Ensemble Kalman Filter Method Research Articles

ParticleDA.jl v.1.0: a distributed particle-filtering data assimilation package

Abstract. Digital twins of physical and human systems informed by real-time data are becoming ubiquitous across weather forecasting, disaster preparedness, and urban planning, but researchers lack the tools to run these models effectively and efficiently, limiting progress. One of the current challenges is to assimilate observations in highly non-linear dynamical systems, as the practical need is often to detect abrupt changes. We have developed a software platform to improve the use of real-time data in non-linear system representations where non-Gaussianity limits the applicability of data assimilation algorithms such as the ensemble Kalman filter and variational methods. Particle-filter-based data assimilation algorithms have been implemented within a user-friendly open-source software platform in Julia – ParticleDA.jl. To ensure the applicability of the developed platform in realistic scenarios, emphasis has been placed on numerical efficiency and scalability on high-performance computing systems. Furthermore, the platform has been developed to be forward-model agnostic, ensuring that it is applicable to a wide range of modelling settings, for instance unstructured and non-uniform meshes in the spatial domain or even state spaces that are not spatially organized. Applications to tsunami and numerical weather prediction demonstrate the computational benefits and ease of using the high-level Julia interface with the package to perform filtering in a variety of complex models.

Regional Comparison of Performance between EnKF and EnOI in the North Pacific

Abstract The North Pacific is divided into different regions based on ocean currents and sea surface temperature (SST) distribution. Data assimilation is a useful tool for generating accurate ocean estimates because of the limited availability of observational data. This study compared the performances of two data assimilation methods, ensemble optimal interpolation (EnOI) and ensemble Kalman filter (EnKF), in various North Pacific subregions using an ocean model configured with the Regional Ocean Modeling System (ROMS). Both methods assimilated spaceborne SST observations, and the simulation results varied by subregion. The study found that EnKF and EnOI methods performed better than the control model in all regions when compared against satellite SST. EnOI reproduced SST as well as EnKF and required fewer computational resources. However, EnOI performed worse than the control model at sea surface height (SSH) in the equatorial region, while EnKF’s performance improved. This was due to the crushed mean state in the EnOI, which used long-term historical data as an ensemble member. El Niño–Southern Oscillation at the equator drove substantial interannual variability that crushed the ensemble mean of SSH in the EnOI. It is crucial to use a suitable assimilation method for the target area, considering the regional properties of ocean variables. Otherwise, the performance of the assimilated model may be even worse than that of the control model. While EnKF is better suited for regions with high variability in ocean variables, EnOI requires fewer computational resources. Thus, it is crucial to use a suitable assimilation method for accurately predicting and understanding the dynamics of the North Pacific.

Data assimilation of soil moisture and leaf area index effectively improves the simulation accuracy of water and carbon fluxes in coupled farmland hydrological model

Real time status of farmland hydrology and crop growth is essential for agricultural management. Data assimilation is a common method to improve the prediction accuracy of the model by fusing observed and simulated data. For the farmland hydrological processes, evapotranspiration (ET) and net ecosystem productivity (NEP) are widely concerned and strongly affected by crop growth and soil moisture. In this study, data assimilation for soil water content (SWC) and leaf area index (LAI) was combined with a coupled farmland hydrological model, and the potential of Kalman filter (KF) and ensemble Kalman filter (EnKF) methods to enhance model accuracy were explored. Furthermore, the impact of observation density of assimilated data and different assimilation strategies (single-factor or dual-factor assimilation) were analyzed. The findings revealed that both KF and EnKF methods effectively improved the simulation ability of SWC and LAI. When assimilation was performed daily, KF could obtain results comparable to EnKF with assimilation efficiency coefficient (Eff) exceeded 70%. However, with a reduced assimilation frequency for LAI to ten-day interval, EnKF exhibited superior applicability, demonstrating a 13% increase in Eff. The assimilation of soil moisture could positively affect the simulation results of ET with Eff close to 10%, and the assimilation of LAI could improve the simulation accuracy of NEP with Eff close to 15%. Overall, dual-factor assimilation proved to have a more substantial impact than single-factor, even reducing the frequency to ten-day interval. The sensitivity analysis showed that the coupling model could resist the influence of the preset observation error in the filter, with data assimilation effectively mitigating the influence of parameter errors in coupling model. These analyses supply an effective basis to deepen the understanding of improve real time simulation accuracy of farmland hydrological model with data assimilation.

