Theoretical analyses and numerical experiments of variational assimilation for one-dimensional ocean temperature model with techniques in inverse problems

There is an international focus on the develop-ments of data assimilation systems for meteorology and physical oceanography models. Data assimilation and inverse methods are normally used for optimal control of poorly known initial boundary conditions and model parameters by taking into account both the information about dynamics of a model and the infor-mation about the true state which is constrained by a set of measurements. The research methodology of parameter estimation in meteorology ca…     

Variational assimilation of meteorological observations in the lower atmosphere: A tutorial on how it works

Data assimilation combines atmospheric measurements with knowledge of atmospheric behavior as codified in computer models, thus producing a “best” estimate of current conditions that is consistent with both information sources. The four major challenges in data assimilation are: (1) to generate an initial state for a computer forecast that has the same mass-wind balance as the assimilating model, (2) to deal with the common problem of highly non-uniform distribution of observations, (3) to exploit the value of proxy observations (of parameters that are not carried explicitly in the model), and (4) to determine the statistical error properties of observing systems and numerical model alike so as to give each information source the proper weight. Variational data assimilation is practiced at major meteorological centers around the world. It is based upon multivariate linear regression, dating back to Gauss, and variational calculus. At the heart of the method is the minimization of a cost function, which guarantees that the analyzed fields will closely resemble both the background field (a short forecast containing a priori information about the atmospheric state) and current observations. The size of the errors in the background and the observations (the latter, arising from measurement and non-representativeness) determine how close the analysis is to each basic source of information. Three-dimensional variational (3DVAR) assimilation provides a logical framework for incorporating the error information (in the form of variances and spatial covariances) and deals directly with the problem of proxy observations. 4DVAR assimilation is an extension of 3DVAR assimilation that includes the time dimension; it attempts to find an evolution of model states that most closely matches observations taken over a time interval measured in hours. Both 3DVAR and, especially, 4DVAR assimilation require very large computing resources. Researchers are trying to find more efficient numerical solutions to these problems. Variational assimilation is applicable in the upper atmosphere, but practical implementation demands accurate modeling of the physical processes that occur at high altitudes and multiple sources of observations.     

Selection of Momentum Variables for a Three-Dimensional Variational Analysis

Three choices of control variables for meteorological variational analysis (3DVAR or 4DVAR) are associated with horizontal wind: (1) streamfunction and velocity potential, (2) eastward and northward velocity, and (3) vorticity and divergence. This study shows theoretical and numerical differences of these variables in practical 3DVAR data assimilation through statistical analysis and numerical experiments. This paper demonstrates that (a) streamfunction and velocity potential could potentially introduce analysis errors; (b) A 3DVAR using velocity or vorticity and divergence provides a natural scale dependent influence radius in addition to the covariance; (c) for a regional analysis, streamfunction and velocity potential are retrieved from the background velocity field with Neumann boundary condition. Improper boundary conditions could result in further analysis errors; (d) a variational data assimilation or an inverse problem using derivatives as control variables yields smoother analyses, for example, a 3DVAR using vorticity and divergence as controls yields smoother wind analyses than those analyses obtained by a 3DVAR using either velocity or streamfunction/velocity potential as control variables; and (e) statistical errors of higher order derivatives of variables are more independent, e.g., the statistical correlation between U and V is smaller than the one between streamfunction and velocity potential, and thus the variables in higher derivatives are more appropriate for a variational system when a cross-correlation between variables is neglected for efficiency or other reasons. In summary, eastward and northward velocity, or vorticity and divergence are preferable control variables for variational systems and the former is more attractive because of its numerical efficiency. Numerical experiments are presented using analytic functions and real atmospheric observations.     

Variational data assimilation in the transport of sediment in river

The variational method of data assimilation is used to solve an inverse problem in the transport of sediment in river, which plays an important role in the change of natural environment. The cost function is defined to measure the error between model predictions and field observations. The adjoint model of IAP river sedimentation model is created to obtain the gradient of the cost function with respect to control variables. The initial conditions are taken as the control variables; their optimal values can be retrieved by minimizing the cost function with limited memory quasi-Newton method (LMQN). The results show that the adjoint method approach can successfully make the model prediction well fit the simulated observations. And it is expected to use this method to solve other inverse problems of river sedimentation. But some numerical problems need to be discussed before applying to real river data. Project partially supported by the State Key Laboratory of Numerical Modelling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences     

An explicit four-dimensional variational data assimilation method based on the proper orthogonal decomposition: Theoretics and evaluation

