Sequential Data Assimilation Research Articles

Sequential data assimilation with multiple models

Data assimilation is an essential tool for predicting the behavior of real physical systems given approximate simulation models and limited observations. For many complex systems, there may exist several models, each with different properties and predictive capabilities. It is desirable to incorporate multiple models into the assimilation procedure in order to obtain a more accurate prediction of the physics than any model alone can provide. In this paper, we propose a framework for conducting sequential data assimilation with multiple models and sources of data. The assimilated solution is a linear combination of all model predictions and data. One notable feature is that the combination takes the most general form with matrix weights. By doing so the method can readily utilize different weights in different sections of the solution state vectors, allow the models and data to have different dimensions, and deal with the case of a singular state covariance. We prove that the proposed assimilation method, termed direct assimilation, minimizes a variational functional, a generalized version of the one used in the classical Kalman filter. We also propose an efficient iterative assimilation method that assimilates two models at a time until all models and data are assimilated. The mathematical equivalence of the iterative method and the direct method is established. Numerical examples are presented to demonstrate the effectiveness of the new method.

On sequential data assimilation for scalar macroscopic traffic flow models

We consider the problem of sequential data assimilation for transportation networks using optimal filtering with a scalar macroscopic traffic flow model. Properties of the distribution of the uncertainty on the true state related to the specific nonlinearity and non-differentiability inherent to macroscopic traffic flow models are investigated, derived analytically and analyzed. We show that nonlinear dynamics, by creating discontinuities in the traffic state, affect the performances of classical filters and in particular that the distribution of the uncertainty on the traffic state at shock waves is a mixture distribution. The non-differentiability of traffic dynamics around stationary shock waves is also proved and the resulting optimality loss of the estimates is quantified numerically. The properties of the estimates are explicitly studied for the Godunov scheme (and thus the Cell-Transmission Model), leading to specific conclusions about their use in the context of filtering, which is a significant contribution of this article. Analytical proofs and numerical tests are introduced to support the results presented. A Java implementation of the classical filters used in this work is available on-line at http://traffic.berkeley.edu for facilitating further efforts on this topic and fostering reproducible research.

Examining the effectiveness and robustness of sequential data assimilation methods for quantification of uncertainty in hydrologic forecasting

In hydrologic modeling, state‐parameter estimation using data assimilation techniques is increasing in popularity. Several studies, using both the ensemble Kalman filter (EnKF) and the particle filter (PF) to estimate both model states and parameters have been published in recent years. Though there is increasing interest and a growing literature in this area, relatively little research has been presented to examine the effectiveness and robustness of these methods to estimate uncertainty. This study suggests that state‐parameter estimation studies need to provide a more rigorous testing of these techniques than has previously been presented. With this in mind, this paper presents a study with multiple calibration replicates and a range of performance measures to test the ability of each technique to calibrate two separate hydrologic models. The results show that the EnKF is consistently overconfident in predicting streamflow, which relates to the assumption of a Gaussian error structure. In addition, the EnKF and PF were found to perform similarly in terms of tracking the observations with an expected value, but the potential for filter divergence in the EnKF is highlighted.

Ensemble smoother with multiple data assimilation

In the last decade, ensemble-based methods have been widely investigated and applied for data assimilation of flow problems associated with atmospheric physics and petroleum reservoir history matching. This paper focuses entirely on the reservoir history-matching problem. Among the ensemble-based methods, the ensemble Kalman filter (EnKF) is the most popular for history-matching applications. However, the recurrent simulation restarts required in the EnKF sequential data assimilation process may prevent the use of EnKF when the objective is to incorporate the history matching in an integrated geo-modeling workflow. In this situation, the ensemble smoother (ES) is a viable alternative. However, because ES computes a single global update, it may not result in acceptable data matches; therefore, the development of efficient iterative forms of ES is highly desirable. In this paper, we propose to assimilate the same data multiple times with an inflated measurement error covariance matrix in order to improve the results obtained by ES. This method is motivated by the equivalence between single and multiple data assimilation for the linear-Gaussian case. We test the proposed method for three synthetic reservoir history-matching problems. Our results show that the proposed method provides better data matches than those obtained with standard ES and EnKF, with a computational cost comparable with the computational cost of EnKF.

