Traditional Ensemble Kalman Filter Research Articles

An Improved Coupled Data Assimilation System with a CGCM Using Multi-Time-Scale High-Efficiency EnOI-Like Filtering

Abstract Coupled data assimilation (CDA), which combines coupled models and observations from multiple Earth system domains, plays a critical role in climate studies by producing a four-dimensional estimation of Earth system states. Traditional ensemble Kalman filter (EnKF) CDA algorithms, while convenient to implement in multiple DA components in a coupled system, are, however, expensive and lack sufficient representativeness for low-frequency background flows. Here, a multi-time-scale high-efficiency approximate filter with ensemble optimal interpolation (MSHea-EnOI) scheme has been implemented with a global fully coupled model. It consists of stationary, low-frequency, and high-frequency filters constructed from the time series of a single-model solution with improved representativeness for low-frequency background error statistics and enhanced computational efficiency. The MSHea-EnOI is evaluated in a biased twin experiment framework with synthetic “observations” produced by another coupled model, and a three-decade coupled reanalysis experiment with real observations. Results show that with increased representativeness on multiscale background flows, while computationally costing only a small fraction of ensemble-based CDA, the MSHea-EnOI shows the potential to improve CDA quality with synthetic observations. The coupled reanalysis experiment with real observations also shows reasonable fittings to observations and comparable results to other reanalysis products using different DA schemes. While reconstructing a close-to-rapid Atlantic meridional overturning circulation, the coupled reanalysis reproduces most of the atmosphere and ocean reanalysis signals such as the Hadley circulation and upper ocean heat content. The MSHea-EnOI could have good application potential in ensemble-based DA systems in terms of its multiscale property and computational efficiency.

Improving Vortex Position Accuracy with a New Multiscale Alignment Ensemble Filter

Abstract A multiscale alignment (MSA) ensemble filtering method was introduced by Ying to reduce nonlinear position errors effectively during data assimilation. The MSA method extends the traditional ensemble Kalman filter (EnKF) to update states from large to small scales sequentially, during which it leverages the displacement vectors derived from the large-scale analysis increments to reduce position errors at smaller scales through warping of the model grid. This study stress tests the MSA method in various scenarios using an idealized vortex model. We show that the MSA improves filter performance as number of scales (Ns) increases in the presence of nonlinear position errors. We tuned localization parameters for the cross-scale EnKF updates to find the best performance when assimilating an observation network. To further reduce the scale mismatch between observations and states, a new option called MSA-O is introduced to decompose observations into scale components during assimilation. Cycling DA experiments show that the MSA-O consistently outperforms the traditional EnKF at equal computational cost. A more challenging scenario for the MSA is identified when the large-scale background flow and the small-scale vortex are incoherent in terms of their errors, making the displacement vectors not effective in reducing vortex position errors. Observation availability for the small scales also limits the use of large Ns for the MSA. Potential remedies for these issues are discussed.

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.