Time-varying characteristics of saturated hydraulic conductivity in grassed swales based on the ensemble Kalman filter algorithm —A case study of two long-running swales in Netherlands

Saturated hydraulic conductivity (Ks) of the filler layer in grassed swales are varying in the changing environment. In most of the hydrological models, Ks is assumed as constant or decrease with a clogging factor. However, the Ks measured on site cannot be the input of the hydrological model directly. Therefore, in this study, an Ensemble Kalman Filter (EnKF) based approach was carried out to estimate the Ks of the whole systems in two monitored grassed swales at Enschede and Utrecht, the Netherlands. The relationship between Ks and possible influencing factors (antecedent dry period, temperature, rainfall, rainfall duration, total rainfall and seasonal factors) were studied and a Multivariate nonlinear function was established to optimize the hydrological model. The results revealed that the EnKF method was satisfying in the Ks estimation, which showed a notable decrease after long-term operation, but revealed a recovery in summer and winter. After the addition of Multivariate nonlinear function of the Ks into hydrological model, 63.8% of the predicted results were optimized among the validation events, and compared with constant Ks. A sensitivity analysis revealed that the effect of each influencing factors on the Ks varies depending on the type of grassed swale. However, these findings require further investigation and data support.

Component-wise iterative ensemble Kalman inversion for static Bayesian models with unknown measurement error covariance

The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a Gaussian likelihood, in a way that is analogous to likelihood tempering sequential Monte Carlo (SMC). These methods are commonly referred to as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is asymptotically biased if the likelihood is non-linear and/or non-Gaussian and if the priors are non-Gaussian. However, it is significantly faster to run. Currently, a large limitation of EKI methods is that the covariance of the measurement error is assumed to be fully known. We develop a new method, which we call component-wise iterative EKI (CW-IEKI), that allows elements of the covariance matrix to be inferred alongside the model parameters at negligible extra cost. This novel method is compared to SMC on a linear Gaussian example as well as four examples with non-linear dynamics (i.e. non-linear function of the model parameters). The non-linear examples include a set of population models applied to synthetic data, a model of nitrogen mineralisation in soil that is based on the Agricultural Production Systems Simulator, a model predicting seagrass decline due to stress from water temperature and light, and a model predicting coral calcification rates. On our examples, we find that CW-IEKI has relatively similar predictive performance to SMC, albeit with greater uncertainty, and it has a significantly faster run time.

Gaussian Variational Approximations for High-dimensional State Space Models

We consider a Gaussian variational approximation of the posterior density in high-dimensional state space models. The number of parameters in the covariance matrix of the variational approximation grows as the square of the number of model parameters, so it is necessary to find simple yet effective parametrisations of the covariance structure when the number of model parameters is large. We approximate the joint posterior density of the state vectors by a dynamic factor model, having Markovian time dependence and a factor covariance structure for the states. This gives a reduced description of the dependence structure for the states, as well as a temporal conditional independence structure similar to that in the true posterior. We illustrate the methodology on two examples. The first is a spatio-temporal model for the spread of the Eurasian collared-dove across North America. Our approach compares favorably to a recently proposed ensemble Kalman filter method for approximate inference in high-dimensional hierarchical spatio-temporal models. Our second example is a Wishart-based multivariate stochastic volatility model for financial returns, which is outside the class of models the ensemble Kalman filter method can handle.

