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In chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real-time knowledge about the error growth could enable strategies to adjust the modelling and forecasting infrastructure on-the-fly to increase accuracy or reduce computation time. One could, e.g., change the ensemble size, or the distribution and type of target observations. Local Lyapunov exponents are known indicators of the rate at which very small prediction errors grow over a finite time interval. However, their computation is very expensive: it requires maintaining and evolving a tangent linear model, orthogonalisation algorithms and storing large matrices. In this feasibility study, we investigate the accuracy of supervised machine learning in estimating the current local Lyapunov exponents, from input of current and recent time steps of the system trajectory, as an alternative to the classical method. Thus machine learning is not used here to emulate a physical model or some of its components, but non intrusively as a complementary tool. We test four popular supervised learning algorithms: regression trees, multilayer perceptrons, convolutional neural networks and long short-term memory networks. Experiments are conducted on two low-dimensional chaotic systems of ordinary differential equations, the R\"ossler and the Lorenz 63 models. We find that on average the machine learning algorithms predict the stable local Lyapunov exponent accurately, the unstable exponent reasonably accurately, and the neutral exponent only somewhat accurately. We show that greater prediction accuracy is associated with local homogeneity of the local Lyapunov exponents on the system attractor. Importantly, the situations in which (forecast) errors grow fastest are not necessarily the same as those where it is more difficult to predict local Lyapunov exponents with machine learning., Comment: 37 pages, 10 Figures

We use complex network theory to better represent and understand the ecosystem connectivity in a shelf-sea environment. The baseline data used for the analysis are obtained from a state-of-the art coupled marine physics-biogeochemistry model simulating the North-West European Shelf (NWES). The complex network built on model outputs is used to identify the functional types of variables behind the biogeochemistry dynamics, suggesting how to simplify our understanding of the complex web of interactions within the shelf-sea ecosystem. We demonstrate that complex networks can be also used to understand spatial ecosystem connectivity, both identifying the (geographically varying) connectivity lengthscales and the clusters of spatial locations that are connected. These clusters indicate geographic regions where there is a substantial flow of information between the degrees of freedom within the ecosystem, while information exchange across the boundaries of these regions is limited. The results of this study help to understand how natural, or antrophogenic, perturbations propagate through the shelf-sea ecosystem, and can be used in multiple future applications such as stochastic noise modelling, data assimilation, or machine learning.

We introduce a proof of concept to parametrise the unresolved subgrid scale of sea-ice dynamics with deep learning techniques. Instead of parametrising single processes, a single neural network is trained to correct all model variables at the same time. This data-driven approach is applied to a regional sea-ice model that accounts exclusively for dynamical processes with a Maxwell elasto-brittle rheology. Driven by an external wind forcing in a 40 km×200 km domain, the model generates examples of sharp transitions between unfractured and fully fractured sea ice. To correct such examples, we propose a convolutional U-Net architecture which extracts features at multiple scales. We test this approach in twin experiments: the neural network learns to correct forecasts from low-resolution simulations towards high-resolution simulations for a lead time of about 10 min. At this lead time, our approach reduces the forecast errors by more than 75 %, averaged over all model variables. As the most important predictors, we identify the dynamics of the model variables. Furthermore, the neural network extracts localised and directional-dependent features, which point towards the shortcomings of the low-resolution simulations. Applied to correct the forecasts every 10 min, the neural network is run together with the sea-ice model. This improves the short-term forecasts up to an hour. These results consequently show that neural networks can correct model errors from the subgrid scale for sea-ice dynamics. We therefore see this study as an important first step towards hybrid modelling to forecast sea-ice dynamics on an hourly to daily timescale.

Advanced data assimilation (DA) methods, widely used in geophysical and climate studies to merge observations with numerical models, can improve state estimates and consequent forecasts. We interface the deterministic ensemble Kalman filter (DEnKF) to the Lagrangian neXt generation Sea Ice Model, neXtSIM. The ensemble is generated by perturbing the atmospheric and oceanic forcing throughout the simulations and randomly initialized ice cohesion. Our ensemble–DA system assimilates sea ice concentration (SIC) from the Ocean and Sea Ice Satellite Application Facility (OSI-SAF) and sea ice thickness (SIT) from the merged CryoSat-2 and SMOS datasets (CS2SMOS). Because neXtSIM is computationally solved on a time-dependent evolving mesh, it is a challenging application for ensemble–DA. As a solution, we perform the DEnKF analysis on a fixed and regular reference mesh, on which model variables are interpolated before the DA and then back to each member's mesh after the DA. We evaluate the impact of assimilating different types of sea ice observations on the model's forecast skills of the Arctic sea ice by comparing satellite observations and a free-run ensemble in an Arctic winter period, 2019–2020. Significant improvements in modeled SIT indicate the importance of assimilating weekly CS2SMOS SIT, while the improvements of SIC and ice extent are moderate but benefit from daily ingestion of the OSI-SAF SIC. For most of the winter, the correlation between SIT and SIC is weaker, which results in little cross-inference between the two variables in the assimilation step. Overall, the ensemble–DA system based on the stand-alone sea ice model demonstrates the feasibility of winter Arctic sea ice prediction with good computational efficiency. These results open the path toward operational implementation and the extension to multi-year assimilation.

