Coupling Ensemble Kalman Filter with Four-dimensional Variational Data Assimilation

This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assimilation. The coupled assimilation scheme （E4DVAR） benefits from using the state-dependent uncertainty provided by EnKF while taking advantage of 4DVAR in preventing filter divergence： the 4DVAR analysis produces posterior maximum likelihood solutions through minimization of a cost function about which the ensemble perturbations are transformed, and the resulting ensemble analysis can be propagated forward both for the next assimilation cycle and as a basis for ensemble forecasting. The feasibility and effectiveness of this coupled approach are demonstrated in an idealized model with simulated observations. It is found that the E4DVAR is capable of outperforming both 4DVAR and the EnKF under both perfect- and imperfect-model scenarios. The performance of the coupled scheme is also less sensitive to either the ensemble size or the assimilation window length than those for standard EnKF or 4DVAR implementations.

Data assimilation using support vector machines and ensemble Kalman filter for multi-layer soil moisture prediction

Hybrid data assimilation （DA） is a method seeing more use in recent hydrology and water resources research. In this study, a DA method coupled with the support vector machines （SVMs） and the ensemble Kalman filter （EnKF） technology was used for the prediction of soil moisture in different soil layers： 0-5 cm, 30 cm, 50 cm, 100 cm, 200 cm, and 300 cm. The SVM methodology was first used to train the ground measurements of soil moisture and meteorological parameters from the Meilin study area, in East China, to construct soil moisture statistical prediction models. Subsequent observations and their statistics were used for predictions, with two approaches： the SVM predictor and the SVM-EnKF model made by coupling the SVM model with the EnKF technique using the DA method. Validation results showed that the proposed SVM-EnKF model can improve the prediction results of soil moisture in different layers, from the surface to the root zone.

Improvement in Background Error Covariances Using Ensemble Forecasts for Assimilation of High-Resolution Satellite Data

Satellite data obtained over synoptic data-sparse regions such as an ocean contribute toward improving the quality of the initial state of limited-area models. Background error covariances are crucial to the proper distribution of satellite-observed information in variational data assimilation. In the NMC （National Meteorological Center） method, background error covariances are underestimated over data-sparse regions such as an ocean because of small differences between different forecast times. Thus, it is necessary to reconstruct and tune the background error covariances so as to maximize the usefulness of the satellite data for the initial state of limited-area models, especially over an ocean where there is a lack of conventional data. In this study, we attempted to estimate background error covariances so as to provide adequate error statistics for data-sparse regions by using ensemble forecasts of optimal perturbations using bred vectors. The background error covariances estimated by the ensemble method reduced the overestimation of error amplitude obtained by the NMC method. By employing an appropriate horizontal length scale to exclude spurious correlations, the ensemble method produced better results than the NMC method in the assimilation of retrieved satellite data. Because the ensemble method distributes observed information over a limited local area, it would be more useful in the analysis of high-resolution satellite data. Accordingly, the performance of forecast models can be improved over the area where the satellite data are assimilated.

Assimilation of Doppler Radar Data with an Ensemble 3DEnVar Approach to Improve Convective Forecasting

An ensemble three-dimensional ensemble-variational(3DEnVar)data assimilation(E3DA)system was developed within the Weather Research and Forecasting model’s 3DVar framework to assimilate radar data to improve convective forecasting.In this system,ensemble perturbations are updated by an ensemble of 3DEnVar and the ensemble forecasts are used to generate the flow-dependent background error covariance.The performance of the E3DA system was first evaluated against one experiment without radar DA and one radar DA experiment with 3DVar,using a severe storm case over southeastern China on 5 June 2009.Results indicated that E3DA improved the quantitative forecast skills of reflectivity and precipitation,as well as their spatial distributions in terms of both intensity and coverage over 3DVar.The root-mean-square error of radial velocity from 3DVar was reduced by E3DA,with stronger low-level wind closer to observation.It was also found that E3DA improved the wind,temperature and water vapor mixing ratio,with the lowest errors at the surface and upper levels.3DVar showed moderate improvements in comparison with forecasts without radar DA.A diagnosis of the analysis revealed that E3DA increased vertical velocity,temperature,and humidity corresponding to the added reflectivity,while 3DVar failed to produce these adjustments,because of the lack of reasonable cross-variable correlations.The performance of E3DA was further verified using two convective cases over southern and southeastern China,and the reflectivity forecast skill was also improved over 3DVar.

