Cycling the representer algorithm for variational data assimilation with a nonlinear reduced gravity ocean model

The representer method was used by [Ngodock, H.E., Jacobs, G.A., Chen, M., 2006. The representer method, the ensemble Kalman filter and the ensemble Kalman smoother: a comparison study using a nonlinear reduced gravity ocean model. Ocean Modelling 12, 378–400] in a comparison study with the ensemble Kalman filter and smoother involving a 1.5 nonlinear reduced gravity idealized ocean model simulating the Loop Current (LC) and the Loop Current eddies (LCE) in the Gulf of Mexico. It was reported that the representer method was more accurate than its ensemble counterparts, yet it had difficulties fitting the data in the last month of the 4-month assimilation window when the data density was significantly decreased. The authors attributed this failure to increased advective nonlinearities in the presence of an eddy shedding causing the tangent linear model (TLM) to become inaccurate. In a separate study [Ngodock, H.E., Smith, S.R., Jacobs, G.A., 2007. Cycling the representer algorithm for variational data assimilation with the Lorenz attractor. Monthly Weather Review 135 (2), 373–386] applied the cycling representer algorithm to the Lorenz attractor and demonstrated that the cycling solution was able to accurately fit the data within each cycle and beyond the range of accuracy of the TLM, once adjustments were made in the early cycles, thus overcoming the difficulties of the non-cycling solution. The cycling algorithm is used here in assimilation experiments with the nonlinear reduced gravity model. It is shown that the cycling solution overcomes the difficulties encountered by the non-cycling solution due to a limited time range of accuracy of the TLM. Thus, for variational assimilation applications where the TLM accuracy is limited in time, the cycling representer becomes a very powerful and attractive alternative, given that its computational cost is significantly lower than that of the non-cycling algorithm.     

Cycling the representer algorithm for variational data assimilation with a nonlinear reduced gravity ocean model

The representer method was used by [Ngodock, H.E., Jacobs, G.A., Chen, M., 2006. The representer method, the ensemble Kalman filter and the ensemble Kalman smoother: a comparison study using a nonlinear reduced gravity ocean model. Ocean Modelling 12, 378–400] in a comparison study with the ensemble Kalman filter and smoother involving a 1.5 nonlinear reduced gravity idealized ocean model simulating the Loop Current (LC) and the Loop Current eddies (LCE) in the Gulf of Mexico. It was reported that the representer method was more accurate than its ensemble counterparts, yet it had difficulties fitting the data in the last month of the 4-month assimilation window when the data density was significantly decreased. The authors attributed this failure to increased advective nonlinearities in the presence of an eddy shedding causing the tangent linear model (TLM) to become inaccurate. In a separate study [Ngodock, H.E., Smith, S.R., Jacobs, G.A., 2007. Cycling the representer algorithm for variational data assimilation with the Lorenz attractor. Monthly Weather Review 135 (2), 373–386] applied the cycling representer algorithm to the Lorenz attractor and demonstrated that the cycling solution was able to accurately fit the data within each cycle and beyond the range of accuracy of the TLM, once adjustments were made in the early cycles, thus overcoming the difficulties of the non-cycling solution. The cycling algorithm is used here in assimilation experiments with the nonlinear reduced gravity model. It is shown that the cycling solution overcomes the difficulties encountered by the non-cycling solution due to a limited time range of accuracy of the TLM. Thus, for variational assimilation applications where the TLM accuracy is limited in time, the cycling representer becomes a very powerful and attractive alternative, given that its computational cost is significantly lower than that of the non-cycling algorithm.     

Ocean current estimation using a Multi-Model Ensemble Kalman Filter during the Grand Lagrangian Deployment experiment (GLAD)

In the summer and fall of 2012, during the GLAD experiment in the Gulf of Mexico, the Consortium for Advanced Research on Transport of Hydrocarbon in the Environment (CARTHE) used several ocean models to assist the deployment of more than 300 surface drifters. The Navy Coastal Ocean Model (NCOM) at 1 km and 3 km resolutions, the US Navy operational NCOM at 3 km resolution (AMSEAS), and two versions of the Hybrid Coordinates Ocean Model (HYCOM) set at 4 km were running daily and delivering 72-h range forecasts. They all assimilated remote sensing and local profile data but they were not assimilating the drifter’s observations. This work presents a non-intrusive methodology named Multi-Model Ensemble Kalman Filter that allows assimilating the local drifter data into such a set of models, to produce improved ocean currents forecasts. The filter is to be used when several modeling systems or ensembles are available and/or observations are not entirely handled by the operational data assimilation process. It allows using generic in situ measurements over short time windows to improve the predictability of local ocean dynamics and associated high-resolution parameters of interest for which a forward model exists (e.g. oil spill plumes). Results can be used for operational applications or to derive enhanced background fields for other data assimilation systems, thus providing an expedite method to non-intrusively assimilate local observations of variables with complex operators. Results for the GLAD experiment show the method can improve water velocity predictions along the observed drifter trajectories, hence enhancing the skills of the models to predict individual trajectories.     

