We present and apply a novel method of describing and modeling complex multivariate datasets in the geosciences and elsewhere. Data-adaptive harmonic (DAH) decomposition identifies narrow-banded, spatio-temporal modes (DAHMs) whose frequencies are not necessarily integer multiples of each other. The evolution in time of the DAH coefficients (DAHCs) of these modes can be modeled using a set of coupled Stuart-Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time and space. This methodology is applied first to a challenging synthetic dataset and then to Arctic sea ice concentration (SIC) data from the US National Snow and Ice Data Center (NSIDC). The 36-year (1979–2014) dataset is parsimoniously and accurately described by our DAHMs. Preliminary results indicate that simulations using our multilayer Stuart-Landau model (MSLM) of SICs are stable for much longer time intervals, beyond the end of the twenty-first century, and exhibit interdecadal variability consistent with past historical records. Preliminary results indicate that this MSLM is quite skillful in predicting September sea ice extent.

%B Advances in Nonlinear Geosciences %I Springer %G eng %U http://doi.org/10.1007/978-3-319-58895-7_10 %0 Journal Article %J Journal of Atmospheric and Solar-Terrestrial Physics, %D 2018 %T Data-adaptive harmonic analysis and modeling of solar wind-magnetosphere coupling %A Kondrashov, Dmitri %A Chekroun, Mickaël D %X The solar wind-magnetosphere coupling is studied by new data-adaptive harmonic decomposition (DAHD) approach for the spectral analysis and inverse modeling of multivariate time observations of complex nonlinear dynamical systems. DAHD identifies frequency-based modes of interactions in the combined dataset of Auroral Electrojet (AE) index and solar wind forcing. The time evolution of these modes can be very efficiently simulated by using systems of stochastic differential equations (SDEs) that are stacked per frequency and formed by coupled Stuart-Landau oscillators. These systems of SDEs capture the modes' frequencies as well as their amplitude modulations, and yield, in turn, an accurate modeling of the AE index' statistical properties. %B Journal of Atmospheric and Solar-Terrestrial Physics, %V 177 %P 179-189 %G eng %U https://www.sciencedirect.com/science/article/pii/S136468261730295X %0 Journal Article %J Dynamics and Statistics of the Climate System %D 2018 %T Data-adaptive harmonic decomposition and prediction of Arctic sea ice extent %A Kondrashov, Dmitri %A Chekroun, Mickaël D. %A Ghil, Michael %X Decline in the Arctic sea ice extent (SIE) is an area of active scientific research with profound socio-economic implications. Of particular interest are reliable methods for SIE forecasting on subseasonal time scales, in particular from early summer into fall, when sea ice coverage in the Arctic reaches its minimum. Here, we apply the recent data-adaptive harmonic (DAH) technique of Chekroun and Kondrashov, (2017), Chaos,The multiscale variability of the ocean circulation due to its nonlinear dynamics remains a big challenge for theoretical understanding and practical ocean modeling. This paper demonstrates how the data-adaptive harmonic (DAH) decomposition and inverse stochastic modeling techniques introduced in (Chekroun and Kondrashov, (2017), Chaos, 27), allow for reproducing with high fidelity the main statistical properties of multiscale variability in a coarse-grained eddy-resolving ocean flow. This fully-data-driven approach relies on extraction of frequency-ranked time-dependent coefficients describing the evolution of spatio-temporal DAH modes (DAHMs) in the oceanic flow data. In turn, the time series of these coefficients are efficiently modeled by a family of low-order stochastic differential equations (SDEs) stacked per frequency, involving a fixed set of predictor functions and a small number of model coefficients. These SDEs take the form of stochastic oscillators, identified as multilayer Stuart–Landau models (MSLMs), and their use is justified by relying on the theory of Ruelle–Pollicott resonances. The good modeling skills shown by the resulting DAH-MSLM emulators demonstrates the feasibility of using a network of stochastic oscillators for the modeling of geophysical turbulence. In a certain sense, the original quasiperiodic Landau view of turbulence, with the amendment of the inclusion of stochasticity, may be well suited to describe turbulence.

%B Fluids %V 3 %P 21 %G eng %U https://www.mdpi.com/2311-5521/3/1/21/html %N 1 %0 Book Section %B The Gap between Weather and Climate Forecasting: Sub-Seasonal to Seasonal Prediction %D 2018 %T Extratropical sub-seasonal–to–seasonal oscillations and multiple regimes: The dynamical systems view %A Ghil, Michael %A Groth, Andreas %A Kondrashov, Dmitri %A Robertson, Andrew W. %E Robertson, Andrew W. %E Vitart, Frederic %XThis chapter considers the sub-seasonal–to–seasonal (S2S) prediction problem as intrinsically more difficult than either short-range weather prediction or interannual–to–multidecadal climate prediction. The difficulty arises from the comparable importance of atmospheric initial states and of parameter values in determining the atmospheric evolution on the S2S time scale. The chapter relies on the theoretical framework of dynamical systems and the practical tools this framework helps provide to low-order modeling and prediction of S2S variability. The emphasis is on mid-latitude variability and the complementarity of the nonlinear-waves vs. multiple-regime points of view in understanding this variability. Empirical model reduction and the forecast skill of the models thus produced in real-time prediction are reviewed.

%B The Gap between Weather and Climate Forecasting: Sub-Seasonal to Seasonal Prediction %7 1 %I Elsevier %P 119-142 %G eng %U https://doi.org/10.1016/B978-0-12-811714-9.00006-1 %& Extratropical sub-seasonal–to–seasonal oscillations and multiple regimes: The dynamical systems view %0 Journal Article %J Chaos %D 2017 %T Data-adaptive harmonic spectra and multilayer Stuart-Landau models %A Chekroun, Mickaël D. %A Kondrashov, Dmitri %XHarmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure.

