Data assimilation with the Ensemble Kalman Filter and the SEIK filter applied to a finite element model of the North Atlantic

Currently there are different approaches to filter algorithms based on the Kalman filter. One of the most used filter algorithms is the Ensemble Kalman Filter (EnKF). It uses a Monte Carlo approach to the filtering problem. Another approach is given by the Singular Evolutive Extended Kalman (SEEK) and Singular Evolutive Interpolated Kalman (SEIK) filters. These filters operate explicitly on a low-dimensional error space which is represented by an ensemble of model states. The EnKF and the SEIK filter have been implemented within a parallel data assimilation framework in the Finite Element Ocean Model FEOM. In order to compare the filter performances of the algorithms, several data assimilation experiments are performed. The filter algorithms have been applied with a model configuration of FEOM for the North Atlantic to assimilate the sea surface height in twin experiments. The dependence of the filter estimates on the represented error subspace is discussed. In the experiments the SEIK algorithm provides better estimates than the EnKF. Furthermore, the SEIK filter is much cheaper in terms of computing time.     

Multigrid state vector for data assimilation in a two-way nested model of the Ligurian Sea

A system of two nested models composed by a coarse resolution model of the Mediterranean Sea, an intermediate resolution model of the Provençal Basin and a high resolution model of the Ligurian Sea is coupled with a Kalman-filter based assimilation method. The state vector for the data assimilation is composed by the temperature, salinity and elevation of the three models. The forecast error is estimated by an ensemble run of 200 members by perturbing initial condition and atmospheric forcings. The 50 dominant empirical orthogonal functions (EOF) are taken as the error covariance of the model forecast. This error covariance is assumed to be constant in time. Sea surface temperature (SST) and sea surface height (SSH) are assimilated in this system.     

Improving assimilation of SeaWiFS data by the application of bias correction with a local SEIK filter

Ocean-biogeochemical models show typically significant errors in the representation of chlorophyll concentrations. The model state can be improved by the assimilation of satellite chlorophyll data with algorithms based on the Kalman filter. However, these algorithms do usually not account for the possibility that the model prediction contains systematic errors in the form of model bias. Accounting explicitly for model biases can improve the assimilation performance. To study the effect of bias estimation on the estimation of surface chlorophyll concentrations, chlorophyll data from the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) are assimilated on a daily basis into the NASA Ocean Biogeochemical Model (NOBM). The assimilation is performed by the ensemble-based SEIK filter combined with an online bias correction scheme. The SEIK filter is simplified here by the use of a static error covariance matrix. The performance of the filter algorithm is assessed by comparison with independent in situ data over the 7-year period 1998–2004. The bias correction results in significant improvements of the surface chlorophyll concentrations compared to the assimilation without bias estimation. With bias estimation, the daily surface chlorophyll estimates from the assimilation show about 3.3% lower error than SeaWiFS data. In contrast, the error in the global surface chlorophyll estimate without bias estimation is 10.9% larger than the error of SeaWiFS data.     

Wave data assimilation with the Kalman filter

A new data assimilation method for ocean waves is presented, based on an efficient low-rank approximation to the Kalman filter. Both the extended Kalman filter and a truncated second-order filter are implemented. In order to explicitly estimate past wind corrections based on current wave measurements, the filter is extended to a fixed-lag Kalman smoother for the wind fields. The filter is tested in a number of synthetic experiments with simple geometries. Propagation experiments with errors in the boundary condition showed that the KF was able to accurately propagate forecast errors, resulting in spatially varying error correlations, which would be impossible to model with time-independent assimilation methods like OI. An explicit comparison with an OI assimilation scheme showed that the KF also is superior in estimating the sea state at some distance from the observations. In experiments with errors in the driving wind, the modeled error estimates were also in agreement with the actual forecast errors. The bias in the state estimate, which is introduced through the nonlinear dependence of the waves on the driving wind field, was largely removed by the second-order filter, even without actually assimilating data. Assimilation of wave observations resulted in an improved wave analysis and in correction of past wind fields. The accuracy of this wind correction depends strongly on the actual place and time of wave generation, which is correctly modeled by the error estimate supplied by the Kalman filter. In summary, the KF approach is shown to be a reliable assimilation scheme in these simple experiments, and has the advantage over other assimilation methods that it supplies explicit dynamical error estimates.     

