Articles | Volume 23, issue 2
https://doi.org/10.5194/npg-23-59-2016
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
https://doi.org/10.5194/npg-23-59-2016
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
Research article
| 
11 Mar 2016
Research article |
| 11 Mar 2016
Hybrid Levenberg–Marquardt and weak-constraint ensemble Kalman smoother method
University of Colorado Denver, Denver, CO 80217-3364, USA
Institute of Computer Science, The Czech Academy of Sciences, 182 07 Prague, Czech Republic
E. Bergou
INRA, MaIAGE, Université Paris-Saclay, 78350 Jouy-en-Josas, France
S. Gürol
CERFACS, 31100 Toulouse, France
S. Gratton
CERFACS, 31100 Toulouse, France
INP-ENSEEIHT, 31071 Toulouse, France
I. Kasanický
Institute of Computer Science, The Czech Academy of Sciences, 182 07 Prague, Czech Republic
Related authors
I. Kasanický, J. Mandel, and M. Vejmelka
Nonlin. Processes Geophys., 22, 485–497, https://doi.org/10.5194/npg-22-485-2015, https://doi.org/10.5194/npg-22-485-2015, 2015
Short summary
Short summary
A new type of ensemble Kalman filter for data assimilation is developed, based on fast Fourier transform and wavelet transform. The method can work with minimal computational resources. We develop variants for several general types of observations, give a rigorous proof that the method improves the approximation of the state covariance, and present computational experiments showing that the new technique works reliably with very small ensembles and is stable over multiple analysis cycles.
J. Mandel, S. Amram, J. D. Beezley, G. Kelman, A. K. Kochanski, V. Y. Kondratenko, B. H. Lynn, B. Regev, and M. Vejmelka
Nat. Hazards Earth Syst. Sci., 14, 2829–2845, https://doi.org/10.5194/nhess-14-2829-2014, https://doi.org/10.5194/nhess-14-2829-2014, 2014
A. K. Kochanski, M. A. Jenkins, J. Mandel, J. D. Beezley, C. B. Clements, and S. Krueger
Geosci. Model Dev., 6, 1109–1126, https://doi.org/10.5194/gmd-6-1109-2013, https://doi.org/10.5194/gmd-6-1109-2013, 2013
Anthony Fillion, Marc Bocquet, and Serge Gratton
Nonlin. Processes Geophys., 25, 315–334, https://doi.org/10.5194/npg-25-315-2018, https://doi.org/10.5194/npg-25-315-2018, 2018
Short summary
Short summary
This study generalizes a paper by Pires et al. (1996) to state-of-the-art data assimilation techniques, such as the iterative ensemble Kalman smoother (IEnKS). We show that the longer the time window over which observations are assimilated, the better the accuracy of the IEnKS. Beyond a critical time length that we estimate, we show that this accuracy finally degrades. We show that the use of the quasi-static minimizations but generalized to the IEnKS yields a significantly improved accuracy.
I. Kasanický, J. Mandel, and M. Vejmelka
Nonlin. Processes Geophys., 22, 485–497, https://doi.org/10.5194/npg-22-485-2015, https://doi.org/10.5194/npg-22-485-2015, 2015
Short summary
Short summary
A new type of ensemble Kalman filter for data assimilation is developed, based on fast Fourier transform and wavelet transform. The method can work with minimal computational resources. We develop variants for several general types of observations, give a rigorous proof that the method improves the approximation of the state covariance, and present computational experiments showing that the new technique works reliably with very small ensembles and is stable over multiple analysis cycles.
J. Mandel, S. Amram, J. D. Beezley, G. Kelman, A. K. Kochanski, V. Y. Kondratenko, B. H. Lynn, B. Regev, and M. Vejmelka
Nat. Hazards Earth Syst. Sci., 14, 2829–2845, https://doi.org/10.5194/nhess-14-2829-2014, https://doi.org/10.5194/nhess-14-2829-2014, 2014
A. K. Kochanski, M. A. Jenkins, J. Mandel, J. D. Beezley, C. B. Clements, and S. Krueger
Geosci. Model Dev., 6, 1109–1126, https://doi.org/10.5194/gmd-6-1109-2013, https://doi.org/10.5194/gmd-6-1109-2013, 2013
Related subject area
Subject: Predictability, probabilistic forecasts, data assimilation, inverse problems | Topic: Climate, atmosphere, ocean, hydrology, cryosphere, biosphere
Robust weather-adaptive post-processing using model output statistics random forests
Comparative study of strongly and weakly coupled data assimilation with a global land–atmosphere coupled model
How far can the statistical error estimation problem be closed by collocated data?