An Adaptive Estimation of Forecast Error Covariance Parameters for Kalman Filtering Data Assimilation

An adaptive estimation of forecast error covariance matrices is proposed for Kalman filtering data assim- ilation. A forecast error covariance matrix is initially estimated using an ensemble of perturbation forecasts. This initially estimated matrix is then adjusted with scale parameters that are adaptively estimated by minimizing -2log-likelihood of observed-minus-forecast residuals. The proposed approach could be applied to Kalman filtering data assimilation with imperfect models when the model error statistics are not known. A simple nonlinear model （Burgers＇ equation model） is used to demonstrate the efficacy of the proposed approach.

Assimilation of temporal-spatial leaf area index into the CERES-Wheat model with ensemble Kalman filter and uncertainty assessment for improving winter wheat yield estimation

To accurately estimate winter wheat yields and analyze the uncertainty in crop model data assimilations, winter wheat yield estimates were obtained by assimilating measured or remotely sensed leaf area index （LAI） values. The performances of the calibrated crop environment resource synthesis for wheat （CERES-Wheat） model for two different assimilation scenarios were compared by employing ensemble Kalman filter （EnKF）-based strategies. The uncertainty factors of the crop model data assimilation was analyzed by considering the observation errors, assimilation stages and temporal-spatial scales. Overalll the results indicated a better yield estimate performance when the EnKF-based strategy was used to comprehen- sively consider several factors in the initial conditions and observations. When using this strategy, an adjusted coefficients of determination （R2） of 0.84, a root mean square error （RMSE） of 323 kg ha-1, and a relative errors （RE） of 4.15% were obtained at the field plot scale and an R2 of 0.81, an RMSE of 362 kg ha-1, and an RE of 4.52% were obtained at the pixel scale of 30 mx30 m. With increasing observation errors, the accuracy of the yield estimates obviously decreased, but an acceptable estimate was observed when the observation errors were within 20%. Winter wheat yield estimates could be improved significantly by assimilating observations from the middle to the end of the crop growing seasons. With decreasing assimilation frequency and pixel resolution, the accuracy of the crop yield estimates decreased; however, the computation time decreased. It is important to consider reasonable temporal-spatial scales and assimilation stages to obtain tradeoffs between accuracy and computation time, especially in operational systems used for regional crop yield estimates.

An Ocean Data Assimilation System in the Indian Ocean and West Pacific Ocean

The development and application of a regional ocean data assimilation system are among the aims of the Global Ocean Data Assimilation Experiment. The ocean data assimilation system in the regions including the Indian and West Pacific oceans is an endeavor motivated by this goal. In this study, we describe the system in detail. Moreover, the reanalysis in the joint area of Asia, the Indian Ocean, and the western Pacific Ocean （hereafter AIPOcean） constructed using multi-year model integration with data assimilation is used to test the performance of this system. The ocean model is an eddy-resolving, hybrid coordinate ocean model. Various types of observations including in-situ temperature and salinity profiles （mechanical bathythermograph, expendable bathythermograph, Array for Real-time Geostrophic Oceanography, Tropical Atmosphere Ocean Array, conductivity-temperature-depth, station data）, remotely-sensed sea surface temperature, and altimetry sea level anomalies, are assimilated into the reanalysis via the ensemble optimal interpolation method. An ensemble of model states sampled from a long-term integration is allowed to change with season, rather than remaining stationary. The estimated background error covariance matrix may reasonably reflect the seasonality and anisotropy. We evaluate the performance of AIPOcean during the period 1993-2006 by comparisons with independent observations, and some reanalysis products. We show that AIPOcean reduces the errors of subsurface temperature and salinity, and reproduces mesoscale eddies. In contrast to ECCO and SODA products, AIPOcean captures the interannual variability and linear trend of sea level anomalies very well. AIPOcean also shows a good consistency with tide gauges.

A Prototype Regional GSI-based EnKF-Variational Hybrid Data Assimilation System for the Rapid Refresh Forecasting System：Dual-Resolution Implementation and Testing Results

A dual-resolution（DR） version of a regional ensemble Kalman filter（EnKF）-3D ensemble variational（3DEnVar） coupled hybrid data assimilation system is implemented as a prototype for the operational Rapid Refresh forecasting system. The DR 3DEnVar system combines a high-resolution（HR） deterministic background forecast with lower-resolution（LR） EnKF ensemble perturbations used for flow-dependent background error covariance to produce a HR analysis. The computational cost is substantially reduced by running the ensemble forecasts and EnKF analyses at LR. The DR 3DEnVar system is tested with 3-h cycles over a 9-day period using a 40/13-km grid spacing combination. The HR forecasts from the DR hybrid analyses are compared with forecasts launched from HR Gridpoint Statistical Interpolation（GSI） 3D variational（3DVar）analyses, and single LR hybrid analyses interpolated to the HR grid. With the DR 3DEnVar system, a 90% weight for the ensemble covariance yields the lowest forecast errors and the DR hybrid system clearly outperforms the HR GSI 3DVar.Humidity and wind forecasts are also better than those launched from interpolated LR hybrid analyses, but the temperature forecasts are slightly worse. The humidity forecasts are improved most. For precipitation forecasts, the DR 3DEnVar always outperforms HR GSI 3DVar. It also outperforms the LR 3DEnVar, except for the initial forecast period and lower thresholds.