Application of a SEEK filter to a 1D biogeochemical model of the Ligurian Sea: Twin experiments and real in-situ data assimilation

The Singular Evolutive Extended Kalman (SEEK) filter has been implemented to assimilate in-situ data in a 1D coupled physical-ecosystem model of the Ligurian Sea. The biogeochemical model describes the partly decoupled nitrogen and carbon cycles of the pelagic food web. The GHER hydrodynamic model (1D version) is used to represent the physical forcings. The data assimilation scheme (SEEK filter) parameterizes the error statistics by means of a set of empirical orthogonal functions (EOFs). Twin experiments are first performed with the aim to choose the suitable experimental protocol (observation and estimation vectors, number of EOFs, frequency of the assimilation,…) and to assess the SEEK filter performances. This protocol is then applied to perform real data assimilation experiments using the DYFAMED data base. By assimilating phytoplankton observations, the method has allowed to improve not only the representation of the phytoplankton community, but also of other variables such as zooplankton and bacteria that evolve with model dynamics and that are not corrected by the data assimilation scheme. The validation of the assimilation method and the improvement of model results are studied by means of suitable error measurements.     

Validation of a hybrid optimal interpolation and Kalman filter scheme for sea surface temperature assimilation

A hybrid data assimilation scheme designed for operational assimilation of satellite sea surface temperatures (SST) into an ocean model has been developed and validated against in-situ observations. The scheme consists of an optimal interpolation (OI) part and a greatly simplified Kalman filter (KF) part.The OI is performed only in the longitudinal and latitudinal directions. A climatological field is used as a background field for the interpolation. It is constructed by fitting daily averages of satellite SST to the annual mean, annual, and semiannual harmonics in a 20 km by 20 km grid. The background error covariance is approximated by a spatially varying two-dimensional exponential covariance model. The parameters of the covariance model are fitted to the deviations of the satellite data from the background field using data from a full year.The simplified KF uses ocean model forecasts as a background field. It is based on the assumption that it is possible to neglect horizontal SST covariances in the filter and that the typical time scale for vertical mixing in the mixed layer is much shorter than the average time between observations. We therefore assume that the error variance in a column of water is evenly spread out throughout the mixed layer. The result of these simplifications is a computationally very efficient KF.A one year validation of the scheme is performed for year 2001 using an operational eddy resolving ocean model covering the North Sea and the Baltic Sea. It is found that assimilation of sea surface temperature data reduces the model root mean square error from 1.13 °C to 0.70 °C. The hybrid scheme is found to reduce the root mean square error slightly more than the simplified KF without OI to 0.66 °C. The inclusion of spatially varying satellite error variances does not improve the performance of the scheme significantly.     

Improving the physics of a coupled physical–biogeochemical model of the North Atlantic through data assimilation: Impact on the ecosystem

Several studies on coupled physical–biogeochemical models have shown that major deficiencies in the biogeochemical fields arise from the deficiencies in the physical flow fields. This paper examines the improvement of the physics through data assimilation, and the subsequent impact on the ecosystem response in a coupled model of the North Atlantic. Sea surface temperature and sea surface height data are assimilated with a sequential method based on the SEEK filter adapted to the coupling needs. The model domain covers the Atlantic from 20°S to 70°N at eddy-permitting resolution. The biogeochemical model is a NPZD-DOM model based on the P3ZD formulation. The results of an annual assimilated simulation are compared with an annual free simulation.With assimilation, the representation of the mixed layer depth is significantly improved in mid latitudes, even though the mixed layer depth is generally overestimated compared to the observations. The representation of the mean and variance of the currents is also significantly improved.The nutrient input in the euphotic zone is used to assess the data assimilation impact on the ecosystem. Data assimilation results in a 50% reduction of the input due to vertical mixing in mid-latitudes, and in a four- to six-fold increase of the advective fluxes in mid-latitudes and subtropics. Averaged zonally, the net impact is a threefold increase for the subtropical gyre, and a moderate (20–30%) decrease at mid and high latitudes.Surface chlorophyll concentration increases along the subtropical gyre borders, but little changes are detected at mid and high latitudes. An increase of the primary production appears along the Gulf Stream path, but it represents only 12% on average for mid and high latitudes. In the subtropical gyre centre, primary production is augmented but stays underestimated (20% of observations). These experiments show the benefits of physical data assimilation in coupled physical–biogeochemical applications.     