Advanced data assimilation (DA) methods, widely used in geophysical and climate studies to merge observations with numerical models, can improve state estimates and consequent forecasts. We interface the deterministic ensemble Kalman filter (DEnKF) to the Lagrangian neXt generation Sea Ice Model, neXtSIM. The ensemble is generated by perturbing the atmospheric and oceanic forcing throughout the simulations and randomly initialized ice cohesion. Our ensemble–DA system assimilates sea ice concentration (SIC) from the Ocean and Sea Ice Satellite Application Facility (OSI-SAF) and sea ice thickness (SIT) from the merged CryoSat-2 and SMOS datasets (CS2SMOS). Because neXtSIM is computationally solved on a time-dependent evolving mesh, it is a challenging application for ensemble–DA. As a solution, we perform the DEnKF analysis on a fixed and regular reference mesh, on which model variables are interpolated before the DA and then back to each member's mesh after the DA. We evaluate the impact of assimilating different types of sea ice observations on the model's forecast skills of the Arctic sea ice by comparing satellite observations and a free-run ensemble in an Arctic winter period, 2019–2020. Significant improvements in modeled SIT indicate the importance of assimilating weekly CS2SMOS SIT, while the improvements of SIC and ice extent are moderate but benefit from daily ingestion of the OSI-SAF SIC. For most of the winter, the correlation between SIT and SIC is weaker, which results in little cross-inference between the two variables in the assimilation step. Overall, the ensemble–DA system based on the stand-alone sea ice model demonstrates the feasibility of winter Arctic sea ice prediction with good computational efficiency. These results open the path toward operational implementation and the extension to multi-year assimilation.

At the height of the COVID-19 pandemic, an international team of mathematicians borrowed techniques from the geosciences to predict the complex and shifting dynamics of the virus’ spread. The technique known as data assimilation combines numerical model data with fresh observational data to deliver more accurate forecasts. Validating the method in eight distinct countries, the team demonstrated the potential to reasonably and accurately predict the short-term impacts of various reopening measures on virus transmission. This method can provide critical information to policymakers to make informed decisions and design effective policies to mitigate the pandemic’s impacts.

Data assimilation for systems possessing many scales of motions is a substantial methodological and technological challenge. Systems with these features are found in many areas of computational physics and are becoming common thanks to increased computational power allowing to resolve finer scales and to couple together several sub-components. Coupled data assimilation (CDA) distinctively appears as a main concern in numerical weather and climate prediction with major efforts put forward by meteo services worldwide. The core issue is the scale separation acting as a barrier that hampers the propagation of the information across model components (e.g. ocean and atmosphere). We provide a brief survey of CDA, and then focus on CDA using the ensemble Kalman filter (EnKF), a widely used Monte Carlo Gaussian method. Our goal is to elucidate the mechanisms behind information propagation across model components. We consider first a coupled system of equations with temporal scale difference, and deduce that: (i) cross components effects are strong from the slow to the fast scale, but, (ii) intra-component effects are much stronger in the fast scale. While observing the slow scale is desirable and benefits the fast, the latter must be observed with high frequency otherwise the error will grow up to affect the slow scale. Numerical experiments are performed using the atmosphere-ocean model, MAOOAM. Six configurations are considered, differing for the strength of the atmosphere-ocean coupling and/or the number of model modes. The performance of the EnKF depends on the model configuration, i.e. on its dynamical features. A comprehensive dynamical characterisation of the model configurations is provided by examining the Lyapunov spectrum, Kolmogorov entropy and Kaplan–Yorke attractor dimension. We also compute the covariant Lyapunov vectors and use them to explain how model instabilities act on different model’s modes according to the coupling strength. The experiments confirm the importance of observing the fast scale, but show also that, despite its slow temporal scale, frequent observations in the ocean are beneficial. The relation between the ensemble size, N, and the unstable subspace dimension, $$n_0$$ n 0 , has been studied. Results largely ratify what known for uncoupled system: the condition $$N\ge n_0$$ N ≥ n 0 is necessary for the EnKF to work satisfactorily. Nevertheless the quasi-degeneracy of the Lyapunov spectrum of MAOOAM, with many near-zero exponents, is potentially the cause of the smooth gradual reduction of the analysis error observed for some model configurations, even when $$N>n_0$$ N > n 0 . Future prospects for the EnKF in the context of coupled ocean-atmosphere systems are finally discussed.

