Deterministic Ensemble Kalman Filter Research Articles

Deterministic ensemble Kalman filter based on two localization techniques for mitigating sampling errors with a quasi-geostrophic model

Deterministic ensemble Kalman filter based on two localization techniques for mitigating sampling errors with a quasi-geostrophic model

Arctic sea ice data assimilation combining an ensemble Kalman filter with a novel Lagrangian sea ice model for the winter 2019–2020

Abstract. Advanced data assimilation (DA) methods, widely used in geophysical and climate studies to merge observations with numerical models, can improve state estimates and consequent forecasts. We interface the deterministic ensemble Kalman filter (DEnKF) to the Lagrangian neXt generation Sea Ice Model, neXtSIM. The ensemble is generated by perturbing the atmospheric and oceanic forcing throughout the simulations and randomly initialized ice cohesion. Our ensemble–DA system assimilates sea ice concentration (SIC) from the Ocean and Sea Ice Satellite Application Facility (OSI-SAF) and sea ice thickness (SIT) from the merged CryoSat-2 and SMOS datasets (CS2SMOS). Because neXtSIM is computationally solved on a time-dependent evolving mesh, it is a challenging application for ensemble–DA. As a solution, we perform the DEnKF analysis on a fixed and regular reference mesh, on which model variables are interpolated before the DA and then back to each member's mesh after the DA. We evaluate the impact of assimilating different types of sea ice observations on the model's forecast skills of the Arctic sea ice by comparing satellite observations and a free-run ensemble in an Arctic winter period, 2019–2020. Significant improvements in modeled SIT indicate the importance of assimilating weekly CS2SMOS SIT, while the improvements of SIC and ice extent are moderate but benefit from daily ingestion of the OSI-SAF SIC. For most of the winter, the correlation between SIT and SIC is weaker, which results in little cross-inference between the two variables in the assimilation step. Overall, the ensemble–DA system based on the stand-alone sea ice model demonstrates the feasibility of winter Arctic sea ice prediction with good computational efficiency. These results open the path toward operational implementation and the extension to multi-year assimilation.

Assimilation of sea surface salinities from SMOS in an Arctic coupled ocean and sea ice reanalysis

Abstract. In the Arctic, the sea surface salinity (SSS) plays a key role in processes related to water mixing and sea ice. However, the lack of salinity observations causes large uncertainties in Arctic Ocean forecasts and reanalysis. Recently the Soil Moisture and Ocean Salinity (SMOS) satellite mission was used by the Barcelona Expert Centre to develop an Arctic SSS product. In this study, we evaluate the impact of assimilating this data in a coupled ocean–ice data assimilation system. Using the deterministic ensemble Kalman filter from July to December 2016, two assimilation runs respectively assimilated two successive versions of the SMOS SSS product on top of a pre-existing reanalysis run. The runs were validated against independent in situ salinity profiles in the Arctic. The results show that the biases and the root-mean-squared differences (RMSD) of SSS are reduced by 10 % to 50 % depending on the area and highlight the importance of assimilating satellite salinity data. The time series of freshwater content (FWC) further shows that its seasonal cycle can be adjusted by assimilation of the SSS products, which is encouraging of the assimilation of SSS in a long-time reanalysis to better reproduce the Arctic water cycle.

Equation‐Free Surrogate Modeling of Geophysical Flows at the Intersection of Machine Learning and Data Assimilation