Weak and strong constraint data assimilation in the inverse Regional Ocean Modeling System (ROMS): Development and application for a baroclinic coastal upwelling system

We describe the development and preliminary application of the inverse Regional Ocean Modeling System (ROMS), a four dimensional variational (4DVAR) data assimilation system for high-resolution basin-wide and coastal oceanic flows. Inverse ROMS makes use of the recently developed perturbation tangent linear (TL), representer tangent linear (RP) and adjoint (AD) models to implement an indirect representer-based generalized inverse modeling system. This modeling framework is modular. The TL, RP and AD models are used as stand-alone sub-models within the Inverse Ocean Modeling (IOM) system described in [Chua, B.S., Bennett, A.F., 2001. An inverse ocean modeling system. Ocean Modell. 3, 137–165.]. The system allows the assimilation of a wide range of observation types and uses an iterative algorithm to solve nonlinear assimilation problems. The assimilation is performed either under the perfect model assumption (strong constraint) or by also allowing for errors in the model dynamics (weak constraints). For the weak constraint case the TL and RP models are modified to include additional forcing terms on the right hand side of the model equations. These terms are needed to account for errors in the model dynamics.Inverse ROMS is tested in a realistic 3D baroclinic upwelling system with complex bottom topography, characterized by strong mesoscale eddy variability. We assimilate synthetic data for upper ocean (0–450 m) temperatures and currents over a period of 10 days using both a high resolution and a spatially and temporally aliased sampling array. During the assimilation period the flow field undergoes substantial changes from the initial state. This allows the inverse solution to extract the dynamically active information from the synthetic observations and improve the trajectory of the model state beyond the assimilation window. Both the strong and weak constraint assimilation experiments show forecast skill greater than persistence and climatology during the 10–20 days after the last observation is assimilated.Further investigation in the functional form of the model error covariance and in the use of the representer tangent linear model may lead to improvement in the forecast skill.     

Asynchronous data assimilation with the EnKF

This study revisits the problem of assimilation of asynchronous observations, or four-dimensional data assimilation, with the ensemble Kalman filter (EnKF). We show that for a system with perfect model and linear dynamics the ensemble Kalman smoother (EnKS) provides a simple and efficient solution for the problem: one just needs to use the ensemble observations (that is, the forecast observations for each ensemble member) from the time of observation during the update, for each assimilated observation. This recipe can be used for assimilating both past and future data; in the context of assimilating generic asynchronous observations we refer to it as the asynchronous EnKF. The asynchronous EnKF is essentially equivalent to the four-dimensional variational data assimilation (4D-Var). It requires only one forward integration of the system to obtain and store the data necessary for the analysis, and therefore is feasible for large-scale applications. Unlike 4D-Var, the asynchronous EnKF requires no tangent linear or adjoint model.     

The representer method, the ensemble Kalman filter and the ensemble Kalman smoother: A comparison study using a nonlinear reduced gravity ocean model

This paper compares contending advanced data assimilation algorithms using the same dynamical model and measurements. Assimilation experiments use the ensemble Kalman filter (EnKF), the ensemble Kalman smoother (EnKS) and the representer method involving a nonlinear model and synthetic measurements of a mesoscale eddy. Twin model experiments provide the “truth” and assimilated state. The difference between truth and assimilation state is a mispositioning of an eddy in the initial state affected by a temporal shift. The systems are constructed to represent the dynamics, error covariances and data density as similarly as possible, though because of the differing assumptions in the system derivations subtle differences do occur. The results reflect some of these differences in the tangent linear assumption made in the representer adjoint and the temporal covariance of the EnKF, which does not correct initial condition errors. These differences are assessed through the accuracy of each method as a function of measurement density. Results indicate that these methods are comparably accurate for sufficiently dense measurement networks; and each is able to correct the position of a purposefully misplaced mesoscale eddy. As measurement density is decreased, the EnKS and the representer method retain accuracy longer than the EnKF. While the representer method is more accurate than the sequential methods within the time period covered by the observations (particularly during the first part of the assimilation time), the representer method is less accurate during later times and during the forecast time period for sparse networks as the tangent linear assumption becomes less accurate. Furthermore, the representer method proves to be significantly more costly (2–4 times) than the EnKS and EnKF even with only a few outer iterations of the iterated indirect representer method.     

