Performance Of Ensemble Kalman Filter Research Articles

Assessing the influence of crop model structure on the performance of data assimilation for sugarcane

Process-base crop models (PBM) are important tools to describe how the agricultural system responds to environmental conditions. Sugarcane represents a major world source of sugar and ethanol and its PBMs had different levels of complexity in terms of structure, i.e. how detailed their processes were described. Yet, literature has widely demonstrated that data assimilation techniques (DA) represent a valuable option for reducing model uncertainty, but the inconsistency between PBM and the assimilated variable can significantly affect the performance of DA. Such limitation is strictly connected to model structure, and a hypothesis that arises from literature is that the use of more complex models would reduce model uncertainty after DA. We accessed the performance of using two different PBMs, one more detailed (DSSAT/SAMUCA, DS) and the other more general (WOFOST, WO), by assimilating leaf area index (LAI) retrieved from Landsat 7 ETM + and 8/OLI, using the Ensemble Kalman Filter (EnKF). Both PBMs were calibrated and evaluated with a robust database of 13 experiments and evaluated against a sugarmill database to evaluate the EnKF performance, compared with simulations without DA (Open-loop, OP). Moreover, the processes involved in LAI simulations were analyzed to assess the EnKF performance. The DS had superior performance in the calibration and evaluation step with EF = 0.907, 0.878, 0.458 for stalk dry mass, stalk fresh yield (SFY), and LAI, while WO showed EF = 0.622, 0.610, 0.417 for the same variables, respectively. The calibration step affected the OP plot simulation, with DS having higher accuracy (RMSE = 31.678 Mg ha−1) and precision (R2 = 0.509), compared with WO (RMSE = 39.593 Mg ha−1; R2 = 0.458). However, after DA, both PBM presented error inconsistency with EnKF, despite the decrease in RMSE (-44.73% and −29.58%) and increase in R2 (22.15% and 36.50%) of DS and WO, respectively. The error inconsistency diverged from each PBM: the OP simulation of DS overestimated the Landsat LAI; after DA, simulated LAI decreased resulting in SFY underestimation (Bias = -11.469 Mg ha−1); WO showed OP simulations for LAI closer to Landsat’s LAI values, despite the positive Bias in SFY estimation, and so EnKF slightly reduced the SFY overestimation (Bias = 22.944 Mg ha−1). Thus, the better descriptions of DS in terms of structure did not inhibit the error inconsistency. We suggested that new studies are required to understand how the assimilated variables impact on the other state variables of the PBM.

Application of particle filter to assess uncertainty for reservoir state and parameter estimation

A bimodal water saturation distribution can be found around the waterfront during water flooding in an oil reservoir, leading to subsequent non-Gaussian distribution of water saturation. Therefore, application of the Gaussian based methods may result in poor history matching and corresponding parameter estimation. For the reliability of the oil reservoir model, it is very important to ensure the realistic update of water saturation during history matching at any phase of production from oil reservoirs. This work employs particle filter (PF) for water flooding reservoir data assimilation, considering the effect of “non-Gaussianity” in water saturation distribution around the shock front during updating model state vector. This paper investigates the performance of the PF in updating state variables, particularly water saturation and efficiency to approximation of posterior probability distribution. In addition, particle perturbation is implemented by applying “ensemble covariance” in resampling stage in order to diverse particles and thereby avoid particle collapse. The methodology is presented stepwise considering a one-dimensional (1D) case first and then two-dimensional (2D) five-spot water flood problem. In both cases, state variables are updated with the application of PF and ensemble kalman filter (EnKF). With respect to history and water saturation matching, and approximation of posterior probability distribution, the performance of EnKF is comparable with PF where the non-Gaussianity is weak. However, in the presence of strong non-Gaussianity, EnKF shows four times higher error compared to PF. Furthermore, between “ensemble covariance” and typical “randomization” cases, “ensemble covariance” case shows a higher number of particles participating in resampling stage in particle perturbation. Therefore, “ensemble covariance” case leads to 5.3% and 7.5% more accurate results in estimating reservoir permeability and porosity, respectively.