The corresponding eigenvalues can be grouped per Fourier frequency, and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey furthermore a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum.

The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coe cients stacked by Fourier frequency which can be e ciently modeled—provided the decay of temporal correlations is su ciently well-resolved—within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators.

Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise, are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.

%B Chaos %V 27 %P 093110 %G eng %U http://dx.doi.org/10.1063/1.4989400 %0 Generic %D 2017 %T Data-adaptive Harmonic Decomposition and Real-time Prediction of 2016 September Arctic Sea Ice Extent %A Kondrashov, Dmitri %XDecline in the Arctic sea ice extent (SIE) has profound socio-economic implications and is a focus of active scientific research. Of particular interest is prediction of SIE on subseasonal time scales, i.e.~from early summer into fall, when sea ice coverage in Arctic reaches its minimum. However, subseasonal forecasting of SIE is very challenging due to the high variability of ocean and atmosphere over Arctic in summer, as well as shortness of observational data and inadequacies of the physics-based models to simulate sea-ice dynamics. The Sea Ice Outlook (SIO) by Sea Ice Prediction Network (SIPN, http://www.arcus.org/sipn) is a collaborative effort to facilitate and improve subseasonal prediction of September SIE by physics-based and data-driven statistical models.

Data-adaptive Harmonic Decomposition (DAH) and Multilayer Stuart-Landau Models (MSLM) techniques [Chekroun and Kondrashov, 2017], have been successfully applied to the nonlinear stochastic modeling, as well as retrospective and real-time forecasting of Multisensor Analyzed Sea Ice Extent (MASIE) dataset in key four Arctic regions. In particular, the real-time DAH-MSLM predictions outperformed most statistical models and physics-based models in 2016 SIO submissions. The key success factors are associated with DAH ability to disentangle complex regional dynamics of MASIE by data-adaptive harmonic spatio-temporal patterns that reduce the data-driven modeling effort to elemental MSLMs stacked per frequency with fixed and small number of model coefficients to estimate.

This is a joint work with Mickael Chekroun (UCLA) and Michael Ghil (UCLA,ENS).

Spectral analyses of the North Atlantic temperature field in the Simple Ocean Data Analysis (SODA) reanalysis identify prominent and statistically significant interannual oscillations along the Gulf Stream front and in large regions of the North Atlantic. A 7–8-yr oscillatory mode is characterized by a basin-wide southwest-to-northeast–oriented propagation pattern in the sea surface temperature (SST) field. This pattern is found to be linked to a seesaw in the meridional-dipole structure of the zonal wind stress forcing (TAUX). In the subpolar gyre, the SST and TAUX fields of this mode are shown to be in phase opposition, which suggests a cooling effect of the wind stress on the upper ocean layer. Over all, this mode’s temperature field is characterized by a strong equivalent-barotropic component, as shown by covariations in SSTs and sea surface heights, and by phase-coherent behavior of temperature layers at depth with the SST field. Recent improvements of multivariate singular spectrum analysis (M-SSA) help separate spatio-temporal patterns. This methodology is developed further and applied to studying the ocean’s response to variability in the atmospheric forcing. Statistical evidence is shown to exist for other mechanisms generating oceanic variability of similar 7–8-yr periodicity in the Gulf Stream region; the latter variability is likewise characterized by a strongly equivalent-barotropic component. Two other modes of biennial variability in the Gulf Stream region are also identified, and it is shown that interannual variability in this region cannot be explained by the ocean’s response to similar variability in the atmospheric forcing alone.

%B Journal of Climate %V 30 %P 2655-2678 %8 21 mar 2017 %G eng %N 7 %R 10.1175/JCLI-D-16-0370.1 %0 Journal Article %J Phys. Rev. E %D 2016 %T Comment on ``Nonparametric forecasting of low-dimensional dynamical systems'' %A Kondrashov, Dmitri %A Chekroun, Mickaël D. %A Ghil, Michael %B Phys. Rev. E %I American Physical Society %V 93 %P 036201 %8 Mar %G eng %U http://link.aps.org/doi/10.1103/PhysRevE.93.036201 %R 10.1103/PhysRevE.93.036201 %0 Journal Article %J Space Weather %D 2016 %T Data assimilation of low-altitude magnetic perturbations into a global magnetosphere model %A Merkin, V. G. %A Kondrashov, Dmitri %A Ghil, Michael %A Anderson, B. J. %K Data assimilation %K global MHD %K integration and fusion %K magnetosphere %K Magnetosphere/ionosphere interactions %K Models %K Numerical modeling %B Space Weather %V 14 %P 165–184 %G eng %U http://dx.doi.org/10.1002/2015SW001330 %N 2 %R 10.1002/2015SW001330 %0 Book Section %B The Universe of Digital Sky Surveys %D 2016 %T Singular Spectrum Analysis for astronomical time series: constructing a parsimonious hypothesis test %A Greco, G %A Kondrashov, Dmitri %A Kobayashi, S %A Ghil, Michael %A Branchesi, M %A Guidorzi, C %A Stratta, G %A Ciszak, M %A Marino, F %A Ortolan, A %B The Universe of Digital Sky Surveys %I Springer %P 105–107 %G eng %U http://link.springer.com/chapter/10.1007%2F978-3-319-19330-4_16 %0 Journal Article %J Journal of Climate %D 2016 %T Diversity, nonlinearity, seasonality and memory effect in ENSO simulation and prediction using empirical model reduction %A Chen, C. %A Cane, M. A. %A Henderson, N. %A Lee, D. Eun %A Chapman, D. %A Kondrashov, Dmitri %A Chekroun, Mickaël D. %XA suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.