The proper orthogonal decomposition (POD) method is used to construct a set of basis functions for spanning the ensemble of data in a certain least squares optimal sense. Compared with the singular value decomposition (SVD), the POD basis functions can capture more energy in the forecast ensemble space and can represent its spatial structure and temporal evolution more effectively. After the analysis variables are expressed by a truncated expansion of the POD basis vectors in the ensemble space, the control variables appear explicitly in the cost function, so that the adjoint model, which is used to derive the gradient of the cost function with respect to the control variables, is no longer needed. The application of this new technique significantly simplifies the data assimilation process. Several assimilation experiments show that this POD-based explicit four-dimensional variational data assimilation method performs much better than the usual ensemble Kalman filter method on both enhancing the assimilation precision and reducing the computation cost. It is also better than the SVD-based explicit four-dimensional assimilation method, especially when the forecast model is not perfect and the forecast error comes from both the noise of the initial filed and the uncertainty of the forecast model. Supported by the National Natural Science Foundation of China (Grant No. 40705035), National High Technology Research and Development Program of China (Grant No. 2007AA12Z144), Knowledge Innovation Project of Chinese Academy of Sciences (Grant Nos. KZCX2-YW-217 and KZCX2-YW-126-2), and National Basic Research Program of China (Grant No. 2005CB321704)     

Estimating the 4DVAR analysis error of GODAE products

We explore the ocean circulation estimates obtained by assimilating observational products made available by the Global Ocean Data Assimilation Experiment (GODAE) and other sources in an incremental, four-dimensional variational data assimilation system for the Intra-Americas Sea. Estimates of the analysis error (formally, the inverse Hessian matrix) are computed during the assimilation procedure. Comparing the impact of differing sea surface height and sea surface temperature products on both the final analysis error and difference between the model state estimates, we find that assimilating GODAE and non-GODAE products yields differences between the model and observations that are comparable to the differences between the observation products themselves. While the resulting analysis error estimates depend on the configuration of the assimilation system, the basic spatial structures of the standard deviations of the ocean circulation estimates are fairly robust and reveal that the assimilation procedure is capable of reducing the circulation uncertainty when only surface data are assimilated.     

Estimation of model errors generated by atmospheric forcings for ocean data assimilation: experiments in a regional model of the Bay of Biscay

The characterization of model errors is an essential step for effective data assimilation into open-ocean and shelf-seas models. In this paper, we propose an experimental protocol to properly estimate the error statistics generated by imperfect atmospheric forcings in a regional model of the Bay of Biscay, nested in a basin-scale North Atlantic configuration. The model used is the Hybrid Coordinate Ocean Model (HYCOM), and the experimental protocol involves Monte Carlo (or ensemble) simulations. The spatial structure of the model error is analyzed using the representer technique, which allows us to anticipate the subsequent impact in data assimilation systems. The results show that the error is essentially anisotropic and inhomogeneous, affecting mainly the model layers close to the surface. Even when the forcings errors are centered around zero, a divergence is observed between the central forecast and the mean forecast of the Monte Carlo simulations as a result of nonlinearities. The 3D structure of the representers characterizes the capacity of different types of measurement (sea level, sea surface temperature, surface velocities, subsurface temperature, and salinity) to control the circulation. Finally, data assimilation experiments demonstrate the superiority of the proposed methodology for the implementation of reduced-order Kalman filters.     

The four dimensional variational data assimilation with multiple regularization parameters as a weak constraint (Tikh-4D-Var) and its preliminary application on typhoon initialization

Traditional variational data assimilation (VDA) with only one regularization parameter constraint cannot produce optimal error tuning for all observations. In this paper, a new data assimilation method of “four dimensional variational data assimilation (4D-Var) with multiple regularization parameters as a weak constraint (Tikh-4D-Var)” is proposed by imposing different regularization parameters for different observations. Meanwhile, a new multiple regularization parameters selection method, which is suitable for actual high-dimensional data assimilation system, is proposed based on the posterior information of 4D-Var system. Compared with the traditional single regularization parameter selection method, computation of the proposed multiple regularization parameters selection method is smaller. Based on WRF3.3.1 4D-Var data assimilation system, initialization and simulation of typhoon Chaba (2010) with the new Tikh-4D-Var method are compared with its counterpart 4D-Var to demonstrate the effectiveness of the new method. Results show that the new Tikh-4D-Var method can accelerate the convergence with less iterations. Moreover, compared with 4D-Var method, the typhoon track, intensity (including center surface pressure and maximum wind speed) and structure prediction are obviously improved with Tikh-4D-Var method for 72-h prediction. In addition, the accuracy of the observation error variances can be reflected by the multiple regularization parameters.     