Implementation of ensemble Kalman filter algorithm for underwater target tracking

Surveillance of underwater for maritime warfare is traditionally being carried out by bearings-only tracking from many decades. The measurements used for state estimation here are nonlinear. Also the noise in the measurements and the process cannot be always Gaussian. The traditional nonlinear filtering algorithms like extended Kalman filter and modified gain extended Kalman filter use the linearisation of the system. The unscented Kalman filter (UKF) uses the sigma point approach based on Gaussian distribution to deal with nonlinearity. The particle filter (PF) uses the randomly generated particles based on the pdf of the state. PF is highly complex to implement and it also suffers from sample impoverishment. Hence, ensemble Kalman filter (EnKF) which is a simplified form of PF will be tried out for bearings-only tracking in this research work. The performance of EnKF is compared with PF and UKF and the results obtained using these filters in Matlab are presented.

Object-Based Verification of GSI EnKF and Hybrid En3DVar Radar Data Assimilation and Convection-Allowing Forecasts within a Warn-on-Forecast Framework

Abstract The Center for Analysis and Prediction of Storms has recently developed capabilities to directly assimilate radar reflectivity and radial velocity data within the GSI-based ensemble Kalman filter (EnKF) and hybrid ensemble three-dimensional variational (En3DVar) system for initializing convective-scale forecasts. To assess the performance of EnKF and hybrid En3DVar with different hybrid weights (with 100%, 20%, and 0% of static background error covariance corresponding to pure 3DVar, hybrid En3DVar, and pure En3DVar) for assimilating radar data in a Warn-on-Forecast framework, a set of data assimilation and forecast experiments using the WRF Model are conducted for six convective storm cases of May 2017. Using an object-based verification approach, forecast objects of composite reflectivity and 30-min updraft helicity swaths are verified against reflectivity and rotation track objects in Multi-Radar Multi-Sensor data on space and time scales typical of National Weather Service warnings. Forecasts initialized by En3DVar or the best performing EnKF ensemble member produce the highest object-based verification scores, while forecasts from 3DVar and the worst EnKF member produce the lowest scores. Averaged across six cases, hybrid En3DVar using 20% static background error covariance does not improve forecasts over pure En3DVar, although improvements are seen in some individual cases. The false alarm ratios of EnKF members for both composite reflectivity and updraft helicity at the initial time are lower than those from variational methods, suggesting that EnKF analysis reduces spurious reflectivity and mesocyclone objects more effectively.

Exploring Assimilation Order to Improve Assimilation with Serial Ensemble Kalman Filter

Serial ensemble Kalman filter (EnKF) is a kind of EnKF which treats observations serially during every assimilation step. The assimilation order can be generated by different rules and has significant impacts on the performance of serial EnKF when localization algorithm is applied. In this study, we seek to examine and better understand the characteristics of various ordering methods when they are applied in the serial EnKF. The results show that different ordering methods demonstrate almost the same changes in analysis as the localization radius changing. Moreover, the optimal parameters of localization radius and forgetting factor of serial EnKF are found varying among different ordering rules. In addition, a novel rule for confirming the assimilation order is proposed to further improve the performance of serial EnKF. The observations are sorted from “better” to “worse” (OBS-BtoW), which are evaluated by estimating the distance of analysis between the prior and observations. Compared with the existing ordering methods, the proposed method can improve the performance at a very small computation cost without needing future forecasts and the truth.

Performance of ensemble Kalman filter and Markov chain Monte Carlo under uncertainty in forecast model

Abstract Ensemble Kalman filter (EnKF) and Markov chain Monte Carlo (MCMC) are popular methods to obtain the posterior distribution of unknown parameters in the reservoir model. However, millions of simulation runs may be required in MCMC for accurate sampling of posterior as subsurface flow problems are highly nonlinear and non-Gaussian. Similarly, EnKF formulated on the basis of linear and Gaussian assumptions may also require a large number of realizations to correctly map the solution space of the unknown model parameters, ultimately resulting in the high computational cost. Data-driven meta/surrogate/proxy models provide an alternative solution to alleviate the issue of high computational cost. Since these models are not as accurate as numerical solutions of partial differential equations (PDE), their implementation may add an uncertainty in the forecast model. In literature, the effect of uncertainty in forecast model on data assimilation is not well studied, especially with field-scale reservoir models. In this work, we take the opportunity to evaluate and compare the performance of EnKF and MCMC using polynomial chaos expansion (PCE) based forecast model. Proposed forecast model relies on reducing parameter space using Karhunen–Loeve (KL) expansion which preserves the two-point statistics of the field. Random variables from KL expansion and orthogonal polynomials corresponding to the prior probability density function (pdf) form the set of input parameters in PCE. Further, non-intrusive probabilistic collocation method (PCM) is used to compute PCE coefficients. PCE forecast model is then used in EnKF and MCMC to calculate the likelihood of the samples in place of high fidelity full physics simulation runs. A case study is performed using a 3D field scale model of a reservoir located near Fort McMurray in northern Alberta, Canada. Performance of EnKF and MCMC are assessed under forecast model uncertainty using rigorous qualitative and quantitative analysis and posterior distribution characterization. Results clearly depict that, although EnKF provided reliable mean and variance estimates of model parameters, MCMC outperformed the former even under the uncertainty associated with PCE metamodel. Inaccurate initial assumptions of model parameters were successfully handled by MCMC, although, with a longer burn-in period. Furthermore, characterization of posterior demonstrated reduced uncertainty in the estimation of model parameters using MCMC as compared to EnKF. Practical implications of the proposed approach and performance assessment under forecast model uncertainty will be consequential in designing accurate and computationally efficient reservoir characterization workflows and hence, improved decision-making in reservoir management.