Development of swing method with sequential data assimilation (SDA-SWING) and its application to groundwater problems in real tunnel construction

Tunnel excavation in Japan is beset by difficulties related to complicated geology and groundwater. Controlling the groundwater inflow to the tunnel during excavation is a challenging issue for both tunnel construction and for environmental assessment. The SWING (System for Water INformation of Ground) method was developed to enable predictions regarding groundwater inflow to the tunnel and also to allow evaluation of the impact of excavation on the surrounding groundwater environment. The SWING method is based on the closed-form solution of Dupuit quasi-uniform flow and can follow the tunnel excavation process by reconstructing the hydraulic conductivity fields using data obtained at construction sites. Herein, the original SWING system was modified by the Ensemble Kalman Filter (EnKF) in order to automatically optimize the hydraulic parameters. The newly developed SWING method, SDA-SWING (Sequential Data Assimilation-SWING), was applied to two different tunnel construction sites in Japan and its applicability was discussed by comparison with the results obtained from the original SWING method. The effects of initial hydraulic conductivity and the frequency of data acquisition on the predicted results were also investigated using the SDA-SWING method. The SDA-SWING method can automatically optimize the hydraulic parameters using EnKF algorism without the need for highly skilled personnel and thus leads to a significant reduction in working times. The estimation of groundwater inflow by the SDA-SWING method generally agrees well with results obtained by the original SWING method and also with observations at the tunnel site.

Ensemble Randomized Maximum Likelihood Method as an Iterative Ensemble Smoother

The ensemble Kalman filter (EnKF) is a sequential data assimilation method that has been demonstrated to be effective for history matching reservoir production data and seismic data. To avoid, however, the expense of repeatedly updating variables and restarting simulation runs, an ensemble smoother (ES) has recently been proposed. Like the EnKF, the ES obtains all information necessary to compute a correction to model variables directly from an ensemble of models without the need of an adjoint code. The success of both methods for history matching reservoir data without iteration is somewhat surprising since traditional gradient-based methods for history matching typically require 10 to 30 iterations to converge to an acceptable minimum. In this manuscript we describe a new iterative ensemble smoother (batch-EnRML) that assimilates all data simultaneously and compare the performance of the iterative smoother with the two non-iterative methods and the previously proposed sequential iterative ensemble filter (seq-EnRML). We discuss some aspects of the use of the ensemble estimate of sensitivity, and show that by sequentially assimilating data, the nonlinearity of the assimilation problem is substantially reduced. Although reasonably good data matches can be obtained using a non-iterative ensemble smoother, iteration was necessary to achieve results comparable to the EnKF for nonlinear problems.

TREATMENT OF THE ERROR DUE TO UNRESOLVED SCALES IN SEQUENTIAL DATA ASSIMILATION

In this paper, a method to account for model error due to unresolved scales in sequential data assimilation, is proposed. An equation for the model error covariance required in the extended Kalman filter update is derived along with an approximation suitable for application with large scale dynamics typical in environmental modeling. This approach is tested in the context of a low order chaotic dynamical system. The results show that the filter skill is significantly improved by implementing the proposed scheme for the treatment of the unresolved scales.

Data assimilation with a local Ensemble Kalman Filter applied to a three-dimensional biological model of the Middle Atlantic Bight

A multivariate sequential data assimilation approach, the Localized Ensemble Kalman Filter (LEnKF), was used to assimilate daily satellite observations of ocean chlorophyll into a three-dimensional physical–biological model of the Middle Atlantic Bight (MAB) for the year 2006. Covariance localization was applied to make the EnKF analysis more effective by removing spurious long-range correlations in the ensemble approximation of the model's covariance. The model is based on the Regional Ocean Modeling System (ROMS) and coupled to a biological nitrogen cycle model, which includes seven state variables: chlorophyll, phytoplankton, nitrate, ammonium, small and large detrital nitrogen, and zooplankton. An ensemble of 20 model simulations, generated by perturbing the biological parameters according to assumed probability distributions, was used. Model fields of chlorophyll, phytoplankton, nitrate and zooplankton were updated at all vertical layers during LEnKF analysis steps, based on their cross-correlations with surface chlorophyll (the observed variable). The performance of the LEnKF scheme, its influence on the model's predictive skill and on surface particulate organic matter concentrations and primary production are investigated. Estimates of surface chlorophyll and particulate organic carbon are improved in the data-assimilative simulation when compared to one without any assimilation, as is the model's predictive skill.

