Valerie N. Livina

Methods for Detecting Early Warnings of Critical Transitions in Time Series Illustrated Using Simulated Ecological Data

Many dynamical systems, including lakes, organisms, ocean circulation patterns, or financial markets, are now thought to have tipping points where critical transitions to a contrasting state can happen. Because critical transitions can occur unexpectedly and are difficult to manage, there is a need for methods that can be used to identify when a critical transition is approaching. Recent theory shows that we can identify the proximity of a system to a critical transition using a variety of so-called ‘early warning signals’, and successful empirical examples suggest a potential for practical applicability. However, while the range of proposed methods for predicting critical transitions is rapidly expanding, opinions on their practical use differ widely, and there is no comparative study that tests the limitations of the different methods to identify approaching critical transitions using time-series data. Here, we summarize a range of currently available early warning methods and apply them to two simulated time series that are typical of systems undergoing a critical transition. In addition to a methodological guide, our work offers a practical toolbox that may be used in a wide range of fields to help detect early warning signals of critical transitions in time series data.

No increase in global temperature variability despite changing regional patterns

Evidence from Greenland ice cores shows that year-to-year temperature variability was probably higher in some past cold periods, but there is considerable interest in determining whether global warming is increasing climate variability at present. This interest is motivated by an understanding that increased variability and resulting extreme weather conditions may be more difficult for society to adapt to than altered mean conditions. So far, however, in spite of suggestions of increased variability, there is considerable uncertainty as to whether it is occurring. Here we show that although fluctuations in annual temperature have indeed shown substantial geographical variation over the past few decades, the time-evolving standard deviation of globally averaged temperature anomalies has been stable. A feature of the changes has been a tendency for many regions of low variability to experience increases, which might contribute to the perception of increased climate volatility. The normalization of temperature anomalies creates the impression of larger relative overall increases, but our use of absolute values, which we argue is a more appropriate approach, reveals little change. Regionally, greater year-to-year changes recently occurred in much of North America and Europe. Many climate models predict that total variability will ultimately decrease under high greenhouse gas concentrations, possibly associated with reductions in sea-ice cover. Our findings contradict the view that a warming world will automatically be one of more overall climatic variation.

Long-term memory in earthquakes and the distribution of interoccurrence times

We study seismic records in regimes of stationary seismic activity in Northern and Southern California. Our analysis suggests that the earthquakes are long-term power law correlated with a correlation exponent ? close to 0.4. We show explicitly that the long-term correlations can explain both the fluctuations of magnitudes and interoccurrence times (between events above a certain magnitude M) and, without any fit parameter, the scaling form of the distribution function of the interoccurrence times in the seismic records, recently obtained by Corral (Phys. Rev. Lett., 92 (2004) 108501).

Using GENIE to study a tipping point in the climate system.

We have used the Grid ENabled Integrated Earth system modelling framework to study the archetypal example of a tipping point in the climate system; a threshold for the collapse of the Atlantic thermohaline circulation (THC). eScience has been invaluable in this work and we explain how we have made it work for us. Two stable states of the THC have been found to coexist, under the same boundary conditions, in a hierarchy of models. The climate forcing required to collapse the THC and the reversibility or irreversibility of such a collapse depends on uncertain model parameters. Automated methods have been used to assimilate observational data to constrain the pertinent parameters. Anthropogenic climate forcing leads to a robust weakening of the THC and increases the probability of crossing a THC tipping point, but some ensemble members collapse readily, whereas others are extremely resistant. Hence, we test general methods that have been developed to directly diagnose, from time-series data, the proximity of a ‘tipping element’, such as the THC to a bifurcation point. In a three-dimensional ocean–atmosphere model exhibiting THC hysteresis, despite high variability in the THC driven by the dynamical atmosphere, some early warning of an approaching tipping point appears possible.