Weather and climate are well known exemplars of chaotic systems exhibiting extreme sensitivity to initial conditions. Initial condition errors are subject to exponential growth on average, but the rate and the characteristic of such growth is highly state dependent. In an ideal setting where the degree of predictability of the system is known in real-time, it may be possible and beneficial to take adaptive measures. For instance a local decrease of predictability may be counteracted by increasing the time- or space-resolution of the model computation or the ensemble size in the context of ensemble-based data assimilation or probabilistic forecasting.Local Lyapunov exponents (LLEs) describe growth rates along a finite-time section of a system trajectory. This makes the LLEs the ideal quantities to measure the local degree of predictability, yet a main bottleneck for their real-time use in operational scenarios is the huge computational cost. Calculating LLEs involves computing a long trajectory of the system, propagating perturbations with the tangent linear model, and repeatedly orthogonalising them. We investigate if machine learning (ML) methods can estimate the LLEs based only on information from the system’s solution, thus avoiding the need to evolve perturbations via the tangent linear model. We test the ability of four algorithms (regression tree, multilayer perceptron, convolutional neural network and long short-term memory network) to perform this task in two prototypical low dimensional chaotic dynamical systems. Our results suggest that the accuracy of the ML predictions is highly dependent upon the nature of the distribution of the LLE values in phase space: large prediction errors occur in regions of the attractor where the LLE values are highly non-smooth. In line with classical dynamical systems studies, the neutral LLE is more difficult to predict. We show that a comparatively simple regression tree can achieve performance that is similar to sophisticated neural networks, and that the success of ML strategies for exploiting the temporal structure of data depends on the system dynamics.

Advanced data assimilation methods can improve the forecast of Arctic sea ice, which has been widely used in climate modeling systems to merge observations into simulations. We apply the deterministic Ensemble Kalman filter (DEnKF) to a Lagrangian sea ice model, neXtSIM for sea ice forecast. neXtSIM is computationally solved on a time-dependent evolving mesh, causing a key challenge for applying the EnKF since the mesh grid number and positions are generally different in each ensemble member. The DEnKF analysis is performed on a fixed reference mesh, where model variables are interpolated between the reference mesh and the individual ensemble meshes before and after the assimilation. An ensemble-DA forecasting system for Arctic sea ice forecast based on neXtSIM is built by assimilating the OSI-SAF sea ice concentration (SIC) and the CS2SMOS sea ice thickness (SIT). The ensemble is generated by perturbing atmospheric and oceanic forcing online throughout the forecast. We evaluate the impact of sea-ice assimilation on the Arctic winter sea-ice forecast skills against the satellite observations and a free run during the 2019-2020 Arctic winter. Significant improvements in modeled SIT indicate the importance of assimilating CS2SMOS thickness. While the improvement of SIC and ice extend are clearly observed only in the case with daily assimilating OSI-SAF SIC, which avoids the constraint of daily loaded ocean variables. We found that assimilating a special observation gives the best forecast skill of the relevant variables. With a proper assimilation strategy, neXtSIM as a stand-alone sea ice model could perform computationally efficiently and maintain good forecast skills compared with coupled models.

In this talk, we propose to use neural networks in a hybrid modelling setup to learn sub-grid-scale dynamics of sea-ice that cannot be resolved by geophysical models. The multifractal and stochastic nature of the sea-ice dynamics create significant obstacles to represent such dynamics with neural networks. Here, we will introduce and screen specific neural network architectures that might be suited for this kind of task. To prove our concept, we perform idealised twin experiments in a simplified Maxwell-Elasto-Brittle sea-ice model which includes only sea-ice dynamics within a channel-like setup. In our experiments, we use high-resolution runs as proxy for the reality, and we train neural networks to correct errors of low-resolution forecast runs.Since we perform the two kind of runs on different grids, we need to define a projection operator from high- to low-resolution. In practice, we compare the low-resolution forecasted state at a given time to the projected state of the high resolution run at the same time. Using a catalogue of these forecasted and projected states, we will learn and screen different neural network architectures with supervised training in an offline learning setting. Together with this simplified training, the screening helps us to select appropriate architectures for the representation of multifractality and stochasticity within the sea-ice dynamics. As a next step, these screened architectures have to be scaled to larger and more complex sea-ice models like neXtSIM.