Using orthogonal vectors to improve the ensemble space of the ensemble Kalman filter and its effect on data assimilation and forecasting
Review article: Towards strongly coupled ensemble data assimilation with additional improvements from machine learning
Reducing manipulations in a control simulation experiment based on instability vectors with the Lorenz-63 model
Control simulation experiments of extreme events with the Lorenz-96 model
Toward a multivariate formulation of the parametric Kalman filter assimilation: application to a simplified chemical transport model
Data-driven reconstruction of partially observed dynamical systems
A range of outcomes: the combined effects of internal variability and anthropogenic forcing on regional climate trends over Europe
Extending ensemble Kalman filter algorithms to assimilate observations with an unknown time offset
Guidance on how to improve vertical covariance localization based on a 1000-member ensemble
Weather pattern dynamics over western Europe under climate change: predictability, information entropy and production
Using a hybrid optimal interpolation–ensemble Kalman filter for the Canadian Precipitation Analysis
Applying prior correlations for ensemble-based spatial localization
A stochastic covariance shrinkage approach to particle rejuvenation in the ensemble transform particle filter
Control simulation experiment with Lorenz's butterfly attractor
Ensemble Riemannian data assimilation: towards large-scale dynamical systems
Inferring the instability of a dynamical system from the skill of data assimilation exercises
Reduced non-Gaussianity by 30 s rapid update in convective-scale numerical weather prediction
Multivariate localization functions for strongly coupled data assimilation in the bivariate Lorenz 96 system
A study of capturing Atlantic meridional overturning circulation (AMOC) regime transition through observation-constrained model parameters
Calibrated ensemble forecasts of the height of new snow using quantile regression forests and ensemble model output statistics
Enhancing geophysical flow machine learning performance via scale separation
Improving the potential accuracy and usability of EURO-CORDEX estimates of future rainfall climate using frequentist model averaging
Ensemble Riemannian data assimilation over the Wasserstein space
An early warning sign of critical transition in the Antarctic ice sheet – a data-driven tool for a spatiotemporal tipping point
Training a convolutional neural network to conserve mass in data assimilation
Behavior of the iterative ensemble-based variational method in nonlinear problems
Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation
A methodology to obtain model-error covariances due to the discretization scheme from the parametric Kalman filter perspective
A method for predicting the uncompleted climate transition process
Statistical postprocessing of ensemble forecasts for severe weather at Deutscher Wetterdienst
Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: reservoir computing, artificial neural network, and long short-term memory network
From research to applications – examples of operational ensemble post-processing in France using machine learning
Correcting for model changes in statistical postprocessing – an approach based on response theory
Brief communication: Residence time of energy in the atmosphere
Simulating model uncertainty of subgrid-scale processes by sampling model errors at convective scales
Data-driven versus self-similar parameterizations for stochastic advection by Lie transport and location uncertainty
Seasonal statistical–dynamical prediction of the North Atlantic Oscillation by probabilistic post-processing and its evaluation
Application of a local attractor dimension to reduced space strongly coupled data assimilation for chaotic multiscale systems
Order of operation for multi-stage post-processing of ensemble wind forecast trajectories
Generalization properties of feed-forward neural networks trained on Lorenz systems
Revising the stochastic iterative ensemble smoother
Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data
Non-Gaussian statistics in global atmospheric dynamics: a study with a 10 240-member ensemble Kalman filter using an intermediate atmospheric general circulation model
Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective
Data assimilation using adaptive, non-conservative, moving mesh models
Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models
A Bayesian approach to multivariate adaptive localization in ensemble-based data assimilation with time-dependent extensions
Thomas Muschinski, Georg J. Mayr, Achim Zeileis, and Thorsten Simon
Nonlin. Processes Geophys., 30, 503–514, https://doi.org/10.5194/npg-30-503-2023, https://doi.org/10.5194/npg-30-503-2023, 2023
Short summary
Short summary
Statistical post-processing is necessary to generate probabilistic forecasts from physical numerical weather prediction models. To allow for more flexibility, there has been a shift in post-processing away from traditional parametric regression models towards modern machine learning methods. By fusing these two approaches, we developed model output statistics random forests, a new post-processing method that is highly flexible but at the same time also very robust and easy to interpret.
Kenta Kurosawa, Shunji Kotsuki, and Takemasa Miyoshi
Nonlin. Processes Geophys., 30, 457–479, https://doi.org/10.5194/npg-30-457-2023, https://doi.org/10.5194/npg-30-457-2023, 2023
Short summary
Short summary
This study aimed to enhance weather and hydrological forecasts by integrating soil moisture data into a global weather model. By assimilating atmospheric observations and soil moisture data, the accuracy of forecasts was improved, and certain biases were reduced. The method was found to be particularly beneficial in areas like the Sahel and equatorial Africa, where precipitation patterns vary seasonally. This new approach has the potential to improve the precision of weather predictions.
Annika Vogel and Richard Ménard
Nonlin. Processes Geophys., 30, 375–398, https://doi.org/10.5194/npg-30-375-2023, https://doi.org/10.5194/npg-30-375-2023, 2023
Short summary
Short summary
Accurate estimation of the error statistics required for data assimilation remains an ongoing challenge, as statistical assumptions are required to solve the estimation problem. This work provides a conceptual view of the statistical error estimation problem in light of the increasing number of available datasets. We found that the total number of required assumptions increases with the number of overlapping datasets, but the relative number of error statistics that can be estimated increases.
Yung-Yun Cheng, Shu-Chih Yang, Zhe-Hui Lin, and Yung-An Lee
Nonlin. Processes Geophys., 30, 289–297, https://doi.org/10.5194/npg-30-289-2023, https://doi.org/10.5194/npg-30-289-2023, 2023
Short summary
Short summary
In the ensemble Kalman filter, the ensemble space may not fully capture the forecast errors due to the limited ensemble size and systematic model errors, which affect the accuracy of analysis and prediction. This study proposes a new algorithm to use cost-free pseudomembers to expand the ensemble space effectively and improve analysis accuracy during the analysis step, without increasing the ensemble size during forecasting.
Eugenia Kalnay, Travis Sluka, Takuma Yoshida, Cheng Da, and Safa Mote
Nonlin. Processes Geophys., 30, 217–236, https://doi.org/10.5194/npg-30-217-2023, https://doi.org/10.5194/npg-30-217-2023, 2023
Short summary
Short summary
Strongly coupled data assimilation (SCDA) generates coherent integrated Earth system analyses by assimilating the full Earth observation set into all Earth components. We describe SCDA based on the ensemble Kalman filter with a hierarchy of coupled models, from a coupled Lorenz to the Climate Forecast System v2. SCDA with a sufficiently large ensemble can provide more accurate coupled analyses compared to weakly coupled DA. The correlation-cutoff method can compensate for a small ensemble size.
Mao Ouyang, Keita Tokuda, and Shunji Kotsuki
Nonlin. Processes Geophys., 30, 183–193, https://doi.org/10.5194/npg-30-183-2023, https://doi.org/10.5194/npg-30-183-2023, 2023
Short summary
Short summary
This research found that weather control would change the chaotic behavior of an atmospheric model. We proposed to introduce chaos theory in the weather control. Experimental results demonstrated that the proposed approach reduced the manipulations, including the control times and magnitudes, which throw light on the weather control in a real atmospheric model.
Qiwen Sun, Takemasa Miyoshi, and Serge Richard
Nonlin. Processes Geophys., 30, 117–128, https://doi.org/10.5194/npg-30-117-2023, https://doi.org/10.5194/npg-30-117-2023, 2023
Short summary
Short summary
This paper is a follow-up of a work by Miyoshi and Sun which was published in NPG Letters in 2022. The control simulation experiment is applied to the Lorenz-96 model for avoiding extreme events. The results show that extreme events of this partially and imperfectly observed chaotic system can be avoided by applying pre-designed small perturbations. These investigations may be extended to more realistic numerical weather prediction models.
Antoine Perrot, Olivier Pannekoucke, and Vincent Guidard
Nonlin. Processes Geophys., 30, 139–166, https://doi.org/10.5194/npg-30-139-2023, https://doi.org/10.5194/npg-30-139-2023, 2023
Short summary
Short summary
This work is a theoretical contribution that provides equations for understanding uncertainty prediction applied in air quality where multiple chemical species can interact. A simplified minimal test bed is introduced that shows the ability of our equations to reproduce the statistics estimated from an ensemble of forecasts. While the latter estimation is the state of the art, solving equations is numerically less costly, depending on the number of chemical species, and motivates this research.