AbstractThere is a growing interest in developing data‐driven reduced‐order models for atmospheric and oceanic flows that are trained on data obtained either from high‐resolution simulations or satellite observations. The data‐driven models are non‐intrusive in nature and offer significant computational savings compared to large‐scale numerical models. These low‐dimensional models can be utilized to reduce the computational burden of generating forecasts and estimating model uncertainty without losing the key information needed for data assimilation (DA) to produce accurate state estimates. This paper aims at exploring an equation‐free surrogate modeling approach at the intersection of machine learning and DA in Earth system modeling. With this objective, we introduce an end‐to‐end non‐intrusive reduced‐order modeling (NIROM) framework equipped with contributions in modal decomposition, time series prediction, optimal sensor placement, and sequential DA. Specifically, we use proper orthogonal decomposition (POD) to identify the dominant structures of the flow, and a long short‐term memory network to model the dynamics of the POD modes. The NIROM is integrated within the deterministic ensemble Kalman filter (DEnKF) to incorporate sparse and noisy observations at optimal sensor locations obtained through QR pivoting. The feasibility and the benefit of the proposed framework are demonstrated for the NOAA Optimum Interpolation Sea Surface Temperature (SST) V2 data set. Our results indicate that the NIROM is stable for long‐term forecasting and can model dynamics of SST with a reasonable level of accuracy. Furthermore, the prediction accuracy of the NIROM gets improved by almost one order of magnitude by the DEnKF algorithm.

Parallel Implementation of the Deterministic Ensemble Kalman Filter for Reservoir History Matching

In this paper, the deterministic ensemble Kalman filter is implemented with a parallel technique of the message passing interface based on our in-house black oil simulator. The implementation is separated into two cases: (1) the ensemble size is greater than the processor number and (2) the ensemble size is smaller than or equal to the processor number. Numerical experiments for estimations of three-phase relative permeabilities represented by power-law models with both known endpoints and unknown endpoints are presented. It is shown that with known endpoints, good estimations can be obtained. With unknown endpoints, good estimations can still be obtained using more observations and a larger ensemble size. Computational time is reported to show that the run time is greatly reduced with more CPU cores. The MPI speedup is over 70% for a small ensemble size and 77% for a large ensemble size with up to 640 CPU cores.

Can assimilation of satellite observations improve subsurface biological properties in a numerical model? A case study for the Gulf of Mexico

Abstract. Given current threats to ocean ecosystem health, there is a growing demand for accurate biogeochemical hindcasts, nowcasts, and predictions. Provision of such products requires data assimilation, i.e., a comprehensive strategy for incorporating observations into biogeochemical models, but current data streams of biogeochemical observations are generally considered insufficient for the operational provision of such products. This study investigates to what degree the assimilation of satellite observations in combination with a priori model calibration by sparse BGC-Argo profiles can improve subsurface biogeochemical properties. The multivariate deterministic ensemble Kalman filter (DEnKF) has been implemented to assimilate physical and biological observations into a three-dimensional coupled physical–biogeochemical model, the biogeochemical component of which has been calibrated by BGC-Argo float data for the Gulf of Mexico. Specifically, observations of sea surface height, sea surface temperature, and surface chlorophyll were assimilated, and profiles of both physical and biological variables were updated based on the surface information. We assessed whether this leads to improved subsurface distributions, especially of biological properties, using observations from five BGC-Argo floats that were not assimilated. An alternative light parameterization that was tuned a priori using BGC-Argo observations was also applied to test the sensitivity of data assimilation impact on subsurface biological properties. Results show that assimilation of the satellite data improves model representation of major circulation features, which translate into improved three-dimensional distributions of temperature and salinity. The multivariate assimilation also improves the agreement of subsurface nitrate through its tight correlation with temperature, but the improvements in subsurface chlorophyll were modest initially due to suboptimal choices of the model's optical module. Repeating the assimilation run by using the alternative light parameterization greatly improved the subsurface distribution of chlorophyll. Therefore, even sparse BGC-Argo observations can provide substantial benefits for biogeochemical prediction by enabling a priori model tuning. Given that, so far, the abundance of BGC-Argo profiles in the Gulf of Mexico and elsewhere has been insufficient for sequential assimilation, updating 3D biological properties in a model that has been well calibrated is an intermediate step toward full assimilation of the new data types.