Validation of interannual simulations from the 1/8° global Navy Coastal Ocean Model (NCOM)

A 1/8° global version of the Navy Coastal Ocean Model (NCOM) is used for simulation of upper-ocean quantities on interannual time scales. The model spans the global ocean from 80°S to a complete Arctic cap, and includes 19 terrain-following σ- and 21 fixed z-levels. The global NCOM assimilates three-dimensional (3D) temperature and salinity fields produced by the Modular Ocean Data Assimilation System (MODAS) which generates synthetic temperature and salinity profiles based on ocean surface observations. Model-data intercomparisons are performed to measure the effectiveness of NCOM in predicting upper-ocean quantities such as sea surface temperature (SST), sea surface salinity (SSS) and mixed layer depth (MLD). Subsurface temperature and salinity are evaluated as well. An extensive set of buoy observations is used for this validation. Where possible, the model validation is performed between year-long time series obtained from the model and time series from the buoys. The statistical analyses include the calculation of dimensionless skill scores (SS), which are positive if statistical skill is shown and equal to one for perfect SST simulations. Model SST comparisons with year-long SST time series from all 83 buoys give a median SS value of 0.82. Model subsurface temperature comparisons with the year-long subsurface temperature time series from 24 buoys showed that the model is able to predict temperatures down to 500 m reasonably well, with positive SS values ranging from 0.18 to 0.97. Intercomparisons of MLD reveal that the model MLD is usually shallower than the buoy MLD by an average of about 15 m. Annual mean SSS and subsurface salinity biases between the model and buoy values are small. A comparison of SST between NCOM and a satellite-based Pathfinder data set demonstrates that the model has a root-mean-square (RMS) SST difference of 0.61 °C over the global ocean. Spatial variations of kinetic energy fields from NCOM show agree with historical observations. Based on these results, it is concluded that the global NCOM presented in this paper is able to predict upper-ocean quantities with reasonable accuracy for both coastal and open ocean locations.     

一维扩散方程整体观测资料下的初值变分同化反演

本文考虑一维扩散方程的反问题,利用变分同化方法通过观测资料来确定方程中的未知初值,通过分析观测误差对于初值误差的影响,证明变分同化初值收敛于原问题的真实参数,并得到了参数的收敛精度。同时将得到的初值代入预报模式中,得到预报解,并分析了预报解的收敛性和预报误差。     

Numerical model of the hydrodynamics of the Black Sea and the Sea of Azov with variational initialization of temperature and salinity

A numerical primitive-equation model of the hydrodynamics of the Black Sea and the Sea of Azov in σ-coordinates is proposed. The model has a resolution of ～4 × 4 km in horizontal coordinates with 40-σ levels in the vertical and includes the four-dimensional variational initialization of temperature and salinity fields. A numerical initialization algorithm combines splitting methods and adjoint equations. Flow, temperature, sea level, and salinity fields driven by atmospheric forcing are calculated for the year 2008. The calculations are made in a variational initialization — prediction regime. Temperature and salinity fields are initialized at the end of each month. The optimality system includes forward and adjoint transport-diffusion equations for heat and salt that are linearized on the assimilation interval. Results of three numerical experiments with different sets of assimilated data in comparison with the prediction obtained from the forward model are discussed.     

Statistical assessment of ocean observing networks – A study of water level measurements in the German Bight

A set of tools for the statistical assessment of ocean observing networks is presented and applied for the analysis of different instrumentation scenarios in the German Bight. An optimal linear estimator is used to re-construct ocean state parameters from observations taking into account both the prior distribution of the state and measurement errors. The proposed method enables a re-construction of any scalar parameter or vector field with linear relationship to the state. The performance of the observing network is quantified in terms of the re-construction quality. Apart from the capability of the network to provide estimates of state parameters at the time of the observations, the potential of the measurements for forecasts is investigated as well. Furthermore, a generic method to compare single measurements with continuous observations is presented. Finally, a technique is described to quantify the relative importance of different components of an observational network.The proposed methods are applied to water level measurements in the German Bight. A numerical model is used to estimate the background statistics. Synthetic measurements provided by tide gauges, satellite altimeters, and HF radar are considered in the analysis. The estimation of the complete water level field in the German Bight is compared for altimeter and tide gauge measurements. It is shown that the orientation of the satellite track with respect to the coastline is of high relevance. The importance of water level measurements taken in deeper water, e.g., at the FINO-1 platform, is demonstrated. It is shown that continuous tide gauge measurements provide more information on the area mean water level in the German Bight than altimeter observations taken by ENVISAT and JASON-1/2. It is furthermore shown how the information provided by a tide gauge propagates with the Kelvin wave. Implications for the design of an assimilation scheme are discussed.     

