Carlos Pires

Beyond Gaussian Statistical Modeling in Geophysical Data Assimilation

This review discusses recent advances in geophysical data assimilation beyond Gaussian statistical modeling, in the fields of meteorology, oceanography, as well as atmospheric chemistry. The non-Gaussian features are stressed rather than the nonlinearity of the dynamical models, although both aspects are entangled. Ideas recently proposed to deal with these non-Gaussian issues, in order to improve the state or parameter estimation, are emphasized. The general Bayesian solution to the estimation problem and the techniques to solve it are first presented, as well as the obstacles that hinder their use in high-dimensional and complex systems. Approximations to the Bayesian solution relying on Gaussian, or on second-order moment closure, have been wholly adopted in geophysical data assimilation (e.g., Kalman filters and quadratic variational solutions). Yet, nonlinear and non-Gaussian effects remain. They essentially originate in the nonlinear models and in the non-Gaussian priors. How these effects are handled within algorithms based on Gaussian assumptions is then described. Statistical tools that can diagnose them and measure deviations from Gaussianity are recalled. The following advanced techniques that seek to handle the estimation problem beyond Gaussianity are reviewed: maximum entropy filter, Gaussian anamorphosis, non-Gaussian priors, particle filter with an ensemble Kalman filter as a proposal distribution, maximum entropy on the mean, or strictly Bayesian inferences for large linear models, etc. Several ideas are illustrated with recent or original examples that possess some features of high-dimensional systems. Many of the new approaches are well understood only in special cases and have difficulties that remain to be circumvented. Some of the suggested approaches are quite promising, and sometimes already successful for moderately large though specific geophysical applications. Hints are given as to where progress might come from.

Non-Gaussianity and Asymmetry of the Winter Monthly Precipitation Estimation from the NAO

Abstract The present work assesses non-Gaussianity and asymmetry within the statistical response of the monthly winter (December–February) precipitation to the North Atlantic Oscillation (NAO) over the North Atlantic–European region (NAE). To evaluate asymmetry, data are split through the median of the NAO index and side correlations are computed for each regime [negative and positive phases of the NAO (NAO− and NAO+, respectively)]. The following statistically significant differences between these correlations are found: (a) near the central North Atlantic, around 40°N, 20°W, and southeast of Iceland, with much stronger correlations in the wet-favorable regime: NAO− in the first location and NAO+ in the second location; (b) around 42°N, 48°W in the west North Atlantic; and (c) south of Greenland and in the west Mediterranean near 36°N, where, in both cases, the correlation is only relevant for the dry-favorable NAO+ regime. Based on the above decomposition, a map of a statistical test of asymmetry, appli...

Tsunami waveform inversion by adjoint methods

An adjoint method for tsunami waveform inversion is proposed, as an alternative to the technique based on Greens functions of the linear long wave model. The method has the advantage of being able to use the nonlinear shallow water equations, or other appropriate equation sets, and to optimize an initial state given as a linear or nonlinear function of any set of free parameters. This last facility is used to perform explicit optimization of the focal fault parameters, characterizing the initial sea surface displacement of tsunamigenic earthquakes. The proposed methodology is validated with experiments using synthetic data, showing the possibility of recovering all relevant details of a tsunami source from tide gauge observations, providing that the adjoint method is constrained in an appropriate manner. It is found, as in other methods, that the inversion skill of tsunami sources increases with the azimuthal and temporal coverage of assimilated tide gauge stations; furthermore, it is shown that the eigenvalue analysis of the Hessian matrix of the cost function provides a consistent and useful methodology to choose the subset of independent parameters that can be inverted with a given dataset of observations and to evaluate the error of the inversion process. The method is also applied to real tide gauge series, from the tsunami of the February 28, 1969, Gorringe Bank earthquake, suggesting some reasonable changes to the assumed focal parameters of that event. It is suggested that the method proposed may be able to deal with transient tsunami sources such as those generated by submarine landslides.

Long-Range Atmospheric Predictability Using Space–Time Principal Components

Abstract The long-term predictability of 70-kPa geopotential heights is examined by a series of hindcast experiments over a validation period of 40 years using empirical models. Only the North Atlantic sector is considered. Significant skill is found up to lead times of one to two months for forecasts of time averages and of weather regime occurrence frequencies. The empirical schemes produce forecasts of the conditional probability of occurrence of a predictand within its natural terciles. These probabilistic forecasts are compared for two sets of predictors. The (spatial) principal components of the Atlantic large-scale flow (S-PCs) and its space–time principal components (ST-PCs) obtained from multichannel singular spectrum analysis (MSSA). These latter predictors achieve a good compromise between explained variance and predictability. In particular, the skill of a one-step model, where predictands conditional probabilities are obtained directly from an analog method, is compared with a two-step model...