%B Journal of Climate %V 29 %P 1809-1830 %8 Mar 2016 %G eng %N 5 %R 10.1175/JCLI-D-15-0372.1 %0 Journal Article %J Journal of Climate %D 2015 %T Predicting critical transitions in ENSO models. Part II: Spatially dependent models %A Mukhin, Dmitry %A Kondrashov, Dmitri %A Loskutov, Evgeny %A Gavrilov, Andrey %A Feigin, Alexander %A Ghil, Michael %X The present paper is the second part of a two-part study on empirical modeling and prediction of climate variability. This paper deals with spatially distributed data, as opposed to the univariate data of Part I. The choice of a basis for effective data compression becomes of the essence. In many applications, it is the set of spatial empirical orthogonal functions that provides the uncorrelated time series of principal components (PCs) used in the learning set. In this paper, the basis of the learning set is obtained instead by applying multichannel singular-spectrum analysis to climatic time series and using the leading spatiotemporal PCs to construct a reduced stochastic model. The effectiveness of this approach is illustrated by predicting the behavior of the Jin–Neelin–Ghil (JNG) hybrid seasonally forced coupled ocean–atmosphere model of El Niño–Southern Oscillation. The JNG model produces spatially distributed and weakly nonstationary time series to which the model reduction and prediction methodology is applied. Critical transitions in the hybrid periodically forced coupled model are successfully predicted on time scales that are substantially longer than the duration of the learning sample. %B Journal of Climate %V 28 %P 1962–1976 %G eng %N 5 %R 10.1175/JCLI-D-14-00240.1 %0 Journal Article %J Physica D: Nonlinear Phenomena %D 2015 %T Data-driven non-Markovian closure models %A Kondrashov, Dmitri %A Chekroun, Mickaël D. %A Ghil, Michael %XThis paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori–Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR–MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR–MSM methodology is first applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an MSM is shown to successfully reproduce the statistics of a partially observed, generalized Lotka–Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up.

%B Physica D: Nonlinear Phenomena %I Elsevier %V 297 %P 33–55 %G eng %R 10.1016/j.physd.2014.12.005 %0 Journal Article %J Geophysical Research Letters %D 2015 %T Stochastic modeling of decadal variability in ocean gyres %A Kondrashov, Dmitri %A Berloff, Pavel S. %B Geophysical Research Letters %V 42 %P 1543–1553 %G eng %R 10.1002/2014gl062871 %0 Journal Article %J Journal of Geophysical Research: Space Physics %D 2014 %T Noise statistics identification for Kalman filtering of the electron radiation belt observations I: Model errors %A Podladchikova, T. V. %A Shprits, Y. Y. %A Kondrashov, Dmitri %A Kellerman, A. C. %K Data assimilation %K Instruments and techniques %K Kalman filter %K model error identification %K Modeling and forecasting %K Prediction %K Radiation belts %K Space weather %B Journal of Geophysical Research: Space Physics %V 119 %P 5700–5724 %G eng %U http://dx.doi.org/10.1002/2014JA019897 %N 7 %R 10.1002/2014JA019897 %0 Journal Article %J Journal of Geophysical Research: Space Physics %D 2014 %T Noise statistics identification for Kalman filtering of the electron radiation belt observations: 2. Filtration and smoothing %A Podladchikova, T. V. %A Shprits, Y. Y. %A Kellerman, A. C. %A Kondrashov, Dmitri %K Data assimilation %K Instruments and techniques %K Kalman filter %K Modeling and forecasting %K observation error identification %K optimal smoothing %K Prediction %K Radiation belts %K Space weather %B Journal of Geophysical Research: Space Physics %V 119 %P 5725–5743 %G eng %U http://dx.doi.org/10.1002/2014JA019898 %N 7 %R 10.1002/2014JA019898 %0 Journal Article %J Geophysical Research Letters %D 2014 %T Reconstruction of gaps in the past history of solar wind parameters %A Kondrashov, Dmitri %A Denton, R. %A Shprits, Y. Y. %A Singer, H. J. %K gaps %K reconstruction %K singular spectrum analysis %K solar wind %K Solar wind plasma %K Time series analysis %B Geophysical Research Letters %V 41 %P 2702–2707 %G eng %U http://dx.doi.org/10.1002/2014GL059741 %N 8 %R 10.1002/2014GL059741 %0 Journal Article %J Journal of Geophysical Research: Space Physics %D 2014 %T Three-dimensional data assimilation and reanalysis of radiation belt electrons: Observations of a four-zone structure using five spacecraft and the VERB code %A Kellerman, A. C. %A Shprits, Y. Y. %A Kondrashov, Dmitri %A Subbotin, D. %A Makarevich, R. A. %A Donovan, E. %A Nagai, T. %K 3-D %K Data assimilation %K Magnetosphere: inner %K Models %K Numerical modeling %K Radiation belts %K Space radiation environment %K storm %B Journal of Geophysical Research: Space Physics %V 119 %P 8764–8783 %G eng %U http://dx.doi.org/10.1002/2014JA020171 %N 11 %R 10.1002/2014JA020171 %0 Journal Article %J Proceedings of the National Academy of Sciences %D 2014 %T Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances %A Chekroun, Mickaël D. %A Neelin, J. David %A Kondrashov, Dmitri %A McWilliams, James C. %A Ghil, Michael %XDespite 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.