Assimilation of altimetric data in the mid-latitude oceans using the Singular Evolutive Extended Kalman filter with an eddy-resolving, primitive equation model

A new data assimilation scheme has been elaborated for ocean circulation models based on the concept of an evolutive, reduced-order Kalman filter. The dimension of the assimilation problem is reduced by expressing the initial error covariance matrix as a truncated series of orthogonal perturbations. This error sub-space evolves during the assimilation so as to capture the growing modes of the estimation error. The algorithm has been formulated in quite a general fashion to make it tractable with a large variety of ocean models and measurement types. In the present paper, we have examined three possible strategies to compute the evolution of the error subspace in the so-called Singular Evolutive Extended Kalman (SEEK) filter: the steady filter considers a time-independent error sub-space, the apprentice filter progressively enriches the error sub-space with the information learned from the innovation vector after each analysis step, and the dynamical filter updates the error sub-space according to the model dynamics. The SEEK filter has been implemented to assimilate synthetic observations of the surface topography in a non-linear, primitive equation model that uses density as vertical coordinate. A simplified box configuration has been adopted to simulate a Gulf Stream-like current and its associated eddies and gyres with a resolution of 20 km in the horizontal, and 4 levels in the vertical. The concept of twin experiments is used to demonstrate that the conventional SEEK filter must be complemented by a learning mechanism in order to model the misrepresented tail of the error covariance matrix. An approach based on the vertical physics of the isopycnal model, is shown particularly robust to control the velocity field in deep layers with surface observations only. The cost of the method makes it a suitable candidate for large-size assimilation problems and operational applications.     

Bayesian statistical data assimilation for ecosystem models using Markov Chain Monte Carlo

This study considers advanced statistical approaches for sequential data assimilation. These are explored in the context of nowcasting and forecasting using nonlinear differential equation based marine ecosystem models assimilating sparse and noisy non-Gaussian multivariate observations. The statistical framework uses a state space model with the goal of estimating the time evolving probability distribution of the ecosystem state. Assimilation of observations relies on stochastic dynamic prediction and Bayesian principles. In this study, a new sequential data assimilation approach is introduced based on Markov Chain Monte Carlo (MCMC). The ecosystem state is represented by an ensemble, or sample, from which distributional properties, or summary statistical measures, can be derived. The Metropolis-Hastings based MCMC approach is compared and contrasted with two other sequential data assimilation approaches: sequential importance resampling, and the (approximate) ensemble Kalman filter (including computational comparisons). A simple illustrative application is provided based on a 0-D nonlinear plankton ecosystem model with multivariate non-Gaussian observations of the ecosystem state from a coastal ocean observatory. The MCMC approach is shown to be straightforward to implement and to effectively characterize the non-Gaussian ecosystem state in both nowcast and forecast experiments. Results are reported which illustrate how non-Gaussian information originates, and how it can be used to characterize ecosystem properties.     

Gaussian anamorphosis extension of the DEnKF for combined state parameter estimation: Application to a 1D ocean ecosystem model

We consider the problem of combined state-parameter estimations in biased nonlinear models with non-Gaussian extensions of the Deterministic Ensemble Kalman Filter (DEnKF). We focus on the particular framework of ocean ecosystem models. Such models present important obstacles to the use of data assimilation methods based on Kalman filtering due to the non-linearity of the models, the constraints of positiveness that apply to the variables and parameters, and the non-Gaussian distribution of the variables in which they result.We present extensions of the DEnKF dealing with these difficulties by introducing a nonlinear change of variables (anamorphosis function) in order to execute the analysis step with Gaussian transformed variables and parameters. Several strategies to build the anamorphosis functions are investigated and compared within the framework of twin experiments realized in a simple 1D ocean ecosystem model. A solution to the problem of the specification of the observation error for transformed observations is suggested. The study highlights the inability of the plain DEnKF with a simple post-processing of the negative values to properly estimate parameters when constraints of positiveness apply to the variables. It goes on to show that the introduction of the Gaussian anamorphosis can remedy these assimilation biases.     