Analysis and design of covariance inflation methods using inflation functions. Part 1: Theoretical framework

AbstractWe propose a unifying theory for covariance inflation (CI) in the Ensemble Kalman Filter (EnKF) that encompasses all existing CI methods and can explain many open problems in CI. Each CI method is identified with an inflation function that alters analysis perturbations through their singular values. Inflation functions are usually considered as functions of singular values of background or analysis perturbations. However, we have shown that it is more fruitful if inflation functions are viewed as functions of reduction factors of background singular values after assimilation. These factors indeed comprise the spectra of linear transformations between background and analysis perturbations. To be an inflation function, a function has to satisfy three conditions: (a) the functional condition: all reduction factors must increase, (b) the no‐observation condition: when no observations are assimilated, analysis perturbations are identical to background perturbations, and (c) the order‐preserving condition: inflated analysis singular values must have the same order as background singular values. If the upper‐bound condition, that is, inflated analysis error variances must be less than observation error variances, is imposed, the resulting inflation functions are shown to be equivalent to prior inflation functions which are functions of singular values of background perturbations. This condition is necessary if we want to inflate analysis increments in posterior CI. It turns out that the relaxation‐to‐prior‐spread method and the relaxation‐to‐prior‐perturbation method belong to the class of linear inflation functions. In this class, we also have constant inflation functions, multiplicative inflation functions and parameter‐varying linear inflation functions. More interesting, the Deterministic EnKF is found to belong to the class of quadratic inflation functions. This quadratic class introduces an elegant form for computing analysis perturbations through the Kalman gain. Higher‐order polynomial and non‐polynomial forms of inflation functions are less appealing in practice due to high computation cost and difficulty in determining free parameters.

A Bayesian approach to multivariate adaptive localization in ensemble-based data assimilation with time-dependent extensions

Abstract. Ever since its inception, the ensemble Kalman filter (EnKF) has elicited many heuristic approaches that sought to improve it. One such method is covariance localization, which alleviates spurious correlations due to finite ensemble sizes by using relevant spatial correlation information. Adaptive localization techniques account for how correlations change in time and space, in order to obtain improved covariance estimates. This work develops a Bayesian approach to adaptive Schur-product localization for the deterministic ensemble Kalman filter (DEnKF) and extends it to support multiple radii of influence. We test the proposed adaptive localization using the toy Lorenz'96 problem and a more realistic 1.5-layer quasi-geostrophic model. Results with the toy problem show that the multivariate approach informs us that strongly observed variables can tolerate larger localization radii. The univariate approach leads to markedly improved filter performance for the realistic geophysical model, with a reduction in error by as much as 33 %.

On the Efficiency of Covariance Localisation of the Ensemble Kalman Filter Using Augmented Ensembles

Localisation is one of the main reasons for the success of the ensemble Kalman filter (EnKF) in high-dimensional geophysical data assimilation problems. It is based on the assumption that correlations between variables of a dynamical system decrease at a fast rate with the physical distance. In the EnKF, two types of localisation methods have emerged: domain localisation and covariance localisation. In this article, we explore possible implementations for covariance localisation in the deterministic EnKF using augmented ensembles in the analysis. We first discuss and compare two different methods to construct the augmented ensemble. The first method, known as modulation, relies on a factorisation property of the background covariance matrix. The second method is based on randomised singular value decomposition (svd) techniques, which has not been previously applied to covariance localisation. The qualitative properties of both methods are illustrated using a simple one-dimensional covariance model. We then show how localisation can be introduced in the perturbation update step using this augmented ensemble framework and we derive a generic ensemble square root Kalman filter with covariance localisation (LEnSRF). Using twin simulations of the Lorenz-1996 model we show that the LEnSRF is numerically more efficient when combined with the randomised svd method than with the modulation method. Finally we introduce a realistic extension of the LEnSRF that uses domain localisation in the horizontal direction and covariance localisation in the vertical direction. Using twin simulations of a multilayer extension of the Lorenz-1996 model, we show that this approach is adequate to assimilate satellite radiances, for which domain localisation alone is insufficient.

