Comparison of sequential data assimilation methods for the Kuramoto–Sivashinsky equation

In this study, we present a probabilistic urban inundation modeling framework that combines high-resolution process-based modeling with observed information via ensemble data assimilation (DA). We investigate the impact of multivariate flood observations on improving urban inundation prediction accuracy through synthetic experiments. The framework leverages diverse flood observations from both the urban surface and sewer system, integrating them into the modeling process using non-Gaussian sequential DA methods, such as particle filtering. The modeling framework employs the H12 model for integrated 1D sewer network and 2D surface inundation analyses, with synthetic experiments conducted in an urban catchment in Osaka, Japan. Prior to implementing DA, a sensitivity analysis is performed to assess the effects of uncertainties in inundation modeling. Major uncertainty components, such as input forcings and storm drain box efficiency, are perturbed to evaluate their influence. The synthetic DA experiments analyze the influence of various types of flood observations, including urban surface inundation depths and sewer water levels, on the posterior distributions of the model ensemble. Additionally, the impact of observation location and density on DA performance is evaluated. This study demonstrates the potential of utilizing diverse flood observations in data assimilation to enhance the accuracy and reliability of urban flood predictions.

In numerical weather prediction, parameterisations are used to simulate missing physics in the model. These can be due to a lack of scientific understanding or a lack of computing power available to address all the known physical processes. Parameterisations are sources of large uncertainty in a model as parameter values used in these parameterisations cannot be measured directly and hence are often not well known; and the parameterisations themselves are also approximations of the processes present in the true atmosphere. Whilst there are many efficient and effective methods for combined state/parameter estimation in data assimilation (DA), such as state augmentation, these are not effective at estimating the structure of parameterisations. A new method of parameterisation estimation is proposed that uses sequential DA methods to estimate errors in the numerical models at each space-time point for each model equation. These errors are then fitted to pre-determined functional forms of missing physics or parameterisations that are based upon prior information. We applied the method to a one-dimensional advection model with additive model error, and it is shown that the method can accurately estimate parameterisations, with consistent error estimates. Furthermore, it is shown how the method depends on the quality of the DA results. The results indicate that this new method is a powerful tool in systematic model improvement.

In the past few years sequential data assimilation (SDA) methods have emerged as the best possible method at hand to properly treat all sources of error in hydrological modeling. However, very few studies have actually implemented SDA methods using realistic input error models for precipitation. In this study we use particle filtering as a SDA method to propagate input errors through a conceptual hydrologic model and quantify the state, parameter and streamflow uncertainties. Recent progress in satellite‐based precipitation observation techniques offers an attractive option for considering spatiotemporal variation of precipitation. Therefore, we use the PERSIANN‐CCS precipitation product to propagate input errors through our hydrologic model. Some uncertainty scenarios are set up to incorporate and investigate the impact of the individual uncertainty sources from precipitation, parameters and also combined error sources on the hydrologic response. Also probabilistic measures are used to quantify the quality of ensemble prediction.

This Special Issue presents efficient formulations and implementations of sequential and variational data assimilation methods. The methods address three important issues in the context of operational data assimilation: efficient implementation of localization methods, sampling methods for approaching posterior ensembles under non-linear model errors, and adjoint-free formulations of four dimensional variational methods.

Bayesian inference for nonlinear diffusions, observed at discrete times, is a challenging task that has prompted the development of a number of algorithms, mainly within the computational statistics community. We propose a new direction, and accompanying methodology—borrowing ideas from statistical physics and computational chemistry—for inferring the posterior distribution of latent diffusion paths and model parameters, given observations of the process. Joint configurations of the underlying process noise and of parameters, mapping onto diffusion paths consistent with observations, form an implicitly defined manifold. Then, by making use of a constrained Hamiltonian Monte Carlo algorithm on the embedded manifold, we are able to perform computationally efficient inference for a class of discretely observed diffusion models. Critically, in contrast with other approaches proposed in the literature, our methodology is highly automated, requiring minimal user intervention and applying alike in a range of settings, including: elliptic or hypo-elliptic systems; observations with or without noise; linear or non-linear observation operators. Exploiting Markovianity, we propose a variant of the method with complexity that scales linearly in the resolution of path discretisation and the number of observation times. Python code reproducing the results is available at http://doi.org/10.5281/zenodo.5796148.

