatic oscillations, and the regime changes in their amplitude are detected. About 10 degrees of freedom are statistically significant in the data. Large noise and insufficient sample length do not allow reliable estimates of the dynamical dimension.

Ghil, Michael, and S. Childress. Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics. Springer-Verlag, New York/Berlin, 1987.
Legras, B., and Michael Ghil. “Persistent anomalies, blocking and variations in atmospheric predictability.” Journal of the Atmospheric Sciences 42, no. 5 (1985): 433–471. Abstract

We consider regimes of low-frequency variability in large-scale atmospheric dynamics. The model used for the study of these regimes is the fully-nonlinear, equivalent-barotropic vorticity equation on the sphere, with simplified forcing, dissipation and topography. Twenty-five modes are retained in a spherical harmonics expansion of the streamfunction. Solutions are studied as a function of the nondimensional intensity of the forcing and dissipation.Multiple stationary solutions are obtained as a result of nonlinear interaction between waves, mean flow and orography. The number of modes retained in the analysis permits these multiple equilibria to appear for realistic values of the forcing. The equilibria exhibit blocked and zonal flow patterns bearing a marked resemblance to synoptically defined zonal and blocked Northern Hemisphere midlatitude flows.Wave-wave interactions influence strongly the stability properties of the equilibria and the time evolution of nonequilibrium solutions. Time-dependent solutions show persistent sequences which occur in the phase-space vicinity of the zonal and blocked equilibria. Composite flow patterns of the persistent sequences are similar to the equilibria nearby, which permits the unambiguous definition of quasi-stationary flow regimes, zonal and blocked, respectively. The number of episodes of blocked or zonal flow decreases monotonically as their duration increases, in agreement with observations.The statistics of transitions between the two types of planetary flow regimes are computed from the model's deterministic dynamics. These transitional called breaks in statistical-synoptic long-range forecasting, are shown to be influenced by changes in model parameters. This influence is discussed in terms of the effect of anomalous boundary conditions on large-scale midlatitude atmospheric flow and on its predictability.

Ghil, Michael, R. Benzi, and G. Parisi, ed. Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland Publ. Co., Amsterdam/New York, 1985.
Ghil, Michael, and John Tavantzis. “Global Hopf Bifurcation in a Simple Climate Model.” Siam Journal on Applied Mathematics 43, no. 5 (1983): 1019–1041. Publisher's Version Abstract
The mathematical structure of a simple climate model is investigated. The model is governed by a system of two nonlinear, autonomous differential equations for the evolution in time of global temperature $T$ and meridional ice-sheet extent $L$. The system's solutions are studied by a combination of qualitative reasoning with explicit calculations, both analytical and numerical. For plausible values of the physical parameters, a branch of periodic solutions obtains, which is both orbitally and structurally stable. The amplitude of the stable periodic solutions in $T$ and $L$ correspond roughly to that obtained from proxy records of Quaternary glaciation cycles. The period of these solutions increases along the branch, until it becomes infinite, while the amplitude of the limiting solution is finite. The limiting solution is a homoclinic orbit formed by the reconnecting separatrix of a saddle. The exchange of stability between the branch of periodic solutions and the steady solution from which it arises is studied by a slight simplification of known methods [20], [21].
Ghil, Michael, S. Cohn, John Tavantzis, K. Bube, and Eugene Isaacson. “Applications of estimation theory to numerical weather prediction.” In Dynamic meteorology: Data assimilation methods, 139–224. Springer, 1981.
Ghil, Michael, and H. Le Treut. “A climate model with cryodynamics and geodynamics.” Journal of Geophysical Research 86 (1981): 5262–5270. Publisher's Version
Ghil, Michael, J. Tavantzis S. Coho, K. Bube, and E. Isaacson. “Dynamic Meteorology: Data Assimilation Methods.” In Applied Mathematical Sciences, edited by L. Bengtsson, Michael Ghil, and E. Källén, 36:139–224. Dynamic Meteorology - Data Assimilation Methods. Springer-Verlag, 1981.