Impact of assimilating a merged sea-ice thickness from CryoSat-2 and SMOS in the Arctic reanalysis

Abstract. Accurately forecasting the sea-ice thickness (SIT) in the Arctic is a major challenge. The new SIT product (referred to as CS2SMOS) merges measurements from the CryoSat-2 and SMOS satellites on a weekly basis during the winter. The impact of assimilating CS2SMOS data is tested for the TOPAZ4 system – the Arctic component of the Copernicus Marine Environment Monitoring Services (CMEMS). TOPAZ4 currently assimilates a large set of ocean and sea-ice observations with the Deterministic Ensemble Kalman Filter (DEnKF). Two parallel reanalyses are conducted without (Official run) and with (Test run) assimilation of CS2SMOS data from 19 March 2014 to 31 March 2015. Since only mapping errors were provided in the CS2SMOS observation, an arbitrary term was added to compensate for the missing errors, but was found a posteriori too large. The SIT bias (too thin) is reduced from 16 to 5 cm and the standard errors decrease from 53 to 38 cm (by 28 %) when compared to the assimilated SIT. When compared to independent SIT observations, the error reduction is 24 % against the ice mass balance (IMB) buoy 2013F and by 12.5 % against SIT data from the IceBridge campaigns. The improvement of sea-ice volume persists through the summer months in the absence of CS2SMOS data. Comparisons to sea-ice drift from the satellites show that dynamical adjustments reduce the drift errors around the North Pole by about 8 %–9 % in December 2014 and February 2015. Finally, using the degrees of freedom for signal (DFS), we find that CS2SMOS makes the prime source of information in the central Arctic and in the Kara Sea. We therefore recommend the assimilation of C2SMOS for Arctic reanalyses in order to improve the ice thickness and the ice drift.

Assimilation of semi‐qualitative observations with a stochastic ensemble Kalman filter

The ensemble Kalman filter assumes observations to be Gaussian random variables with a pre‐specified mean and variance. In practice, observations may also have detection limits, for instance when a gauge has a minimum or maximum value. In such cases, most data assimilation schemes discard out‐of‐range values, treating them as “not a number,” with the loss of possibly useful qualitative information.The current work focuses on the development of a data assimilation scheme that tackles observations with a detection limit. We present the Ensemble Kalman Filter Semi‐Qualitative (EnKF‐SQ) and test its performance against the Partial Deterministic Ensemble Kalman Filter (PDEnKF) of Borup et al. Both are designed to assimilate out‐of‐range observations explicitly: the out‐of‐range values are qualitative by nature (inequalities), but one can postulate a probability distribution for them and then update the ensemble members accordingly. The EnKF‐SQ is tested within the framework of twin experiments, using both linear and nonlinear toy models. Different sensitivity experiments are conducted to assess the influence of the ensemble size, observation detection limit and number of observations on the performance of the filter. Our numerical results show that assimilating qualitative observations using the proposed scheme improves the overall forecast mean, making it viable for testing on more realistic applications such as sea‐ice models.

Insights on multivariate updates of physical and biogeochemical ocean variables using an Ensemble Kalman Filter and an idealized model of upwelling