Can intra-seasonal wind stress forcing strongly affect spring predictability barrier for ENSO in Zebiak–Cane model?

The influence of the uncertainties of intra-seasonal wind stress forcing on Spring Predictability Barrier (SPB) in El Niño–Southern Oscillation (ENSO) prediction is studied with the Zebiak–Cane model and observational wind data which are analyzed with Continuous Wavelet Transform (CWT) and utilized to extract intra-seasonal wind stress signals as external forcing. The observational intra-seasonal wind stress forcing are joined into Zebiak–Cane model to get the Zebiak–Cane-add model and subsequently with the Conditional Nonlinear Optimal Perturbation (CNOP) method, the evolutions of the optimal initial errors (i.e., CNOPs), model errors caused by intra-seasonal wind stress uncertainties, and their joint errors based on ENSO events are calculated. By investigating their error growth rates and prediction errors of Niño-3 indices, the effect of observational intra-seasonal wind stress forcing on seasonal error growth of ENSO is explored and the impact of initial error and model error on ENSO predictability is compared quantitatively. The results show that the model errors led by observational intra-seasonal wind stress forcing could scarcely cause a significant SPB whereas the initial errors and their joint errors can do; hence, the initial errors are most likely the main error source of SPB. In fact, this result emphasizes the primary influence of initial errors on ENSO predictability and lays the basis of adaptive data assimilation for ENSO forecast.     

Improvement of Monsoon Depressions Forecast with Assimilation of Indian DWR Data Using WRF-3DVAR Analysis System

An attempt is made to evaluate the impact of Doppler Weather Radar (DWR) radial velocity and reflectivity in Weather Research and Forecasting (WRF)-3D variational data assimilation (3DVAR) system for prediction of Bay of Bengal (BoB) monsoon depressions (MDs). Few numerical experiments are carried out to examine the individual impact of the DWR radial velocity and the reflectivity as well as collectively along with Global Telecommunication System (GTS) observations over the Indian monsoon region. The averaged 12 and 24 h forecast errors for wind, temperature and moisture at different pressure levels are analyzed. This evidently explains that the assimilation of radial velocity and reflectivity collectively enhanced the performance of the WRF-3DVAR system over the Indian region. After identifying the optimal combination of DWR data, this study has also investigated the impact of assimilation of Indian DWR radial velocity and reflectivity data on simulation of the four different summer MDs that occurred over BoB. For this study, three numerical experiments (control no assimilation, with GTS and GTS along with DWR) are carried out to evaluate the impact of DWR data on simulation of MDs. The results of the study indicate that the assimilation of DWR data has a positive impact on the prediction of the location, propagation and development of rain bands associated with the MDs. The simulated meteorological parameters and tracks of the MDs are reasonably improved after assimilation of DWR observations as compared to the other experiments. The root mean square errors (RMSE) of wind fields at different pressure levels, equitable skill score and frequency bias are significantly improved in the assimilation experiments mainly in DWR assimilation experiment for all MD cases. The mean Vector Displacement Errors (VDEs) are significantly decreased due to the assimilation of DWR observations as compared to the CNTL and 3DV_GTS experiments. The study clearly suggests that the performance of the model simulation for the intense convective system which influences the large scale monsoonal flow is significantly improved after assimilation of the Indian DWR data from even one coastal locale within the MDs track.     

Tidal motions in the Dover Straits as a variational inverse of the sea level and surface velocity data

The M₂ tidal component of the flow in the Dover Straits is reconstructed using a natural combination of two independent data sources: HF Ocean Surface Current Radar (HF OSCR) system and coastal tidal measurements. The method used is the variational data assimilation technique into a quasi-linearized finite element tidal model. The model uses triangular elements with horizontal resolution varying from 800 to 1200 m. It is controlled by the boundary conditions at open boundaries, which are adjusted to fit the available data in an optimal way. A “weak” formulation of the dynamical constraints is used. The interpolation scheme allows small (0.01%) deviations from the exact dynamics specified by the model. The optimal state of M₂ parameters (sea surface elevation and depth-averaged velocities) is used to map both the M₂ tidal flux through the strait and the M₂ energy flux. The respective values obtained are the tidal flux amplitude 1.18±0.09×10⁶ m³ s⁻¹, the net residual transport (Stoke's drift) 40±3×10³ m³ s⁻¹, and the net energy flux 1.19±0.09×10¹⁰ W. These figures within the statistically estimated error band are in the close agreement with those obtained by Prandle et al., 1993. A rigorous error analysis is performed using an explicit inversion of the Hessian matrix, associated with the assimilation scheme. As a result, error charts for M₂ velocities and sea surface elevation are obtained. It is shown that OSCR data combined with coastal level measurements and constrained by dynamics is able to provide quite accurate velocity estimates whose errors vary within the range of 0.05–0.45 m s⁻¹ depending upon the location. Error maps also enable us to determine areas requiring better coverage by data, thus forming a basis of optimization approach to the design of the HFR measurements. The use of variational assimilation technique in providing integrated interpolation patterns from various sources of data demonstrates its capabilities in relation to future oceanographic monitoring systems of shelf circulation.     