Chaos is ubiquitous in physical systems. The associated sensitivity to initial conditions is a significant obstacle in forecasting the weather and other geophysical fluid flows. Data assimilation is the process whereby the uncertainty in initial conditions is reduced by the astute combination of model predictions and real-time data. This chapter reviews recent findings from investigations on the impact of chaos on data assimilation methods: for the Kalman filter and smoother in linear systems, analytic results are derived; for their ensemble-based versions and nonlinear dynamics, numerical results provide insights. The focus is on characterizing the asymptotic statistics of the Bayesian posterior in terms of the dynamical instabilities, differentiating between deterministic and stochastic dynamics. We also present two novel results. Firstly, we study the functioning of the ensemble Kalman filter in the context of a chaotic, coupled, atmosphere-ocean model with a quasi-degenerate spectrum of Lyapunov exponents, showing the importance of having sufficient ensemble members to track all of the near-null modes. Secondly, for the fully non-Gaussian method of the particle filter, numerical experiments are conducted to test whether the curse of dimensionality can be mitigated by discarding observations in the directions of little dynamical growth of uncertainty. The results refute this option, most likely because the particles already embody this information on the chaotic system. The results also suggest that it is the rank of the unstable-neutral subspace of the dynamics, and not that of the observation operator, that determines the required number of particles. We finally discuss how knowledge of the random attractor can play a role in the development of future data assimilation schemes for chaotic multiscale systems with large scale separation.

As an alternative to using the standard multi-model ensemble (MME) approach to combine the output of different models to improve prediction skill, models can also be combined dynamically to form a so-called supermodel. The supermodel approach enables a quicker correction of the model errors. In this study we connect different versions of SPEEDO, a global atmosphere-ocean-land model of intermediate complexity, into a supermodel. We focus on a weighted supermodel, in which the supermodel state is a weighted superposition of different imperfect model states. The estimation, “the training”, of the optimal weights of this combination is a critical aspect in the construction of a supermodel. In our previous works two algorithms were developed: (i) cross pollination in time (CPT)-based technique and (ii) a synchronization-based learning rule (synch rule). Those algorithms have so far been applied under the assumption of complete and noise-free observations. Here we go beyond and consider the more realistic case of noisy data that do not cover the full system's state and are not taken at each model's computational time step. We revise the training methods to cope with this observational scenario, while still being able to estimate accurate weights. In the synch rule an additional term is introduced to maintain physical balances, while in CPT nudging terms are added to let the models stay closer to the observations during training. Furthermore, we propose a novel formulation of the CPT method allowing the weights to be negative. This makes it possible for CPT to deal with cases in which the individual model biases have the same sign, a situation that hampers constructing a skillfully weighted supermodel based on positive weights. With these developments, both CPT and the synch rule have been made suitable to train a supermodel consisting of state of the art weather and climate models.

In alternative to using the standard multi-model ensemble (MME) approach to combine the output of different models to improve prediction skill, models can also be combined dynamically to form a so-called supermodel. The supermodel approach allows for a quicker correction of the model errors. In this study we focus on weighted supermodels, in which the supermodel state is a weighted superposition of different imperfect model states. The estimation, “the training”, of the optimal weights of this combination is a critical aspect in the construction of a supermodel. In our previous works two algorithms were developed: (i) cross pollination in time (CPT-based technique), and, (ii) a synchronization based learning rule (synch rule). Those algorithms have been so far applied under the assumption of complete and noise-free observations. Here we go beyond and consider the more realistic case of noisy data that do not cover the full system's state and are not taken at each model's computational time step. We revise the training methods to cope with this observational scenario, while still being able to estimate accurate weights. In the synch rule an additional term is introduced to maintain physical balances, while in CPT nudging terms are added to let the models stay closer to the observations during training. Furthermore, we propose a novel formulation of the CPT method allowing for the weights to be negative. This makes it possible for CPT to deal with cases in which the individual model biases have the same sign, a situation that hampers constructing a skilful weighted supermodel based on positive weights. With these developments, both CPT and the synch rule have been made suitable to train a supermodel consisting of state-of-the-art weather or climate models.