Pierre Tandeo, Pierre Ailliot, and Florian Sévellec
Nonlin. Processes Geophys., 30, 129–137, https://doi.org/10.5194/npg-30-129-2023, https://doi.org/10.5194/npg-30-129-2023, 2023
Short summary
Short summary
The goal of this paper is to obtain probabilistic predictions of a partially observed dynamical system without knowing the model equations. It is illustrated using the three-dimensional Lorenz system, where only two components are observed. The proposed data-driven procedure is low-cost, is easy to implement, uses linear and Gaussian assumptions and requires only a small amount of data. It is based on an iterative linear Kalman smoother with a state augmentation.
Clara Deser and Adam S. Phillips
Nonlin. Processes Geophys., 30, 63–84, https://doi.org/10.5194/npg-30-63-2023, https://doi.org/10.5194/npg-30-63-2023, 2023
Short summary
Short summary
Past and future climate change at regional scales is a result of both human influences and natural (internal) variability. Here, we provide an overview of recent advances in climate modeling and physical understanding that has led to new insights into their respective roles, illustrated with original results for the European climate. Our findings highlight the confounding role of internal variability in attribution, climate model evaluation, and accuracy of future projections.
Elia Gorokhovsky and Jeffrey L. Anderson
Nonlin. Processes Geophys., 30, 37–47, https://doi.org/10.5194/npg-30-37-2023, https://doi.org/10.5194/npg-30-37-2023, 2023
Short summary
Short summary
Older observations of the Earth system sometimes lack information about the time they were taken, posing problems for analyses of past climate. To begin to ameliorate this problem, we propose new methods of varying complexity, including methods to estimate the distribution of the offsets between true and reported observation times. The most successful method accounts for the nonlinearity in the system, but even the less expensive ones can improve data assimilation in the presence of time error.
Tobias Necker, David Hinger, Philipp Johannes Griewank, Takemasa Miyoshi, and Martin Weissmann
Nonlin. Processes Geophys., 30, 13–29, https://doi.org/10.5194/npg-30-13-2023, https://doi.org/10.5194/npg-30-13-2023, 2023
Short summary
Short summary
This study investigates vertical localization based on a convection-permitting 1000-member ensemble simulation. We derive an empirical optimal localization (EOL) that minimizes sampling error in 40-member sub-sample correlations assuming 1000-member correlations as truth. The results will provide guidance for localization in convective-scale ensemble data assimilation systems.
Stéphane Vannitsem
Nonlin. Processes Geophys., 30, 1–12, https://doi.org/10.5194/npg-30-1-2023, https://doi.org/10.5194/npg-30-1-2023, 2023
Short summary
Short summary
The impact of climate change on weather pattern dynamics over the North Atlantic is explored through the lens of information theory. These tools allow the predictability of the succession of weather patterns and the irreversible nature of the dynamics to be clarified. It is shown that the predictability is increasing in the observations, while the opposite trend is found in model projections. The irreversibility displays an overall increase in time in both the observations and the model runs.
Dikraa Khedhaouiria, Stéphane Bélair, Vincent Fortin, Guy Roy, and Franck Lespinas
Nonlin. Processes Geophys., 29, 329–344, https://doi.org/10.5194/npg-29-329-2022, https://doi.org/10.5194/npg-29-329-2022, 2022
Short summary
Short summary
This study introduces a well-known use of hybrid methods in data assimilation (DA) algorithms that has not yet been explored for precipitation analyses. Our approach combined an ensemble-based DA approach with an existing deterministically based DA. Both DA scheme families have desirable aspects that can be leveraged if combined. The DA hybrid method showed better precipitation analyses in regions with a low rate of assimilated surface observations, which is typically the case in winter.
Chu-Chun Chang and Eugenia Kalnay
Nonlin. Processes Geophys., 29, 317–327, https://doi.org/10.5194/npg-29-317-2022, https://doi.org/10.5194/npg-29-317-2022, 2022
Short summary
Short summary
This study introduces a new approach for enhancing the ensemble data assimilation (DA), a technique that combines observations and forecasts to improve numerical weather predictions. Our method uses the prescribed correlations to suppress spurious errors, improving the accuracy of DA. The experiments on the simplified atmosphere model show that our method has comparable performance to the traditional method but is superior in the early stage and is more computationally efficient.
Andrey A. Popov, Amit N. Subrahmanya, and Adrian Sandu
Nonlin. Processes Geophys., 29, 241–253, https://doi.org/10.5194/npg-29-241-2022, https://doi.org/10.5194/npg-29-241-2022, 2022
Short summary
Short summary
Numerical weather prediction requires the melding of both computational model and data obtained from sensors such as satellites. We focus on one algorithm to accomplish this. We aim to aid its use by additionally supplying it with data obtained from separate models that describe the average behavior of the computational model at any given time. We show that our approach outperforms the standard approaches to this problem.
Takemasa Miyoshi and Qiwen Sun
Nonlin. Processes Geophys., 29, 133–139, https://doi.org/10.5194/npg-29-133-2022, https://doi.org/10.5194/npg-29-133-2022, 2022
Short summary
Short summary
The weather is chaotic and hard to predict, but the chaos implies an effective control where a small control signal grows rapidly to make a big difference. This study proposes a control simulation experiment where we apply a small signal to control
naturein a computational simulation. Idealized experiments with a low-order chaotic system show successful results by small control signals of only 3 % of the observation error. This is the first step toward realistic weather simulations.
Sagar K. Tamang, Ardeshir Ebtehaj, Peter Jan van Leeuwen, Gilad Lerman, and Efi Foufoula-Georgiou
Nonlin. Processes Geophys., 29, 77–92, https://doi.org/10.5194/npg-29-77-2022, https://doi.org/10.5194/npg-29-77-2022, 2022
Short summary
Short summary
The outputs from Earth system models are optimally combined with satellite observations to produce accurate forecasts through a process called data assimilation. Many existing data assimilation methodologies have some assumptions regarding the shape of the probability distributions of model output and observations, which results in forecast inaccuracies. In this paper, we test the effectiveness of a newly proposed methodology that relaxes such assumptions about high-dimensional models.