Site-specific (Multi-scenario) validation of ensemble Kalman filter-based source inversion through multi-direction wind tunnel experiments

Source inversion uses air dispersion models and environmental measurements to determine the atmospheric radionuclide release rate, which is critical in formulating an emergency response to nuclear incidents. Because source inversion methods are vulnerable to multiple uncertainties, site-specific validations that consider multiple air dispersion scenarios are important in ensuring their correct implementation. To comprehensively evaluate the ensemble Kalman filter (EnKF) for source inversion, a site-specific validation based on six wind tunnel experiments was performed for a highly heterogeneous nuclear power plant site in China. The six experiments cover the typical meteorology of the site and various topography types, providing abundant air dispersion scenarios for validation. The sensitivity of the EnKF to the initial guess, inflation factor, and position/number of measurements is also investigated. The results demonstrate that EnKF offers stable convergence and a reasonably bounded error in all experiments. Furthermore, the EnKF is insensitive to the initial guess, inflation factor, and number of measurements. However, it is sensitive to the position of the measurements and the air dispersion scenario. This sensitivity results from the complicated biases in air dispersion models, which highlight the key to improving the performance of EnKF.

Dynamic Fracture Characterization From Tracer-Test and Flow-Rate Data With Ensemble Kalman Filter

Summary Hydraulic fracturing is performed to enable production from low-permeability and organic-rich shale-oil/gas reservoirs by stimulating the rock to increase its permeability. Characterization and imaging of hydraulically induced fractures is critical for accurate prediction of production and of the stimulated reservoir volume (SRV). Recorded tracer concentrations during flowback and historical production data can reveal important information about fracture and matrix properties, including fracture geometry, hydraulic conductivity, and natural-fracture density. However, the complexity and uncertainty in fracture and reservoir descriptions, coupled with data limitations, complicate the estimation of these properties. In this paper, tracer-test and production data are used for dynamic characterization of important parameters of hydraulically fractured reservoirs, including matrix permeability and porosity, planar-fracture half-length and hydraulic conductivity, discrete-fracture-network (DFN) density and conductivity, and fracture-closing (conductivity-decline) rate during production. The ensemble Kalman filter (EnKF) is used to update uncertain model parameters by sequentially assimilating first the tracer-test data and then the production data. The results indicate that the tracer-test and production data have complementary information for estimating fracture half-length and conductivity, with the former being more sensitive to hydraulic conductivity and the latter being more affected by fracture half-length. For characterization of DFN, a stochastic representation is adopted and the parameters of the stochastic model along with matrix and hydraulic-fracture properties are updated. Numerical examples are presented to investigate the sensitivity of the observed production and tracer-test data to fracture and matrix properties and to evaluate the EnKF performance in estimating these parameters.

