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The monthly anomaly sea surface temperature field over the global ocean exhibit probabilistic dependencies between remote points and lagged times, which are explained eventually by some oceanic or atmospheric bridge of information transfer. Despite much of the bivariate SST dependencies appear to be linear, others are characterized by robust and statistically significant nonlinear correlations. In order to enhance that, we present a general method of extracting bivariate (X,Y) dependencies, seeking for pairs of polynomials P(X) and Q(Y) which are maximally correlated. The method relies on a Canonical correlation Analysis (CCA) between sets of standardized monomials of X and Y, up to a certain (low) degree (e.g. 4). Polynomial coefficients are obtained from the leading CCA eigenvector. Polynomials are calibrated and validated over independent periods, being afterwards subjected to marginal Gaussian anamorphoses. The bivariate non-Gaussianity in the space of marginally Gaussianized polynomials remains residual because of the correlation concentration and maximization. Consequently, the bivariate Gaussian pdf or in alternative, a copula pdf in the space of maximally correlated polynomials can accurately approximate the bivariate dependency. That probabilistic model is then used to determine conditional pdfs, cdfs and probabilities of extremes.The method is applied to various (X,Y) pairs. In the first example, X is an optimized polynomial of the El-Niño 3.4 index while Y is that index lagged to the future. For lags between 6 and 18 months, the nonlinear El-Niño forecast clearly surpasses the linear one, contributing to lower the El-Niño seasonal predictability barrier. In the second example, we relate El-Niño (X) with the lagged Atlantic multidecadal oscillation index (Y). Nonlinear, robust correlations appear, both for positive and negative lags up to 5 years putting in evidence Pacific-Atlantic basin oceanic teleconnections.The above probabilistic (polynomial based) model appears to be a good candidate tool for the statistical (seasonal up to decadal) forecast of regime probabilities (e.g. dry/wet) and extremes, given certain antecedent precursors.This work was funded by the Portuguese Fundação para a Ciência e a Tecnologia (FCT) I.P./MCTES through national funds (PIDDAC) – UIDB/50019/2020- IDL and the project JPIOCEANS/0001/2019 (ROADMAP: ’The Role of ocean dynamics and Ocean–Atmosphere interactions in Driving cliMAte variations and future Projections of impact–relevant extreme events’). Acknowledgements are also due to the International Meteorological Institute (IMI) at Stockholm University.

The ability to find and recognize patterns in high-dimensional geophysical data is fundamental to climate science and critical for meaningful interpretation of weather and climate processes. Archetypal analysis (AA) is one technique that has recently gained traction in the geophysical science community for its ability to find patterns based on extreme conditions. While traditional empirical orthogonal function (EOF) analysis can reveal patterns based on data covariance, AA seeks patterns from the points located at the edges of the data distribution. The utility of any objective pattern method depends on the properties of the data to which it is applied and the choices made in implementing the method. Given the relative novelty of the application of AA in geophysics it is important to develop experience in applying the method. We provide an assessment of the method, implementation, sensitivity, and interpretation of AA with respect to geophysical data. As an example for demonstration, we apply AA to a 39-yr sea surface temperature (SST) reanalysis dataset. We show that the decisions made to implement AA can significantly affect the interpretation of results, but also, in the case of SST, that the analysis is exceptionally robust under both spatial and temporal coarse graining. Significance Statement Archetypal analysis (AA), when applied to geophysical fields, is a technique designed to find typical configurations or modes in underlying data. This technique is relatively new to the geophysical science community and has been shown to be beneficial to the interpretation of extreme modes of the climate system. The identification of extreme modes of variability and their expression in day-to-day weather or state of the climate at longer time scales may help in elucidating the interplay between major teleconnection drivers and their evolution in a changing climate. The purpose of this work is to bring together a comprehensive report of the AA methodology using an SST reanalysis for demonstration. It is shown that the AA results are significantly affected by each implementation decision, but also can be resilient to spatiotemporal averaging. Any application of AA should provide a clear documentation of the choices made in applying the method.

The low-frequency variability of the extratropical atmosphere involves hemispheric-scale recurring, often persistent, states known as teleconnection patterns or regimes, which can have a profound impact on predictability on intra-seasonal and longer timescales. However, reliable data-driven identification and dynamical representation of such states are still challenging problems in modeling the dynamics of the atmosphere. We present a new method, which allows us both to detect recurring regimes of atmospheric variability and to obtain dynamical variables serving as an embedding for these regimes. The method combines two approaches from nonlinear data analysis: partitioning a network of recurrent states with studying its properties by the recurrence quantification analysis and the kernel principal component analysis. We apply the method to study teleconnection patterns in a quasi-geostrophical model of atmospheric circulation over the extratropical hemisphere as well as to reanalysis data of geopotential height anomalies in the mid-latitudes of the Northern Hemisphere atmosphere in the winter seasons from 1981 to the present. It is shown that the detected regimes as well as the obtained set of dynamical variables explain large-scale weather patterns, which are associated, in particular, with severe winters over Eurasia and North America. The method presented opens prospects for improving empirical modeling and long-term forecasting of large-scale atmospheric circulation regimes.