Yumeng Chen, Alberto Carrassi, and Valerio Lucarini
Nonlin. Processes Geophys., 28, 633–649, https://doi.org/10.5194/npg-28-633-2021, https://doi.org/10.5194/npg-28-633-2021, 2021
Short summary
Short summary
Chaotic dynamical systems are sensitive to the initial conditions, which are crucial for climate forecast. These properties are often used to inform the design of data assimilation (DA), a method used to estimate the exact initial conditions. However, obtaining the instability properties is burdensome for complex problems, both numerically and analytically. Here, we suggest a different viewpoint. We show that the skill of DA can be used to infer the instability properties of a dynamical system.
Juan Ruiz, Guo-Yuan Lien, Keiichi Kondo, Shigenori Otsuka, and Takemasa Miyoshi
Nonlin. Processes Geophys., 28, 615–626, https://doi.org/10.5194/npg-28-615-2021, https://doi.org/10.5194/npg-28-615-2021, 2021
Short summary
Short summary
Effective use of observations with numerical weather prediction models, also known as data assimilation, is a key part of weather forecasting systems. For precise prediction at the scales of thunderstorms, fast nonlinear processes pose a grand challenge because most data assimilation systems are based on linear processes and normal distribution errors. We investigate how, every 30 s, weather radar observations can help reduce the effect of nonlinear processes and nonnormal distributions.
Zofia Stanley, Ian Grooms, and William Kleiber
Nonlin. Processes Geophys., 28, 565–583, https://doi.org/10.5194/npg-28-565-2021, https://doi.org/10.5194/npg-28-565-2021, 2021
Short summary
Short summary
In weather forecasting, observations are incorporated into a model of the atmosphere through a process called data assimilation. Sometimes observations in one location may impact the weather forecast in another faraway location in undesirable ways. The impact of distant observations on the forecast is mitigated through a process called localization. We propose a new method for localization when a model has multiple length scales, as in a model spanning both the ocean and the atmosphere.
Zhao Liu, Shaoqing Zhang, Yang Shen, Yuping Guan, and Xiong Deng
Nonlin. Processes Geophys., 28, 481–500, https://doi.org/10.5194/npg-28-481-2021, https://doi.org/10.5194/npg-28-481-2021, 2021
Short summary
Short summary
A general methodology is introduced to capture regime transitions of the Atlantic meridional overturning circulation (AMOC). The assimilation models with different parameters simulate different paths for the AMOC to switch between equilibrium states. Constraining model parameters with observations can significantly mitigate the model deviations, thus capturing AMOC regime transitions. This simple model study serves as a guideline for improving coupled general circulation models.
Guillaume Evin, Matthieu Lafaysse, Maxime Taillardat, and Michaël Zamo
Nonlin. Processes Geophys., 28, 467–480, https://doi.org/10.5194/npg-28-467-2021, https://doi.org/10.5194/npg-28-467-2021, 2021
Short summary
Short summary
Forecasting the height of new snow is essential for avalanche hazard surveys, road and ski resort management, tourism attractiveness, etc. Météo-France operates a probabilistic forecasting system using a numerical weather prediction system and a snowpack model. It provides better forecasts than direct diagnostics but exhibits significant biases. Post-processing methods can be applied to provide automatic forecasting products from this system.
Davide Faranda, Mathieu Vrac, Pascal Yiou, Flavio Maria Emanuele Pons, Adnane Hamid, Giulia Carella, Cedric Ngoungue Langue, Soulivanh Thao, and Valerie Gautard
Nonlin. Processes Geophys., 28, 423–443, https://doi.org/10.5194/npg-28-423-2021, https://doi.org/10.5194/npg-28-423-2021, 2021
Short summary
Short summary
Machine learning approaches are spreading rapidly in climate sciences. They are of great help in many practical situations where using the underlying equations is difficult because of the limitation in computational power. Here we use a systematic approach to investigate the limitations of the popular echo state network algorithms used to forecast the long-term behaviour of chaotic systems, such as the weather. Our results show that noise and intermittency greatly affect the performances.
Stephen Jewson, Giuliana Barbato, Paola Mercogliano, Jaroslav Mysiak, and Maximiliano Sassi
Nonlin. Processes Geophys., 28, 329–346, https://doi.org/10.5194/npg-28-329-2021, https://doi.org/10.5194/npg-28-329-2021, 2021
Short summary
Short summary
Climate model simulations are uncertain. In some cases this makes it difficult to know how to use them. Significance testing is often used to deal with this issue but has various shortcomings. We describe two alternative ways to manage uncertainty in climate model simulations that avoid these shortcomings. We test them on simulations of future rainfall over Europe and show they produce more accurate projections than either using unadjusted climate model output or statistical testing.
Sagar K. Tamang, Ardeshir Ebtehaj, Peter J. van Leeuwen, Dongmian Zou, and Gilad Lerman
Nonlin. Processes Geophys., 28, 295–309, https://doi.org/10.5194/npg-28-295-2021, https://doi.org/10.5194/npg-28-295-2021, 2021
Short summary
Short summary
Data assimilation aims to improve hydrologic and weather forecasts by combining available information from Earth system models and observations. The classical approaches to data assimilation usually proceed with some preconceived assumptions about the shape of their probability distributions. As a result, when such assumptions are invalid, the forecast accuracy suffers. In the proposed methodology, we relax such assumptions and demonstrate improved performance.
Abd AlRahman AlMomani and Erik Bollt
Nonlin. Processes Geophys., 28, 153–166, https://doi.org/10.5194/npg-28-153-2021, https://doi.org/10.5194/npg-28-153-2021, 2021
Short summary
Short summary
This paper introduces a tool for data-driven discovery of early warning signs of critical transitions in ice shelves from remote sensing data. Our directed spectral clustering method considers an asymmetric affinity matrix along with the associated directed graph Laplacian. We applied our approach to reprocessing the ice velocity data and remote sensing satellite images of the Larsen C ice shelf.
Yvonne Ruckstuhl, Tijana Janjić, and Stephan Rasp
Nonlin. Processes Geophys., 28, 111–119, https://doi.org/10.5194/npg-28-111-2021, https://doi.org/10.5194/npg-28-111-2021, 2021
Short summary
Short summary
The assimilation of observations using standard algorithms can lead to a violation of physical laws (e.g. mass conservation), which is shown to have a detrimental impact on the system's forecast. We use a neural network (NN) to correct this mass violation, using training data generated from expensive algorithms that can constrain such physical properties. We found that, in an idealized set-up, the NN can match the performance of these expensive algorithms at negligible computational costs.