Ensemble Kalman Filtering with One-Step-Ahead Smoothing

The ensemble Kalman filter (EnKF) is widely used for sequential data assimilation. It operates as a succession of forecast and analysis steps. In realistic large-scale applications, EnKFs are implemented with small ensembles and poorly known model error statistics. This limits their representativeness of the background error covariances and, thus, their performance. This work explores the efficiency of the one-step-ahead (OSA) smoothing formulation of the Bayesian filtering problem to enhance the data assimilation performance of EnKFs. Filtering with OSA smoothing introduces an updated step with future observations, conditioning the ensemble sampling with more information. This should provide an improved background ensemble in the analysis step, which may help to mitigate the suboptimal character of EnKF-based methods. Here, the authors demonstrate the efficiency of a stochastic EnKF with OSA smoothing for state estimation. They then introduce a deterministic-like EnKF-OSA based on the singular evolutive interpolated ensemble Kalman (SEIK) filter. The authors show that the proposed SEIK-OSA outperforms both SEIK, as it efficiently exploits the data twice, and the stochastic EnKF-OSA, as it avoids observational error undersampling. They present extensive assimilation results from numerical experiments conducted with the Lorenz-96 model to demonstrate SEIK-OSA’s capabilities.

State estimation comparison for a high-dimensional nonlinear system by particle-based filtering methods

The sequential filtering scheme provides a suitable framework for estimating and tracking geophysical states of systems as new data become available online. Mathematical foundations of sequential Bayesian filtering are reviewed with emphasis on practical issues for both particle filters and Kalman-based filters. In this study, we further investigate the study of Kim (2005) such that the sequential Importance resampling method (SIR), Ensemble Kalman Filter (EnKF), and the Maximum Entropy Filter (MEF) are tested in a relatively high dimensional ocean model that conceptually represents the Atlantic thermohaline circulation. The model exhibits large-amplitude transitions between strong (thermo-dominated) and weak (salinity-dominated) circulations that represent climate states between ice-age and normal climate.The performance of the particle-based schemes is compared with the convergent results from SIR based on measurement errors, observation locations, and particle sizes in various sets of twin experiments. The sensitivity analysis shows strength and weakness of each filtering method when applied to multimodal non-linear systems. As the number of particles is increased, SIR achieves the convergent results that are mathematically optimal solutions. EnKF shows suboptimal results regardless of sample sizes, and MEF achieves the optimal solution even with a small sample size. Both EnKF, and MEF produces robust results with a relatively small sample size or increased measurement locations. Small measurement errors or short intervals of observations (or, more frequent observations) significantly improve the performances of SIR and EnKF, and MEF still show robust results even with a relatively small sample size or sparse measurement locations when the system experiences the transition between one region to the other region.

Data assimilation strategies for volcano geodesy

Ground deformation observed using near-real time geodetic methods, such as InSAR and GPS, can provide critical information about the evolution of a magma chamber prior to volcanic eruption. Rapid advancement in numerical modeling capabilities has resulted in a number of finite element models targeted at better understanding the connection between surface uplift associated with magma chamber pressurization and the potential for volcanic eruption. Robust model-data fusion techniques are necessary to take full advantage of the numerical models and the volcano monitoring observations currently available. In this study, we develop a 3D data assimilation framework using the Ensemble Kalman Filter (EnKF) approach in order to combine geodetic observations of surface deformation with geodynamic models to investigate volcanic unrest. The EnKF sequential assimilation method utilizes disparate data sets as they become available to update geodynamic models of magma reservoir evolution. While the EnKF has been widely applied in hydrologic and climate modeling, the adaptation for volcano monitoring is in its initial stages. As such, our investigation focuses on conducting a series of sensitivity tests to optimize the EnKF for volcano applications and on developing specific strategies for assimilation of geodetic data. Our numerical experiments illustrate that the EnKF is able to adapt well to the spatial limitations posed by GPS data and the temporal limitations of InSAR, and that specific strategies can be adopted to enhance EnKF performance to improve model forecasts. Specifically, our numerical experiments indicate that: (1) incorporating additional iterations of the EnKF analysis step is more efficient than increasing the number of ensemble members; (2) the accuracy of the EnKF results are not affected by initial parameter assumptions; (3) GPS observations near the center of uplift improve the quality of model forecasts; (4) occasionally shifting continuous GPS stations to provide variability in the locations of observations results in better model predictions than utilizing fixed locations when the number of available instruments is limited; (5) spotty InSAR data coverage on the flanks of a volcano due to topographic shadows and/or atmospheric interference does not adversely impact the effectiveness of EnKF if the available coverage is >50%; and (6) snow or glacial obstruction in the center of uplift can adversely impact EnKF forecasts. By utilizing these strategies, we conclude that the EnKF is an effective sequential model-data fusion technique for assimilating multiple geodetic observations to forecast volcanic activity at restless volcanoes.