Comparison of Data Fusion Methods in Fusing Satellite Products and Model Simulations for Estimating Soil Moisture on Semi-Arid Grasslands

In arid and semi-arid areas, soil moisture (SM) plays a crucial role in land-atmosphere interactions, hydrological processes, and ecosystem sustainability. SM data at large scales are critical for related climatic, hydrological, and ecohydrological research. Data fusion based on satellite products and model simulations is an important way to obtain SM data at large scales; however, little has been reported on the comparison of the data fusion methods in different categories. Here, we compared the performance of two widely used data fusion methods, the Ensemble Kalman Filter (EnKF) and the Back-Propagation Artificial Neural Network (BPANN), in the degraded grassland site (DGS) and the alpine grassland site (AGS). The SM data from the Community Land Model 5.0 (CLM5.0) and the Soil Moisture Active and Passive (SMAP) were fused and validated against the observations of the Cosmic-Ray Neutron Sensor (CRNS) to avoid the impacts of scale-mismatch. Results show that compared with the original data sets at both sites, the RMSE of the fused data by BPANN (FD-BPANN) and EnKF (FD-EnKF) had improved by more than 50% and 31%, respectively. Overall, the FD-BPANN performs better than the FD-EnKF because the BPANN method assigned higher weights to input data with better performance and the EnKF method is affected by the strong variabilities of both the fused CLM5.0 and SMAP data and the CRNS data. However, in terms of the percentile range, the FD-BPANN showed the worst performance, with overestimations in the low SM range of 25th percentile (<Q25), because the BPANN method tends to be trapped in a local minimum. The BPANN method performed better in humid areas, then followed by semi-humid areas, and finally arid and semi-arid areas. Moreover, compared with the previous studies in arid and semi-arid areas, the BPANN method in this study performed better.

Review article: Towards strongly coupled ensemble data assimilation with additional improvements from machine learning

Abstract. We assessed different coupled data assimilation strategies with a hierarchy of coupled models, ranging from a simple coupled Lorenz model to the state-of-the-art coupled general circulation model CFSv2 (Climate Forecast System version 2). With the coupled Lorenz model, we assessed the analysis accuracy by strongly coupled ensemble Kalman filter (EnKF) and 4D-Variational (4D-Var) methods with varying assimilation window lengths. The analysis accuracy of the strongly coupled EnKF with a short assimilation window is comparable to that of 4D-Var with a long assimilation window. For 4D-Var, the strongly coupled approach with the coupled model produces more accurate ocean analysis than the Estimating the Circulation and Climate of the Ocean (ECCO)-like approach using the uncoupled ocean model. Experiments with the coupled quasi-geostrophic model conclude that the strongly coupled approach outperforms the weakly coupled and uncoupled approaches for both the full-rank EnKF and 4D-Var, with the strongly coupled EnKF and 4D-Var showing a similar level of accuracy higher than other coupled data assimilation approaches such as outer-loop coupling. A strongly coupled EnKF software framework is developed and applied to the intermediate-complexity coupled model SPEEDY-NEMO and the state-of-the-art operational coupled model CFSv2. Experiments assimilating synthetic or real atmospheric observations into the ocean through strongly coupled EnKF show that the strongly coupled approach improves the analysis of the atmosphere and upper ocean but degrades observation fits in the deep ocean, probably due to the unreliable error correlation estimated by a small ensemble. The correlation-cutoff method is developed to reduce the unreliable error correlations between physically irrelevant model states and observations. Experiments with the coupled Lorenz model demonstrate that strongly coupled EnKF informed by the correlation-cutoff method produces more accurate coupled analyses than the weakly coupled and plain strongly coupled EnKF regardless of the ensemble size. To extend the correlation-cutoff method to operational coupled models, a neural network approach is proposed to systematically acquire the observation localization functions for all pairs between the model state and observation types. The following strongly coupled EnKF experiments with an intermediate-complexity coupled model show promising results with this method.