Shin'ya Nakano
Nonlin. Processes Geophys., 28, 93–109, https://doi.org/10.5194/npg-28-93-2021, https://doi.org/10.5194/npg-28-93-2021, 2021
Short summary
Short summary
The ensemble-based variational method is a method for solving nonlinear data assimilation problems by using an ensemble of multiple simulation results. Although this method is derived based on a linear approximation, highly uncertain problems, in which system nonlinearity is significant, can also be solved by applying this method iteratively. This paper reformulated this iterative algorithm to analyze its behavior in high-dimensional nonlinear problems and discuss the convergence.
Sangeetika Ruchi, Svetlana Dubinkina, and Jana de Wiljes
Nonlin. Processes Geophys., 28, 23–41, https://doi.org/10.5194/npg-28-23-2021, https://doi.org/10.5194/npg-28-23-2021, 2021
Short summary
Short summary
To infer information of an unknown quantity that helps to understand an associated system better and to predict future outcomes, observations and a physical model that connects the data points to the unknown parameter are typically used as information sources. Yet this problem is often very challenging due to the fact that the unknown is generally high dimensional, the data are sparse and the model can be non-linear. We propose a novel approach to address these challenges.
Olivier Pannekoucke, Richard Ménard, Mohammad El Aabaribaoune, and Matthieu Plu
Nonlin. Processes Geophys., 28, 1–22, https://doi.org/10.5194/npg-28-1-2021, https://doi.org/10.5194/npg-28-1-2021, 2021
Short summary
Short summary
Numerical weather prediction involves numerically solving the mathematical equations, which describe the geophysical flow, by transforming them so that they can be computed. Through this transformation, it appears that the equations actually solved by the machine are then a modified version of the original equations, introducing an error that contributes to the model error. This work helps to characterize the covariance of the model error that is due to this modification of the equations.
Pengcheng Yan, Guolin Feng, Wei Hou, and Ping Yang
Nonlin. Processes Geophys., 27, 489–500, https://doi.org/10.5194/npg-27-489-2020, https://doi.org/10.5194/npg-27-489-2020, 2020
Short summary
Short summary
A system transiting from one stable state to another has to experience a period. Can we predict the end moment (state) if the process has not been completed? This paper presents a method to solve this problem. This method is based on the quantitative relationship among the parameters, which is used to describe the transition process of the abrupt change. By using the historical data, we extract some parameters for predicting the uncompleted climate transition process.
Reinhold Hess
Nonlin. Processes Geophys., 27, 473–487, https://doi.org/10.5194/npg-27-473-2020, https://doi.org/10.5194/npg-27-473-2020, 2020
Short summary
Short summary
Forecasts of ensemble systems are statistically aligned to synoptic observations at DWD in order to provide support for warning decision management. Motivation and design consequences for extreme and rare meteorological events are presented. Especially for probabilities of severe wind gusts global logistic parameterisations are developed that generate robust statistical forecasts for extreme events, while local characteristics are preserved.
Ashesh Chattopadhyay, Pedram Hassanzadeh, and Devika Subramanian
Nonlin. Processes Geophys., 27, 373–389, https://doi.org/10.5194/npg-27-373-2020, https://doi.org/10.5194/npg-27-373-2020, 2020
Short summary
Short summary
The performance of three machine-learning methods for data-driven modeling of a multiscale chaotic Lorenz 96 system is examined. One of the methods is found to be able to predict the future evolution of the chaotic system well from just knowing the past observations of the large-scale component of the multiscale state vector. Potential applications to data-driven and data-assisted surrogate modeling of complex dynamical systems such as weather and climate are discussed.
Maxime Taillardat and Olivier Mestre
Nonlin. Processes Geophys., 27, 329–347, https://doi.org/10.5194/npg-27-329-2020, https://doi.org/10.5194/npg-27-329-2020, 2020
Short summary
Short summary
Statistical post-processing of ensemble forecasts is now a well-known procedure in order to correct biased and misdispersed ensemble weather predictions. But practical application in European national weather services is in its infancy. Different applications of ensemble post-processing using machine learning at an industrial scale are presented. Forecast quality and value are improved compared to the raw ensemble, but several facilities have to be made to adjust to operational constraints.
Jonathan Demaeyer and Stéphane Vannitsem
Nonlin. Processes Geophys., 27, 307–327, https://doi.org/10.5194/npg-27-307-2020, https://doi.org/10.5194/npg-27-307-2020, 2020
Short summary
Short summary
Postprocessing schemes used to correct weather forecasts are no longer efficient when the model generating the forecasts changes. An approach based on response theory to take the change into account without having to recompute the parameters based on past forecasts is presented. It is tested on an analytical model and a simple model of atmospheric variability. We show that this approach is effective and discuss its potential application for an operational environment.
Carlos Osácar, Manuel Membrado, and Amalio Fernández-Pacheco
Nonlin. Processes Geophys., 27, 235–237, https://doi.org/10.5194/npg-27-235-2020, https://doi.org/10.5194/npg-27-235-2020, 2020
Short summary
Short summary
We deduce that after a global thermal perturbation, the Earth's
atmosphere would need about a couple of months to come back to equilibrium.
Michiel Van Ginderachter, Daan Degrauwe, Stéphane Vannitsem, and Piet Termonia
Nonlin. Processes Geophys., 27, 187–207, https://doi.org/10.5194/npg-27-187-2020, https://doi.org/10.5194/npg-27-187-2020, 2020
Short summary
Short summary
A generic methodology is developed to estimate the model error and simulate the model uncertainty related to a specific physical process. The method estimates the model error by comparing two different representations of the physical process in otherwise identical models. The found model error can then be used to perturb the model and simulate the model uncertainty. When applying this methodology to deep convection an improvement in the probabilistic skill of the ensemble forecast is found.
Valentin Resseguier, Wei Pan, and Baylor Fox-Kemper
Nonlin. Processes Geophys., 27, 209–234, https://doi.org/10.5194/npg-27-209-2020, https://doi.org/10.5194/npg-27-209-2020, 2020
Short summary
Short summary
Geophysical flows span a broader range of temporal and spatial scales than can be resolved numerically. One way to alleviate the ensuing numerical errors is to combine simulations with measurements, taking account of the accuracies of these two sources of information. Here we quantify the distribution of numerical simulation errors without relying on high-resolution numerical simulations. Specifically, small-scale random vortices are added to simulations while conserving energy or circulation.