Minimum Mutual Information and Non-Gaussianity Through the Maximum Entropy Method: Theory and Properties

Abstract: The application of the Maximum Entropy (ME) principle leads to a minimum of the Mutual Information (MI), I(X,Y) , between random variables X , Y , which is compatible with prescribed joint expectations and given ME marginal distributions. A sequence of sets of joint constraints leads to a hierarchy of lower MI bounds increasingly approaching the true MI. In particular, using standard bivariate Gaussian marginal distributions, it allows for the MI decomposition into two positive terms: the Gaussian MI ( I g ), depending upon the Gaussian correlation or the correlation between ‘Gaussianized variables’, and a non-Gaussian MI ( I ng ), coinciding with joint negentropy and depending upon nonlinear correlations. Joint moments of a prescribed total order p are bounded within a compact set defined by Schwarz-like inequalities, where I ng grows from zero at the ‘Gaussian manifold’ where moments are those of Gaussian distributions, towards infinity at the set’s boundary where a deterministic relationship holds. Sources of joint non-Gaussianity have been systematized by estimating

Minimum Mutual Information and Non-Gaussianity through the Maximum Entropy Method: Estimation from Finite Samples

The Minimum Mutual Information (MinMI) Principle provides the least committed, maximum-joint-entropy (ME) inferential law that is compatible with prescribed marginal distributions and empirical cross constraints. Here, we estimate MI bounds (the MinMI values) generated by constraining sets Tcr comprehended by mcr linear and/or nonlinear joint expectations, computed from samples of N iid outcomes. Marginals (and their entropy) are imposed by single morphisms of the original random variables. N-asymptotic formulas are given both for the distribution of cross expectations estimation errors, the MinMI estimation bias, its variance and distribution. A growing Tcr leads to an increasing MinMI, converging eventually to the total MI. Under N-sized samples, the MinMI increment relative to two encapsulated sets Tcr1  Tcr2 (with numbers of constraints 12

Separation of the atmospheric variability into non-Gaussian multidimensional sources by projection pursuit techniques

We develop an expansion of space-distributed time series into statistically independent uncorrelated subspaces (statistical sources) of low-dimension and exhibiting enhanced non-Gaussian probability distributions with geometrically simple chosen shapes (projection pursuit rationale). The method relies upon a generalization of the principal component analysis that is optimal for Gaussian mixed signals and of the independent component analysis (ICA), optimized to split non-Gaussian scalar sources. The proposed method, supported by information theory concepts and methods, is the independent subspace analysis (ISA) that looks for multi-dimensional, intrinsically synergetic subspaces such as dyads (2D) and triads (3D), not separable by ICA. Basically, we optimize rotated variables maximizing certain nonlinear correlations (contrast functions) coming from the non-Gaussianity of the joint distribution. As a by-product, it provides nonlinear variable changes ‘unfolding’ the subspaces into nearly Gaussian scalars of easier post-processing. Moreover, the new variables still work as nonlinear data exploratory indices of the non-Gaussian variability of the analysed climatic and geophysical fields. The method (ISA, followed by nonlinear unfolding) is tested into three datasets. The first one comes from the Lorenz’63 three-dimensional chaotic model, showing a clear separation into a non-Gaussian dyad plus an independent scalar. The second one is a mixture of propagating waves of random correlated phases in which the emergence of triadic wave resonances imprints a statistical signature in terms of a non-Gaussian non-separable triad. Finally the method is applied to the monthly variability of a high-dimensional quasi-geostrophic (QG) atmospheric model, applied to the Northern Hemispheric winter. We find that quite enhanced non-Gaussian dyads of parabolic shape, perform much better than the unrotated variables in which concerns the separation of the four model’s centroid regimes (positive and negative phases of the Arctic Oscillation and of the North Atlantic Oscillation). Triads are also likely in the QG model but of weaker expression than dyads due to the imposed shape and dimension. The study emphasizes the existence of nonlinear dyadic and triadic nonlinear teleconnections.

Independent Subspace Analysis of the Sea Surface Temperature Variability: Non-Gaussian Sources and Sensitivity to Sampling and Dimensionality

We propose an expansion of multivariate time-series data into maximally independent source subspaces. The search is made among rotations of prewhitened data which maximize non-Gaussianity of candidate sources. We use a tensorial invariant approximation of the multivariate negentropy in terms of a linear combination of squared coskewness and cokurtosis. By solving a high-order singular value decomposition problem, we extract the axes associated with most non-Gaussianity. Moreover, an estimate of the Gaussian subspace is provided by the trailing singular vectors. The independent subspaces are obtained through the search of “quasi-independent” components within the estimated non-Gaussian subspace, followed by the identification of groups with significant joint negentropies. Sources result essentially from the coherency of extremes of the data components. The method is then applied to the global sea surface temperature anomalies, equatorward of 65°, after being tested with non-Gaussian surrogates consistent with the data anomalies. The main emerging independent components and subspaces, supposedly generated by independent forcing, include different variability modes, namely, The East-Pacific, the Central Pacific, and the Atlantic Ninos, the Atlantic Multidecadal Oscillation, along with the subtropical dipoles in the Indian, South Pacific, and South-Atlantic oceans. Benefits and usefulness of independent subspaces are then discussed.

Adjoint Inversion of the Source Parameters of Near-Shore Tsunamigenic Earthquakes

The adjoint method may be designed to perform the direct optimization of tsunami fault parameters, from tide-gauge data, leading to a substantial enhancement of the signal-to-noise ratio, when compared with the classical technique based on Greens functions of the linear long-wave model. A 4-step inversion procedure, which can be fully automated, consists (i) in the source area delimitation, (ii) adjoint optimization of the initial sea state in the grid space, (iii) non-linear adjustment of the fault model and (iv) final adjoint optimization in the fault parameter space. Results with an idealized bathymetry show that the method works well in the presence of reasonable amounts of error and provides, as a by-product, a resolution matrix that contains information on the inversion error, identifying the combinations of source parameters that are best and worst resolved by the inversion. A detailed analysis of inversion sensitivity and of the fine structure of the inversion error is used to identify problems and limitations of the inversion procedure.