Research Spotlights

The first Research Spotlights article in this issue is concerned with filtering, a task of paramount importance in a great many applications such as numerical weather prediction and geophysical data assimilation. Authors Alessio Spantini, Ricardo Baptista, and Youssef M. Marzouk, in their article “Coupling Techniques for Nonlinear Ensemble Filtering,” describe discrete-time filtering as the act of characterizing the sequence of conditional distributions of the latent field at observation times, given all currently available measurements. Despite the existing literature on filtering, issues such as high-dimensional state spaces and sparse (in both space and time) observations still prove formidable in practice. The traditional approach of ensemble-based data assimilation is the ensemble Kalman filter (EnKF), involving a prediction (forecasting) step followed by an analysis step. However, the authors note an intrinsic bias of EnKF due to the linearity of the transformation, estimated under Gaussian assumptions, that is used in the analysis step, which limits its accuracy. To overcome this, they propose two non-Gaussian generalizations of the EnKF---the so-called stochastic and deterministic map filters---using nonlinear transformations derived from couplings between the forecast distribution and the filtering distribution. What is crucial is that the transformations “can be estimated efficiently...perhaps using only convex optimization,” that they “are easy to `localize' in high dimensions,” and that their computation “should not become increasingly challenging as the variance of the observation noise decreases.” Following a comprehensive description of their new approaches, the authors demonstrate numerically the superiority of their stochastic map filter approach over traditional EnKF. The subsequent discussion offers the reader several jumping off points for future research. Recovery of a sparse solution to a large-scale optimization problem is another ubiquitous problem arising in many applications such as image reconstruction, signal processing, and machine learning. The cost functional typically includes a regularization term in the form of an $\ell_1$ norm term on the solution and/or regularized solution to enforce sparsity. Designing suitable algorithms for such recovery problems is the subject of our second Research Spotlights article. In “Sparse Approximations with Interior Point Methods,” authors Valentina De Simone, Daniela di Serafino, Jacek Gondzio, Spyridon Pougkakiotis, and Marco Viola set out to correct the misconception that first-order methods are to be preferred over second-order methods out of hand. Through case studies, they offer evidence that interior point methods (IPMs) which are constructed to “exploit special features of the problems in the linear algebra of IPMs” and which are designed “to take advantage of the expected sparsity of the optimal solution” can in fact be the method of choice for solving this class of optimization problems. The key to their approach is a reformulation of the original sparse approximation problem to one which is seemingly larger but which has properties upon which one can capitalize for computational gain. For each of four representative applications, the authors show how to take computational advantage of the problem-specific structure of the underlying linear systems involved at each iteration. These efforts are complemented by leveraging the expected sparsity: employing heuristics to drop near zero variables, thereby replacing very large, ill-conditioned intermediate systems by better conditioned, smaller systems. Their conclusion is that time invested in tailoring solvers to structure admitted by the reformulated variant, and in taking advantage of expected sparsity, may be well spent, since their demonstrations have shown it is possible for IPMs to have a “noticeable advantage” over state-of-the-art first-order methods for sparse approximation problems.

Geologic Modeling and Ensemble-Based History Matching for Evaluating CO2 Sequestration Potential in Point bar Reservoirs

The target reservoirs in many CO2 projects exhibit point bar geology characterized by the presence of shale drapes that act as barriers preventing the leakage of CO2. However, the extent of the flow barriers can also impede the displacement of CO2 in such reservoirs and restrict the storage volume. Therefore, developing a framework for modeling point bars and their associated heterogeneities is crucial. Yet, for the point bar model to be geologically realistic and reliable for evaluating CO2 sequestration potential, it should be calibrated to reflect historical data (e.g., CO2 injection data). This study is therefore in two parts. The first part focusses on the modeling of point bar heterogeneities (i.e., lateral accretions and inclined heterolithic stratifications). To ensure that the heterogeneities are preserved, we implemented a gridding scheme that generates curvilinear grids representative of the point bar curvilinear geometry. We subsequently incorporated a grid transformation scheme to facilitate geostatistical modeling of reservoir property distributions. The second part of this study is a model calibration step, where the point bar model is updated by assimilating CO2 injection data, in an ensemble framework. Ensemble-Kalman Filter was used first to update ensembles of point bar geometries, to select the geometry that yields the closest match to observed data. Within this geometry, indicator-based ensemble data assimilation was used to perform updates to the ensemble of point bar permeability models. The indicator approach overcomes the Gaussian limitation of the traditional ensemble Kalman filter. The workflow was run on the Cranfield, Mississippi CO2 injection dataset. It was observed, after model calibration, that the final updated ensemble of models yields a reasonable match with the historical data. The updated models were run in a forecast mode to predict the long-term CO2 sequestration potential of the Cranfield point bar reservoir. Results demonstrate that 1) preserving the heterogeneities in the point bar modeling process, and 2) constraining the point bar model to historical data (e.g., CO2 injection data) are essential for accurately evaluating the CO2 sequestration potential in point bar reservoirs.

Position control for an autonomous underwater vehicle under noisy conditions

Position tracking of an autonomous underwater vehicle is not trivial as its accuracy is affected by process noise, measurement noise, and uncertain hydrodynamic parameters. Thus, in this article, we studied the position control of an autonomous underwater vehicle that operates under largely unbounded system uncertainties and large noise levels and proposed a method that reduces the interference and thereby enhances the overall autonomous underwater vehicle position estimation. Our technique extended the ensemble Kalman filter by combining it with a time-delay estimator to compensate against extended uncertainty of the system dynamics which cannot be solved by the traditional ensemble Kalman filter. Our synthetic simulations demonstrated the effectiveness of the proposed controller, highlighting its appealing position-control accuracy under simultaneous noise and uncertain hydrodynamics parameters. In addition, our simulation results showed that the proposed controller outperforms the conventional time-delay controller by a percentage range of approximately 30.8%–92.6% in terms of root-mean-square error and requires on average less than 88.2% calculation time than the conventional model predictive control.