André Düsterhus
Nonlin. Processes Geophys., 27, 121–131, https://doi.org/10.5194/npg-27-121-2020, https://doi.org/10.5194/npg-27-121-2020, 2020
Short summary
Short summary
Seasonal prediction of the of the North Atlantic Oscillation (NAO) has been improved in recent years by improving dynamical models and ensemble predictions. One step therein was the so-called sub-sampling, which combines statistical and dynamical predictions. This study generalises this approach and makes it much more accessible. Furthermore, it presents a new verification approach for such predictions.
Courtney Quinn, Terence J. O'Kane, and Vassili Kitsios
Nonlin. Processes Geophys., 27, 51–74, https://doi.org/10.5194/npg-27-51-2020, https://doi.org/10.5194/npg-27-51-2020, 2020
Short summary
Short summary
This study presents a novel method for reduced-rank data assimilation of multiscale highly nonlinear systems. Time-varying dynamical properties are used to determine the rank and projection of the system onto a reduced subspace. The variable reduced-rank method is shown to succeed over other fixed-rank methods. This work provides implications for performing strongly coupled data assimilation with a limited number of ensemble members on high-dimensional coupled climate models.
Nina Schuhen
Nonlin. Processes Geophys., 27, 35–49, https://doi.org/10.5194/npg-27-35-2020, https://doi.org/10.5194/npg-27-35-2020, 2020
Short summary
Short summary
We present a new way to adaptively improve weather forecasts by incorporating last-minute observation information. The recently measured error, together with a statistical model, gives us an indication of the expected future error of wind speed forecasts, which are then adjusted accordingly. This new technique can be especially beneficial for customers in the wind energy industry, where it is important to have reliable short-term forecasts, as well as providers of extreme weather warnings.
Sebastian Scher and Gabriele Messori
Nonlin. Processes Geophys., 26, 381–399, https://doi.org/10.5194/npg-26-381-2019, https://doi.org/10.5194/npg-26-381-2019, 2019
Short summary
Short summary
Neural networks are a technique that is widely used to predict the time evolution of physical systems. For this the past evolution of the system is shown to the neural network – it is
trained– and then can be used to predict the evolution in the future. We show some limitations in this approach for certain systems that are important to consider when using neural networks for climate- and weather-related applications.
Patrick Nima Raanes, Andreas Størksen Stordal, and Geir Evensen
Nonlin. Processes Geophys., 26, 325–338, https://doi.org/10.5194/npg-26-325-2019, https://doi.org/10.5194/npg-26-325-2019, 2019
Short summary
Short summary
A popular variational ensemble smoother for data assimilation and history matching is simplified. An exact relationship between ensemble linearizations (linear regression) and adjoints (analytic derivatives) is established.
Fei Lu, Nils Weitzel, and Adam H. Monahan
Nonlin. Processes Geophys., 26, 227–250, https://doi.org/10.5194/npg-26-227-2019, https://doi.org/10.5194/npg-26-227-2019, 2019
Short summary
Short summary
ll-posedness of the inverse problem and sparse noisy data are two major challenges in the modeling of high-dimensional spatiotemporal processes. We present a Bayesian inference method with a strongly regularized posterior to overcome these challenges, enabling joint state-parameter estimation and quantifying uncertainty in the estimation. We demonstrate the method on a physically motivated nonlinear stochastic partial differential equation arising from paleoclimate construction.
Keiichi Kondo and Takemasa Miyoshi
Nonlin. Processes Geophys., 26, 211–225, https://doi.org/10.5194/npg-26-211-2019, https://doi.org/10.5194/npg-26-211-2019, 2019
Short summary
Short summary
This study investigates non-Gaussian statistics of the data from a 10240-member ensemble Kalman filter. The large ensemble size can resolve the detailed structures of the probability density functions (PDFs) and indicates that the non-Gaussian PDF is caused by multimodality and outliers. While the outliers appear randomly, large multimodality corresponds well with large analysis error, mainly in the tropical regions and storm track regions where highly nonlinear processes appear frequently.
Frank Kwasniok
Nonlin. Processes Geophys., 26, 195–209, https://doi.org/10.5194/npg-26-195-2019, https://doi.org/10.5194/npg-26-195-2019, 2019
Short summary
Short summary
The stability properties as characterized by finite-time Lyapunov exponents are investigated in an intermediate-complexity atmospheric model. Firstly, the dominant patterns of collective excitation are identified by an empirical orthogonal function analysis of the fluctuation field of all of the finite-time Lyapunov exponents. Secondly, a large-deviation principle is established for all of the Lyapunov exponents and the large-deviation rate functions are estimated.
Ali Aydoğdu, Alberto Carrassi, Colin T. Guider, Chris K. R. T Jones, and Pierre Rampal
Nonlin. Processes Geophys., 26, 175–193, https://doi.org/10.5194/npg-26-175-2019, https://doi.org/10.5194/npg-26-175-2019, 2019
Short summary
Short summary
Computational models involving adaptive meshes can both evolve dynamically and be remeshed. Remeshing means that the state vector dimension changes in time and across ensemble members, making the ensemble Kalman filter (EnKF) unsuitable for assimilation of observational data. We develop a modification in which analysis is performed on a fixed uniform grid onto which the ensemble is mapped, with resolution relating to the remeshing criteria. The approach is successfully tested on two 1-D models.
Marc Bocquet, Julien Brajard, Alberto Carrassi, and Laurent Bertino
Nonlin. Processes Geophys., 26, 143–162, https://doi.org/10.5194/npg-26-143-2019, https://doi.org/10.5194/npg-26-143-2019, 2019
Short summary
Short summary
This paper describes an innovative way to use data assimilation to infer the dynamics of a physical system from its observation only. The method can operate with noisy and partial observation of the physical system. It acts as a deep learning technique specialised to dynamical models without the need for machine learning tools. The method is successfully tested on chaotic dynamical systems: the Lorenz-63, Lorenz-96, and Kuramoto–Sivashinski models and a two-scale Lorenz model.
Andrey A. Popov and Adrian Sandu
Nonlin. Processes Geophys., 26, 109–122, https://doi.org/10.5194/npg-26-109-2019, https://doi.org/10.5194/npg-26-109-2019, 2019
Short summary
Short summary
This work has to do with a small part of existing algorithms that are used in applications such as predicting the weather. We provide empirical evidence that our new technique works well on small but representative models. This might lead to creation of a better weather forecast and potentially save lives as in the case of hurricane prediction.
Cited articles
Bell, B.: The Iterated Kalman Smoother as a Gauss-Newton Method, SIAM
J. Optim., 4, 626–636, https://doi.org/10.1137/0804035, 1994.
Bergou, E., Gratton, S., and Mandel, J.: On the Convergence of a Non-linear
Ensemble Kalman Smoother, arXiv:1411.4608, submitted, 2014.