Dansgaard–Oeschger events: bifurcation points in the climate system

Dansgaard-Oeschger events are a prominent mode of variability in the records of the last glacial cycle. Various prototype models have been proposed to explain these rapid climate fluctuations, and no agreement has emerged on which may be the more correct for describing the paleoclimatic signal. In this work, we assess the bimodality of the system reconstructing the topology of the multi--dimensional attractor over which the climate system evolves. We use high-resolution ice core isotope data to investigate the statistical properties of the climate fluctuations in the period before the onset of the abrupt change. We show that Dansgaard-Oeschger events have weak early warning signals if the ensemble of events is considered. We find that the statistics are consistent with the switches between two different climate equilibrium states in response to a changing external forcing (e.g. solar, ice sheets...), either forcing directly the transition or pacing it through stochastic resonance. These findings are most consistent with a model that associates Dansgaard-Oeschger with changing boundary conditions, and with the presence of a bifurcation point.

Seasonality Effects on Nonlinear Properties of Hydrometeorological Records

Climatic time series, in general, and hydrological time series, in particular, exhibit pronounced annual periodicity. This periodicity and its corresponding harmonics affect the nonlinear properties of the relevant time series (i.e. the long-term volatility correlations and the width of the multifractal spectrum) and thus have to be filtered out before studying fractal and volatility properties. We compare several filtering techniques and find that in order to eliminate the periodicity effects on the nonlinear properties of the hydrological time series, it is necessary to filter out the seasonal standard deviation in addition to the filtering of the seasonal mean, with conservation of linear two-point correlations . We name the proposed filtering technique “phase substitution”, because it employs the Fourier phases of the series. The obtained results still indicate nonlinearity of the river data, its strength being weaker than under previously used techniques.

Forecasting the underlying potential governing the time series of a dynamical system

We introduce a technique of time series analysis, potential forecasting, which is based on dynamical propagation of the probability density of time series. We employ polynomial coefficients of the orthogonal approximation of the empirical probability distribution and extrapolate them in order to forecast the future probability distribution of data. The method is tested on artificial data, used for hindcasting observed climate data, and then applied to forecast Arctic sea-ice time series. The proposed methodology completes a framework for ‘potential analysis’ of tipping points which altogether serves anticipating, detecting and forecasting nonlinear changes including bifurcations using several independent techniques of time series analysis. Although being applied to climatological series in the present paper, the method is very general and can be used to forecast dynamics in time series of any origin.

A recent bifurcation in Arctic sea-ice cover

Arctic sea-ice has experienced striking reductions in areal coverage, especially in recent summers, and summer ice cover is forecast to disappear later this century. This has fuelled debate over whether Arctic sea-ice has already passed a ‘tipping point’, or whether it will do so in future, with several recent model studies arguing that there is no bifurcation involved because the loss of summer sea ice is highly reversible. Recently developed methods can detect and sometimes forewarn of bifurcations in time-series data, hence we applied them to satellite data for Arctic sea-ice area. Here we show that a new low ice cover state has appeared from 2007 onwards, which is distinct from the normal state of seasonal sea ice variation, indicating a bifurcation has occurred (from one attractor to two). There was no robust early warning signal of critical slowing down16 prior to this bifurcation, indeed the normal state of sea-ice cover became more stable in the decade beforehand. Internal climate variability is likely responsible for recent transitions between the two ice cover states. However, there are signals of increasing instability since 2007. Several positive feedbacks between the atmosphere, ocean and sea-ice cover could be contributing to separating the two states for Arctic sea-ice cover, as they have done at a regional scale in the past. Our results provide direct evidence of a recent bifurcation in Arctic sea-ice cover and the ongoing destabilisation suggests that further abrupt changes may lie ahead.

Bifurcation analysis of geophysical time series

We propose a general synthetic framework, combining analytical and experimental techniques, for studying climatic bifurcations and transitions by means of the time series analysis. The method employs three major techniques: (i) derivation of potential from time series using unscented Kalman Filter (UKF); (ii) studying possible bifurcations and transitions of the obtained potential; (iii) projection of the time series according to the estimated perturbation. The method is tested on artificial data and then applied to observed records, in particular, a Greenland temperature proxy.