Effective data assimilation methods for incorporating observations into marine biogeochemical models are required to improve hindcasts, nowcasts and forecasts of the ocean's biogeochemical state. Recent assimilation efforts have shown that updating model physics alone can degrade biogeochemical fields while only updating biogeochemical variables may not improve a model's predictive skill when the physical fields are inaccurate. Here we systematically investigate whether multivariate updates of physical and biogeochemical model states are superior to only updating either physical or biogeochemical variables. We conducted a series of twin experiments in an idealized ocean channel that experiences wind-driven upwelling. The forecast model was forced with biased wind stress and perturbed biogeochemical model parameters compared to the model run representing the “truth”. Taking advantage of the multivariate nature of the deterministic Ensemble Kalman Filter (DEnKF), we assimilated different combinations of synthetic physical (sea surface height, sea surface temperature and temperature profiles) and biogeochemical (surface chlorophyll and nitrate profiles) observations. We show that when biogeochemical and physical properties are highly correlated (e.g., thermocline and nutricline), multivariate updates of both are essential for improving model skill and can be accomplished by assimilating either physical (e.g., temperature profiles) or biogeochemical (e.g., nutrient profiles) observations. In our idealized domain, the improvement is largely due to a better representation of nutrient upwelling, which results in a more accurate nutrient input into the euphotic zone. In contrast, assimilating surface chlorophyll improves the model state only slightly, because surface chlorophyll contains little information about the vertical density structure. We also show that a degradation of the correlation between observed subsurface temperature and nutrient fields, which has been an issue in several previous assimilation studies, can be reduced by multivariate updates of physical and biogeochemical fields.

Improvement of the Snow Depth in the Common Land Model by Coupling a Two-Dimensional Deterministic Ensemble Model with a Variational Hybrid Snow Cover Fraction Data Assimilation Scheme and a New Observation Operator

Abstract During snow cover fraction (SCF) data assimilation (DA), the simplified observation operator and presence of cloud cover cause large errors in the assimilation results. To reduce these errors, a new snow cover depletion curve (SDC), known as an observation operator in the DA system, is statistically fitted to in situ snow depth (SD) observations and Moderate Resolution Imaging Spectroradiometer (MODIS) SCF data from January 2004 to October 2008. Using this new SDC, a two-dimensional deterministic ensemble–variational hybrid DA (2DEnVar) method of integrating the deterministic ensemble Kalman filter (DEnKF) and a two-dimensional variational DA (2DVar) is proposed. The proposed 2DEnVar is then used to assimilate the MODIS SCF into the Common Land Model (CoLM) at five sites in the Altay region of China for data from November 2008 to March 2009. The analysis performance of the 2DEnVar is compared with that of the DEnKF. The results show that the 2DEnVar outperforms the DEnKF as it effectively reduces the bias and root-mean-square error during the snow accumulation and ablation periods at all sites except for the Qinghe site. In addition, the 2DEnVar, with more assimilated MODIS SCF observations, produces more innovations (observation minus forecast) than the DEnKF, with only one assimilated MODIS SCF observation. The problems of cloud cover and overestimation are addressed by the 2DEnVar.

Four-dimensional ensemble variational data assimilation and the unstable subspace

The performance of (ensemble) Kalman filters used for data assimilation in the geosciences critically depends on the dynamical properties of the evolution model. A key aspect is that the error covariance matrix is asymptotically supported by the unstable–neutral subspace only, i.e. it is spanned by the backward Lyapunov vectors with non-negative exponents. The analytic proof of such a property for the Kalman filter error covariance has been recently given, and in particular that of its confinement to the unstable–neutral subspace. In this paper, we first generalize those results to the case of the Kalman smoother in a linear, Gaussian and perfect model scenario. We also provide square-root formulae for the filter and smoother that make the connection with ensemble formulations of the Kalman filter and smoother, where the span of the error covariance is described in terms of the ensemble deviations from the mean. We then discuss how this neat picture is modified when the dynamics are nonlinear and chaotic, and for which analytic results are precluded or difficult to obtain. A numerical investigation is carried out to study the approximate confinement of the anomalies for both a deterministic ensemble Kalman filter (EnKF) and a four-dimensional ensemble variational method, the iterative ensemble Kalman smoother (IEnKS), in a perfect model scenario. The confinement is characterized using geometrical angles that determine the relative position of the anomalies with respect to the unstable–neutral subspace. The alignment of the anomalies and of the unstable–neutral subspace is more pronounced when observation precision or frequency, as well as the data assimilation window length for the IEnKS, are increased. These results also suggest that the IEnKS and the deterministic EnKF realize in practice (albeit implicitly) the paradigm behind the approach of Anna Trevisan and co-authors known as the assimilation in the unstable subspace.