This chapter describes the idea behind, and develops the theory of empirical orthogonal functions (EOFs) along with a historical perspective. It also shows different ways to obtain EOFs and provides examples from climate and discusses their physical interpretation. Strength and weaknesses of EOFs are also mentioned.

This chapter describes patterns obtained based on proximity or similarity measures, i.e. multidimensional scaling (MDS). Conventional EOFs correspond to the case of quadratic distance. In this chapter other forms of similarities are discussed, with climate application, and which can yield structures that cannot be revealed by classical MDS.

Weather and Climate data contain a myriad of processes including oscillating and propagating features. In general EOF method is not suited to identify propagating patterns. In this chapter describes a spectral method based on Hilbert transform to identify propagating features, with application to the stratospheric quasi-biennial oscillation.

Weather and climate data are in general discrete and result from sampling a continuous system. This chapter attempts to take this into account when computing EOFs. The first part of the chapter describes methods to construct EOFs/PCs of profiles with application to oceanography. The second part of the chapter describes regularised EOFs with application to reanalysis data.

EOF method is essentially an exploratory method to analyse the modes of variability of multivariate weather and climate data, with no model is involved. This chapter describes a different method, Principal Oscillation Pattern (POP) analysis, that seeks the simplest dynamical system that can explain the main features of the space–time data. The chapter also provides further extension of POPs by including nonlinearity. Examples from climate data are also given.

Hilbert EOFs presented in Chap. 5 are based on a spectral method to identify propagating or oscillating features. This chapter describes a time domain method, the extended EOFs, to identify propagating patterns from spatio-temporal data sets. The method is similar to the EOF method except that the spatial dimension is extended to include lagged information. Examples from the Madden–Julian oscillation are also provided.

This chapter introduces some basic terminologies that are used in subsequent chapters. It also presents some basic summary statistics of data sets and reviews basic methods of data filtering and smoothing.

Conventional EOFs yield orthogonal spatial patterns and uncorrelated time series. Non-correlation does not necessarily yield independence, which is a strong constraint compared to non-correlation. This chapter discusses the concept of independence, and its relation to non-normality and describes different ways to obtain independent components. The chapter also discusses the application to climate data.

This chapter describes further the drawbacks of EOFs mentioned in Chap. 3. It also provides different ways to overcome those drawbacks, including EOF rotation and simplified EOFs. A number of applications to climate data are also provided.

Previous chapters focussed mostly on single fields, such as EOFs of seal level pressure. This chapter is an extension of previous methods. It describes different methods that mostly deal with two fields to identify coupled patterns that covary coherently. The chapter discusses both the conventional and regularised problems. It also explores the predictive power of coupled pattern analysis.

Previous chapters discuss methods with no particular predictive power. This chapter describes further advanced linear methods that are more related to prediction. Three related methods, involving extrapolation (or prediction) and interpolation, with climate applications are discussed in this chapter.

We studied the inter-annual variability of the vertically integrated zonal/meridional moisture transport in the lower troposphere over the northern Indian Ocean using observed data from 1971-2016 f ...

Understanding the effects of climate change and global warming on the South Asian summer monsoon rainfall trend is critically important for millions of inhabitants of this region. This study invest ...

Conventional analysis methods in weather and climate science (e.g., EOF analysis) exhibit a number of drawbacks including scaling and mixing. These methods focus mostly on the bulk of the probability distribution of the system in state space and overlook its tail. This paper explores a different method, the archetypal analysis (AA), which focuses precisely on the extremes. AA seeks to approximate the convex hull of the data in state space by finding “corners” that represent “pure” types or archetypes through computing mixture weight matrices. The method is quite new in climate science, although it has been around for about two decades in pattern recognition. It encompasses, in particular, the virtues of EOFs and clustering. The method is presented along with a new manifold-based optimization algorithm that optimizes for the weights simultaneously, unlike the conventional multistep algorithm based on the alternating constrained least squares. The paper discusses the numerical solution and then applies it to the monthly sea surface temperature (SST) from HadISST and to the Asian summer monsoon (ASM) using sea level pressure (SLP) from ERA-40 over the Asian monsoon region. The application to SST reveals, in particular, three archetypes, namely, El Niño, La Niña, and a third pattern representing the western boundary currents. The latter archetype shows a particular trend in the last few decades. The application to the ASM SLP anomalies yields archetypes that are consistent with the ASM regimes found in the literature. Merits and weaknesses of the method along with possible future development are also discussed.

The anomalous decadal warming of the subpolar North Atlantic Ocean (SPNA), and the northward spreading of this warm water, has been linked to rapid Arctic sea ice loss and more frequent cold Europe ...