Reconstructing tephra fall deposits via ensemble-based data assimilation techniques

Abstract. In recent years, there has been a growing interest in ensemble approaches for modelling the atmospheric transport of volcanic aerosol, ash, and lapilli (tephra). The development of such techniques enables the exploration of novel methods for incorporating real observations into tephra dispersal models. However, traditional data assimilation algorithms, including ensemble Kalman filter (EnKF) methods, can yield suboptimal state estimates for positive-definite variables such as those related to volcanic aerosols and tephra deposits. This study proposes two new ensemble-based data assimilation techniques for semi-positive-definite variables with highly skewed uncertainty distributions, including aerosol concentrations and tephra deposit mass loading: the Gaussian with non-negative constraints (GNC) and gamma inverse-gamma (GIG) methods. The proposed methods are applied to reconstruct the tephra fallout deposit resulting from the 2015 Calbuco eruption using an ensemble of 256 runs performed with the FALL3D dispersal model. An assessment of the methodologies is conducted considering two independent datasets of deposit thickness measurements: an assimilation dataset and a validation dataset. Different evaluation metrics (e.g. RMSE, MBE, and SMAPE) are computed for the validation dataset, and the results are compared to two references: the ensemble prior mean and the EnKF analysis. Results show that the assimilation leads to a significant improvement over the first-guess results obtained from the simple ensemble forecast. The evidence from this study suggests that the GNC method was the most skilful approach and represents a promising alternative for assimilation of volcanic fallout data. The spatial distributions of the tephra fallout deposit thickness and volume according to the GNC analysis are in good agreement with estimations based on field measurements and isopach maps reported in previous studies. On the other hand, although it is an interesting approach, the GIG method failed to improve the EnKF analysis.

Numerical Methods for Fractional Optimal Control and Estimation

Fractional calculus has become a valuable mathematical tool for modeling various physical phenomena exhibiting anomalous dynamics such as memory and hereditary properties. However, the fractional operators lead to difficulties in analysis, optimization, and estimation that limit the application of fractional models. This paper develops numerical methods to solve fractional optimal control and estimation problems with Caputo derivatives of arbitrary order.
 First, fractional Pontryagin's maximum principle is used to formulate first-order necessary conditions for fractional optimal control problems. A fractional collocation method using polynomial basis functions is then proposed to discretize the resulting boundary value problems. This allows transforming an infinite-dimensional optimal control problem into a finite nonlinear programming problem.
 Second, for fractional estimation, a novel ensemble Kalman filter is proposed based on a Monte Carlo approach to propagate the fractional state dynamics. This provides a recursive fractional state estimator analogous to the classical Kalman filter.
 The capabilities of the proposed collocation and ensemble Kalman filter methods are demonstrated through applications including fractional epidemic control, thermomechanical oscillator control, and state estimation of viscoelastic mechanical systems. The results illustrate improved accuracy over prior discretization schemes along with the ability to handle complex system dynamics.
 This work provides a comprehensive framework for numerical solution of fractional optimal control and estimation problems. The methods enable applying fractional calculus to address challenges in robotics, biomedicine, mechanics, and other fields where systems exhibit non-classical dynamics.

The optimization of SO2 emissions by the 4DVAR and EnKF methods and its application in WRF-Chem

Emissions are essential for forecasting air quality and pollution control, but traditional emissions are often not real-time by the statistics of “bottom-up” approach due to high human resource demand. The four-dimensional variational method (4DVAR) and the ensemble Kalman filter (EnKF) are generally used to optimize emissions based on chemical transport models by assimilating observations. Although the two methods solve similar estimation problems, different functions have been developed to address the process of converting the emissions to concentrations. In this paper, we evaluated the performance of the 4DVAR and EnKF methods in optimizing SO2 emissions over China during 23–29 January 2020. The emissions optimized by the 4DVAR and EnKF methods showed a similar spatiotemporal distribution in most regions of China during the study period, suggesting that both methods are useful in reducing uncertainties in the prior emissions. Three forecast experiments with different emissions were conducted. Compared with the forecasts with prior emissions, the root-mean-square error of the forecasts with the emissions optimized by the 4DVAR and EnKF methods decreased by 45.7 % and 40.4 %. This indicates that the 4DVAR method was slightly more effective than the EnKF method in optimizing emissions and improves the accuracy of forecasts. Furthermore, it is found that the 4DVAR method performed better than the EnKF method when the spatial and/or temporal distribution of SO2 observations with strong local characteristics, The EnKF method showed a better performance for the condition of the large difference between prior emissions and real emissions. The results may help to design suitable assimilation algorithms for optimizing emissions and improving model forecasts. The advance data assimilation systems are beneficial for the understanding the effectiveness and value of emission inventories and air quality model.