We study the response of the Lagrangian sea ice model neXtSIM to the uncertainty in the sea surface wind and sea ice cohesion. The ice mechanics in neXtSIM is based on a brittle-like rheological framework. The study considers short-term ensemble forecasts of the Arctic sea ice from January to April 2008. Ensembles are generated by perturbing the wind inputs and ice cohesion field both separately and jointly. The resulting uncertainty in the probabilistic forecasts is evaluated statistically based on the analysis of Lagrangian sea ice trajectories as sampled by virtual drifters seeded in the model to cover the Arctic Ocean and using metrics borrowed from the search-and-rescue literature. The comparison among the different ensembles indicates that wind perturbations dominate the forecast uncertainty i.e. the absolute spread of the ensemble, while the inhomogeneities in the ice cohesion field significantly increase the degree of anisotropy in the spread i.e. trajectories drift differently in different directions. We suggest that in order to get a full flavor of uncertainties in a sea ice model with brittle-like rheologies, to predict sea ice drift and trajectories, one should consider using ensemble-based simulations where both wind forcing and sea ice cohesion are perturbed., Comment: 18 pages, 10 figures

In recent years, machine learning (ML) has been proposed to devise data-driven parametrizations of unresolved processes in dynamical numerical models. In most cases, the ML training leverages high-resolution simulations to provide a dense, noiseless target state. Our goal is to go beyond the use of high-resolution simulations and train ML-based parametrization using direct data, in the realistic scenario of noisy and sparse observations. The algorithm proposed in this work is a two-step process. First, data assimilation (DA) techniques are applied to estimate the full state of the system from a truncated model. The unresolved part of the truncated model is viewed as a model error in the DA system. In a second step, ML is used to emulate the unresolved part, a predictor of model error given the state of the system. Finally, the ML-based parametrization model is added to the physical core truncated model to produce a hybrid model. The DA component of the proposed method relies on an ensemble Kalman filter while the ML parametrization is represented by a neural network. The approach is applied to the two-scale Lorenz model and to MAOOAM, a reduced-order coupled ocean-atmosphere model. We show that in both cases, the hybrid model yields forecasts with better skill than the truncated model. Moreover, the attractor of the system is significantly better represented by the hybrid model than by the truncated model. This article is part of the theme issue ‘Machine learning for weather and climate modelling’.

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This work shows how one can use iterative ensemble smoothers to effectively estimate parameters of an SEIR model with age-classes and compartments of sick, hospitalized, and dead. The data conditioned on are the daily numbers of accumulated deaths and the number of hospitalized. Also, it is possible to condition on the number of cases obtained from testing. We start from a wide prior distribution for the model parameters; then, the ensemble conditioning leads to a posterior ensemble of estimated parameters leading to model predictions in close agreement with the observations. The updated ensemble of model simulations have predictive capabilities and include uncertainty estimates. In particular, we estimate the effective reproductive number as a function of time, and we can assess the impact of different intervention measures. By starting from the updated set of model parameters, we can make accurate short-term predictions of the epidemic development given knowledge of the future effective reproductive number. Also, the model system allows for the computation of long-term scenarios of the epidemic under different assumptions. We have applied the model system on data sets from several countries with vastly different developments of the epidemic, and we can accurately model the development of the COVID-19 outbreak in these countries. We realize that more complex models, e.g., with regional compartments, may be desirable, and we suggest that the approach used here should be applicable also for these models.

Can we build a machine learning parametrization in a numerical model using sparse and noisy observations?In recent years, machine learning (ML) has been proposed to devise data-driven parametrizations of unresolved processes in dynamical numerical models. In most of the cases, ML is trained by coarse-graining high-resolution simulations to provide a dense, unnoisy target state (or even the tendency of the model).Our goal is to go beyond the use of high-resolution simulations and train ML-based parametrization using direct data. Furthermore, we intentionally place ourselves in the realistic scenario of noisy and sparse observations.The algorithm proposed in this work derives from the algorithm presented by the same authors in https://arxiv.org/abs/2001.01520.The principle is to first apply data assimilation (DA) techniques to estimate the full state of the system from a non-parametrized model, referred hereafter as the physical model. The parametrization term to be estimated is viewed as a model error in the DA system. In a second step, ML is used to define the parametrization, e.g., a predictor of the model error given the state of the system. Finally, the ML system is incorporated within the physical model to produce a hybrid model, combining a physical core with a ML-based parametrization.The approach is applied to dynamical systems from low to intermediate complexity. The DA component of the proposed approach relies on an ensemble Kalman filter/smoother while the parametrization is represented by a convolutional neural network. We show that the hybrid model yields better performance than the physical model in terms of both short-term (forecast skill) and long-term (power spectrum, Lyapunov exponents) properties. Sensitivity to the noise and density of observation is also assessed.