The study of regional rainfall trends over South Asia is critically important for food security and economy, as both these factors largely depend on the availability of water. In this study, South Asian summer monsoon rainfall trends on seasonal and monthly (June–September) time scales have been investigated using three observational data sets. Our analysis identify a dipole-type structure in rainfall trends over the region north of the Indo–Pak subcontinent, with significant increasing trends over the core monsoon region of Pakistan and significant decreasing trends over the central-north India and adjacent areas. The dipole is also evident in monthly rainfall trend analyses, which is more prominent in July and August. We show, in particular, that the strengthening of northward moisture transport over the Arabian Sea is a likely reason for the significant positive trend of rainfall in the core monsoon region of Pakistan. In contrast, over the central-north India region, the rainfall trends are significantly decreasing due to the weakening of northward moisture transport over the Bay of Bengal. The leading empirical orthogonal functions clearly show the strengthening (weakening) patterns of vertically integrated moisture transport over the Arabian Sea (Bay of Bengal) in seasonal and monthly interannual time scales. The regression analysis between the principal components and rainfall confirm the dipole pattern over the region. Our results also suggest that the extra-tropical phenomena could influence the mean monsoon rainfall trends over Pakistan by enhancing the cross-equatorial flow of moisture into the Arabian Sea.

The Asian summer monsoon is a high-dimensional and highly nonlinear phenomenon involving considerable moisture transport towards land from the ocean, and is critical for the whole region. We have used daily ECMWF reanalysis (ERA-40) sea-level pressure (SLP) anomalies on the seasonal cycle, over the region 50–145° E, 20° S–35° N, to study the nonlinearity of the Asian monsoon using Isomap. We have focused on the two-dimensional embedding of the SLP anomalies for ease of interpretation. Unlike the unimodality obtained from tests performed in empirical orthogonal function space, the probability density function, within the two-dimensional Isomap space, turns out to be bimodal. But a clustering procedure applied to the SLP data reveals support for three clusters, which are identified using a three-component bivariate Gaussian mixture model. The modes are found to appear similar to active and break phases of the monsoon over South Asia in addition to a third phase, which shows active conditions over the western North Pacific. Using the low-level wind field anomalies, the active phase over South Asia is found to be characterised by a strengthening and an eastward extension of the Somali jet. However during the break phase, the Somali jet is weakened near southern India, while the monsoon trough in northern India also weakens. Interpretation is aided using the APHRODITE gridded land precipitation product for monsoon Asia. The effect of large-scale seasonal mean monsoon and lower boundary forcing, in the form of ENSO, is also investigated and discussed. The outcome here is that ENSO is shown to perturb the intraseasonal regimes, in agreement with conceptual ideas.

The extra-tropical atmosphere is characterized by robust circulations which have time scales longer than that associated with developing baroclinic systems but shorter than a season. Such low frequency variability is governed to a large extent by non-linear dynamics, and hence is chaotic. A useful aspect of this low-frequency circulation is that it can often be described by just a few quasi-stationary regime states, broadly defined as recurrent or persistent large scale structures, that exert a significant impact on the probability of experiencing extreme surface weather conditions. We review a variety of techniques for identifying circulation regimes from reanalysis and numerical model output. While various techniques often yield similar regime circulation patterns, they offer different perspectives on the regimes. The regimes themselves are manifest in planetary scale patterns. They affect the structure of synoptic scale patterns. Extra-tropical flow regimes have been identified in simplified atmospheric models and comprehensive coupled climate models and in reanalysis data sets. It is an ongoing challenge to accurately model these regime states and high horizontal resolutions are often needed to accurately reproduce them. The regime paradigm helps to understand the response to external forcing on a variety of time scales, has been helpful in categorizing a large number of weather types and their effect on local conditions, and is useful in downscaling. Despite their usefulness, there is a debate on the \non-equivocal" and systematic existence of these nonlinear circulation regimes. We review our current understanding of the nonlinear and regime paradigms and suggest future research.

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 Niños, 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.

A systematic analysis of the winter North Atlantic eddy-driven jet stream latitude and wind speed from 52 model integrations, taken from the coupled model intercomparison project phase 3, is carried out and compared to results obtained from the ERA-40 reanalyses. We consider here a control simulation, twentieth century simulation, and two time periods (2046-2065 and 2081-2100) from a twenty-first century, high-emission A2 forced simulation. The jet wind speed seasonality is found to be similar between the twentieth century simulations and the ERA-40 reanalyses and also between the control and forced simulations although nearly half of the models overestimate the amplitude of the seasonal cycle. A systematic equatorward bias of the models jet latitude seasonality, by up to 7°, is observed, and models additionally overestimate the seasonal cycle of jet latitude about the mean, with the majority of the models showing equatorward and poleward biases during the cold and warm seasons respectively. A main finding of this work is that no GCM under any forcing scenario considered here is able to simulate the trimodal behaviour of the observed jet latitude distribution. The models suffer from serious problems in the structure of jet variability, rather than just quantitiative errors in the statistical moments. © 2012 Springer-Verlag Berlin Heidelberg.