High-resolution synthetic monitoring by a 4-dimensional variational data assimilation system in the northwestern North Pacific

A data assimilation system is applied to the integrated monitoring of oceanic states in the northwestern North Pacific by combining a high resolution ocean general circulation model with an adjoint method. A comparison of assimilation results with observations shows that the system is better able to represent synoptic features of ocean circulation than do models or data alone. Furthermore, meso-scale features associated with frontal structures and eddies, which are often seen in the Kuroshio and Oyashio extension regions and the Sea of Japan, are better defined in the assimilation results. These features suggest that our 4D-VAR high-resolution data assimilation system is capable of providing time series data which satisfy the model physics and fit the observations, and hence the ocean state derived from our system has greater information content than that obtained from earlier methods.     

Assimilation of sea surface temperature in the POL Coastal Ocean Modelling System

A one-dimensional scheme is used to assimilate satellite Sea Surface Temperature data into the Proudman Oceanographic Laboratory Coastal Ocean Modelling System, set up in the Irish Sea with a fine resolution ( 1.8 km). The capabilities of the assimilation scheme are investigated using two different sets of satellite data, of lower and similar resolution to that of the model respectively. Comparison of results with independent data show that assimilation improves the modelled Sea Surface Temperature, but does not address model representation of the temperature vertical structure. It is concluded that for the Irish Sea and at the scales resolved by the model, the assimilation problem cannot be approached in a one-dimensional framework. It is also pointed out that forecast error needs to account explicitly for errors in the representation of the vertical structure of the thermal field.Three-dimensional methods that are suited for coastal systems are then suggested.     

Application of an Ensemble Kalman filter to a 1-D coupled hydrodynamic-ecosystem model of the Ligurian Sea

The Ensemble Kalman filter (EnKF) has been applied to a 1-D complex ecosystem model coupled with a hydrodynamic model of the Ligurian Sea. In order to improve the performance of the EnKF, an ensemble subsampling strategy has been used to better represent the covariance matrices and a pre-analysis step for correcting the non-normality of the members distribution has been implemented. Twin experiments have been realized to assess the performance of the developed tool and a real data assimilation experiment has been conducted to hindcast the ecosystem at the Dyfamed site during the year 2000. Finally the performance of the EnKF has been compared with a Singular Evolutive Extended Kalman (SEEK) filter with a fixed basis. We conclude that, on one hand, there is a benefit in using the subsampling strategy and the lognormal transformation with the EnKF, and on the other hand, this filter presents better performance than the fixed basis version of the SEEK filter. However, it also incurs a large computational cost.     

卡尔曼滤波在潜艇垂直面运动控制器设计中的应用

潜艇低噪声操纵存在无效用舵现象。简要介绍了滤波信号的表示形式,在潜艇垂直面线性运动方程的基础上,得出了滤波信号的系统模型,并对其进行了离散化处理;将均方差最小线性递推滤波算法引入到输入信号的控制过程中,通过仿真证明对输入信号进行卡尔曼滤波能够提高输入信号的品质,用于工程应用。     

基于卡尔曼滤波的磁偶极子目标张量定位

采用现有单点磁梯度张量定位方法定位磁偶极子目标时,易出现定位结果受测量噪声影响而发散的现象。针对该方法的不足,根据磁偶极子磁感应强度与磁场梯度张量的基本关系式,建立张量定位的状态空间模型,并采用卡尔曼滤波器对状态向量进行估计,实现目标定位。设计数值算例对定位方法进行检验,定位结果表明,基于卡尔曼滤波的定位方法具有较高的精度和稳定性,可用于航空磁探测和水下小目标磁场探测。     

Real time assimilation of HF radar currents into a coastal ocean model

A real time assimilation and forecasting system for coastal currents is presented. The purpose of the system is to deliver current analyses and forecasts on based on assimilation of high-frequency radar surface current measurements. The local Vessel Traffic Service monitoring the ship traffic to two oil terminals on the coast of Norway received the analyses and forecasts in real time.A new assimilation method based on optimal interpolation is presented where spatial covariances derived from an ocean model are used instead of simplified mathematical formulations. An array of high frequency radar antennae provides the current measurements. A suite of nested ocean models comprises the model system. The observing system is found to yield good analyses and short range forecasts that are significantly improved compared to a model twin without assimilation. The system is fast, analysis and 6-h forecasts are ready at the Vessel Traffic Service 45 min after acquisition of radar measurements.     