Evaluation of Ocean Data Assimilation in CAS-ESM-C:Constraining the SST Field

A weakly coupled assimilation system, in which SST observations are assimilated into a coupled climate model （CAS- ESM-C） through an ensemble optimal interpolation scheme, was established. This system is a useful tool for historical climate simulation, showing substantial advantages, including maintaining the atmospheric feedback, and keeping the oceanic tields from drifting far away from the observation, among others. During the coupled model integration, the bias of both surface and subsurface oceanic fields in the analysis can be reduced compared to unassimilated fields. Based on 30 model years of ot.tput fiom the system, the climatology and imerannual variability of the climate system were evaluated. The results showed that the system can reasonably reproduce the climatological global precipitation and SLP, bul it still sutters from the double ITCZ problem. Besides, the ENSO footprint, which is revealed by ENSO-related surface air temperature, geopotential height and precipitation during El Nifio evolution, is basically reproduced by the system. The system can also simulate the observed SST-rainfall relationships well on both interannual and intraseasonal timescales in the western North Pacific region, in which atmospheric feedback is crucial for climate simulation.

Using Analysis State to Construct a Forecast Error Covariance Matrix in Ensemble Kalman Filter Assimilation

Correctly estimating the forecast error covariance matrix is a key step in any data assimilation scheme. If it is not correctly estimated, the assimilated states could be far from the true states. A popular method to address this problem is error covariance matrix inflation. That is, to multiply the forecast error covariance matrix by an appropriate factor. In this paper, analysis states are used to construct the forecast error covariance matrix and an adaptive estimation procedure associated with the error covariance matrix inflation technique is developed. The proposed assimilation scheme was tested on the Lorenz-96 model and 2D Shallow Water Equation model, both of which are associated with spatially correlated observational systems. The experiments showed that by introducing the proposed structure of the forecast error eovariance matrix and applying its adaptive estimation procedure, the assimilation results were further improved.

Research on data assimilation strategy of turbulent separated flow over airfoil

In order to increase the accuracy of turbulence field reconstruction,this paper combines experimental observation and numerical simulation to develop and establish a data assimilation framework,and apply it to the study of S809 low-speed and high-angle airfoil flow.The method is based on the ensemble transform Kalman filter(ETKF)algorithm,which improves the disturbance strategy of the ensemble members and enhances the richness of the initial members by screening high flow field sensitivity constants,increasing the constant disturbance dimensions and designing a fine disturbance interval.The results show that the pressure distribution on the airfoil surface after assimilation is closer to the experimental value than that of the standard Spalart-Allmaras(S-A)model.The separated vortex estimated by filtering is fuller,and the eddy viscosity field information is more abundant,which is physically consistent with the observation information.Therefore,the data assimilation method based on the improved ensemble strategy can more accurately and effectively describe complex turbulence phenomena.

A Homogeneous Linear Estimation Method for System Error in Data Assimilation

In this paper, a new bias estimation method is proposed and applied in a regional ensemble Kalman filter （EnKF） based on the Weather Research and Forecasting （WRF） Model. The method is based on a homogeneous linear bias model, and the model bias is estimated using statistics at each assimilation cycle, which is different from the state augmentation methods proposed in pre- vious literatures. The new method provides a good estimation for the model bias of some specific variables, such as sea level pres- sure （SLP）. A series of numerical experiments with EnKF are performed to examine the new method under a severe weather condi- tion. Results show the positive effect of the method on the forecasting of circulation pattern and meso-scale systems, and the reduc- tion of analysis errors. The background error covarianee structures of surface variables and the effects of model system bias on EnKF are also studied under the error covariance structures and a new concept ＇correlation scale＇ is introduced. However, the new method needs further evaluation with more cases of assimilation.