Bocquet, M. and Sakov, P.: Combining inflation-free and iterative ensemble
Kalman filters for strongly nonlinear systems, Nonlin. Processes Geophys., 19,
383–399, https://doi.org/10.5194/npg-19-383-2012, 2012.
Bocquet, M. and Sakov, P.: Joint state and parameter estimation with an
iterative ensemble Kalman smoother, Nonlin. Processes Geophys., 20,
803–818, https://doi.org/10.5194/npg-20-803-2013, 2013.
Bocquet, M. and Sakov, P.: An iterative ensemble Kalman smoother,
Q. J. Roy. Meteor. Soc., 140, 1521–1535,
https://doi.org/10.1002/qj.2236, 2014.
Brusdal, K., Brankart, J. M., Halberstadt, G., Evensen, G., Brasseur, P., van
Leeuwen, P. J., Dombrowsky, E., and Verron, J.: A demonstration of ensemble
based assimilation methods with a layered OGCM from the perspective of
operational ocean forecasting systems, J. Mar. Syst., 40–41,
253–289, https://doi.org/10.1016/S0924-7963(03)00021-6, 2003.
Butala, M. D.: A localized ensemble Kalman smoother, in: 2012 IEEE
Statistical Signal Processing Workshop (SSP), 21–24, IEEE,
https://doi.org/10.1109/SSP.2012.6319665, 2012.
Chen, Y. and Oliver, D.: Levenberg-Marquardt forms of the iterative
ensemble smoother for efficient history matching and uncertainty
quantification, Comp. Geosci., 17, 689–703,
https://doi.org/10.1007/s10596-013-9351-5, 2013.
Cosme, E., Brankart, J.-M., Verron, J., Brasseur, P., and Krysta, M.:
Implementation of a reduced rank square-root smoother for high resolution
ocean data assimilation, Ocean Model., 33, 87–100,
https://doi.org/10.1016/j.ocemod.2009.12.004, 2010.
Courtier, P., Thépaut, J.-N., and Hollingsworth, A.: A strategy for
operational implementation of 4D-Var, using an incremental approach,
Q. J. Roy. Meteor. Soc., 120, 1367–1387,
https://doi.org/10.1002/qj.49712051912, 1994.
Desroziers, G., Camino, J.-T., and Berre, L.: 4DEnVar: link with 4D state
formulation of variational assimilation and different possible
implementations, Q. J. Roy. Meteor. Soc., 140,
2097–2110, https://doi.org/10.1002/qj.2325, 2014.
Developmental Testbed Center: NOAA Ensemble Kalman Filter Beta Release
v1.0, http://www.dtcenter.org/com-GSI/users/docs (last access: March 2015),
2015.
Dietrich, C. R. and Newsam, G. N.: Fast and Exact Simulation of Stationary
Gaussian Processes through Circulant Embedding of the Covariance Matrix, SIAM
J. Sci. Comp., 18, 1088–1107, 1997.
Evensen, G.: Data Assimilation: The Ensemble Kalman Filter, Springer, 2nd
edn., xxiv+307, https://doi.org/10.1007/978-3-642-03711-5, 2009.
Fandry, C. and Leslie, L.: A Two-Layer Quasi-Geostrophic Model of Summer
Trough Formation in the Australian Subtropical Easterlies, J. Atmos. Sci., 41, 807–817, 1984.
Fisher, M., Tr'emolet, Y., Auvinen, H., Tan, D., and Poli, P.:
Weak-Constraint
and Long-Window 4D-Var, Tech. rep., European Centre for Medium-Range Weather
Forecasts, 2011.
Fisher, M., Leutbecher, M., and Kelly, G. A.: On the equivalence between
Kalman smoothing and weak-constraint four-dimensional variational data
assimilation, Q. J. Roy. Meteor. Soc., 131,
3235–3246, https://doi.org/10.1256/qj.04.142, 2005.
Furrer, R. and Bengtsson, T.: Estimation of high-dimensional prior and
posterior covariance matrices in Kalman filter variants, J. Multivar.
Anal., 98, 227–255, https://doi.org/10.1016/j.jmva.2006.08.003, 2007.
Gill, P. E. and Murray, W.: Algorithms for the solution of the nonlinear
least-squares problem, SIAM J. Numer. Anal., 15, 977–992,
https://doi.org/10.1137/0715063, 1978.
Gratton, S., Gürol, S., and Toint, P.: Preconditioning and globalizing
conjugate gradients in dual space for quadratically penalized nonlinear-least
squares problems, Computational Opt. Appl., 54, 1–25,
https://doi.org/10.1007/s10589-012-9478-7, 2013.
Gu, Y. and Oliver, D.: An iterative ensemble Kalman filter for multiphase
fluid flow data assimilation, SPE J., 12, 438–446,
https://doi.org/10.2118/108438-PA, 2007.
Hamill, T. M. and Snyder, C.: A Hybrid Ensemble Kalman Filter–3D
Variational Analysis Scheme, Mon. Weather Rev., 128, 2905–2919,
https://doi.org/10.1175/1520-0493(2000)128<2905:AHEKFV>2.0.CO;2, 2000a.
Hamill, T. M. and Snyder, C.: A Hybrid Ensemble Kalman Filter–3D
Variational Analysis Scheme, Mon. Weather Rev., 128, 2905–2919,
https://doi.org/10.1175/1520-0493(2000)128<2905:AHEKFV>2.0.CO;2,
2000b.
Johns, C. J. and Mandel, J.: A Two-Stage Ensemble Kalman Filter for Smooth
Data Assimilation, Environ. Ecol. Stat., 15, 101–110,
https://doi.org/10.1007/s10651-007-0033-0, 2008.
Kasanický, I., Mandel, J., and Vejmelka, M.: Spectral diagonal ensemble
Kalman filters, Nonlin. Processes Geophys., 22, 485–497,
https://doi.org/10.5194/npg-22-485-2015, 2015.
Khare, S. P., Anderson, J. L., Hoar, T. J., and Nychka, D.: An investigation
into the application of an ensemble Kalman smoother to high-dimensional
geophysical systems, Tellus A, 60, 97–112,
https://doi.org/10.1111/j.1600-0870.2007.00281.x, 2008.
Levenberg, K.: A method for the solution of certain non-linear problems in
least squares, Q. Appl. Math., 2, 164–168, 1944.