Use of remote-sensing reflectance to constrain a data assimilating marine biogeochemical model of the Great Barrier Reef

Abstract. Skillful marine biogeochemical (BGC) models are required to understand a range of coastal and global phenomena such as changes in nitrogen and carbon cycles. The refinement of BGC models through the assimilation of variables calculated from observed in-water inherent optical properties (IOPs), such as phytoplankton absorption, is problematic. Empirically derived relationships between IOPs and variables such as chlorophyll-a concentration (Chl a), total suspended solids (TSS) and coloured dissolved organic matter (CDOM) have been shown to have errors that can exceed 100 % of the observed quantity. These errors are greatest in shallow coastal regions, such as the Great Barrier Reef (GBR), due to the additional signal from bottom reflectance. Rather than assimilate quantities calculated using IOP algorithms, this study demonstrates the advantages of assimilating quantities calculated directly from the less error-prone satellite remote-sensing reflectance (RSR). To assimilate the observed RSR, we use an in-water optical model to produce an equivalent simulated RSR and calculate the mismatch between the observed and simulated quantities to constrain the BGC model with a deterministic ensemble Kalman filter (DEnKF). The traditional assumption that simulated surface Chl a is equivalent to the remotely sensed OC3M estimate of Chl a resulted in a forecast error of approximately 75 %. We show this error can be halved by instead using simulated RSR to constrain the model via the assimilation system. When the analysis and forecast fields from the RSR-based assimilation system are compared with the non-assimilating model, a comparison against independent in situ observations of Chl a, TSS and dissolved inorganic nutrients (NO3, NH4 and DIP) showed that errors are reduced by up to 90 %. In all cases, the assimilation system improves the simulation compared to the non-assimilating model. Our approach allows for the incorporation of vast quantities of remote-sensing observations that have in the past been discarded due to shallow water and/or artefacts introduced by terrestrially derived TSS and CDOM or the lack of a calibrated regional IOP algorithm.

Global Soil Moisture Estimation by Assimilating AMSR-E Brightness Temperatures in a Coupled CLM4–RTM–DART System

Abstract Very few frameworks exist that estimate global-scale soil moisture through microwave land data assimilation (DA). Toward this goal, such a framework has been developed by linking the Community Land Model, version 4 (CLM4), and a microwave radiative transfer model (RTM) with the Data Assimilation Research Testbed (DART). The deterministic ensemble adjustment Kalman filter (EAKF) within DART is utilized to estimate global multilayer soil moisture by assimilating brightness temperature observations from the Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E). A 40-member ensemble of Community Atmosphere Model, version 4.0 (CAM4.0), reanalysis is adopted to drive CLM4 simulations. Space-specific, time-invariant microwave parameters are precalibrated to minimize uncertainties in RTM. Besides, various methods are designed to upscale AMSR-E observations for computational efficiency and time shift CAM4.0 forcing to facilitate global daily assimilations. A series of experiments are conducted to quantify the DA sensitivity to microwave parameters, choice of assimilated observations, and different CLM4 updating schemes. Evaluation results indicate that the newly established CLM4–RTM–DART framework improves the open-loop CLM4-simulated soil moisture. Precalibrated microwave parameters, rather than their default values, can ensure a more robust global-scale performance. In addition, updating near-surface soil moisture is capable of improving soil moisture in deeper layers (0–30 cm), while simultaneously updating multilayer soil moisture fails to obtain intended improvements. Future work is needed to address the systematic bias in CLM4 that cannot be fully covered through the ensemble spread in CAM4.0 reanalysis.