A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter. The unbounded error growth found in the extended Kalman filter, which is caused by an overly simplified closure in the error covariance equation, is completely eliminated. Open boundaries can be handled as long as the ocean model is well posed. Well‐known numerical instabilities associated with the error covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method should be feasible also for more sophisticated primitive equation models. The computational load for reasonable accuracy is only a fraction of what is required for the extended Kalman filter and is given by the storage of, say, 100 model states for an ensemble size of 100 and thus CPU requirements of the order of the cost of 100 model integrations. The proposed method can therefore be used with realistic nonlinear ocean models on large domains on existing computers, and it is also well suited for parallel computers and clusters of workstations where each processor integrates a few members of the ensemble.

We apply information theory within an ensemble‐based data assimilation approach and define information matrix in ensemble subspace. The information matrix in ensemble subspace employs a flow‐dependent forecast error covariance and it is of relatively small dimensions (equal to the ensemble size). The information matrix in ensemble subspace can be directly linked to the information matrix typically used in non‐ensemble‐based data assimilation methods, such as the Kalman Filter (KF) and the three‐dimensional variational (3D‐Var) methods, which provides a framework for consistent comparisons of information measures between different data assimilation methods.We evaluate information measures, such as degrees of freedom for signal, within the Maximum Likelihood Ensemble Filter (MLEF) data assimilation approach and compare them with those obtained using the KF approach and the 3D‐Var approach. We assimilate model‐simulated observations and use the Goddard Earth Observing System Single Column Model (GEOS‐5 SCM) as a dynamical forecast model.The experimental results demonstrate that the proposed framework is useful for comparing information measures obtained in different data assimilation approaches. These comparisons indicate that using a flow‐dependent forecast error covariance matrix (e.g. as in the KF and the MLEF experiments) is fundamentally important for adequately describing prior knowledge about the true model state when calculating information measures of assimilated observations. We also demonstrate that data assimilation results obtained using the KF and the MLEF approach (when ensemble size is larger than 10 ensemble members) are superior to the results of the 3D‐Var approach. Copyright © 2007 Royal Meteorological Society

Quantifying continental-scale river discharge is essential to understanding the terrestrial water cycle but is susceptible to errors caused by a lack of observations and the limitations of hydrodynamic modeling. Data assimilation (DA) methods are increasingly used to estimate river discharge in combination with emerging river-related remote sensing products (e.g., water surface elevation [WSE], water surface slope, river width, and flood extent). However, directly comparing simulated WSE to satellite altimetry data remains challenging (e.g., because of large biases between simulations and observations or uncertainties in parameters), and large errors can be introduced when satellite observations are assimilated into hydrodynamic models. In this study we performed direct, anomaly, and normalized value assimilation experiments to investigate the capacity of DA to improve river discharge within the current limitations of hydrodynamic modeling. We performed hydrological DA using a physically-based empirical localization method applied to the Amazon Basin. We used satellite altimetry data from ENVISAT, Jason 1, and Jason 2. Direct DA was the baseline assimilation method and was subject to errors due to biases in the simulated WSE. To overcome these errors, we used anomaly DA as an alternative to direct DA. We found that the modeled and observed WSE distributions differed considerably (e.g., differences in amplitude, seasonal flow variation, and a skewed distribution due to limitations of the hydrodynamic models). Therefore, normalized value DA was performed to improve discharge estimation. River discharge estimates were improved at 24 %, 38 %, and 62 % of stream gauges in the direct, anomaly, and normalized value assimilations relative to simulations without DA. Normalized value assimilation performed best for estimating river discharge given the current limitations of hydrodynamic models. Most gauges within the river reaches covered by satellite observations accurately estimated river discharge, with Nash-Sutcliffe efficiency (NSE) > 0.6. The amplitudes of WSE variation were improved in the normalized DA experiment. Furthermore, in the Amazon Basin, normalized assimilation (median NSE = 0.50) improved river discharge estimation compared to open-loop simulation with the global hydrodynamic model (median NSE = 0.42). River discharge estimation using direct DA methods was improved by 7 % with calibration of river bathymetry based on NSE. The direct DA approach outperformed the other DA approaches when runoff was considerably biased, but anomaly DA performed best when the river bathymetry was erroneous. The uncertainties in hydrodynamic modeling (e.g., river bottom elevation, river width, simplified floodplain dynamics, and the rectangular cross-section assumption) should be improved to fully realize the advantages of river discharge DA through the assimilation of satellite altimetry. This study contributes to the development of a global river discharge reanalysis product that is consistent spatially and temporally.