Robust Ensemble Filtering and Its Relation to Covariance Inflation in the Ensemble Kalman Filter

Abstract A robust ensemble filtering scheme based on the H∞ filtering theory is proposed. The optimal H∞ filter is derived by minimizing the supremum (or maximum) of a predefined cost function, a criterion different from the minimum variance used in the Kalman filter. By design, the H∞ filter is more robust than the Kalman filter, in the sense that the estimation error in the H∞ filter in general has a finite growth rate with respect to the uncertainties in assimilation, except for a special case that corresponds to the Kalman filter. The original form of the H∞ filter contains global constraints in time, which may be inconvenient for sequential data assimilation problems. Therefore a variant is introduced that solves some time-local constraints instead, and hence it is called the time-local H∞ filter (TLHF). By analogy to the ensemble Kalman filter (EnKF), the concept of ensemble time-local H∞ filter (EnTLHF) is also proposed. The general form of the EnTLHF is outlined, and some of its special cases are discussed. In particular, it is shown that an EnKF with certain covariance inflation is essentially an EnTLHF. In this sense, the EnTLHF provides a general framework for conducting covariance inflation in the EnKF-based methods. Some numerical examples are used to assess the relative robustness of the TLHF–EnTLHF in comparison with the corresponding KF–EnKF method.

A data assimilation tool for the Pagasitikos Gulf ecosystem dynamics: Methods and benefits

Within the framework of the European INSEA project, an advanced assimilation system has been implemented for the Pagasitikos Gulf ecosystem. The system is based on a multivariate sequential data assimilation scheme that combines satellite ocean sea color (chlorophyll-a) data with the predictions of a three-dimensional coupled physical–biochemical model of the Pagasitikos Gulf ecosystem presented in a companion paper. The hydrodynamics are solved with a very high resolution (1/100°) implementation of the Princeton Ocean Model (POM). This model is nested within a coarser resolution model of the Aegean Sea which is part of the Greek POSEIDON forecasting system. The forecast of the Aegean Sea model, itself nested and initialized from a Mediterranean implementation of POM, is also used to periodically re-initalize the Pagatisikos hydrodynamics model using variational initialization techniques. The ecosystem dynamics of Pagasitikos are tackled with a stand-alone implementation of the European Seas Ecosystem Model (ERSEM). The assimilation scheme is based on the Singular Evolutive Extended Kalman (SEEK) filter, in which the error statistics are parameterized by means of a suitable set of Empirical Orthogonal Functions (EOFs).The assimilation experiments were performed for year 2003 and additionally for a 9-month period over 2006 during which the physical model was forced with the POSEIDON-ETA 6-hour atmospheric fields. The assimilation system is validated by assessing the relevance of the system in fitting the data, the impact of the assimilation on non-observed biochemical processes and the overall quality of the forecasts. Assimilation of either GlobColour in 2003 or SeaWiFS in 2006 chlorophyll-a data enhances the identification of the ecological state of the Pagasitikos Gulf. Results, however, suggest that subsurface ecological observations are needed to improve the controllability of the ecosystem in the deep layers.

Estimation of Unsaturated Soil Hydraulic Parameters Using the Ensemble Kalman Filter

The parameters of soil hydraulic functions are essential to the accurate simulation of soil moisture based on the Richards equation. Optimal values of these parameters can be calibrated by inverse modeling, in which the incomplete consideration of various errors may influence the parameter estimation results, thus further limiting the accuracy of modeling and forecasting. The ensemble Kalman filter (EnKF) is believed to be a flexible and effective sequential data assimilation method that provides a framework of explicit consideration of the various sources of uncertainty and is suitable for real‐time, updated observations. The objective of this study was to extend the use of the EnKF to parameter estimation in vadose zone hydrology to improve the treatment of uncertainty in the calibration process. The parameters of soil hydraulic functions were estimated by assimilating observations of soil water pressure dynamics using EnKF with an augmentation technique. The results of the synthetic experiments on 12 soils with different textures indicated that EnKF estimates can quickly approach stable estimates. In contrast to the batch calibration process that used a simple least squares objective function, the EnKF reduced the risk of obtaining suboptimal estimates. The EnKF also performed well in the multiparameter estimation scenarios with synthetic observations and in its application in a heterogeneous soil profile with in situ field observations from a previous study. We further explored the factors that may influence estimation results, including the initial estimate, the ensemble size, the observation error and the model error, the assimilation interval, the water regime, and the variability of the estimated parameters. The result of this study indicates that the EnKF scheme is an effective method for parameter estimation in vadose zone hydrology.