Joint DEnKF-albedo assimilation scheme that considers the common land model subgrid heterogeneity and a snow density-based observation operator for improving snow depth simulations

For the large-area snow depth (SD) data sets with high spatial resolution in the Altay region of Northern Xinjiang, China, we present a deterministic ensemble Kalman filter (DEnKF)-albedo assimilation scheme that considers the common land model (CoLM) subgrid heterogeneity. In the albedo assimilation of DEnKF-albedo, the assimilated albedos over each subgrid tile are estimated with the MCD43C1 bidirectional reflectance distribution function (BRDF) parameters product and CoLM calculated solar zenith angle. The BRDF parameters are hypothesized to be consistent over all subgrid tiles within a specified grid. In the SCF assimilation of DEnKF-albedo, a DEnKF combining a snow density-based observation operator considers the effects of the CoLM subgrid heterogeneity and is employed to assimilate MODIS SCF to update SD states over all subgrid tiles. The MODIS SCF over a grid is compared with the area-weighted sum of model predicted SCF over all the subgrid tiles within the grid. The results are validated with <italic<in situ</italic< SD measurements and AMSR-E product. Compared with the simulations, the DEnKF-albedo scheme can reduce errors of SD simulations and accurately simulate the seasonal variability of SD. Furthermore, it can improve simulations of SD spatiotemporal distribution in the Altay region, which is more accurate and shows more detail than the AMSR-E product.

On the prediction of storm surges during super cyclone Phailin using a deterministic ensemble Kalman filter

Deterministic ensemble Kalman filter has been used to assimilate data from the tracks of three existing altimeters in a state-of-the-art hydrodynamic model configured for the north Indian Ocean. The assimilation period has been chosen to cover the entire time history of the super cyclone Phailin occurring in the Bay of Bengal basin of the north Indian Ocean in mid-October 2013. The filter was proposed in a previous study in which the authors showed its advantage over the existing ensemble-based filter techniques. Different combinations of localization radii and inflation factors have been tried to find the optimum combination. Assimilation with this combination has been found to exhibit significant positive impact on basin-scale simulations of sea level. This conclusion has been further corroborated by comparing the surges simulated by the model with tide-gauge data acquired during the occurrence of Phailin. Assimilation has been also found to enhance the predictive capability of the model.

Covariance correction for estimating groundwater level using deterministic Ensemble Kalman Filter

The main problem in developing a groundwater model is to determine model parameters, particularly hydrogeologic coefficients, in a precise way. In this research, Deterministic Ensemble Kalman Filter (DEnKF) is described as a modern sequential method for data assimilation and a localization scheme within the framework of DEnKF is applied. Najafabad aquifer (in Iran) with area of 1150 km2, is modeled in the time window of Oct. 2000 to Sept. 2007 to obtain water table level data when its values of hydrogeologic coefficients calibrated and verified. DEnKF assimilated 45 observations of true run into the model with 2, 5, and 10 times of calibrated values of hydraulic conductivity and specific yield. This filter has been run both with and without use of localization. Results show easily-implemented localized DEnKF is favorably robust in groundwater flow modeling.

Mitigating Observation Perturbation Sampling Errors in the Stochastic EnKF

Abstract The stochastic ensemble Kalman filter (EnKF) updates its ensemble members with observations perturbed with noise sampled from the distribution of the observational errors. This was shown to introduce noise into the system and may become pronounced when the ensemble size is smaller than the rank of the observational error covariance, which is often the case in real oceanic and atmospheric data assimilation applications. This work introduces an efficient serial scheme to mitigate the impact of observations’ perturbations sampling in the analysis step of the EnKF, which should provide more accurate ensemble estimates of the analysis error covariance matrices. The new scheme is simple to implement within the serial EnKF algorithm, requiring only the approximation of the EnKF sample forecast error covariance matrix by a matrix with one rank less. The new EnKF scheme is implemented and tested with the Lorenz-96 model. Results from numerical experiments are conducted to compare its performance with the EnKF and two standard deterministic EnKFs. This study shows that the new scheme enhances the behavior of the EnKF and may lead to better performance than the deterministic EnKFs even when implemented with relatively small ensembles.