The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically.

We present a sensitivity analysis and discuss the probabilistic forecast capabilities of the novel sea ice model neXtSIM used in hindcast mode. The study pertains to the response of the model to the uncertainty on winds using probabilistic forecasts of ice trajectories. neXtSIM is a continuous Lagrangian numerical model that uses an elasto-brittle rheology to simulate the ice response to external forces. The sensitivity analysis is based on a Monte Carlo sampling of 12 members. The response of the model to the uncertainties is evaluated in terms of simulated ice drift distances from their initial positions, and from the mean position of the ensemble, over the mid-term forecast horizon of 10 days. The simulated ice drift is decomposed into advective and diffusive parts that are characterised separately both spatially and temporally and compared to what is obtained with a free-drift model, that is, when the ice rheology does not play any role in the modelled physics of the ice. The seasonal variability of the model sensitivity is presented and shows the role of the ice compactness and rheology in the ice drift response at both local and regional scales in the Arctic. Indeed, the ice drift simulated by neXtSIM in summer is close to the one obtained with the free-drift model, while the more compact and solid ice pack shows a significantly different mechanical and drift behaviour in winter. For the winter period analysed in this study, we also show that, in contrast to the free-drift model, neXtSIM reproduces the sea ice Lagrangian diffusion regimes as found from observed trajectories. The forecast capability of neXtSIM is also evaluated using a large set of real buoy's trajectories and compared to the capability of the free-drift model. We found that neXtSIM performs significantly better in simulating sea ice drift, both in terms of forecast error and as a tool to assist search and rescue operations, although the sources of uncertainties assumed for the present experiment are not sufficient for complete coverage of the observed IABP positions.

Relatively little attention has been given to the impact of discretization error on twin experiments in the stochastic form of the Lorenz-96 equations when the dynamics are fully resolved but random. We study a simple form of the stochastically forced Lorenz-96 equations that is amenable to higher order time-discretization schemes in order to investigate these effects. We provide numerical benchmarks for the overall discretization error, in the strong and weak sense, for several commonly used integration schemes and compare these methods for biases introduced into ensemble-based statistics and filtering performance. Focus is given to the distinction between strong and weak convergence of the numerical schemes, highlighting which of the two concepts is relevant based on the problem at hand. Using the above analysis, we suggest a mathematically consistent framework for the treatment of these discretization errors in ensemble forecasting and data assimilation twin experiments for unbiased and computationally efficient benchmark studies. Pursuant to this, we provide a novel derivation of the order 2.0 strong Taylor scheme for numerically generating the truth-twin in the stochastically perturbed Lorenz-96 equations.

Is it possible to emulate a numerical model from noisy and sparse observations? How realistic and skilful can it be? Recent progress in machine learning has shown how to forecast a model from observations. We will show that by leveraging on data assimilation techniques, it is possible to produce realistic and skilful surrogate models of the underlying dynamics given sparse and noisy observations. The approach is tested with several chaotic systems. The surrogate model shows both forecast skills and abilities to reproduce the “climate” (i.e. spectral properties and statistical moments) of the underlying dynamical model on long-term simulations.