An effective method based on dynamic sampling for data assimilation in a global wave model

The ensemble Kalman filter (EnKF) performs well because that the covariance of background error is varying along time. It provides a dynamic estimate of background error and represents the reasonable statistic characters of background error. However, high computational cost due to model ensemble in EnKF is employed. In this study, two methods referred as static and dynamic sampling methods are proposed to obtain a good performance and reduce the computation cost. Ensemble adjustment Kalman filter (EAKF) method is used in a global surface wave model to examine the performance of EnKF. The 24-h interval difference of simulated significant wave height (SWH) within 1 year is used to compose the static samples for ensemble errors, and these errors are used to construct the ensemble states at each time the observations are available. And then, the same method of updating the model states in the EAKF is applied for the ensemble states constructed by a static sampling method. The dynamic sampling method employs a similar method to construct the ensemble states, but the period of the simulated SWH is changing with time. Here, 7 days before and after the observation time is used as this period. To examine the performance of three schemes, EAKF, static, or dynamic sampling method, observations from satellite Jason-2 in 2014 are assimilated into a global wave model, and observations from satellite Saral are used for validation. The results indicate that the EAKF performs best, while the static sampling method is relatively worse. The dynamic sampling method improves an assimilation effect dramatically compared to the static sampling method, and its overall performance is closed to the EAKF. In low latitudes, the dynamic sampling method has a slight advantage over the EAKF. In the dynamic or static sampling methods, only one wave model is required to run and their computational cost is reduced sharply. According to the performance of these three methods, the dynamic sampling method can treated as an effective alternative of EnKF, which could reduce the computational cost and provide a good performance of data assimilation.

Impact of coastal radar observability on the forecast of the track and rainfall of Typhoon Morakot (2009) using WRF-based ensemble Kalman filter data assimilation

This study explored the impact of coastal radar observability on the forecast of the track and rainfall of Typhoon Morakot (2009) using a WRF-based ensemble Kalman filter (EnKF) data assimilation (DA) system. The results showed that the performance of radar EnKF DA was quite sensitive to the number of radars being assimilated and the DA timing relative to the landfall of the tropical cyclone (TC). It was found that assimilating radial velocity (Vr) data from all the four operational radars during the 6 h immediately before TC landfall was quite important for the track and rainfall forecasts after the TC made landfall. The TC track forecast error could be decreased by about 43% and the 24-h rainfall forecast skill could be almost tripled. Assimilating Vr data from a single radar outperformed the experiment without DA, though with less improvement compared to the multiple-radar DA experiment. Different forecast performances were obtained by assimilating different radars, which was closely related to the first-time wind analysis increment, the location of moisture transport, the quasi-stationary rainband, and the local convergence line. However, only assimilating Vr data when the TC was farther away from making landfall might worsen TC track and rainfall forecasts. Besides, this work also demonstrated that Vr data from multiple radars, instead of a single radar, should be used for verification to obtain a more reliable assessment of the EnKF performance.

Fast linearized forecasts for subsurface flow data assimilation with ensemble Kalman filter

Ensemble methods present a practical framework for parameter estimation, performance prediction, and uncertainty quantification in subsurface flow and transport modeling. In particular, the ensemble Kalman filter (EnKF) has received significant attention for its promising performance in calibrating heterogeneous subsurface flow models. Since an ensemble of model realizations is used to compute the statistical moments needed to perform the EnKF updates, large ensemble sizes are needed to provide accurate updates and uncertainty assessment. However, for realistic problems that involve large-scale models with computationally demanding flow simulation runs, the EnKF implementation is limited to small-sized ensembles. As a result, spurious numerical correlations can develop and lead to inaccurate EnKF updates, which tend to underestimate or even eliminate the ensemble spread. Ad hoc practical remedies, such as localization, local analysis, and covariance inflation schemes, have been developed and applied to reduce the effect of sampling errors due to small ensemble sizes. In this paper, a fast linear approximate forecast method is proposed as an alternative approach to enable the use of large ensemble sizes in operational settings to obtain more improved sample statistics and EnKF updates. The proposed method first clusters a large number of initial geologic model realizations into a small number of groups. A representative member from each group is used to run a full forward flow simulation. The flow predictions for the remaining realizations in each group are approximated by a linearization around the full simulation results of the representative model (centroid) of the respective cluster. The linearization can be performed using either adjoint-based or ensemble-based gradients. Results from several numerical experiments with two-phase and three-phase flow systems in this paper suggest that the proposed method can be applied to improve the EnKF performance in large-scale problems where the number of full simulation is constrained.