A nonparametric sequential data assimilation scheme for soil moisture flow

Various of data assimilation methods such as the ensemble Kalman filter (EnKF) have been established in physical science and engineering for the fusion of observed data with physically motivated models. However, there are cases where a physically motivated model is not available. In particular, the unsaturated flow model is often difficult to build due to the intrinsic nonlinearity and heterogeneity of soil flow process. In this paper, a nonparametric sequential data assimilation scheme (Kalman-GP) is introduced based on the filtering equations of EnKF and data-driven modeling with Gaussian process (GP). The method replaces the physical model with GP constructed directly from available multiple-source data and observations of state variables of interest, while using the Kalman update formulation to reconcile real-time observations. We tested the proposed Kalman-GP method in a real-world case study of soil moisture profile simulation. The performance of Kalman-GP was compared with two different data assimilation methods, i.e. the traditional EnKF with a physical model (Kalman-physics) and a hybrid filter which integrated a dynamic GP error model into physics-based EnKF (Kalman-physics-GP, Zhang et al., 2019). Results showed that, without knowledge of any governing equation, the Kalman-GP filter was able to reconstruct soil moisture dynamics to a level comparable with the parametric EnKF, even exhibit superior performance. The Kalman-physics-GP led to a better soil moisture estimation than Kalman-GP, however at the expense of dependence on the underlying physical model and more requirements of prior knowledge. In contrast, the proposed Kalman-GP requires only easy-to-obtain meteorological data and can be a better alternative to achieve good tradeoff between effectiveness and efficiency, especially in areas short of hydrogeological data. When less prior physical knowledge is available, the advantage of the proposed Kalman-GP over the Kalman-Physics was enhanced, while the superiority of the hybrid filter over the Kalman-GP was weakened.

A dynamic data-driven method for dealing with model structural error in soil moisture data assimilation

Attributing to the flexibility in considering various types of observation error and model error, data assimilation has been increasingly applied to dynamically improve soil moisture modeling in many hydrological practices. However, accurate characterization of model error, especially the part caused by defective model structure, presents a significant challenge to the successful implementation of data assimilation. Model structural error has received limited attention relative to parameter and input errors, mainly due to our poor understanding of structural inadequacy and the difficulties in parameterizing structural error. In this paper, we present a dynamic data-driven approach to estimate the model structural error in soil moisture data assimilation without the need for identifying error generation mechanism or specifying particular form for the error model. The error model is based on the Gaussian process regression and then integrated into the ensemble Kalman filter (EnKF) to form a hybrid method for dealing with multi-source model errors. Two variants of the hybrid method in terms of two different error correction manners are proposed. The effectiveness of the proposed method is tested through a suit of synthetic cases and a real-world case. Results demonstrate the potential of the proposed hybrid method for estimating model structural error and providing improved model predictions. Compared to the traditional EnKF without explicitly considering the model structural error, parameter compensation issue is obviously reduced and soil moisture retrieval is substantially improved.

Ensemble-Based Assimilation of Nonlinearly Related Dynamic Data in Reservoir Models Exhibiting Non-Gaussian Characteristics

Inverse modeling techniques for estimating reservoir parameters (e.g., transmissivity, permeability) utilize some dynamic (secondary) information (e.g., hydraulic head or production data at well locations). Ensemble based data assimilation methods are one such class of inverse modeling techniques. Ensemble Kalman filter (EnKF) in particular is built around the basic framework of linearly updating a state vector $${\psi ^f}$$ to $${\psi ^a}$$ , based on observed dynamic data. Components of the state vector are reservoir model parameters such as transmissivity, permeability, storativity, porosity, hydraulic head, and phase-saturation. Although EnKF is able to update a large number of parameters successively as data become available, it has some shortcomings. It is optimal only in the cases where multivariate joint distribution describing the relationship between components in the state vector is Gaussian. The relationship between the model parameters and the observed data space is quantified in terms of a covariance function, which is adequate to describe a multi-Gaussian distribution. These assumptions and simplifications may result in updated models that are inconsistent with the prior geology (exhibiting non-Gaussian characteristics) and that, in turn, may yield inaccurate predictions of reservoir performance. The aim of this research work is to propose a novel method for data assimilation which is free from the Gaussian assumptions. This new method can be used to assimilate dynamic data into reservoir models using an ensemble-based approach. Model updates are performed after transforming the model parameters into indicator variables. The indicator transform is insensitive to non-linear operations, and it thus allows us to overcome some of the limitations of the traditional EnKF.