An Ensemble Kalman Filter Data Assimilation Method for the Sea Surface Temperature in the China Seas: Implementation and Simulation Experiments

Data assimilation refers to a method of integrating observation data in the dynamic operation of numerical models on the basis of considering the temporal and spatial distribution of data and the error of observation field and background field. The Ensemble Kalman filter (EnKF) as a technology that has been widely used in the field of atmosphere and ocean has been applied to the ROMS (the Regional Ocean Modeling System) for predicting Sea surface temperature in Yellow, and East China Seas. In order to explore the applicability and effectiveness of the EnKF method for improving the accuracy of marine numerical model, the Sea surface temperature (SST) gained from buoy were used to conduct data assimilation process with EnKF method. Twin experiments have been performed to analysis the sensitivity of this system to the ensemble size and errors in model simulation and observations and a real data assimilation scheme has been conducted to hindcast the SST at the Yellow, and East China Seas during the July of year 2014. The updated results after data assimilation indicate that the model simulation fits observation better when the forecast was updated by observations. The result show that EnKF can effectively reduce the simulation error of complex numerical marine models.

BI-IEnKF coupling model for effective source term estimation of natural gas leakage in urban utility tunnels

As an effective way to facilitate the increasing demand for reliable infrastructure, energy supply and sustainable urban development, underground utility tunnels have been developed rapidly in recent years. Due to the widespread distribution of utility tunnels, the safe operation of natural gas pipelines accommodated in utility tunnels has caused great concern considering fire, explosion, and other coupling consequences induced by the gas pipeline leakage. However, the limited information on leakage source terms in accidental leakage scenarios could preclude timely consequence assessment and effective emergency response. In this study, a BI-IEnKF coupling source term estimation (STE) model is developed, with the combination of gas dispersion model, Bayesian inference (BI) and iterative ensemble Kalman filter (IEnKF) method, to achieve the effective source term estimation (including leakage location and leakage rate) and gas concentration distribution prediction. The newly developed model is first evaluated by the twin experiment with good reliability and accuracy. Furthermore, three contributing factors affecting the performance of the developed BI-IEnKF coupling STE model were investigated to assist parameter selection for practical use. Additionally, the novel application of mobile sensors serving as an alternative for fixed sensors is explored, and an application framework is sequentially given to guide the deployment of the developed coupling model in utility tunnels. The results show that the developed model has great performance in accuracy, efficiency and robustness, as well as the potential to be applied in actual utility tunnel scenarios. This study can provide technical supports for safety control and emergency response in the case of natural gas pipeline leakage accidents in utility tunnels. Also, it could be helpful to reasonable references for gas lekage monitoring system design.

Improvement of transition prediction model in hypersonic boundary layer based on field inversion and machine learning framework

The classical four-equation γ−Reθ transition model has presented excellent accuracy in low-speed boundary layer transition prediction. However, once the incoming flow reaches hypersonic speed, the original model is no longer applicable due to the compressibility problem and the appearance of multiple instability modes. Recently, there has been widespread interest in data-driven modeling for quantifying uncertainty or improving model prediction accuracy. In this paper, a data-driven framework based on field inversion and machine learning is performed to extend the prediction capability of the original γ−Reθ transition model for the hypersonic boundary layer transition. First, the iterative regularized ensemble Kalman filter method is applied to obtain the spatial distribution of the perturbation correction term β for the switching function Fonset1, and the effectiveness of this method is initially verified in the hypersonic flat plate case. Then, the random forest algorithm is adopted to construct a mapping from the average flow features to β. The generalizability of the well-trained learning model is fully validated in the blunt cone cases with different unit Reynolds numbers, free-stream flow temperature, and bluntness. The simulation results indicate that the performance of the original γ−Reθ transition model in the hypersonic boundary layer transition prediction is significantly improved, and the boundary layer transition onset location and the length of transition zone can be correctly obtained. In addition, the machine learning model investigates the importance of the input features and confirms that the effective length scale plays a significant role in the numerical simulation of the hypersonic boundary layer transition.