Impacts of different types of measurements on estimating unsaturated flow parameters

Summary This paper assesses the value of different types of measurements for estimating soil hydraulic parameters. A numerical method based on ensemble Kalman filter (EnKF) is presented to solely or jointly assimilate point-scale soil water head data, point-scale soil water content data, surface soil water content data and groundwater level data. This study investigates the performance of EnKF under different types of data, the potential worth contained in these data, and the factors that may affect estimation accuracy. Results show that for all types of data, smaller measurements errors lead to faster convergence to the true values. Higher accuracy measurements are required to improve the parameter estimation if a large number of unknown parameters need to be identified simultaneously. The data worth implied by the surface soil water content data and groundwater level data is prone to corruption by a deviated initial guess. Surface soil moisture data are capable of identifying soil hydraulic parameters for the top layers, but exert less or no influence on deeper layers especially when estimating multiple parameters simultaneously. Groundwater level is one type of valuable information to infer the soil hydraulic parameters. However, based on the approach used in this study, the estimates from groundwater level data may suffer severe degradation if a large number of parameters must be identified. Combined use of two or more types of data is helpful to improve the parameter estimation.

Towards predictive data-driven simulations of wildfire spread – Part I: Reduced-cost Ensemble Kalman Filter based on a Polynomial Chaos surrogate model for parameter estimation

Abstract. This paper is the first part in a series of two articles and presents a data-driven wildfire simulator for forecasting wildfire spread scenarios, at a reduced computational cost that is consistent with operational systems. The prototype simulator features the following components: an Eulerian front propagation solver FIREFLY that adopts a regional-scale modeling viewpoint, treats wildfires as surface propagating fronts, and uses a description of the local rate of fire spread (ROS) as a function of environmental conditions based on Rothermel's model; a series of airborne-like observations of the fire front positions; and a data assimilation (DA) algorithm based on an ensemble Kalman filter (EnKF) for parameter estimation. This stochastic algorithm partly accounts for the nonlinearities between the input parameters of the semi-empirical ROS model and the fire front position, and is sequentially applied to provide a spatially uniform correction to wind and biomass fuel parameters as observations become available. A wildfire spread simulator combined with an ensemble-based DA algorithm is therefore a promising approach to reduce uncertainties in the forecast position of the fire front and to introduce a paradigm-shift in the wildfire emergency response. In order to reduce the computational cost of the EnKF algorithm, a surrogate model based on a polynomial chaos (PC) expansion is used in place of the forward model FIREFLY in the resulting hybrid PC-EnKF algorithm. The performance of EnKF and PC-EnKF is assessed on synthetically generated simple configurations of fire spread to provide valuable information and insight on the benefits of the PC-EnKF approach, as well as on a controlled grassland fire experiment. The results indicate that the proposed PC-EnKF algorithm features similar performance to the standard EnKF algorithm, but at a much reduced computational cost. In particular, the re-analysis and forecast skills of DA strongly relate to the spatial and temporal variability of the errors in the ROS model parameters.

Regional scale ozone data assimilation using an ensemble Kalman filter and the CHIMERE chemical transport model

Abstract. An ensemble Kalman filter (EnKF) has been coupled to the CHIMERE chemical transport model in order to assimilate ozone ground-based measurements on a regional scale. The number of ensembles is reduced to 20, which allows for future operational use of the system for air quality analysis and forecast. Observation sites of the European ozone monitoring network have been classified using criteria on ozone temporal variability, based on previous work by Flemming et al. (2005). This leads to the choice of specific subsets of suburban, rural and remote sites for data assimilation and for evaluation of the reference run and the assimilation system. For a 10-day experiment during an ozone pollution event over Western Europe, data assimilation allows for a significant improvement in ozone fields: the RMSE is reduced by about a third with respect to the reference run, and the hourly correlation coefficient is increased from 0.75 to 0.87. Several sensitivity tests focus on an a posteriori diagnostic estimation of errors associated with the background estimate and with the spatial representativeness of observations. A strong diurnal cycle of both these errors with an amplitude up to a factor of 2 is made evident. Therefore, the hourly ozone background error and the observation error variances are corrected online in separate assimilation experiments. These adjusted background and observational error variances provide a better uncertainty estimate, as verified by using statistics based on the reduced centered random variable. Over the studied 10-day period the overall EnKF performance over evaluation stations is found relatively unaffected by different formulations of observation and simulation errors, probably due to the large density of observation sites. From these sensitivity tests, an optimal configuration was chosen for an assimilation experiment extended over a three-month summer period. It shows a similarly good performance as the 10-day experiment.

