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We propose a certified reduced basis approach for the strong- and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the…

Optimization and Control · Mathematics 2018-02-08 Mark Kärcher , Sébastien Boyaval , Martin A. Grepl , Karen Veroy

Methods to deal with systematic model errors are an increasingly important component of modern data assimilation systems and their effectiveness has increased in recent years thanks to advances in methodology and the quality and density of…

Applications · Statistics 2022-09-26 Massimo Bonavita , Patrick Laloyaux

Variational data assimilation in continuous time is revisited. The central techniques applied in this paper are in part adopted from the theory of optimal nonlinear control. Alternatively, the investigated approach can be considered as a…

Atmospheric and Oceanic Physics · Physics 2015-05-18 Jochen Bröcker

Dynamic downscaling typically involves using numerical weather prediction (NWP) solvers to refine coarse data to higher spatial resolutions. Data-driven models such as FourCastNet have emerged as a promising alternative to the traditional…

Atmospheric and Oceanic Physics · Physics 2025-03-05 Philip Dinenis , Vishwas Rao , Mihai Anitescu

For many years, strongly and weakly constrained approaches were the only options to deal with errors in four-dimensional variational data assimilation (4DVar), with the aim of balancing the degrees of freedom and model constraints. Strong…

Atmospheric and Oceanic Physics · Physics 2022-12-21 Xiangjun Tian , Hongqin Zhang , Zhe Jin , Min Zhao , Yilong Wang , Yinhai Luo , Ziqing Zhang , Yanyan Tan

Four-dimensional variational data assimilation (4D-Var) on a seasonal-to-interdecadal time scale under the existence of unstable modes can be viewed as an optimization problem of synchronized, coupled chaotic systems. The problem is tackled…

Data Analysis, Statistics and Probability · Physics 2015-11-17 Nozomi Sugiura , Shuhei Masuda , Yosuke Fujii , Masafumi Kamachi , Yoichi Ishikawa , Toshiyuki Awaji

In this paper, we propose a reduced order approach for 3D variational data assimilation governed by parametrized partial differential equations. In contrast to the classical 3D-VAR formulation that penalizes the measurement error directly,…

Numerical Analysis · Mathematics 2019-05-16 Nicole Aretz-Nellesen , Martin A. Grepl , Karen Veroy

This work introduces a new class of four-dimensional variational data assimilation (4D-Var) methods grounded in data-consistent inversion (DCI) theory. The methods extend classical 4D-Var by incorporating a predictability-aware…

Numerical Analysis · Mathematics 2025-11-04 Rylan Spence , Troy Butler , Clint Dawson

This paper presents a reduced-order approach for four-dimensional variational data assimilation, based on a prior EO F analysis of a model trajectory. This method implies two main advantages: a natural model-based definition of a mul…

Variational data assimilation estimates the dynamical system states by minimizing a cost function that fits the numerical models with the observational data. Although four-dimensional variational assimilation (4D-Var) is widely used, it…

Machine Learning · Computer Science 2025-06-16 Yiming Yang , Xiaoyuan Cheng , Daniel Giles , Sibo Cheng , Yi He , Xiao Xue , Boli Chen , Yukun Hu

Using a dynamical model to make predictions about a system has many sources of error. These can include errors in how the model was initialised but also errors in the dynamics of the model itself. For many applications in data assimilation,…

Numerical Analysis · Mathematics 2025-07-07 P. A. Browne

Recent studies have demonstrated that it is possible to combine machine learning with data assimilation to reconstruct the dynamics of a physical model partially and imperfectly observed. Data assimilation is used to estimate the system…

Machine Learning · Statistics 2022-10-26 Alban Farchi , Marcin Chrust , Marc Bocquet , Patrick Laloyaux , Massimo Bonavita

The performance of ensemble-based data assimilation techniques that estimate the state of a dynamical system from partial observations depends crucially on the prescribed uncertainty of the model dynamics and of the observations. These are…

Computation · Statistics 2021-02-24 Tadeo Javier Cocucci , Manuel Pulido , Magdalena Lucini , Pierre Tandeo

Model error covariances play a central role in the performance of data assimilation methods applied to nonlinear state-space models. However, these covariances are largely unknown in most of the applications. A misspecification of the model…

Computation · Statistics 2019-11-06 María Magdalena Lucini , Peter Jan van Leeuwen , Manuel Pulido

This paper discusses the practical use of the saddle variational formulation for the weakly-constrained 4D-VAR method in data assimilation. It is shown that the method, in its original form, may produce erratic results or diverge because of…

Numerical Analysis · Mathematics 2021-05-31 S. Gratton , S. Gürol , E. Simon , Ph. L. Toint

Data assimilation method consists in combining all available pieces of information about a system to obtain optimal estimates of initial states. The different sources of information are weighted according to their accuracy by the means of…

Data Analysis, Statistics and Probability · Physics 2014-04-30 Angélique Ponçot , Jean-Philippe Argaud , Bertrand Bouriquet , Patrick Erhard , Serge Gratton , Olivier Thual

Data assimilation of atmospheric observations traditionally relies on variational and Kalman filter methods. Here, an alternative neural-network data assimilation (NNDA) with variational autoencoder (VAE) is proposed. The three-dimensional…

Atmospheric and Oceanic Physics · Physics 2024-04-29 Boštjan Melinc , Žiga Zaplotnik

In data assimilation, the model may be subject to uncertainties and errors. The weak-constraint data assimilation framework enables incorporating model uncertainty in the dynamics of the governing equations. We propose a new framework for…

Numerical Analysis · Mathematics 2025-12-23 Alen Alexanderian , Hugo Díaz , Vishwas Rao , Arvind K. Saibaba

Four-dimensional weak-constraint variational data assimilation estimates a state given partial noisy observations and dynamical model by minimizing a cost function that takes into account both discrepancy between the state and observations…

Dynamical Systems · Mathematics 2023-04-13 Nazanin Abedini , Svetlana Dubinkina

Using a high degree of parallelism is essential to perform data assimilation efficiently. The state formulation of the incremental weak constraint four-dimensional variational data assimilation method allows parallel calculations in the…

Numerical Analysis · Mathematics 2021-08-19 Ieva Daužickaitė , Amos S. Lawless , Jennifer A. Scott , Peter Jan van Leeuwen
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