卫星高度计波高数据同化对西北太平洋海浪数值预报的影响评估

通过数值试验验证卫星高度计波高数据同化对西北太平洋3 d海浪预报的改进效果。驱动海浪模式的强迫场采用国家海洋环境预报中心基于MM5模式预报的风场,波高数据同化使用的观测数据是Jason-1卫星高度计有效波高。用最优插值数据同化方法获得海浪有效波高的最优估计并重构相应的海浪方向谱,以此为初始场进行为期3 d的数值预报试验。与没有同化的预报进行了比较和分析,结果表明卫星高度计海浪数据同化对0~72 h预报有不同程度的明显改善,改进程度随预报时效的增加而减少。     

基于地球系统模式的局地化粒子滤波器与集合卡尔曼滤波器同化实验

粒子滤波器(PF)是一种非常具有应用前景的非线性资料同化方法。但由于其算法本身存在的粒子退化问题,目前尚未被广泛地应用于大型地球物理模式。目前主流的集合同化系统仍然倾向于使用集合卡尔曼滤波器(EnKF)及其衍生方法。一种新近被提出的局地化粒子滤波器(LPF)在经典的粒子滤波器算法中引入局地化技术,可以使用较小的计算成本有效地避免退化问题,具有非常大的业务应用潜力。本文在全耦合的通用地球系统模式中开展了LPF和EnKF的同化实验,同化资料为模拟的卫星海表温度资料。着重考察了不同局地化参数对两种方法的不同影响,对比了局地化粒子滤波器与集合卡尔曼滤波器的同化效果差异。比较的结果表明,LPF的同化效果对于局地化参数的选择非常敏感,在使用最优局地化参数的条件下,LPF能达到与EnKF相当甚至优于后者的同化效果,并具有较大的改进空间。     

Temporal evolution of innovation and residual statistics in the ECMWF variational data assimilation systems

The temporal evolution of innovation and residual statistics of the ECMWF 3D‐ and 4D‐Var data assimilation systems have been studied. First, the observational method is applied on an hourly basis to the innovation sequences in order to partition the perceived forecast error covariance into contributions from observation and background errors. The 4D‐Var background turns out to be ignificantly more accurate than the background in the 3D‐Var. The estimated forecast error variance associated with the 4D‐Var background trajectory increases over the assimilation window. There is also a marked broadening of the horizontal error covariance length scale over the assimilation window. Second, the standard deviation of the residuals, i.e., the fit of observations to the analysis is studied on an hourly basis over the assimilation window. This fit should, in theory, reveal the effect of model error in a strong constraint variational problem. A weakly convex curve is found for this fit implying that the perfect model assumption of 4D‐Var may be violated with as short an assimilation window as six hours. For improving the optimality of variational data assimilation systems, a sequence of retunes are needed, until the specified and diagnosed error covariances agree.     

4DVAR data assimilation in the Intra-Americas Sea with the Regional Ocean Modeling System (ROMS)

We present the background, development, and preparation of a state-of-the-art 4D variational (4DVAR) data assimilation system in the Regional Ocean Modeling System (ROMS) with an application in the Intra-Americas Sea (IAS). This initial application with a coarse model shows the efficacy of the 4DVAR methodology for use within complex ocean environments, and serves as preparation for deploying an operational, real-time assimilation system onboard the Royal Caribbean Cruise Lines ship Explorer of the Seas. Assimilating satellite sea surface height and temperature observations with in situ data from the ship in 14 day cycles over 2 years from January 2005 through March 2007, reduces the observation-model misfit by over 75%. Using measures of the Loop Current dynamics, we show that the assimilated solution is consistent with observed statistics.     

4DVAR data assimilation in the Intra-Americas Sea with the Regional Ocean Modeling System (ROMS)

We present the background, development, and preparation of a state-of-the-art 4D variational (4DVAR) data assimilation system in the Regional Ocean Modeling System (ROMS) with an application in the Intra-Americas Sea (IAS). This initial application with a coarse model shows the efficacy of the 4DVAR methodology for use within complex ocean environments, and serves as preparation for deploying an operational, real-time assimilation system onboard the Royal Caribbean Cruise Lines ship Explorer of the Seas. Assimilating satellite sea surface height and temperature observations with in situ data from the ship in 14 day cycles over 2 years from January 2005 through March 2007, reduces the observation-model misfit by over 75%. Using measures of the Loop Current dynamics, we show that the assimilated solution is consistent with observed statistics.     