Ensuring a reliable water supply in the face of changing conditions and growing demand is a critical global challenge. Water distribution networks (WDNs) are essential infrastructure, but traditional modeling methods based on historical data often struggle to adapt in real-time and integrate new information. In order to lower model errors in WDN models, this study explores the use of a Data Assimilation (DA) method that makes use of the Ensemble Kalman Filter. The study explores the effectiveness of the DA method, and a novel Greedy Algorithm (GA) for optimizing the location of sensors. The study shows that the DA method improves the WDN model and the GA is able to successfully determine optimal sensor locations. It was observed that increasing the number of sensors in the WDN increased the effectiveness of the DA method. This study highlights the potential of data assimilation to improve WDN modeling. Water utilities can gain from more precise predictions, greater system performance, and efficient decision-making for the management and maintenance of water distribution networks by allowing models to adapt and improve dynamically

A demonstration study of three advanced, sequential data assimilation methods, applied with the nonlinear Miami Isopycnic Coordinate Ocean Model (MICOM), has been performed within the European Commission-funded DIADEM project. The data assimilation techniques considered are the Ensemble Kalman Filter (EnKF), the Ensemble Kalman Smoother (EnKS) and the Singular Evolutive Extended Kalman (SEEK) Filter, which all in different ways resemble the original Kalman Filter. In the EnKF and EnKS an ensemble of model states is integrated forward in time according to the model dynamics, and statistical moments needed at analysis time are calculated from the ensemble of model states. The EnKS, as opposed to the EnKF, update the analysis also backward in time whenever new observations are available, thereby improving the estimated states at the previous analysis times. The SEEK filter reduces the computational burden of the error propagation by representing the errors in a subspace which is initially calculated from a truncated EOF analysis. A hindcast experiment, where sea-level anomaly and sea-surface temperature data are assimilated, has been conducted in the North Atlantic for the time period July until September 1996. In this paper, we describe the implementation of ensemble-based assimilation methods with a common theoretical framework, we present results from hindcast experiments achieved with the EnKF, EnKS and SEEK filter, and we discuss the relative merits of these methods from the perspective of operational marine monitoring and forecasting systems. We found that the three systems have similar performances, and they can be considered feasible technologically for building preoperational prototypes.

Challenges for data assimilation – from convective‐scale to climate

Reconstructing spatiotemporal systems from sparse observations remains a long-standing challenge in several domains, including geoscience, air pollution and fluid dynamics. While various data assimilation (DA) and machine learning (ML) methods have shown potential, they still face significant challenges (see [1]): The computational burden of conventional DA algorithms (including error covariance specification), particularly for multivariate, high-dimensional systems. Sparse and movable sensor placement, which makes conventional ML models (typically requiring fixed and regularly distributed input data) cumbersome. The ill-defined nature of the sparse reconstruction problem, which poses significant risks of overfitting. We will present our recent works aimed at addressing these challenges. More specifically, we developed latent DA algorithms [2] to reduce the computational burden of variational DA methods. These algorithms demonstrate great potential in efficiently assimilating sparse observations within a reduced-order latent space constructed by neural networks, thanks to the TorchDA library [3]. The latter enables GPU implementation of mainstream data assimilation methods and supports non-explicit state-observation transformation functions, provided they can be learned by a neural network.We have also employed advanced deep learning techniques, including Voronoi-tessellation CNNs [4] and Vision Transformer-based autoencoders [5], to learn mappings from sparse observations to the complete physical space. These approaches effectively address challenges such as movable sensor placements and varying sensor numbers. Their integration with DA algorithms has also been evaluated.Finally, our recent work explores [6] the utility of generative AI techniques, particularly denoising diffusion models, for field reconstruction from sparse observations. Generative AI methods offer two main advantages: first, they produce a sample from a probability distribution rather than predicting the mean as a fixed output, which can help mitigate overfitting caused by the illy-defined problem. Second, they inherently function as ensemble predictors by generating several samples, facilitating uncertainty quantification, which is essential in data assimilation. The numerical results tested on cases ranging from fluid dynamics benchmarks to semi-operational air pollution simulations will also be discussed.[1] Cheng, S., Quilodrán-Casas, C., Ouala, S., Farchi, A., Liu, C., Tandeo, P., Fablet, R., Lucor, D., Iooss, B., Brajard, J., Xiao, D., Janjic, T., Ding, W., Guo, Y., Carrassi, A., Bocquet, M. and Arcucci, R, 2023. Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review. IEEE/CAA Journal of Automatica Sinica[2] Cheng, S., Chen, J., Anastasiou, C., Angeli, P., Matar, O.K., Guo, Y.K., Pain, C.C. and Arcucci, R., 2023. Generalised latent assimilation in heterogeneous reduced spaces with machine learning surrogate models. Journal of Scientific Computing[3] Cheng, S., Min, J., Liu, C. and Arcucci, R., 2025. TorchDA: A Python package for performing data assimilation with deep learning forward and transformation functions. Computer Physics Communications[4] Cheng, S., Liu, C., Guo, Y. and Arcucci, R., 2024. Efficient deep data assimilation with sparse observations and time-varying sensors. Journal of Computational Physics[5] Fan, H., Cheng, S., de Nazelle, A.J. and Arcucci, R., 2024. ViTAE-SL: a vision transformer-based autoencoder and spatial interpolation learner for field reconstruction. Computer Physics Communications [6] Zhuang, Y., Cheng, S. and Duraisamy, K., 2025. Spatially-aware diffusion models with cross-attention for global field reconstruction with sparse observations. Computer Methods in Applied Mechanics and Engineering