%B Proceedings of the National Academy of Sciences %V 111 %P 1684-1690 %G eng %N 5 %R 10.1073/pnas.1321816111 %0 Journal Article %J Geophysical Research Letters %D 2013 %T Application of a new data operator-splitting data assimilation technique to the 3-D VERB diffusion code and CRRES measurements %A Shprits, Yuri %A Kellerman, Adam %A Kondrashov, Dmitri %A Subbotin, Dmitriy %K Data assimilation %K modeling %K Modeling and forecasting %K Particle acceleration %K Radiation belts %K Wave/particle interactions %B Geophysical Research Letters %V 40 %P 4998–5002 %G eng %U http://dx.doi.org/10.1002/grl.50969 %N 19 %R 10.1002/grl.50969 %0 Journal Article %J Geophysical Research Letters %D 2013 %T Low-order stochastic model and ``past-noise forecasting" of the Madden-Julian oscillation %A Kondrashov, Dmitri %A Chekroun, Mickaël D. %A Robertson, Andrew W. %A Ghil, Michael %B Geophysical Research Letters %V 40 %P 5305–5310 %G eng %R 10.1002/grl.50991 %0 Journal Article %J Advances in Space Research %D 2011 %T Reanalysis of radiation belt electron phase space density using various boundary conditions and loss models %A M. Daae %A Shprits, Y. Y. %A B. Ni %A J. Koller %A Kondrashov, Dmitri %A Y. Chen %B Advances in Space Research %V 48 %P 1327 - 1334 %G eng %U http://www.sciencedirect.com/science/article/pii/S0273117711004741 %N 8 %R http://dx.doi.org/10.1016/j.asr.2011.07.001 %0 Journal Article %J Ocean Science %D 2011 %T An empirical stochastic model of sea-surface temperatures and surface winds over the Southern Ocean %A Kravtsov, Sergey %A Kondrashov, Dmitri %A Kamenkovich, I. %A Ghil, Michael %XThis study employs NASA's recent satellite measurements of sea-surface temperatures (SSTs) and sea-level winds (SLWs) with missing data filled-in by Singular Spectrum Analysis (SSA), to construct empirical models that capture both intrinsic and SST-dependent aspects of SLW variability. The model construction methodology uses a number of algorithmic innovations that are essential in providing stable estimates of the model's propagator. The best model tested herein is able to faithfully represent the time scales and spatial patterns of anomalies associated with a number of distinct processes. These processes range from the daily synoptic variability to interannual signals presumably associated with oceanic or coupled dynamics. Comparing the simulations of an SLW model forced by the observed SST anomalies with the simulations of an SLW-only model provides preliminary evidence for the ocean driving the atmosphere in the Southern Ocean region.

%B Ocean Science %V 7 %P 755–770 %G eng %U http://www.ocean-sci.net/7/755/2011/ %N 6 %R 10.5194/os-7-755-2011 %0 Journal Article %J Nonlinear Processes in Geophysics %D 2011 %T Extreme events: dynamics, statistics and prediction %A Ghil, Michael %A P. Yiou %A Hallegatte, S. %A Malamud, B. D. %A Naveau, P. %A Soloviev, A. %A Friederichs, P. %A Keilis-Borok, V. %A Kondrashov, Dmitri %A Kossobokov, V. %A Mestre, O. %A Nicolis, C. %A Rust, H. W. %A Shebalin, P. %A Vrac, M. %A Witt, A. %A Zaliapin, I. %XWe review work on extreme events, their causes and consequences, by a group of Euro- pean and American researchers involved in a three-year project on these topics. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deteministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction.

%B Nonlinear Processes in Geophysics %V 18 %P 295–350 %G eng %N 3 %R 10.5194/npg-18-295-2011 %0 Journal Article %J Space Weather %D 2011 %T Lognormal Kalman filter for assimilating phase space density data in the radiation belts %A Kondrashov, Dmitri %A Ghil, Michael %A Y. Shprits %B Space Weather %I Wiley Online Library %V 9 %G eng %N 11 %R 10.1029/2011sw000726 %0 Journal Article %J Proceedings of the National Academy of Sciences %D 2011 %T Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation %A Chekroun, Mickaël D. %A Kondrashov, Dmitri %A Ghil, Michael %XInterannual and interdecadal prediction are major challenges of climate dynamics. In this article we develop a prediction method for climate processes that exhibit low-frequency variability (LFV). The method constructs a nonlinear stochastic model from past observations and estimates a path of the “weather” noise that drives this model over previous finite-time windows. The method has two steps: (i) select noise samples—or “snippets”—from the past noise, which have forced the system during short-time intervals that resemble the LFV phase just preceding the currently observed state; and (ii) use these snippets to drive the system from the current state into the future. The method is placed in the framework of pathwise linear-response theory and is then applied to an El Niño–Southern Oscillation (ENSO) model derived by the empirical model reduction (EMR) methodology; this nonlinear model has 40 coupled, slow, and fast variables. The domain of validity of this forecasting procedure depends on the nature of the system’s pathwise response; it is shown numerically that the ENSO model’s response is linear on interannual time scales. As a result, the method’s skill at a 6- to 16-month lead is highly competitive when compared with currently used dynamic and statistic prediction methods for the Niño-3 index and the global sea surface temperature field.

%B Proceedings of the National Academy of Sciences %V 108 %P 11766–11771 %G eng %N 29 %R 10.1073/pnas.1015753108 %0 Book Section %B Stochastic physics and climate modeling. Cambridge University Press, Cambridge %D 2010 %T Empirical model reduction and the modelling hierarchy in climate dynamics and the geosciences %A Kravtsov, Sergey %A Kondrashov, Dmitri %A Ghil, Michael %E P. Williams %E T. Palmer %B Stochastic physics and climate modeling. Cambridge University Press, Cambridge %I Cambridge University Press %P 35–72 %G eng %0 Journal Article %J Geophysical Research Letters %D 2010 %T Gap Filling of Solar Wind Data by Singular Spectrum Analysis %A Kondrashov, Dmitri %A Shprits, Yuri %A Ghil, Michael %XObservational data sets in space physics often contain instrumental and sampling errors, as well as large gaps. This is both an obstacle and an incentive for research, since continuous data sets are typically needed for model formulation and validation. For example, the latest global empirical models of Earth's magnetic field are crucial for many space weather applications, and require time continuous solar wind and interplanetary magnetic field (IMF) data; both of these data sets have large gaps before 1994. Singular spectrum analysis (SSA) reconstructs missing data by using an iteratively inferred, smooth “signal” that captures coherent modes, while “noise” is discarded. In this study, we apply SSA to fill in large gaps in solar wind and IMF data, by combining it with geomagnetic indices that are time continuous, and generalizing it to multivariate geophysical data consisting of gappy “driver” and continuous “response” records. The reconstruction error estimates provide information on the physics of co variability between particular solar wind parameters and geomagnetic indices.