Applying sequential Monte Carlo methods into a distributed hydrologic model: lagged particle filtering approach with regularization

Abstract. Data assimilation techniques have received growing attention due to their capability to improve prediction. Among various data assimilation techniques, sequential Monte Carlo (SMC) methods, known as "particle filters", are a Bayesian learning process that has the capability to handle non-linear and non-Gaussian state-space models. In this paper, we propose an improved particle filtering approach to consider different response times of internal state variables in a hydrologic model. The proposed method adopts a lagged filtering approach to aggregate model response until the uncertainty of each hydrologic process is propagated. The regularization with an additional move step based on the Markov chain Monte Carlo (MCMC) methods is also implemented to preserve sample diversity under the lagged filtering approach. A distributed hydrologic model, water and energy transfer processes (WEP), is implemented for the sequential data assimilation through the updating of state variables. The lagged regularized particle filter (LRPF) and the sequential importance resampling (SIR) particle filter are implemented for hindcasting of streamflow at the Katsura catchment, Japan. Control state variables for filtering are soil moisture content and overland flow. Streamflow measurements are used for data assimilation. LRPF shows consistent forecasts regardless of the process noise assumption, while SIR has different values of optimal process noise and shows sensitive variation of confidential intervals, depending on the process noise. Improvement of LRPF forecasts compared to SIR is particularly found for rapidly varied high flows due to preservation of sample diversity from the kernel, even if particle impoverishment takes place.

Sequential variations of phytoplankton growth and mortality in an NPZ model: A remote-sensing-based assessment

Radiometric data in the visible domain acquired by satellite remote sensing have proven to be powerful for monitoring the states of the ocean, both physical and biological. With the help of these data it is possible to understand certain variations in biological responses of marine phytoplankton on ecological time scales. Here, we implement a sequential data-assimilation technique to estimate from a conventional nutrient–phytoplankton–zooplankton (NPZ) model the time variations of observed and unobserved variables. In addition, we estimate the time evolution of two biological parameters, namely, the specific growth rate and specific mortality of phytoplankton. Our study demonstrates that: (i) the series of time-varying estimates of specific growth rate obtained by sequential data assimilation improves the fitting of the NPZ model to the satellite-derived time series: the model trajectories are closer to the observations than those obtained by implementing static values of the parameter; (ii) the estimates of unobserved variables, i.e., nutrient and zooplankton, obtained from an NPZ model by implementation of a pre-defined parameter evolution can be different from those obtained on applying the sequences of parameters estimated by assimilation; and (iii) the maximum estimated specific growth rate of phytoplankton in the study area is more sensitive to the sea-surface temperature than would be predicted by temperature-dependent functions reported previously. The overall results of the study are potentially useful for enhancing our understanding of the biological response of phytoplankton in a changing environment.

Impact of non‐smooth observation operators on variational and sequential data assimilation for a limited‐area shallow‐water equation model

AbstractWe investigate the issue of variational and sequential data assimilation with nonlinear and non‐smooth observation operators using a two‐dimensional limited‐area shallow‐water equation model and its adjoint. The performance of the four‐dimensional variational approach (4D‐Var: two dimensions plus time) compared with that of the maximum‐likelihood ensemble filter (MLEF), a hybrid ensemble/variational method, is tested in the presence of non‐smooth observation operators.Following the work of Lewis & Overton and Karmitsa, we investigate minimization of the data‐assimilation cost functional using the limited‐memory Broyden–Fletcher–Goldfarb–Shanno (L‐BFGS) quasi‐Newton algorithm originally intended for smooth optimization and the limited‐memory bundle method (LMBM) algorithm specifically designed to address large‐scale non‐smooth minimization problems.Numerical results obtained for the MLEF method show that the LMBM algorithm yields results superior to the L‐BFGS method. Results for 4D‐Var suggest that L‐BFGS performs well when the non‐smoothness is not extreme, but fails for non‐smooth functions with large Lipschitz constants. The LMBM method is found to be a suitable choice for large‐scale non‐smooth optimization, although additional work is needed to improve its numerical stability. Finally, the results and methodologies of 4D‐Var and MLEF are compared and contrasted. Copyright © 2011 Royal Meteorological Society

Sequential data assimilation in fine-resolution models using error-subspace emulators: Theory and preliminary evaluation