The REMO Ocean Data Assimilation System into HYCOM(RODAS_H):General Description and Preliminary Results

The first version of the Brazilian Oceano- graphic Modeling and Observation Network （REMO） ocean data assimilation system into the Hybrid Coordi- nate Ocean Model （HYCOM） （RODAS H） has recently been constructed for research and operational purposes. The system is based on a multivariate Ensemble Optimal Interpolation （EnOI） scheme and considers the high fre- quency variability of the model error co-variance matrix. The EnOl can assimilate sea surface temperature （SST）, satellite along-track and gridded sea level anomalies （SLA）, and vertical profiles of temperature （T） and salinity （S） from Argo. The first observing system experiment was carried out over the Atlantic Ocean （78°S-50°N, 100°W-20°E） with HYCOM forced with atmospheric reanalysis from 1 January to 30 June 2010. Five integra- tions were performed, including the control run without assimilation. In the other four, different observations were assimilated： SST only （A SST）; Argo T-S profiles only （AArgo）; along-track SLA only （A_SLA）; and all data employed in the previous runs （A_All）. The A_SST, A_Argo, and A_SLA runs were very effective in improv- ing the representation of the assimilated variables, but they had relatively little impact on the variables that were not assimilated. In particular, only the assimilation of S was able to reduce the deviation of S with respect to ob- servations. Overall, the A_All run produced a good analy- sis by reducing the deviation of SST, T, and S with respect to the control run by 39%, 18%, and 30%, respectively, and by increasing the correlation of SLA by 81%.

The Impact of "Bad" Argo Profiles on Ocean Data Assimilation

Recent studies have found cold biases in a fraction of Argo profiles (hereinafter referred to as bad Array for Real-time Geostrophic Oceanography (Argo) profiles) due to the pressure drifts during 2003 and 2006. These bad Argo profiles have had an important impact on in situ observation-based global ocean heat content esti- mates. This study investigated the impact of bad Argo profiles on ocean data assimilation results that were based on observations from diverse ocean observation systems, such as in situ profiles (e.g., Argo, expendable bathy- thermograph (XBT), and Tropical Atmosphere Ocean (TAO), remote-sensing sea surface temperature products and satellite altimetry between 2004 and 2006. Results from this work show that the upper ocean heat content analysis is vulnerable to bad Argo profiles and demon- strate a cooling trend in the studied period despite the multiple independent data types that were assimilated. When the bad Argo profiles were excluded from the as- similation, the decreased heat content disappeared and a warming occurred. Combination of satellite altimetry and mass variation data from gravity satellite demonstrated an increase, which agrees well with the increased heat con- tent. Additionally, when an additional Argo profile quality control procedure was utilized that simply removed the profiles that presented static unstable water columns, the results were very similar to those obtained when the bad Argo profiles were excluded from the assimilation. This indicates that an ocean data assimilation that uses multiple data sources with improved quality control could be less vulnerable to a major observation system failure, such as a bad Argo event.

Implementation of a One-Dimensional Enthalpy Sea-Ice Model in a Simple Pycnocline Prediction Model for Sea-Ice Data Assimilation Studies

To further explore enthalpy-based sea-ice assimilation, a one-dimensional （1D） enthalpy sea-ice model is implemented into a simple pycnocline prediction model. The 1D enthalpy sea-ice model includes the physical processes such as brine expulsion, flushing, and salt diffusion. After being coupled with the atmosphere and ocean components, the enthalpy sea-ice model can be integrated stably and serves as an important modulator of model variability. Results from a twin experiment show that the sea-ice data assimilation in the enthalpy space can produce smaller root-mean-square errors of model variables than the traditional scheme that assimilates the observations of ice concentration, especially for slow-varying states. This study provides some insights into the improvement of sea-ice data assimilation in a coupled general circulation model.

Data Assimilation Method Based on the Constraints of Confidence Region

The ensemble Kalman filter （EnKF） is a distinguished data assimilation method that is widely used and studied in various fields including methodology and oceanography. However, due to the limited sample size or imprecise dynamics model, it is usually easy for the forecast error variance to be underestimated, which further leads to the phenomenon of filter divergence. Additionally, the assimilation results of the initial stage are poor if the initial condition settings differ greatly from the true initial state. To address these problems, the variance inflation procedure is usually adopted. In this paper, we propose a new method based on the constraints of a confidence region constructed by the observations, called EnCR, to estimate the inflation parameter of the forecast error variance of the EnKF method. In the new method, the state estimate is more robust to both the inaccurate forecast models and initial condition settings. The new method is compared with other adaptive data assimilation methods in the Lorenz-63 and Lorenz-96 models under various model parameter settings. The simulation results show that the new method performs better than the competing methods.