太湖叶绿素a同化系统敏感性分析

太湖叶绿素a同化系统对于不同参数的敏感性将直接影响到该系统能否精确的估算太湖叶绿素a的浓度分布.利用2009年4月21日环境一号卫星(HJ-1B CCD2)影像数据反演太湖叶绿素a浓度场信息.以此作为背景场信息,结合基于集合均方根滤波的太湖叶绿素a同化系统,分析和评价了样本数目、同化时长、背景场误差、观测误差和模型误差对于同化系统性能的影响.结果表明:从计算成本、系统运行时间和同化效果等方面分析,当集合样本数目达到30~40左右时同化系统取得了较好的结果;同化系统对于背景场误差的估计变化不是很敏感,即初始场的估计是否准确对于同化系统的性能影响不是很大;同化系统对于模型误差和观测误差的变化较为敏感,不同的测试点位由于水体动力学性质不一,其敏感性的表现形式有所差异;利用数据同化方法可以有效地估算太湖叶绿素a浓度.     

An explicit four-dimensional variational data assimilation method

A new data assimilation method called the explicit four-dimensional variational (4DVAR) method is proposed. In this method, the singular value decomposition (SVD) is used to construct the orthogonal basis vectors from a forecast ensemble in a 4D space. The basis vectors represent not only the spatial structure of the analysis variables but also the temporal evolution. After the analysis variables are ex-pressed by a truncated expansion of the basis vectors in the 4D space, the control variables in the cost function appear explicitly, so that the adjoint model, which is used to derive the gradient of cost func-tion with respect to the control variables, is no longer needed. The new technique significantly simpli-fies the data assimilation process. The advantage of the proposed method is demonstrated by several experiments using a shallow water numerical model and the results are compared with those of the conventional 4DVAR. It is shown that when the observation points are very dense, the conventional 4DVAR is better than the proposed method. However, when the observation points are sparse, the proposed method performs better. The sensitivity of the proposed method with respect to errors in the observations and the numerical model is lower than that of the conventional method.     

A Genetic Algorithm Variational Approach to Data Assimilation and Application to Volcanic Emissions

Variational data assimilation methods optimize the match between an observed and a predicted field. These methods normally require information on error variances of both the analysis and the observations, which are sometimes difficult to obtain for transport and dispersion problems. Here, the variational problem is set up as a minimization problem that directly minimizes the root mean squared error of the difference between the observations and the prediction. In the context of atmospheric transport and dispersion, the solution of this optimization problem requires a robust technique. A genetic algorithm (GA) is used here for that solution, forming the GA-Variational (GA-Var) technique. The philosophy and formulation of the technique is described here. An advantage of the technique includes that it does not require observation or analysis error covariances nor information about any variables that are not directly assimilated. It can be employed in the context of either a forward assimilation problem or used to retrieve unknown source or meteorological information by solving the inverse problem. The details of the method are reviewed. As an example application, GA-Var is demonstrated for predicting the plume from a volcanic eruption. First the technique is employed to retrieve the unknown emission rate and the steering winds of the volcanic plume. Then that information is assimilated into a forward prediction of its transport and dispersion. Concentration data are derived from satellite data to determine the observed ash concentrations. A case study is made of the March 2009 eruption of Mount Redoubt in Alaska. The GA-Var technique is able to determine a wind speed and direction that matches the observations well and a reasonable emission rate.     