%B Geophysical Research Letters %I CiteSeerX - Scientific Literature Digital Library and Search Engine [http://citeseerx.ist.psu.edu/oai2] (United States) %V 37 %P L15101 %G eng %R 10.1029/2010GL044138 %0 Journal Article %J Physica D: Nonlinear Phenomena %D 2010 %T Reduced models of atmospheric low-frequency variability: Parameter estimation and comparative performance %A Strounine, K. %A Kravtsov, Sergey %A Kondrashov, Dmitri %A Ghil, Michael %XLow-frequency variability (LFV) of the atmosphere refers to its behavior on time scales of 10–100 days, longer than the life cycle of a mid-latitude cyclone but shorter than a season. This behavior is still poorly understood and hard to predict. The present study compares various model reduction strategies that help in deriving simplified models of LFV. Three distinct strategies are applied here to reduce a fairly realistic, high-dimensional, quasi-geostrophic, 3-level (QG3) atmospheric model to lower dimensions: (i) an empirical–dynamical method, which retains only a few components in the projection of the full QG3 model equations onto a specified basis, and finds the linear deterministic and the stochastic corrections empirically as in Selten (1995) [5]; (ii) a purely dynamics-based technique, employing the stochastic mode reduction strategy of Majda et al. (2001) [62]; and (iii) a purely empirical, multi-level regression procedure, which specifies the functional form of the reduced model and finds the model coefficients by multiple polynomial regression as in Kravtsov et al. (2005) [3]. The empirical–dynamical and dynamical reduced models were further improved by sequential parameter estimation and benchmarked against multi-level regression models; the extended Kalman filter was used for the parameter estimation. Overall, the reduced models perform better when more statistical information is used in the model construction. Thus, the purely empirical stochastic models with quadratic nonlinearity and additive noise reproduce very well the linear properties of the full QG3 model’s LFV, i.e. its autocorrelations and spectra, as well as the nonlinear properties, i.e. the persistent flow regimes that induce non-Gaussian features in the model’s probability density function. The empirical–dynamical models capture the basic statistical properties of the full model’s LFV, such as the variance and integral correlation time scales of the leading LFV modes, as well as some of the regime behavior features, but fail to reproduce the detailed structure of autocorrelations and distort the statistics of the regimes. Dynamical models that use data assimilation corrections do capture the linear statistics to a degree comparable with that of empirical–dynamical models, but do much less well on the full QG3 model’s nonlinear dynamics. These results are discussed in terms of their implications for a better understanding and prediction of LFV.