Numerical models solved on adaptive moving meshes have become increasingly prevalent in recent years. Motivating problems include the study of fluids in a Lagrangian frame and the presence of highly localized structures such as shock waves or interfaces. In the former case, Lagrangian solvers move the nodes of the mesh with the dynamical flow; in the latter, mesh resolution is increased in the proximity of the localized structure. Mesh adaptation can include remeshing, a procedure that adds or removes mesh nodes according to specific rules reflecting constraints in the numerical solver. In this case, the number of mesh nodes will change during the integration and, as a result, the dimension of the model's state vector will not be conserved. This work presents a novel approach to the formulation of ensemble data assimilation (DA) for models with this underlying computational structure. The challenge lies in the fact that remeshing entails a different state space dimension across members of the ensemble, thus impeding the usual computation of consistent ensemble-based statistics. Our methodology adds one forward and one backward mapping step before and after the ensemble Kalman filter (EnKF) analysis, respectively. This mapping takes all the ensemble members onto a fixed, uniform reference mesh where the EnKF analysis can be performed. We consider a high-resolution (HR) and a low-resolution (LR) fixed uniform reference mesh, whose resolutions are determined by the remeshing tolerances. This way the reference meshes embed the model numerical constraints and are also upper and lower uniform meshes bounding the resolutions of the individual ensemble meshes. Numerical experiments are carried out using 1-D prototypical models: Burgers and Kuramoto–Sivashinsky equations and both Eulerian and Lagrangian synthetic observations. While the HR strategy generally outperforms that of LR, their skill difference can be reduced substantially by an optimal tuning of the data assimilation parameters. The LR case is appealing in high dimensions because of its lower computational burden. Lagrangian observations are shown to be very effective in that fewer of them are able to keep the analysis error at a level comparable to the more numerous observers for the Eulerian case. This study is motivated by the development of suitable EnKF strategies for 2-D models of the sea ice that are numerically solved on a Lagrangian mesh with remeshing.

We describe a new approach that allows for systematic causal attribution of weather and climate-related events, in near-real time. The method is designed so as to facilitate its implementation at meteorological centers by relying on data and methods that are routinely available when numerically forecasting the weather. We thus show that causal attribution can be obtained as a by-product of data assimilation procedures run on a daily basis to update numerical weather prediction (NWP) models with new atmospheric observations; hence, the proposed methodology can take advantage of the powerful computational and observational capacity of weather forecasting centers. We explain the theoretical rationale of this approach and sketch the most prominent features of a “data assimilation–based detection and attribution” (DADA) procedure. The proposal is illustrated in the context of the classical three-variable Lorenz model with additional forcing. The paper concludes by raising several theoretical and practical questions that need to be addressed to make the proposal operational within NWP centers. Fil: Hannart, Alexis. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmosfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmosfera; Argentina Fil: Carrasi, A.. Mohn-Sverdrup Center; Noruega Fil: Bocquet, M.. Université Paris-Est; Francia Fil: Ghil, M.. Ecole Normale Supérieure; Francia. University of California at Los Angeles; Estados Unidos Fil: Naveau, P.. Centre National de la Recherche Scientifique; Francia Fil: Pulido, M.. Universidad Nacional del Nordeste; Argentina Fil: Ruiz, Juan Jose. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmosfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmosfera; Argentina Fil: Tandeo, P.. Télécom Bretagne; Francia

A new nudging method for data assimilation, delay‐coordinate nudging, is presented. Delay‐coordinate nudging makes explicit use of present and past observations in the formulation of the forcing driving the model evolution at each time step. Numerical experiments with a low‐order chaotic system show that the new method systematically outperforms standard nudging in different model and observational scenarios, also when using an unoptimized formulation of the delay‐nudging coefficients. A connection between the optimal delay and the dominant Lyapunov exponent of the dynamics is found based on heuristic arguments and is confirmed by the numerical results, providing a guideline for the practical implementation of the algorithm. Delay‐coordinate nudging preserves the easiness of implementation, the intuitive functioning and the reduced computational cost of the standard nudging, making it a potential alternative especially in the field of seasonal‐to‐decadal predictions with large Earth system models that limit the use of more sophisticated data assimilation procedures.

This work demonstrates the efficiency of using iterative ensemble smoothers to estimate the parameters of an SEIR model. We have extended a standard SEIR model with age-classes and compartments of sick, hospitalized, and dead. The data conditioned on are the daily numbers of accumulated deaths and the number of hospitalized. Also, it is possible to condition the model on the number of cases obtained from testing. We start from a wide prior distribution for the model parameters; then, the ensemble conditioning leads to a posterior ensemble of estimated parameters yielding model predictions in close agreement with the observations. The updated ensemble of model simulations has predictive capabilities and include uncertainty estimates. In particular, we estimate the effective reproductive number as a function of time, and we can assess the impact of different intervention measures. By starting from the updated set of model parameters, we can make accurate short-term predictions of the epidemic development assuming knowledge of the future effective reproductive number. Also, the model system allows for the computation of long-term scenarios of the epidemic under different assumptions. We have applied the model system on data sets from several countries, i.e., the four European countries Norway, England, The Netherlands, and France; the province of Quebec in Canada; the South American countries Argentina and Brazil; and the four US states Alabama, North Carolina, California, and New York. These countries and states all have vastly different developments of the epidemic, and we could accurately model the SARS-CoV-2 outbreak in all of them. We realize that more complex models, e.g., with regional compartments, may be desirable, and we suggest that the approach used here should be applicable also for these models.