基于四维变分同化方法的南海中尺度涡后报实验

海洋中尺度涡在本质上是属于满足准地转平衡的大尺度运动,因此理论上,其在短时间内的运动将主要受到准地转平衡关系的约束,而外部强迫场的影响在短期内不会明显改变其运动特征。基于上述思想,我们提出了一种基于四维变分同化初始场的中尺度涡旋预报方案。为了检验该方案的可行性,本文使用区域海洋模式(regional ocean modeling system, ROMS)和其内建的增量强约束四维变分同化(incremental strong constraint four dimensional variational, I4D-Var)模块,建立了一个南海海洋同化模拟系统。首先,通过I4D-Var方法将AVISO卫星高度计资料同化到海洋数值模拟中,获得了理想的中尺度涡同化模拟结果。同化、模式模拟和观测三者的中尺度涡统计结果表明,该同化系统模拟的南海中尺度涡的路径、半径、海表高度异常和振幅等特征信息与AVISO(Archiving ValidationandInterpolationofSatelliteOceanographicData)观测结果高度吻合,同时在深度上的分析表明,涡旋对应的温度、盐度和密度均得到有效的调整。然后,将该同化系统的模拟结果做为初始场,对某一特定时段的南海中尺度涡进行了后报模拟和结果的定量化分析。通过比较后报模拟与观测资料中对应涡旋的海表面高度异常(sea surface height anomalies, SSHA)相关系数、涡心差距和半径绝对误差,证明该方案的中尺度涡后报时效至少可达10 d以上。后报实验结果验证了该中尺度涡预报方案的可行性,从而为中尺度涡的预报提供一定的理论基础和可行性方案。     

变分同化方法反演海气耦合模型参数的研究

采用变分资料同化技术,结合最优控制思想,对一个海气耦合模型的模式参数和强迫项进行了反演,结果表明,采用该方法对模式进行优化,既可以补偿模式参数不准确性给预报带来的误差,又可以对模式参数本身进行修正和估计,为将来在实际应用中改善更复杂的预报模式、提高预报准确率提供了一个可借鉴的思路。     

上海邻近海域风暴潮数据同化与特征分析

风暴潮是一种复杂的对众多因素敏感又备受关注的海洋现象.本文基于协方差局地化的集合卡尔曼滤波方法(EnKF),选择201810号台风"安比"登陆上海的风暴潮过程,首次将海洋站和FVCOM数值模拟的不同来源、不同误差信息、不同时空分辨率的风暴潮进行数据同化融合,获得了逐72 h的上海海域风暴潮的最优解,进行了同化结果评估验...     

Kalman滤波风暴潮数值预报四维同化模式研究进展

本文首先介绍了Kalman滤波在风暴潮数值预报中的应用，特别介绍了近年来国际上发展的一些在实际中可行的次优化Kalman滤波算法。并通过一个稳态Kalman滤波风暴潮数值预报模式的实例表明，使用资料同化可以明显改进风暴潮后报结果；资料同化能够提供更为合理的预报初始场，对风暴潮的短期预报有较明显的改进。一旦没有资料同化到模式中去，预报结果很快接近确定性模式。     

Sensitivity of large-scale atmospheric analyses to humidity observations and its impact on the global water cycle and tropical and extratropical weather systems in ERA40

Reanalysis data obtained from data assimilation are increasingly used for diagnostic studies of the general circulation of the atmosphere, for the validation of modelling experiments and for estimating energy and water fluxes between the Earth surface and the atmosphere. Because fluxes are not specifically observed, but determined by the data assimilation system, they are not only influenced by the utilized observations but also by model physics and dynamics and by the assimilation method. In order to better understand the relative importance of humidity observations for the determination of the hydrological cycle, in this paper we describe an assimilation experiment using the ERA40 reanalysis system where all humidity data have been excluded from the observational data base. The surprising result is that the model, driven by the time evolution of wind, temperature and surface pressure, is able to almost completely reconstitute the large-scale hydrological cycle of the control assimilation without the use of any humidity data. In addition, analysis of the individual weather systems in the extratropics and tropics using an objective feature tracking analysis indicates that the humidity data have very little impact on these systems. We include a discussion of these results and possible consequences for the way moisture information is assimilated, as well as the potential consequences for the design of observing systems for climate monitoring. It is further suggested, with support from a simple assimilation study with another model, that model physics and dynamics play a decisive role for the hydrological cycle, stressing the need to better understand these aspects of model parametrization.