Performance comparisons of the three data assimilation methods for improved predictability of PM2·5: Ensemble Kalman filter, ensemble square root filter, and three-dimensional variational methods

To improve the predictability of concentrations of atmospheric particulate matter, a data assimilation (DA) system using ensemble square root filter (EnSRF) has been developed for the Community Multiscale Air Quality (CMAQ) model. The EnSRF DA method is a deterministic variant of the ensemble Kalman filter (EnKF) method, which means that unlike the EnKF method, it does not add random noise to the observations. To compare the performances of the EnSRF with those of other DA methods, such as EnKF and 3DVAR (three-dimensional variational), these three methods were applied to the same CMAQ model simulations with identical experimental settings. This is the first attempt in the field of chemical DA to compare the EnKF and EnSRF methods. An identical set of surface fine particulate matter (PM2.5) were assimilated every 6 h by all the DA methods over a CMAQ domain of East Asia, during the period from 01 May to 11 June 2016. In parallel with ‘reanalysis experiments’, we also carried out ‘48 h prediction experiments’ using the optimized initial conditions produced by the three DA methods. Detailed analyses among the three DA methods were then carried out by comparing both the reanalysis and the prediction outputs with the observed surface PM2.5 over four regions (i.e., South Korea, the Beijing–Tianjin–Hebei (BTH) region, Shandong province, and Liaoning province). The comparison results revealed that the EnSRF produced the best reanalysis and prediction fields in terms of several statistical metrics. For example, when the 3DVAR, EnKF, and EnSRF methods were used, averaged normalized mean biases (NMBs) decreased by (57.6, 85.6, and 91.8) % in reanalyses and (39.7, 87.6, and 91.5) % in first-day predictions, compared to the CMAQ control experiment (i.e., without DA) over South Korea, respectively. Also, over the three Chinese regions, the EnSRF method outperformed the EnKF and 3DVAR methods.

Joint identification of contaminant source based on the ensemble Kalman filter integrated with relation coefficient

To reduce the adverse influence of sudden water pollution accidents, it is essential to estimate the unknown contaminant source information (normally including the source location, initial release time, and total release mass) as soon as possible. The ensemble Kalman filter (EnKF) has been proven to be an effective algorithm for such an inverse problem. This paper proposes a new method based on EnKF to identify contaminant source information. The method we called RC-EnKF uses the relation coefficient of concentration instead of timely concentration as a state variable in the assimilation process. The advantage lies in the release source mass that can be decoupled from the parameter group of unknown contaminant information to improve the assimilation speed and reduce interference with the accuracy of assimilation. Two categories of cases are employed for validating the applicability and testing the performance compared with the traditional EnKF method in detail. Sensitivity analyses are carried out with different observation errors, number of observation sites, number of ensemble realizations, and model grid size. The results demonstrate that with the uncertain error of observation data, the RC-EnKF works nicely and shows superiority to the traditional EnKF method reflected in the strong immunity to interference from observation data errors and elevated efficiency with the requirement of fewer observation sites, ensemble realizations, and model grids. It illustrates that the RC-EnKF is a more efficient and robust method for estimating unknown contaminant source information.