%B Physica D: Nonlinear Phenomena %I Elsevier %V 239 %P 145–166 %G eng %N 3 %R 10.1016/j.physd.2009.10.013 %0 Journal Article %J Journal of the Atmospheric Sciences %D 2010 %T Signatures of nonlinear dynamics in an idealized atmospheric model %A Kondrashov, Dmitri %A Kravtsov, Sergey %A Ghil, Michael %B Journal of the Atmospheric Sciences %V 68 %P 3–12 %G eng %N 1 %R 10.1175/2010jas3524.1 %0 Journal Article %J Journal of Geophysical Research: Space Physics %D 2009 %T Reanalyses of the radiation belt electron phase space density using nearly equatorial CRRES and polar-orbiting Akebono satellite observations %A Ni, Binbin %A Shprits, Yuri %A Nagai, Tsugunobu %A Thorne, Richard %A Chen, Yue %A Kondrashov, Dmitri %A Kim, Hee-jeong %K Charged particle motion and acceleration %K Data assimilation %K data assimilation technique %K Magnetic storms and substorms %K phase space density %K radiation belt electrons %K Radiation belts %K Space radiation environment %B Journal of Geophysical Research: Space Physics %V 114 %P n/a–n/a %G eng %U http://dx.doi.org/10.1029/2008JA013933 %N A5 %R 10.1029/2008JA013933 %0 Journal Article %J Stochastic physics and climate modelling. Cambridge University Press, Cambridge %D 2009 %T Empirical model reduction and the modelling hierarchy in climate dynamics and the geosciences %A Kravtsov, Sergey %A Kondrashov, Dmitri %A Ghil, Michael %X Modern climate dynamics uses a two-fisted approach in attacking and solving the problems of atmospheric and oceanic flows. The two fists are: (i) observational analyses; and (ii) simulations of the geofluids, including the coupled atmosphere–ocean system, using a hierarchy of dynamical models. These models represent interactions between many processes that act on a broad range of spatial and time scales, from a few to tens of thousands of kilometers, and from diurnal to multidecadal, respectively. The evolution of virtual climates simulated by the most detailed and realistic models in the hierarchy is typically as difficult to interpret as that of the actual climate system, based on the available observations thereof. Highly simplified models of weather and climate, though, help gain a deeper understanding of a few isolated processes, as well as giving clues on how the interaction between these processes and the rest of the climate system may participate in shaping climate variability. Finally, models of intermediate complexity, which resolve well a subset of the climate system and parameterise the remainder of the processes or scales of motion, serve as a conduit between the models at the two ends of the hierarchy. We present here a methodology for constructing intermediate mod- els based almost entirely on the observed evolution of selected climate fields, without reference to dynamical equations that may govern this evolution; these models parameterise unresolved processes as multi- variate stochastic forcing. This methodology may be applied with equal success to actual observational data sets, as well as to data sets resulting from a high-end model simulation. We illustrate this methodology by its applications to: (i) observed and simulated low-frequency variability of atmospheric flows in the Northern Hemisphere; (ii) observed evo- lution of tropical sea-surface temperatures; and (iii) observed air–sea interaction in the Southern Ocean. Similar results have been obtained for (iv) radial-diffusion model simulations of Earth’s radiation belts, but are not included here because of space restrictions. In each case, the reduced stochastic model represents surprisingly well a variety of linear and nonlinear statistical properties of the resolved fields. Our methodology thus provides an efficient means of constructing reduced, numerically inexpensive climate models. These models can be thought of as stochastic–dynamic prototypes of more complex deterministic models, as in examples (i) and (iv), but work just as well in the situation when the actual governing equations are poorly known, as in (ii) and (iii). These models can serve as competitive prediction tools, as in (ii), or be included as stochastic parameterisations of certain processes within more complex climate models, as in (iii). Finally, the methodology can be applied, with some modifications, to geophysical problems outside climate dynamics, as illustrated by (iv). %B Stochastic physics and climate modelling. Cambridge University Press, Cambridge %P 35–72 %G eng %0 Journal Article %J Monthly Weather Review %D 2008 %T Data Assimilation for a Coupled Ocean–Atmosphere Model. Part II: Parameter Estimation %A Kondrashov, Dmitri %A Sun, Chaojiao %A Ghil, Michael %XThe parameter estimation problem for the coupled ocean–atmosphere system in the tropical Pacific Ocean is investigated using an advanced sequential estimator [i.e., the extended Kalman filter (EKF)]. The intermediate coupled model (ICM) used in this paper consists of a prognostic upper-ocean model and a diagnostic atmospheric model. Model errors arise from the uncertainty in atmospheric wind stress. First, the state and parameters are estimated in an identical-twin framework, based on incomplete and inaccurate observations of the model state. Two parameters are estimated by including them into an augmented state vector. Model-generated oceanic datasets are assimilated to produce a time-continuous, dynamically consistent description of the model’s El Niño–Southern Oscillation (ENSO). State estimation without correcting erroneous parameter values still permits recovering the true state to a certain extent, depending on the quality and accuracy of the observations and the size of the discrepancy in the parameters. Estimating both state and parameter values simultaneously, though, produces much better results. Next, real sea surface temperatures observations from the tropical Pacific are assimilated for a 30-yr period (1975–2004). Estimating both the state and parameters by the EKF method helps to track the observations better, even when the ICM is not capable of simulating all the details of the observed state. Furthermore, unobserved ocean variables, such as zonal currents, are improved when model parameters are estimated. A key advantage of using this augmented-state approach is that the incremental cost of applying the EKF to joint state and parameter estimation is small relative to the cost of state estimation alone. A similar approach generalizes various reduced-state approximations of the EKF and could improve simulations and forecasts using large, realistic models.

%B Monthly Weather Review %V 136 %P 5062–5076 %G eng %R 10.1175/2008MWR2544.1 %0 Journal Article %J Journal of Geophysical Research: Space Physics %D 2007 %T A Kalman filter technique to estimate relativistic electron lifetimes in the outer radiation belt %A Kondrashov, Dmitri %A Y. Shprits %A Ghil, Michael %A Thorne, R. %B Journal of Geophysical Research: Space Physics %I Wiley Online Library %V 112 %G eng %N A10 %R 10.1029/2007ja012583 %0 Journal Article %J Climate Dynamics %D 2007 %T Predicting weather regime transitions in Northern Hemisphere datasets %A Kondrashov, Dmitri %A Shen, Jie %A Berk, Richard %A D'Andrea, Fabio %A Ghil, Michael %B Climate Dynamics %I Springer %V 29 %P 535–551 %G eng %N 5 %R 10.1007/s00382-007-0293-2 %0 Journal Article %J Journal of Geophysical Research: Space Physics %D 2007 %T Reanalysis of relativistic radiation belt electron fluxes using CRRES satellite data, a radial diffusion model, and a Kalman filter %A Shprits, Yuri %A Kondrashov, Dmitri %A Chen, Yue %A Thorne, Richard %A Ghil, Michael %A Friedel, Reiner %A Reeves, Geoff %B Journal of Geophysical Research: Space Physics %I Wiley Online Library %V 112 %G eng %N A12 %R 10.1029/2007ja012579 %0 Journal Article %J Nonlinear Processes in Geophysics %D 2007 %T Reply to T. Schneider's comment on "Spatio-temporal filling of missing points in geophysical data sets" %A Kondrashov, Dmitri %A Ghil, Michael %B Nonlinear Processes in Geophysics %V 14 %P 3–4 %G eng %N 1 %R 10.5194/npg-14-3-2007 %0 Journal Article %J Journal of the Atmospheric Sciences %D 2006 %T Empirical Mode Reduction in a Model of Extratropical Low-Frequency Variability %A Kondrashov, Dmitri %A Kravtsov, S. %A M. Ghil %B Journal of the Atmospheric Sciences %V 63 %P 1859-1877 %G eng %U http://dx.doi.org/10.1175/JAS3719.1 %N 7 %R 10.1175/JAS3719.1 %0 Journal Article %J Journal of the Atmospheric Sciences %D 2006 %T Empirical mode reduction in a model of extratropical low-frequency variability %A Kondrashov, Dmitri %A Kravtsov, S %A Ghil, M %XThis paper constructs and analyzes a reduced nonlinear stochastic model of extratropical low-frequency variability. To do so, it applies multilevel quadratic regression to the output of a long simulation of a global baroclinic, quasigeostrophic, three-level (QG3) model with topography; the model's phase space has a dimension of O(104). The reduced model has 45 variables and captures well the non-Gaussian features of the QG3 model's probability density function (PDF). In particular, the reduced model's PDF shares with the QG3 model its four anomalously persistent flow patterns, which correspond to opposite phases of the Arctic Oscillation and the North Atlantic Oscillation, as well as the Markov chain of transitions between these regimes. In addition, multichannel singular spectrum analysis identifies intraseasonal oscillations with a period of 35–37 days and of 20 days in the data generated by both the QG3 model and its low-dimensional analog. An analytical and numerical study of the reduced model starts with the fixed points and oscillatory eigenmodes of the model's deterministic part and uses systematically an increasing noise parameter to connect these with the behavior of the full, stochastically forced model version. The results of this study point to the origin of the QG3 model's multiple regimes and intraseasonal oscillations and identify the connections between the two types of behavior.