A novel technique for nonlinear sequential data assimilation in computationally expensive fine-resolution models is introduced. The technique involves basis function approximation for dimension reduction and Gaussian Process Modelling for simulation speedup. The basis function approximation is carried out via the Singular Value Decomposition (SVD) of the model ensemble. The Gaussian Process Models propagate the model solution in the error-subspace defined by a finite set of basis functions. The developed technique can also be considered approximate Particle Filtering with two classes of particles: model-particles representing an ensemble of computationally expensive model solutions, and emulator-particles representing an ensemble of fast and cheap model approximations. The algorithm was tested by assimilating synthetic data into a two-dimensional (one spatial dimension plus time) sediment transport model in an idealised vertically-resolved benthic–pelagic system. The assimilation algorithm updates 2 spatially varying state variables and 3 unknown parameters. Numerical experiments illustrate robust performance of the technique for a wide range of the assimilation settings. The capabilities and limitations of the approach are discussed, and further developments are outlined.

Hydrological real-time modelling in the Zambezi river basin using satellite-based soil moisture and rainfall data

Abstract. Reliable real-time forecasts of the discharge can provide valuable information for the management of a river basin system. For the management of ecological releases even discharge forecasts with moderate accuracy can be beneficial. Sequential data assimilation using the Ensemble Kalman Filter provides a tool that is both efficient and robust for a real-time modelling framework. One key parameter in a hydrological system is the soil moisture, which recently can be characterized by satellite based measurements. A forecasting framework for the prediction of discharges is developed and applied to three different sub-basins of the Zambezi River Basin. The model is solely based on remote sensing data providing soil moisture and rainfall estimates. The soil moisture product used is based on the back-scattering intensity of a radar signal measured by a radar scatterometer. These soil moisture data correlate well with the measured discharge of the corresponding watershed if the data are shifted by a time lag which is dependent on the size and the dominant runoff process in the catchment. This time lag is the basis for the applicability of the soil moisture data for hydrological forecasts. The conceptual model developed is based on two storage compartments. The processes modeled include evaporation losses, infiltration and percolation. The application of this model in a real-time modelling framework yields good results in watersheds where soil storage is an important factor. The lead time of the forecast is dependent on the size and the retention capacity of the watershed. For the largest watershed a forecast over 40 days can be provided. However, the quality of the forecast increases significantly with decreasing prediction time. In a watershed with little soil storage and a quick response to rainfall events, the performance is relatively poor and the lead time is as short as 10 days only.

Hydraulic parameter estimation by remotely-sensed top soil moisture observations with the particle filter

► Top soil moisture observations for estimation of hydraulic parameters. ► Simultaneous update of model states (soil moisture) and hydraulic parameters. ► SIR-PF for propagation of non-Gaussian distributions through a nonlinear model. ► Estimation of hydr. parameters driven by non linearity between SM and pressure head. In a synthetic study we explore the potential of using surface soil moisture measurements obtained from different satellite platforms to retrieve soil moisture profiles and soil hydraulic properties using a sequential data assimilation procedure and a 1D mechanistic soil water model. Four different homogeneous soil types were investigated including loamy sand, loam, silt, and clayey soils. The forcing data including precipitation and potential evapotranspiration were taken from the meteorological station of Aachen (Germany). With the aid of the forward model run, a synthetic data set was designed and observations were generated. The virtual top soil moisture observations were then assimilated to update the states and hydraulic parameters of the model by means of a particle filtering data assimilation method. Our analyses include the effect of assimilation strategy, measurement frequency, accuracy in surface soil moisture measurements, and soils differing in textural and hydraulic properties. With this approach we were able to assess the value of periodic spaceborne observations of top soil moisture for soil moisture profile estimation and identify the adequate conditions (e.g. temporal resolution and measurement accuracy) for remotely sensed soil moisture data assimilation. Updating of both hydraulic parameters and state variables allowed better predictions of top soil moisture contents as compared with updating of states only. An important conclusion is that the assimilation of remotely-sensed top soil moisture for soil hydraulic parameter estimation generates a bias depending on the soil type. Results indicate that the ability of a data assimilation system to correct the soil moisture state and estimate hydraulic parameters is driven by the non linearity between soil moisture and pressure head.