A modified iterative ensemble Kalman filter data assimilation method

High nonlinearity is a typical characteristic associated with data assimilation systems. Additionally, iterative ensemble based methods have attracted a large amount of research attention, which has been focused on dealing with nonlinearity problems. To solve the local convergence problem of the iterative ensemble Kalman filter, a modified iterative ensemble Kalman filter algorithm was put forward, which was based on a global convergence strategy from the perspective of a Gauss-Newton iteration. Through self-adaption, the step factor was adjusted to enable every iteration to approach expected values during the process of the data assimilation. A sensitivity experiment was carried out in a low dimensional Lorenz-63 chaotic system, as well as a Lorenz-96 model. The new method was tested via ensemble size, observation variance, and inflation factor changes, along with other aspects. Meanwhile, comparative research was conducted with both a traditional ensemble Kalman filter and an iterative ensemble Kalman filter. The results showed that the modified iterative ensemble Kalman filter algorithm was a data assimilation method that was able to effectively estimate a strongly nonlinear system state.

Development of integrated approaches for hydrological data assimilation through combination of ensemble Kalman filter and particle filter methods

This study improved hydrologic data assimilation through integrating the capabilities of particle filter (PF) and ensemble Kalman filter (EnKF) methods, leading to two integrated data assimilation schemes: the coupled EnKF and PF (CEnPF) and parallelized EnKF and PF (PEnPF) approaches. The applicability and usefulness of CEnPF and PEnPF were demonstrated using a conceptual rainfall-runoff model. The performance of two new developed data assimilation methods and traditional EnKF and PF approaches was tested through a synthetic experiment and two real-world cases with one located in the Jing River basin and one located in the Yangtze River basin. The results show that both PEnPF and CEnPF approaches have more opportunities to provide better results for both deterministic and probabilistic predictions than traditional EnKF and PF approaches. Moreover, the computational time of the two integrated methods is manageable. But the proposed PEnPF may need much more time for some large-scale or time-consuming hydrologic models since it generally needs three times of model runs used by EnKF, PF and CEnPF.

Assimilating temperature and salinity profiles using Ensemble Kalman Filter with an adaptive observation error and T-S constraint

Temperature (T) and salinity (S) profiles from conductivity-temperature-depth data collected during the Northern South China Sea Open Cruise from August 16 to September 13, 2008 are assimilated using Ensemble Kalman Filter (EnKF). An adaptive observational error strategy is used to prevent filter from diverging. In the meantime, aiming at the limited improvement in some sites caused by the T and S biases in the model, a T-S constraint scheme is adopted to improve the assimilation performance, where T and S are separately updated at these locations. Validation is performed by comparing assimilated outputs with independent in situ data (satellite remote sensing sea level anomaly (SLA), the OSCAR velocity product and shipboard ADCP). The results show that the new EnKF assimilation scheme can significantly reduce the root mean square error (RMSE) of oceanic T and S compared with the control run and traditional EnKF. The system can also improve the simulation of circulations and SLA.

History matching using traditional and finite size ensemble Kalman filter

Generally reservoir simulation results show considerable distance from actual behavior of system which is mainly due to poor initial estimates of model parameters. The problem of improving the property estimates using reservoir observations is called history matching which is an essential and inseparable part of reservoir studies. In recent years a promising tool from control engineering called ensemble filtering is used in history matching problems which could be used to quantify the uncertainty in predictions.This paper focuses on comparison of traditional ensemble Kalman filter (EnKF) and a relatively new form of ensemble filters called finite size ensemble transform Kalman filter on a small size benchmark model. This study tries to estimate the porosity and permeability fields by assimilating production data in 13 time steps. A weighting approach is used as modification to filter procedure. The results show traditional ensemble Kalman filter with localization has a better performance in a problem with a small ensemble but this superiority comes with more computational cost.