Numerical Comparison of Iterative Ensemble Kalman Filters for Unsaturated Flow Inverse Modeling

This study evaluated three algorithms of the iterative ensemble Kalman filter (EnKF). They are Confirming EnKF, Restart EnKF, and modified Restart EnKF developed to resolve the inconsistency problem (i.e., updated model parameters and state variables do not follow the Richards equation) in vadose zone data assimilation due to model nonlinearity. While Confirming and Restart EnKF were adapted from literature, modified Restart EnKF was developed in this study to reduce computational costs by calculating only the mean simulation, not all the ensemble realizations, from time t = 0. A total of 11 cases were designed to investigate the performance of EnKF, Confirming EnKF, Restart EnKF, and modified Restart EnKF with different types and spatial configurations of observations (pressure head and water content) and different values of observation error variance, initial guess of ensemble mean and variance, ensemble size, and damping factor. The numerical study showed that Confirming EnKF produced considerable inconsistency for the nonlinear unsaturated flow problem, which differs from the apparent consensus opinion that Confirming EnKF can resolve the inconsistency problem. In contrast, Restart EnKF and its modification can resolve the inconsistency problem. Restart EnKF and its modification outperformed EnKF and Confirming EnKF in the various cases considered in this study. It ws also found that combining different types of observations can achieve better assimilation results, which is useful for monitoring network design.

Rapid Updating of Stochastic Models by Use of an Ensemble-Filter Approach

Summary Applying an ensemble Kalman filter (EnKF) is an effective method for reservoir history matching. The underlying principle is that an initial ensemble of stochastic models can be progressively updated to reflect measured values as they become available. The EnKF performance is only optimal, however, if the prior-state vector is linearly related to the predicted data and if the joint distribution of the prior-state vector is multivariate Gaussian. Therefore, it is challenging to implement the filtering scheme for non-Gaussian random fields, such as channelized reservoirs, in which the continuity of permeability extremes is well-preserved. In this paper, we develop a methodology by combining model classification with multidimensional scaling (MDS) and the EnKF to create rapidly updating models of a channelized reservoir. A dissimilarity matrix is computed by use of the dynamic responses of ensemble members. This dissimilarity matrix is transformed into a lower-dimensional space by use of MDS. Responses mapped in the lower-dimension space are clustered, and on the basis of the distances between the models in a cluster and the actual observed response, the closest models to the observed response are retrieved. Model updates within the closest cluster are performed with EnKF equations. The results of an update are used to resample new models for the next step. Two-dimensional, waterflooding examples of channelized reservoirs are provided to demonstrate the applicability of the proposed method. The obtained results demonstrate that the proposed algorithm is viable both for sequentially updating reservoir models and for preserving channel features after the data-assimilation process.

A Regional GSI-Based Ensemble Kalman Filter Data Assimilation System for the Rapid Refresh Configuration: Testing at Reduced Resolution

Abstract A regional ensemble Kalman filter (EnKF) system is established for potential Rapid Refresh (RAP) operational application. The system borrows data processing and observation operators from the gridpoint statistical interpolation (GSI), and precalculates observation priors using the GSI. The ensemble square root Kalman filter (EnSRF) algorithm is used, which updates both the state vector and observation priors. All conventional observations that are used in the operational RAP GSI are assimilated. To minimize computational costs, the EnKF is run at ⅓ of the operational RAP resolution or about 40-km grid spacing, and its performance is compared to the GSI using the same datasets and resolution. Short-range (up to 18 h, the RAP forecast length) forecasts are verified against soundings, surface observations, and precipitation data. Experiments are run with 3-hourly assimilation cycles over a 9-day convectively active retrospective period from spring 2010. The EnKF performance was improved by extensive tuning, including the use of height-dependent covariance localization scales and adaptive covariance inflation. When multiple physics parameterization schemes are employed by the EnKF, forecast errors are further reduced, especially for relative humidity and temperature at the upper levels and for surface variables. The best EnKF configuration produces lower forecast errors than the parallel GSI run. Gilbert skill scores of precipitation forecasts on the 13-km RAP grid initialized from the 3-hourly EnKF analyses are consistently better than those from GSI analyses.