Data assimilation with the EnKF in a 1-D numerical model of a North Sea station

A series of numerical experiments for data assimilation with the Ensemble Kalman Filter (EnKF) in a shallow water model are reported. Temperature profiles measured at a North Sea location, 55°30ˊ North and 0°55ˊ East (referred to as the CS station of the NERC North Sea project), are assimilated in 1-D simulations. Comparison of simulations without assimilation to model results obtained when assimilating data with the EnKF allows us to assess the filter performance in reproducing features of the observations not accounted for by the model. The quality of the model error sampling is tested as well as the validity of the Gaussian hypothesis underlying the analysis scheme of the EnKF. The influence of the model error parameters and the frequency of the data assimilation are investigated and discussed. From these experiments, a set of optimal parameters for the model error sampling are deduced and used to test the behavior of the EnKF when propagating surface information into the water column.     

Assimilation of sea surface temperature into CECOM by flux correction

Sea surface temperature (SST) from a near real-time data set produced from satellites data has been assimilated into a coupled ice–ocean forecasting model (Canadian East Coast Ocean Model) using an efficient data assimilation method. The method is based on an optimal interpolation scheme by which SST is melded into the model through the adjustment of surface heat flux. The magnitude and space–time variation of the adjustment depend on the depth of heat diffusion into the water column in response to changes in surface flux, the correlation time scale of the data, and model and data errors. The diffusion depth is scaled by the eddy diffusivity for temperature. The ratio of the model and data errors is treated as an adjustable parameter. To evaluate the quality of the assimilation, the results from the model with and without assimilation are compared to independent ship data from the Atlantic Zone Monitoring Program and the World Ocean Circulation Experiment. It is shown that the assimilation has a significant impact on the modeled SST, reducing the root mean square difference (RMSD) between the model SST and the ship SST by 0.63°C or 37%. The RMSD of the assimilated SST is smaller than that of the satellite SST by 0.23°C. This suggests that model simulations or predictions with data assimilation can provide the best estimate of the true SST. A sensitivity study is performed to examine the change of the model RMSD with the adjustable parameter in the assimilation equation. The results show that there is an optimal value of the parameter and the model SST is not very sensitive to the parameter.     

Comparison of reduced-order, sequential and variational data assimilation methods in the tropical Pacific Ocean

This paper presents a comparison of two reduced-order, sequential, and variational data assimilation methods: the singular evolutive extended Kalman filter (SEEK) and the reduced 4D-Var (R-4D-Var). A hybridization of the two, combining the variational framework and the sequential evolution of covariance matrices, is also preliminarily investigated and assessed in the same experimental conditions. The comparison is performed using the twin-experiment approach on a model of the tropical Pacific domain. The assimilated data are simulated temperature profiles at the locations of the TAO/TRITON array moorings. It is shown that, in a quasilinear regime, both methods produce similarly good results. However, the hybrid approach provides slightly better results and thus appears as potentially fruitful. In a more nonlinear regime, when tropical instability waves develop, the global nature of the variational approach helps control model dynamics better than the sequential approach of the SEEK filter. This aspect is probably enhanced by the context of the experiments in that there is a limited amount of assimilated data and no model error.     

Terrestrial carbon cycle model-data fusion: Progress and challenges

The terrestrial carbon cycle is an important component of global biogeochemical cycling and is closely related to human well-being and sustainable development. However, large uncertainties exist in carbon cycle simulations and observations.Model-data fusion is a powerful technique that combines models and observational data to minimize the uncertainties in terrestrial carbon cycle estimation. In this paper, we comprehensively overview the sources and characteristics of the uncertainties in terrestrial carbon cycle models and observations. We present the mathematical principles of two model-data fusion methods, i.e., data assimilation and parameter estimation, both of which essentially achieve the optimal fusion of a model with observational data while considering the respective errors in the model and in the observations. Based upon reviewing the progress in carbon cycle models and observation techniques in recent years, we have highlighted the major challenges in terrestrial carbon cycle model-data fusion research, such as the "equifinality" of models, the identifiability of model parameters,the estimation of representativeness errors in surface fluxes and remote sensing observations, the potential role of the posterior probability distribution of parameters obtained from sensitivity analysis in determining the error covariance matrixes of the models, and opportunities that emerge by assimilating new remote sensing observations, such as solar-induced chlorophyll fluorescence. It is also noted that the synthesis of multisource observations into a coherent carbon data assimilation system is by no means an easy task, yet a breakthrough in this bottleneck is a prerequisite for the development of a new generation of global carbon data assimilation systems. This article also highlights the importance of carbon cycle data assimilation systems to generate reliable and physically consistent terrestrial carbon cycle reanalysis data products with high spatial resolution and longterm time series. These products are critical to the accurate estimation of carbon cycles at the global and regional scales and will help future carbon management strategies meet the goals of carbon neutrality.