Liu, C. and Xiao, Q.: An Ensemble-Based Four-Dimensional Variational Data
Assimilation Scheme. Part III: Antarctic Applications with Advanced
Research WRF Using Real Data, Mon. Weather Rev., 141, 2721–2739,
https://doi.org/10.1175/MWR-D-12-00130.1, 2013.
Liu, C., Xiao, Q., and Wang, B.: An Ensemble-Based Four-Dimensional
Variational
Data Assimilation Scheme. Part I: Technical Formulation and Preliminary
Test, Mon. Weather Rev., 136, 3363–3373, https://doi.org/10.1175/2008MWR2312.1,
2008.
Liu, C., Xiao, Q., and Wang, B.: An Ensemble-Based Four-Dimensional
Variational
Data Assimilation Scheme. Part II: Observing System Simulation Experiments
with Advanced Research WRF (ARW), Mon. Weather Rev., 137, 1687–1704,
https://doi.org/10.1175/2008MWR2699.1, 2009.
Lorenc, A. C., Bowler, N. E., Clayton, A. M., Pring, S. R., and Fairbairn,
D.:
Comparison of Hybrid-4DEnVar and Hybrid-4DVar Data Assimilation Methods
for Global NWP, Mon. Weather Rev., 143, 212–229,
https://doi.org/10.1175/MWR-D-14-00195.1, 2014.
Lorenz, E. N.: Deterministic Nonperiodic Flow, J. Atmos. Sci., 20, 130–141,
https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2, 1963.
Mandel, J., Beezley, J. D., Coen, J. L., and Kim, M.: Data Assimilation for
Wildland Fires: Ensemble Kalman filters in coupled atmosphere-surface
models, IEEE Contr. Syst. Mag., 29, 47–65,
https://doi.org/10.1109/MCS.2009.932224, 2009.
Marquardt, D. W.: An Algorithm for Least-Squares Estimation of Nonlinear
Parameters, Journal of the Society for Industrial and Applied Mathematics,
11, 431–441, https://doi.org/10.1137/0111030, 1963.
Metref, S., Cosme, E., Snyder, C., and Brasseur, P.: A non-Gaussian
analysis scheme using rank histograms for ensemble data assimilation,
Nonlin. Processes Geophys., 21, 869–885, https://doi.org/10.5194/npg-21-869-2014, 2014.
Nowak, W., Tenkleve, S., and Cirpka, O.: Efficient Computation of Linearized
Cross-Covariance and Auto-Covariance Matrices of Interdependent Quantities,
Math. Geol., 35, 53–66, 2003.
Osborne, M. R.: Nonlinear least squares – the Levenberg algorithm
revisited, J. Austr. Math. Soc. B, 19, 343–357,
https://doi.org/10.1017/S033427000000120X, 1976.
Ott, E., Hunt, B. R., Szunyogh, I., Zimin, A. V., Kostelich, E. J., Corazza,
M., Kalnay, E., Patil, D., and Yorke, J. A.: A Local Ensemble Kalman Filter
for Atmospheric Data Assimilation, Tellus, 56A, 415–428,
https://doi.org/10.1111/j.1600-0870.2004.00076.x, 2004.
Pedlosky, J.: Geophysical Fluid Dynamics, Springer, xii+625,
1979.
Rauch, H. E., Tung, F., and Striebel, C. T.: Maximum likelihood estimates of
linear dynamic systems, AIAA J., 3, 1445–1450, 1965.
Sakov, P. and Bertino, L.: Relation Between Two Common Localisation Methods
for the EnKF, Comp. Geosci., 10, 225–237,
https://doi.org/10.1007/s10596-010-9202-6, 2011.
Sakov, P., Oliver, D. S., and Bertino, L.: An Iterative EnKF for Strongly
Nonlinear Systems, Mon. Weather Rev., 140, 1988–2004,
https://doi.org/10.1175/MWR-D-11-00176.1, 2012.
Strang, G. and Borre, K.: Linear Algebra, Geodesy, and GPS,
Wellesley-Cambridge Press, 624 pp., 1997.
Stroud, J. R., Stein, M. L., Lesht, B. M., Schwab, D. J., and Beletsky, D.:
An
Ensemble Kalman Filter and Smoother for Satellite Data Assimilation,
J. Am. Stat. Assoc., 105, 978–990,
https://doi.org/10.1198/jasa.2010.ap07636, 2010.
Trémolet, Y.: Model-error estimation in 4D-Var, Q. J. Roy. Meteor. Soc.
Roy. Meteor. Soc., 133, 1267–1280, https://doi.org/10.1002/qj.94, 2007.
Trémolet, Y.: Object-Oriented Prediction System, http://www.data-assimilation.net/Events/Year3/OOPS.pdf (last access: 19 February 2016), 2013.
Tshimanga, J., Gratton, S., Weaver, A. T., and Sartenaer, A.: Limited-memory
preconditioners, with application to incremental four-dimensional variational
data assimilation, Q. J. Roy. Meteor. Soc.,
134, 751–769, https://doi.org/10.1002/qj.228, 2008.
Wang, X.: Incorporating Ensemble Covariance in the Gridpoint Statistical
Interpolation Variational Minimization: A Mathematical Framework, Mon.
Weather Rev., 138, 2990–2995, https://doi.org/10.1175/2010MWR3245.1, 2010.
Wright, S. J. and Holt, J. N.: An inexact Levenberg-Marquardt method for
large sparse nonlinear least squares, J. Austr. Math. Soc. Ser. B, 26,
387–403, https://doi.org/10.1017/S0334270000004604, 1985.
Zhang, F., Zhang, M., and Hansen, J.: Coupling ensemble Kalman filter with
four-dimensional variational data assimilation, Adv. Atmos.
Sci., 26, 1–8, https://doi.org/10.1007/s00376-009-0001-8, 2009.
Zupanski, M.: Maximum Likelihood Ensemble Filter: Theoretical Aspects,
Mon. Weather Rev., 133, 1710–1726, https://doi.org/10.1175/MWR2946.1, 2005.
Short summary
A stochastic method, the ensemble Kalman smoother (EnKS), is proposed as a linear solver in four-dimensional variational data assimilation (4DVAR). The method approaches 4DVAR for large ensembles. Regularization provides global convergence, and it is implemented as an additional artificial observation. Since the EnKS is uncoupled from the insides of the 4DVAR, any version of EnKS can be used.
A stochastic method, the ensemble Kalman smoother (EnKS), is proposed as a linear solver in...