基于灰色 Markov模型的我国远洋船舶货运总量预测*

对我国远洋船舶货运总量的预测关系到我国港口建设和航运发展的一系列科学规划,也关系到远洋运输在综合交通运输中的比重和国际海上运输的资源分配。单纯的灰色模型对明显单调的序列往往预测得较好,而对波动的序列则会出现精度不够理想的问题。在结合改进后的灰色预测模型和M arkov模型的理论优势后,先用改进后的灰色模型来预测远洋船舶货运总量的总体走势,再用M arkov模型来确定各种状态间的转移方式,以此提高对波动序列的预测精度,具有较高的实用价值。     

基于实船与模型试验的船舶操纵性相关性分析

操纵性相关性分析一直以来是受到国内外学者的关注,但是由于其复杂性,至今未能很好地解决.目前,模型试验的数据往往是直接预报到实船的操纵性能,这有可能导致很大的偏差,影响船舶的航行安全.本文针对育鲲轮,进行了实船试验与模型试验的相关性分析.首先进行了实船回转试验和z形试验,测试了该船的操纵性.通过试验数据分析获得了纵距、横距、战术直径、回转直径、超越角等操纵性特征参数,试验过程中也进行了船舶姿态,如纵、横摇以及主机功率的测试.通过实船回转试验发现,主机功率比直航时增大了大约15%,这给船舶设计提供了参考.然后,采用6m长左右的模型也进行了相同试验的测试,电机转速与实船主机转速相似,且一直保持不变.最后,分析比较了模型所测数据与实船数据,发现特征参数误差基本都在10%以内.说明模型长度、螺旋桨的模拟等模型试验方案基本可行.     

Estimates of vertical mixing during a Lagrangian experiment off the Galician coast

An adjoint 1-D model was used to determine vertical diffusivity coefficients from temperature profiles collected within a filament escaping from the Galician coast following an upwelling event. The optimisation scheme ended with relatively high diffusivity values within the thermocline (9×10⁻⁵ m² s⁻¹). Such high values are relevant for biogeochemical exchanges between surface and deep waters in stratified areas.The optimised values were several orders of magnitude higher than the bulk of diffusivity measurements recorded with a free-falling device; however, the optimisation solution was consistent with the arithmetic mean of the measurements in the thermocline (7.7×10⁻⁵ m² s⁻¹), giving more weight to the few largest values. Below the thermocline, the data assimilation method failed because of the three-dimensional nature of the advective field of the upwelling system. Ignoring this advective forcing in the model led to estimates that were two orders of magnitude too high.The results suggest that turbulent mixing is a random process where a few intense events determine the average mixing that drives the long-term evolution of the water column structure. This statistical property is very important when one wants to use instantaneous diffusivity measurements for modelling purposes.     

The estimation of wave elevation and wave disturbance caused by the wave orbital motion of a fully submerged hydrofoil craft

The current control system of a fully submerged hydrofoil craft has manual input of fore-foil depth and control mode selection to improve the performance of the control system. However, the manual input needs skillful human operation and observation of waves the encountered to work well over a wide range of waves. In order to use information about the waves encountered in the control system, we considered the estimation of wave elevation and wave disturbance which was caused by the orbital motion of the waves in irregular waves. First, we investigated the wave disturbance by a fully submerged hydrofoil craft, in a state-space model of wave disturbance, and in hydrofoil craft motion, etc. We than considered estimations of the wave elevation and wave disturbance using a shaping filter, a Kalman filter, an autoregressive (AR) model, etc. Finally, we confirmed through simulations that the estimation results and estimation error of wave elevation and wave disturbance were valid.