%B Journal of the Atmospheric Sciences %V 63 %P 1859–1877 %G eng %N 7 %R 10.1175/JAS3719.1 %0 Journal Article %J Nonlinear Processes in Geophysics %D 2006 %T Spatio-temporal filling of missing points in geophysical data sets %A Kondrashov, Dmitri %A Ghil, Michael %XThe majority of data sets in the geosciences are obtained from observations and measurements of natural systems, rather than in the laboratory. These data sets are often full of gaps, due to to the conditions under which the measurements are made. Missing data give rise to various problems, for example in spectral estimation or in specifying boundary conditions for numerical models. Here we use Singular Spectrum Analysis (SSA) to fill the gaps in several types of data sets. For a univariate record, our procedure uses only temporal correlations in the data to fill in the missing points. For a multivariate record, multi-channel SSA (M-SSA) takes advantage of both spatial and temporal correlations. We iteratively produce estimates of missing data points, which are then used to compute a self-consistent lag-covariance matrix; cross-validation allows us to optimize the window width and number of dominant SSA or M-SSA modes to fill the gaps. The optimal parameters of our procedure depend on the distribution in time (and space) of the missing data, as well as on the variance distribution between oscillatory modes and noise. The algorithm is demonstrated on synthetic examples, as well as on data sets from oceanography, hydrology, atmospheric sciences, and space physics: global sea-surface temperature, flood-water records of the Nile River, the Southern Oscillation Index (SOI), and satellite observations of relativistic electrons.

%B Nonlinear Processes in Geophysics %V 13 %P 151–159 %G eng %N 2 %R 10.5194/npg-13-151-2006 %0 Journal Article %J Journal of climate %D 2005 %T A hierarchy of data-based ENSO models %A Kondrashov, Dmitri %A Kravtsov, S %A Robertson, Andrew W. %A Ghil, Michael %XGlobal sea surface temperature (SST) evolution is analyzed by constructing predictive models that best describe the dataset’s statistics. These inverse models assume that the system’s variability is driven by spatially coherent, additive noise that is white in time and are constructed in the phase space of the dataset’s leading empirical orthogonal functions. Multiple linear regression has been widely used to obtain inverse stochastic models; it is generalized here in two ways. First, the dynamics is allowed to be nonlinear by using polynomial regression. Second, a multilevel extension of classic regression allows the additive noise to be correlated in time; to do so, the residual stochastic forcing at a given level is modeled as a function of variables at this level and the preceding ones. The number of variables, as well as the order of nonlinearity, is determined by optimizing model performance. The two-level linear and quadratic models have a better El Niño–Southern Oscillation (ENSO) hindcast skill than their one-level counterparts. Estimates of skewness and kurtosis of the models’ simulated Niño-3 index reveal that the quadratic model reproduces better the observed asymmetry between the positive El Niño and negative La Niña events. The benefits of the quadratic model are less clear in terms of its overall, cross-validated hindcast skill; this model outperforms, however, the linear one in predicting the magnitude of extreme SST anomalies. Seasonal ENSO dependence is captured by incorporating additive, as well as multiplicative forcing with a 12-month period into the first level of each model. The quasi-quadrennial ENSO oscillatory mode is robustly simulated by all models. The “spring barrier” of ENSO forecast skill is explained by Floquet and singular vector analysis, which show that the leading ENSO mode becomes strongly damped in summer, while nonnormal optimum growth has a strong peak in December.

%B Journal of climate %V 18 %P 4425–4444 %G eng %N 21 %R 10.1175/JCLI3567.1 %0 Journal Article %J Journal of Climate %D 2005 %T Multilevel regression modeling of nonlinear processes: Derivation and applications to climatic variability %A Kravtsov, S %A Kondrashov, Dmitri %A Ghil, M %XPredictive models are constructed to best describe an observed field’s statistics within a given class of nonlinear dynamics driven by a spatially coherent noise that is white in time. For linear dynamics, such inverse stochastic models are obtained by multiple linear regression (MLR). Nonlinear dynamics, when more appropriate, is accommodated by applying multiple polynomial regression (MPR) instead; the resulting model uses polynomial predictors, but the dependence on the regression parameters is linear in both MPR and MLR. The basic concepts are illustrated using the Lorenz convection model, the classical double-well problem, and a three-well problem in two space dimensions. Given a data sample that is long enough, MPR successfully reconstructs the model coefficients in the former two cases, while the resulting inverse model captures the three-regime structure of the system’s probability density function (PDF) in the latter case. A novel multilevel generalization of the classic regression procedure is introduced next. In this generalization, the residual stochastic forcing at a given level is subsequently modeled as a function of variables at this level and all the preceding ones. The number of levels is determined so that the lag-0 covariance of the residual forcing converges to a constant matrix, while its lag-1 covariance vanishes. This method has been applied to the output of a three-layer, quasigeostrophic model and to the analysis of Northern Hemisphere wintertime geopotential height anomalies. In both cases, the inverse model simulations reproduce well the multiregime structure of the PDF constructed in the subspace spanned by the dataset’s leading empirical orthogonal functions, as well as the detailed spectrum of the dataset’s temporal evolution. These encouraging results are interpreted in terms of the modeled low-frequency flow’s feedback on the statistics of the subgrid-scale processes.