Importance of Distributed Temperature Sensor Data for Steam Assisted Gravity Drainage Reservoir Characterization and History Matching Within Ensemble Kalman Filter Framework

Distributed temperature sensing (DTS), an optical fiber down-hole monitoring technique, provides a continuous and permanent well temperature profile. In steam assisted gravity drainage (SAGD) reservoirs, the DTS plays an important role to provide depth-and-time continuous temperature measurement for steam management and production optimization. These temperature observations provide useful information for reservoir characterization and shale detection in SAGD reservoirs. However, use of these massive data for automated SAGD reservoir characterization has not been investigated. The ensemble Kalman filter (EnKF), a parameter estimation approach using these real-time temperature observations, provides a highly attractive algorithm for automatic history matching and quantitative reservoir characterization. Due to its complex geological nature, the shale barrier exhibits as a different facies in sandstone reservoirs. In such reservoirs, due to non-Gaussian distributions, the traditional EnKF underestimates the uncertainty and fails to obtain a good production data match. We implemented discrete cosine transform (DCT) to parameterize the facies labels with EnKF. Furthermore, to capture geologically meaningful and realistic facies distribution in conjunction with matching observed data, we included fiber-optic sensor temperature data. Several case studies with different facies distribution and well configurations were conducted. In order to investigate the effect of temperature observations on SAGD reservoir characterization, the number of DTS observations and their locations were varied for each study. The qualities of the history-matched models were assessed by comparing the facies maps, facies distribution, and the root mean square error (RMSE) of the predicted data mismatch. Use of temperature data in conjunction with production data demonstrated significant improvement in facies detection and reduced uncertainty for SAGD reservoirs. The RMSE of the predicted data is also improved. The results indicate that the assimilation of DTS data from nearby steam chamber location has a significant potential in significant reduction of uncertainty in steam chamber propagation and production forecast.

Data Assimilation for Strongly Nonlinear Problems by Transformed Ensemble Kalman Filter

Summary The ensemble Kalman filter (EnKF) has been widely used for data assimilation. It is challenging, however, when the relation of state and observation is strongly nonlinear. For example, near the flooding front in an immiscible flow, directly updating the saturation by use of the EnKF may lead to nonphysical results. One possible solution, which may be referred to as the restarted EnKF (REnKF), is to update the static state (e.g., permeability and porosity) and rerun the forward model from the initial time to obtain the updated dynamic state (e.g., pressure and saturation). However, it may become time-consuming, especially when the number of assimilation steps is large. In this study, we develop a transformed EnKF (TEnKF), in which the state is represented by displacement as an alternative variable. The displacement is first transformed from the forecasted state, then updated, and finally transformed back to obtain the updated state. Because the relation between displacement and observation is relatively linear, this new method provides a physically meaningful updated state without resolving the forward model. The TEnKF is tested in the history matching of multiphase flow in a 1D homogeneous medium, a 2D heterogeneous reservoir, and a 3D PUNQ-S3 model. The case studies show that the TEnKF produces physical results without the oscillation problem that occurs in the traditional EnKF, whereas the computational effort is reduced compared with the REnKF.

EnKF coupled with groundwater flow moment equations applied to Lauswiesen aquifer, Germany

Summary We describe a field application of a new data assimilation method recently proposed by Panzeri et al. (2013, 2014). The method couples a modified Ensemble Kalman Filter (EnKF) algorithm with stochastic moment equations (MEs) governing space–time variations of (theoretical ensemble) mean and covariance values of groundwater flow state variables (hydraulic heads and fluxes). Whereas traditional EnKF entails Monte Carlo (MC) simulations and suffers from inbreeding, our approach obviates both. Synthetic case studies have shown the ME-based approach to be computationally efficient and accurate when compared to MC-based results. Here we use our ME-based method to assimilate drawdown data recorded during cross-hole pumping tests in the heterogeneous alluvial Lauswiesen aquifer near Tubingen, Germany. Our results include an estimate of log transmissivity distribution throughout the aquifer and corresponding measures of estimation error. We validate our calibrated model by using it to predict drawdowns recorded during another pumping test at the site and compare its performance with that of standard MC-based EnKF.

Facies estimation through data assimilation and structure parameterization

Data assimilation in a reservoir with facies description using the ensemble Kalman filter (EnKF) is challenging. An important reason is that probability density functions for pixel-based representations of facies fields seldom follow the unimodal Gaussian assumption underlying traditional EnKF implementations. Different approaches for identification of facies fields, aiming to overcome this challenge, have been proposed within the EnKF framework. Level set (LS) representations of the facies field have been reported to alleviate the problems of multimodality. Several authors have, however, pointed out that the most commonly applied LS representation suffers from topological constraints that can create difficulties in an estimation setting. An alternative LS representation, that overcomes these topological constraints, leads to instabilities in the assimilated ensemble members. To overcome topological constraints, the recently proposed hierarchical LS representation is applied in an estimation setting for the first time in this paper. To improve stability and to alleviate challenges associated with model nonlinearities, we apply regularization by reduced representation of the LS functions and adjustable smoothing of the LS representation. The resolution of the reduced LS representation is selected based on the variability of the initial ensemble, aiming at preserving enough flexibility to disclose unexpected features. 2D and 3D estimation results demonstrate that the hierarchical LS representation does avoid topological constraints and that instabilities are avoided. The results suggest that the method is capable of handling estimation of facies fields while preserving geological plausibility.