Recent progress in machine learning has shown how to forecast and, to some extent, learn the dynamics of a model from its output, resorting in particular to neural networks and deep learning techniques. We will show how the same goal can be directly achieved using data assimilation techniques without leveraging on machine learning software libraries, with a view to high-dimensional models. The dynamics of a model are learned from its observation and an ordinary differential equation (ODE) representation of this model is inferred using a recursive nonlinear regression. Because the method is embedded in a Bayesian data assimilation framework, it can learn from partial and noisy observations of a state trajectory of the physical model. Moreover, a space-wise local representation of the ODE system is introduced and is key to cope with high-dimensional models. It has recently been suggested that neural network architectures could be interpreted as dynamical systems. Reciprocally, we show that our ODE representations are reminiscent of deep learning architectures. Furthermore, numerical analysis considerations on stability shed light on the assets and limitations of the method. The method is illustrated on several chaotic discrete and continuous models of various dimensions, with or without noisy observations, with the goal to identify or improve the model dynamics, build a surrogate or reduced model, or produce forecasts from mere observations of the physical model.

In recent years, there has been increased interest in applying data assimilation (DA) methods, originally designed for state estimation, to the model selection problem. In this setting, previous studies introduced the contextual formulation of model evidence, or contextual model evidence (CME), and showed that CME can be efficiently computed using a hierarchy of ensemble‐based DA procedures. Although these studies analysed the DA methods most commonly used for operational atmospheric and oceanic prediction worldwide, they did not study these methods in conjunction with localization to a specific domain. Yet, any application of ensemble DA methods to realistic, very high‐dimensional geophysical models requires the implementation of some form of localization. The present study extends CME estimation to ensemble DA methods with domain localization. Domain‐localized CME (DL‐CME) developed in this article is tested for model selection with two models: (a) the Lorenz 40‐variable midlatitude atmospheric dynamics model (Lorenz‐95); and (b) the simplified global atmospheric SPEEDY model. CME is compared to the root‐mean‐square error (RMSE) as a metric for model selection. The experiments show that CME systematically outperforms RMSE in model selection skill, and that this skill improvement is further enhanced by applying localization to the CME estimate using DL‐CME. The potential use and range of applications of CME and DL‐CME as a model selection metric are also discussed.

International audience; This paper studies multiplicative inflation: the complementary scaling of the state covariance in the ensemble Kalman filter (EnKF). Firstly, error sources in the EnKF are catalogued and discussed in relation to inflation; nonlinearity is given particular attention as a source of sampling error. In response, the "finite-size" refinement known as the EnKF-N is re-derived via a Gaussian scale mixture, again demonstrating how it yields adaptive inflation. Existing methods for adaptive inflation estimation are reviewed, and several insights are gained from a comparative analysis. One such adaptive inflation method is selected to complement the EnKF-N to make a hybrid that is suitable for contexts where model error is present and imperfectly parameterized. Benchmarks are obtained from experiments with the two-scale Lorenz model and its slow-scale truncation. The proposed hybrid EnKF-N method of adaptive inflation is found to yield systematic accuracy improvements in comparison with the existing methods, albeit to a moderate degree.

Recent progress in machine learning has shown how to forecast and, to some extent, learn the dynamics of a model from its output, resorting in particular to neural networks and deep learning techniques. We will show how the same goal can be directly achieved using data assimilation techniques without leveraging on machine learning software libraries, with a view to high-dimensional models. The dynamics of a model are learned from its observation and an ordinary differential equation (ODE) representation of this model is inferred using a recursive nonlinear regression. Because the method is embedded in a Bayesian data assimilation framework, it can learn from partial and noisy observations of a state trajectory of the physical model. Moreover, a space-wise local representation of the ODE system is introduced and is key to coping with high-dimensional models. It has recently been suggested that neural network architectures could be interpreted as dynamical systems. Reciprocally, we show that our ODE representations are reminiscent of deep learning architectures. Furthermore, numerical analysis considerations of stability shed light on the assets and limitations of the method. The method is illustrated on several chaotic discrete and continuous models of various dimensions, with or without noisy observations, with the goal of identifying or improving the model dynamics, building a surrogate or reduced model, or producing forecasts solely from observations of the physical model.