Sequential data assimilation (DA) was performed on three-dimensional flow fields of a circular jet measured by tomography particle image velocimetry (tomo-PIV). The work focused on an in-depth analysis of the flow enhancement and the pressure determination from volumetric flow measurement data. The jet was issued from a circular nozzle with an inner diameter of D= 20 mm. A split-screen configuration including two high-speed cameras was used to capture the particle images from four different views for a tomography reconstruction of the voxels in the tomo-PIV measurement. Planar PIV was also performed to obtain the benchmark two-dimensional velocity fields for validation. The adjoint-based sequential DA scheme was used with the measurement uncertainty implanted using a threshold function to recover the flow fields with high fidelity and fewer measurement errors. The pressure was determined by either the direct mode, with implementation directly in the DA solver, or by the separate mode, which included solving the Poisson equation on the DA-recovered flow fields. Sequential DA recovered high signal-to-noise flow fields that had piecewise-smooth temporal variations due to the intermittent constraints of the observations, while only the temporal sequence of the fields at the observational instances was selected as the DA output. Errors were significantly reduced, and DA improved the divergence condition of the three-dimensional flow fields. DA also enhanced the dynamical features of the vortical structures, and the pressure determined by both modes successfully captured the downstream convection signatures of the vortex rings.

A new sequential data assimilation method named “Monte Carlo H∞ filter” is introduced based on H∞ filter technique and Monte Carlo method in this paper. This method applies to nonlinear systems in condition of lacking the statistical properties of observational errors. In order to compare the assimilation capability of Monte Carlo H∞ filter with that of the ensemble Kalman filter (EnKF) in solving practical problems caused by temporal correlation or spatial correlation of observational errors, two numerical experiments are performed by using Lorenz (1963) system and shallow-water equations respectively. The result is that the assimilation capability of the new method is better than that of EnKF method. It is also shown that Monte Carlo H∞ filter assimilation method is effective and suitable to nonlinear systems in that it does not depend on the statistical properties of observational errors and has better robustness than EnKF method when the statistical properties of observational errors are varying. In addition, for the new method, the smallest level factor founded by search method is flow-dependent.

This paper investigates the applicability of the Vaganov–Shashkin–Lite (VSL) forward model for tree-ring-width chronologies as observation operator within a proxy data assimilation (DA) setting. Based on the principle of limiting factors, VSL combines temperature and moisture time series in a nonlinear fashion to obtain simulated TRW chronologies. When used as observation operator, this modelling approach implies three compounding, challenging features: (1) time averaging, (2) “switching recording” of 2 variables and (3) bounded response windows leading to “thresholded response”. We generate pseudo-TRW observations from a chaotic 2-scale dynamical system, used as a cartoon of the atmosphere-land system, and attempt to assimilate them via ensemble Kalman filtering techniques. Results within our simplified setting reveal that VSL’s nonlinearities may lead to considerable loss of assimilation skill, as compared to the utilization of a time-averaged (TA) linear observation operator. In order to understand this undesired effect, we embed VSL’s formulation into the framework of fuzzy logic (FL) theory, which thereby exposes multiple representations of the principle of limiting factors. DA experiments employing three alternative growth rate functions disclose a strong link between the lack of smoothness of the growth rate function and the loss of optimality in the estimate of the TA state. Accordingly, VSL’s performance as observation operator can be enhanced by resorting to smoother FL representations of the principle of limiting factors. This finding fosters new interpretations of tree-ring-growth limitation processes.