An ensemble adjustment Kalman filter study for Argo data

An ensemble adjustment Kalman filter system is developed to assimilate Argo profiles into the Northwest Pacific MASNUM wave-circulation coupled model, which is based on the Princeton Ocean Model (POM). This model was recoded in FORTRAN-90 style, and some new data types were defined to improve the efficiency of system design and execution. This system is arranged for parallel computing by using UNIX shell scripts: it is easier with single models running separately with the required information exchanged through input/output files. Tests are carried out to check the performance of the system: one for checking the ensemble spread and another for the performance of assimilation of the Argo data in 2005. The first experiment shows that the assimilation system performs well. The comparison with the Satellite derived sea surface temperature (SST) shows that modeled SST errors are reduced after assimilation; at the same time, the spatial correlation between the simulated SST anomalies and the satellite data is improved because of Argo assimilation. Furthermore, the temporal evolution/trend of SST becomes much better than those results without data assimilation. The comparison against GTSPP profiles shows that the improvement is not only in the upper layers of ocean, but also in the deeper layers. All these results suggest that this system is potentially capable of reconstructing oceanic data sets that are of high quality and are temporally and spatially continuous.

Review on Mathematical Perspective for Data Assimilation Methods: Least Square Approach

Environmental systems including our atmosphere oceans, biological… etc. can be modeled by mathematical equations to estimate their states. These equations can be solved with numerical methods. Initial and boundary conditions are needed for such of these numerical methods. Predication and simulations for different case studies are major sources for the great importance of these models. Satellite data from different wide ranges of sensors provide observations that indicate system state. So both numerical models and satellite data provide estimation of system states, and between the different estimations it is required the best estimate for system state. Assimilation of observations in numerical weather models with data assimilation techniques provide an improved estimate of system states. In this work, highlights on the mathematical perspective for data assimilation methods are introduced. Least square estimation techniques are introduced because it is considered the basic mathematical building block for data assimilation methods. Stochastic version of least square is included to handle the error in both model and observation. Then the three and four dimensional variational assimilation 3dvar and 4dvar respectively will be handled. Kalman filters and its derivatives Extended, (KF, EKF, ENKF) and hybrid filters are introduced.

Time-Expanded Sampling for Ensemble-Based Filters:Assimilation Experiments with Real Radar Observations

By sampling perturbed state vectors from each ensemble prediction run at properly selected time levels in the vicinity of the analysis time, the recently proposed time-expanded sampling approach can enlarge the ensemble size without increasing the number of prediction runs and, hence, can reduce the computational cost of an ensemble-based filter. In this study, this approach is tested for the first time with real radar data from a tornadic thunderstorm. In particular, four assimilation experiments were performed to test the time-expanded sampling method against the conventional ensemble sampling method used by ensemble- based filters. In these experiments, the ensemble square-root filter （EnSRF） was used with 45 ensemble members generated by the time-expanded sampling and conventional sampling from 15 and 45 prediction runs, respectively, and quality-controlled radar data were compressed into super-observations with properly reduced spatial resolutions to improve the EnSRF performances. The results show that the time-expanded sampling approach not only can reduce the computational cost but also can improve the accuracy of the analysis, especially when the ensemble size is severely limited due to computational constraints for real-radar data assimilation. These potential merits are consistent with those previously demonstrated by assimilation experiments with simulated data.

Estimating the Soil Moisture Profile by Assimilating Near-Surface Observations with the Ensemble Kalman Filter (EnKF)

The paper investigates the ability to retrieve the true soil moisture profile by assimilating near-surface soil moisture into a soil moisture model with an ensemble Kalman filter （EnKF） assimilation scheme, including the effect of ensemble size, update interval and nonlinearities in the profile retrieval, the required time for full retrieval of the soil moisture profiles, and the possible influence of the depth of the soil moisture observation. These questions are addressed by a desktop study using synthetic data. The ＂true＂ soil moisture profiles are generated from the soil moisture model under the boundary condition of 0.5 cm d^-1 evaporation. To test the assimilation schemes, the model is initialized with a poor initial guess of the soil moisture profile, and different ensemble sizes are tested showing that an ensemble of 40 members is enough to represent the covariance of the model forecasts. Also compared are the results with those from the direct insertion assimilation scheme, showing that the EnKF is superior to the direct insertion assimilation scheme, for hourly observations, with retrieval of the soil moisture profile being achieved in 16 h as compared to 12 days or more. For daily observations, the true soil moisture profile is achieved in about 15 days with the EnKF, but it is impossible to approximate the true moisture within 18 days by using direct insertion. It is also found that observation depth does not have a significant effect on profile retrieval time for the EnKF. The nonlinearities have some negative influence on the optimal estimates of soil moisture profile but not very seriously.