Mesoscale Data Assimilation of Myanmar Cyclone Nargis Part II: Assimilation of GPS-Derived Precipitable Water Vapor

Four-dimensional variational (4D-Var) data assimilation (DA) experiments using Global Positioning System (GPS)-derived precipitable water vapor (PWV) were conducted for the tropical cyclone (TC) Nargis in 2008. In order to analyze the initial field at 1200 UTC 30 April 2008, 12, 24, 36, and 48 h sequential DA experiments with 3 h assimilation windows were performed. The initial fields made by these DA experiments were applied to subsequent forecast experiments using a nonhydrostatic model (NHM) with a horizontal resolution of 10 km. NHM predictions using initial fields produced by DA experiments that used only ordinary observational data (without GPS PWV) exhibited a large variation of predicted maximum TC intensity (958 to 983 hPa) for each experiment. In these experiments, a longer assimilation period did not necessarily result in better prediction. The DA of GPS PWV yielded a smaller variation of predicted maximum TC intensity (964 to 974 hPa), and a longer assimilation period tended to bring deeper depression of TC central pressure. Overall, TC intensities determined by DA experiments with GPS data were closer to the best track produced by the Regional Specialized Meteorological Centre (RSMC) New Delhi than the DA experiments without GPS data. The 48 h DA without GPS PWV resulted in the weakest prediction of TC development with the deepest TC central pressure of 983 hPa, while 48 h DA with GPS PWV successfully predicted rapid TC development with the deepest pressure of 967 hPa. One cause of the incomplete development of Nargis in the 48 h DA experiment without GPS PWV was insufficient observations in the Bay of Bengal, especially in the first 12 h. Underestimation of precipitation was conspicuous in the first 12 h of the DA. Implementation of GPS PWV into the DA contributed to increasing the precipitation and changed the fields of pressure and wind in the bay. In the first several hours, modifications of the fields of pressure and wind around the Andaman Islands were conspicuous. These affected areas extended with time and created a more favorable environment for TC development.

From the Ensemble Kalman Filter to the Particle Filter: a comparative study in rainfall-runoff models

AbstractRainfall-runoff models are hydrologic models and they play a very important role in flood forecasting. Hydrologic systems are featured by a non-linear behaviour, non-Gaussian distributions and the presence of large uncertainties in the model itself and in the input data. Therefore, the use of non-linear/non-Gaussian state estimation or data assimilation techniques is important in order to improve the model predictions. The objective of this paper is to present a comparative study between the performances of two sequential data assimilation techniques, which are based on Monte Carlo (MC) methods. The advantages and drawbacks of the well known ensemble Kalman Filter (EnKF) and the standard particle filter will be discussed in this study. Results show that both filters perform similar with the surface soil moisture as the estimated variable and using synthetic generated data.

A comparison of sequential assimilation schemes for ocean prediction with the HYbrid Coordinate Ocean Model (HYCOM): Twin experiments with static forecast error covariances

We assess and compare four sequential data assimilation methods developed for HYCOM in an identical twin experiment framework. The methods considered are Multi-variate Optimal Interpolation (MVOI), Ensemble Optimal Interpolation (EnOI), the fixed basis version of the Singular Evolutive Extended Kalman Filter (SEEK) and the Ensemble Reduced Order Information Filter (EnROIF). All methods can be classified as statistical interpolation but differ mainly in how the forecast error covariances are modeled. Surface elevation and temperature data sampled from an 1/12° Gulf of Mexico HYCOM simulation designated as the truth are assimilated into an identical model starting from an erroneous initial state, and convergence of assimilative runs towards the truth is tracked. Sensitivity experiments are first performed to evaluate the impact of practical implementation choices such as the state vector structure, initialization procedures, correlation scales, covariance rank and details of handling multivariate datasets, and to identify an effective configuration for each assimilation method. The performance of the methods are then compared by examining the relative convergence of the assimilative runs towards the truth. All four methods show good skill and are able to enhance consistency between the assimilative and truth runs in both observed and unobserved model variables. Prediction errors in observed variables are typically less than the errors specified for the observations, and the differences between the assimilated products are small compared to the observation errors. For unobserved variables, RMS errors are reduced by 50% relative to a non-assimilative run and differ between schemes on average by about 5%. Dynamical consistency between the updated state space variables in the data assimilation algorithm, and the data adequately sampling significant dynamical features are the two crucial components for reliable predictions. The experiments presented here suggest that practical implementation details can have at least as much an impact on the accuracy of the assimilated product as the choice of assimilation technique itself. We also present a discussion of the numerical implementation and the computational requirements for the use of these methods in large scale applications.