Assimilation of surface currents into a regional model over Qingdao coastal waters of China

Surface currents measured by high frequency (HF) radar arrays are assimilated into a regional ocean model over Qingdao coastal waters based on Kalman filter method. A series of numerical experiments are performed to evaluate the performance of the data assimilation schemes. In order to optimize the analysis procedure in the traditional ensemble Kalman filter (ENKF), a different analysis scheme called quasiensemble Kaman filter (QENKF) is proposed. The comparisons between the ENKF and the QENKF suggest that both them can improve the simulated error and the spatial structure. The estimations of the background error covariance (BEC) are also assessed by comparing three different methods: Monte Carlo method; Canadian quick covariance (CQC) method and data uncertainty engine (DUE) method. A significant reduction of the root-mean-square (RMS) errors between model results and the observations shows that the CQC method is able to better reproduce the error statistics for this coastal ocean model and the corresponding external forcing. In addition, the sensibility of the data assimilation system to the ensemble size is also analyzed by means of different scales of the ensemble size used in the experiments. It is found that given the balance of the computational cost and the forecasting accuracy, the ensemble size of 50 will be an appropriate choice in the Qingdao coastal waters.

An Evapotranspiration Assimilation Method Based on Ensemble Kalman Filter and À Trous Wavelet

It is challenging to assimilate the evapotranspiration product (EP) retrieved from satellite data into land surface models (LSMs). In this paper, a perturbed ensemble Kalman filter (PEKF) and à trous wavelet transform (AWT) integrated method are proposed to implement the evapotranspiration assimilation. In this method, the AWT is used to decompose the EPs into multiple channels since it is very powerful in fusing high frequency spatial information of multisource data, and then the Kalman filter is performed in the AWT domain. The proposed method combines the advantages of the PEKF that is capable of accommodating model error and observation error, and the AWT can effectively perform multiresolution fusion. Assimilation experiment conducted with the Noah model and the EP retrieved from the MODIS data shows that the proposed method performs better than the traditional ensemble Kalman filter (EnKF) and PEKF methods. The analysis results fit well with the evapotranspiration observation at two field sites with different land surface conditions. These indicate that the proposed method is promising for assimilating regional scale satellite retrieved EP into LSMs.

Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems

Abstract. The finite-size ensemble Kalman filter (EnKF-N) is an ensemble Kalman filter (EnKF) which, in perfect model condition, does not require inflation because it partially accounts for the ensemble sampling errors. For the Lorenz '63 and '95 toy-models, it was so far shown to perform as well or better than the EnKF with an optimally tuned inflation. The iterative ensemble Kalman filter (IEnKF) is an EnKF which was shown to perform much better than the EnKF in strongly nonlinear conditions, such as with the Lorenz '63 and '95 models, at the cost of iteratively updating the trajectories of the ensemble members. This article aims at further exploring the two filters and at combining both into an EnKF that does not require inflation in perfect model condition, and which is as efficient as the IEnKF in very nonlinear conditions. In this study, EnKF-N is first introduced and a new implementation is developed. It decomposes EnKF-N into a cheap two-step algorithm that amounts to computing an optimal inflation factor. This offers a justification of the use of the inflation technique in the traditional EnKF and why it can often be efficient. Secondly, the IEnKF is introduced following a new implementation based on the Levenberg-Marquardt optimisation algorithm. Then, the two approaches are combined to obtain the finite-size iterative ensemble Kalman filter (IEnKF-N). Several numerical experiments are performed on IEnKF-N with the Lorenz '95 model. These experiments demonstrate its numerical efficiency as well as its performance that offer, at least, the best of both filters. We have also selected a demanding case based on the Lorenz '63 model that points to ways to improve the finite-size ensemble Kalman filters. Eventually, IEnKF-N could be seen as the first brick of an efficient ensemble Kalman smoother for strongly nonlinear systems.