%B Journal of Climate %V 18 %P 4404–4424 %G eng %N 21 %R 10.1175/JCLI3544.1 %0 Journal Article %J Geophysical Research Letters %D 2005 %T Oscillatory modes of extended Nile River records (A.D. 622–1922) %A Kondrashov, Dmitri %A Feliks, Yizhak %A Ghil, Michael %XThe historical records of the low- and high-water levels of the Nile River are among the longest climatic records that have near-annual resolution. There are few gaps in the first part of the records (A.D. 622-1470) and larger gaps later (A.D. 1471-1922). We apply advanced spectral methods, Singular-Spectrum Analysis (SSA) and the Multi-Taper Method (MTM), to fill the gaps and to locate interannual and interdecadal periodicities. The gap filling uses a novel, iterative version of SSA. Our analysis reveals several statistically significant features of the records: a nonlinear, data-adaptive trend that includes a 256-year cycle, a quasi-quadriennial (4.2-year) and a quasi-biennial (2.2-year) mode, as well as additional periodicities of 64, 19, 12, and, most strikingly, 7 years. The quasi-quadriennial and quasi-biennial modes support the long-established connection between the Nile River discharge and the El-Niño/Southern Oscillation (ENSO) phenomenon in the Indo-Pacific Ocean. The longest periods might be of astronomical origin. The 7-year periodicity, possibly related to the biblical cycle of lean and fat years, seems to be due to North Atlantic influences.

%B Geophysical Research Letters %I AGU %V 32 %P L10702 %8 may %G eng %N 10 %R 10.1029/2004GL022156 %0 Journal Article %J Journal of the Atmospheric Sciences %D 2004 %T Weather regimes and preferred transition paths in a three-level quasigeostrophic model %A Kondrashov, Dmitri %A Ide, K. %A Ghil, Michael %B Journal of the Atmospheric Sciences %V 61 %P 568–587 %G eng %N 5 %R 10.1175/1520-0469(2004)061<0568:wraptp>2.0.co;2 %0 Conference Paper %B Proceedings ECMWF/CLIVAR Workshop on Simulation and Prediction of Intra-Seasonal Variability with Emphasis on the MJO %D 2003 %T Intraseasonal oscillations in the mid-latitudes: observations, theory, and GCM results %A Ghil, Michael %A Kondrashov, Dmitri %A Lott, F. %A Robertson, Andrew W. %B Proceedings ECMWF/CLIVAR Workshop on Simulation and Prediction of Intra-Seasonal Variability with Emphasis on the MJO %P 3–6 %G eng %0 Journal Article %J Reviews of Geophysics %D 2002 %T Advanced spectral methods for climatic time series %A Ghil, Michael %A M. R. Allen %A M. D. Dettinger %A Ide, Kayo %A Kondrashov, Dmitri %A M. E. Mann %A Robertson, Andrew W. %A A. Saunders %A Y. Tian %A Varadi, Ferenc %A Yiou, Pascal %B Reviews of Geophysics %V 40 %P 1–41 %G eng %N 1 %R 10.1029/2000RG000092 %0 Conference Paper %B AMS Symposium on Observations, Data Assimilation, and Probabilistic Prediction %D 2002 %T Data Assimilation and Weather Regimes in a Three-Level Quasi-Geostrophic Model %A Kondrashov, Dmitri %A Ghil, Michael %A Ide, K. %A Todling, R. %B AMS Symposium on Observations, Data Assimilation, and Probabilistic Prediction %G eng %0 Journal Article %J Journal of Plasma Physics %D 1999 %T Three-dimensional deformable-grid electromagnetic particle-in-cell for parallel computers %A J. WANG %A Kondrashov, Dmitri %A P. C. LIEWER %A S. R. KARMESIN %XWe describe a new parallel, non-orthogonal-grid, three-dimensional electromagnetic particle-in-cell (EMPIC) code based on a finite-volume formulation. This code uses a logically Cartesian grid of deformable hexahedral cells, a discrete surface integral (DSI) algorithm to calculate the electromagnetic field, and a hybrid logical‚Äìphysical space algorithm to push particles. We investigate the numerical instability of the DSI algorithm for non-orthogonal grids, analyse the accuracy for EMPIC simulations on non-orthogonal grids, and present performance benchmarks of this code on a parallel supercomputer. While the hybrid particle push algorithm has a second-order accuracy in space, the accuracy of the DSI field solve algorithm is between first and second order for non-orthogonal grids. The parallel implementation of this code, which is almost identical to that of a Cartesian-grid EMPIC code using domain decomposition, achieved a high parallel efficiency of over 96% for large-scal" # "e simulations.

%B Journal of Plasma Physics %I Cambridge University Press %V 61 %P 367-389 %8 04 %G eng %U https://www.cambridge.org/core/article/three-dimensional-deformable-grid-electromagnetic-particle-in-cell-for-parallel-computers/162F88C631B4741CB569FA2BE4BBD14A %N 3