Estimation of water momentum and propeller velocity in bow thruster model of autonomous surface vehicle using modified Kalman filter

<span lang="EN-US">Autonomous surface vehicle (ASV) is a vehicle in the form of an unmanned on-water surface vessel that can move automatically. As such, an automatic control system is essentially required. The bow thruster system functions as a propulsion control device in its operations. In this research, the water momentum and propeller velocity were estimated based on the dynamic bow thruster model. The estimation methods used is the Kalman filter (KF) and ensemble Kalman filter (EnKF). There are two scenarios: tunnel thruster condition and open-bladed thruster condition. The estimation results in the tunnel thruster condition showed that the root mean square error (RMSE) by the EnKF method was relatively smaller, that is, 0.7920 and 0.1352, while the estimation results in the open-bladed thruster condition showed that the RMSE by the KF method was relatively smaller, that is, 1.9957 and 2.0609.</span>

A combined search method based on a deep learning combined surrogate model for groundwater DNAPL contamination source identification

Ensemble Kalman filter (EnKF) and optimization methods are two mainstream methods in groundwater contamination source identification, but most researchers usually only use or improve on one method. However, each method has strengths and weaknesses, and using one method alone may cause inaccuracy of the identification results for complex situations. Therefore, it is necessary to explore the combination of methods. In this paper, for the optimization method, to further enhance the solution precision of the optimization model for groundwater dense non-aqueous phase liquid (DNAPL) contamination source identification (GDCSI), we first constructed an improved butterfly optimization algorithm by introducing the dynamic switching probability mechanism in the previous butterfly optimization algorithm. Next, the EnKF method and optimization method (based on the improved butterfly optimization algorithm) were used respectively for GDCSI, to assess the strengths and weaknesses of the methods. Then, according to the strengths and weaknesses of the EnKF and optimization methods, the two methods were merged to build a more robust and practical combined search method (CSM) for GDCSI, to further enhance the identification accuracy and effectiveness. In addition, we innovatively applied the EnKF method to combine two single deep learning surrogate models: the deep belief neural network surrogate model and the deep residual neural network surrogate model. And a deep learning combined surrogate model was established, which further enhanced the approximation precision and applicability to the groundwater DNAPL contamination multiphase flow numerical simulation model. Finally, the results of CSM were compared with those of two single methods to analyze the former's effectiveness. The results showed that the CSM significantly improved the identification accuracy and effectiveness of GDCSI compared with any single method.

Phase-resolved ocean wave forecast with simultaneous current estimation through data assimilation

In Wang & Pan (J. Fluid Mech., vol. 918, 2021, A19), the authors developed the first ensemble-based data assimilation (DA) capability for the reconstruction and forecast of ocean surface waves, namely the EnKF-HOS method coupling an ensemble Kalman filter (EnKF) and the high-order spectral (HOS) method. In this work, we continue to enrich the method by allowing it to simultaneously estimate the ocean current field, which is in general not known a priori and can (slowly) vary in both space and time. To achieve this goal, we incorporate the effect of ocean current (as unknown parameters) on waves to build the HOS-C method as the forward prediction model, and obtain a simultaneous estimation of (current) parameters and (wave) states via an iterative EnKF (IEnKF) method that is necessary to handle the complexity in this DA problem. The new algorithm, named the IEnKF-HOS-C method, is first tested in synthetic problems with various forms (steady/unsteady, uniform/non-uniform) of current. It is shown that the IEnKF-HOS-C method is able to not only estimate the current field accurately, but also boost the prediction accuracy of the wave field (even) relative to the state-of-the-art EnKF-HOS method. Finally, using real data from a shipborne radar, we show that the IEnKF-HOS-C method successfully recovers the current speed that matches the in situ measurement by a floating buoy.

Multi-index ensemble Kalman filtering

In this work we combine ideas from multi-index Monte Carlo and ensemble Kalman filtering (EnKF) to produce a highly efficient filtering method called multi-index EnKF (MIEnKF). MIEnKF is based on independent samples of four-coupled EnKF estimators on a multi-index hierarchy of resolution levels, and it may be viewed as an extension of the multilevel EnKF (MLEnKF) method developed by the same authors in 2020. Multi-index here refers to a two-index method, consisting of a hierarchy of EnKF estimators that are coupled in two degrees of freedom: time discretization and ensemble size. Under certain assumptions, when strong coupling between solutions on neighboring numerical resolutions is attainable, the MIEnKF method is proven to be more tractable than EnKF and MLEnKF. Said efficiency gains are also verified numerically in a series of test problems.