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.     

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.     

Numerical modeling of ocean hydrodynamics with variational assimilation of observational data

Models and methods of the numerical modeling of ocean hydrodynamics dating back to the pioneering works of A.S. Sarkisyan are considered, with emphasis on the formulation of problems and algorithms of mathematical modeling and the four-dimensional variational assimilation of observational data. An algorithm is proposed for studying the sensitivity of the optimal solution to observational data errors in a seasurface temperature assimilation problem in order to retrieve heat fluxes on the surface. An example of a solution of the optimal problem of the World Ocean hydrodynamics with the assimilation of climatic temperature and salinity observations is offered.     

A reduced-dynamics variational approach for the assimilation of altimeter data into eddy-resolving ocean models

A new method of assimilating sea surface height (SSH) data into ocean models is introduced and tested. Many features observable by satellite altimetry are approximated by the first baroclinic mode over much of the ocean, especially in the lower (but non-equatorial) and mid latitude regions. Based on this dynamical trait, a reduced-dynamics adjoint technique is developed and implemented with a three-dimensional model using vertical normal mode decomposition. To reduce the complexity of the variational data assimilation problem, the adjoint equations are based on a one-active-layer reduced-gravity model, which approximates the first baroclinic mode, as opposed to the full three-dimensional model equations. The reduced dimensionality of the adjoint model leads to lower computational cost than a traditional variational data assimilation algorithm. The technique is applicable to regions of the ocean where the SSH variability is dominated by the first baroclinic mode. The adjustment of the first baroclinic mode model fields dynamically transfers the SSH information to the deep ocean layers. The technique is developed in a modular fashion that can be readily implemented with many three-dimensional ocean models. For this study, the method is tested with the Navy Coastal Ocean Model (NCOM) configured to simulate the Gulf of Mexico.     

Analysis of the response of a non-linear data assimilation algorithm in a one-dimensional model of the upper ocean

A one-dimensional (1D) integral dynamico-stochastic model of the upper ocean with a non-linear assimilation algorithm is considered. The accuracy of computing the characteristics of the upper layer depends essentially on the values of the empirical coefficients. A numerical experiment was carried out which verified the efficiency of the model's adaptive mechanism operation when different values of the empirical coefficients and their variances were preset. Recommendations on assigment of the model's initial parameters are derived.Translated by Mikhail M. Trufanov.     

Towards a true 4‐dimensional data assimilation algorithm: application of a cycling representer algorithm to a simple transport problem

3‐dimensional variational algorithms are widely used for atmospheric data assimilation at the present time, particularly on the synoptic and global scales. However, mesoscale and convective scale phenomena are considerably more chaotic and intermittent and it is clear that true 4‐dimensional data assimilation algorithms will be required to properly analyze these phenomena. In its most general form, the data assimilation problem can be posed as the minimization of a 4‐dimensional cost function with the forecast model as a weak constraint. This is a much more difficult problem than the widely discussed 4DVAR algorithm where the model is a strong constraint. Bennett and collaborators have considered a method of solution to the weak constraint problem, based on representer theory. However, their method is not suitable for the numerical weather prediction problem, because it does not cycle in time. In this paper, the representer method is modified to permit cycling in time, in a manner which is entirely internally consistent. The method was applied to a simple 1‐dimensional constituent transport problem where the signal was sampled (perfectly and imperfectly) with various sparse observation network configurations. The cycling representer algorithm discussed here successfully extracted the signal from the noisy, sparse observations     

Comparison of methods for argo drifters data assimilation into a hydrodynamical model of the ocean

Different data assimilation methods such as an extended Kalman filter, the optimal interpolation method, and a method based on the Fokker-Planck equation applications are considered. Data from the ARGO drifters are assimilated into the HYCOM shallow water model (University of Miami, USA). Throughout the study, the schemes and methods of parallel computations with an MPI library are used. The results of the computations with assimilations are compared between themselves and with independent observations. The method based on the Fokker-Planck equation and the extended Kalman filter are preferable because they give better results than the optimal interpolation scheme. The various model characteristics of the ocean, such as the heat content fields and others, are analyzed after the data assimilation.     

Problems of variational assimilation of observational data for ocean general circulation models and methods for their solution

Problems of the variational assimilation of satellite observational data on the temperature and level of the ocean surface, as well as data on the temperature and salinity of the ocean from the ARGO system of buoys, are formulated with the use of the global three-dimensional model of ocean thermodynamics developed at the Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS). Algorithms for numerical solutions of the problems are developed and substantiated, and data assimilation blocks are developed and incorporated into the global three-dimensional model. Numerical experiments are performed with the use of the Indian Ocean or the entire World Ocean as examples. These numerical experiments support the theoretical conclusions and demonstrate that the use of a model with an assimilation block of operational observational data is expedient.     

Corrections to ocean surface forcing in the California Current System using 4D variational data assimilation

The option for surface forcing correction, recently developed in the 4D-variational (4DVAR) data assimilation systems of the Regional Ocean Model System (ROMS), is presented. Assimilation of remotely-sensed (satellite sea surface height anomaly and sea surface temperature) and in situ (from mechanical and expendable bathythermographs, Argo floats and CTD profiles) oceanic observations has been applied in a realistic, high resolution configuration of the California Current System (CCS) to sequentially correct model initial conditions and surface forcing, using the Incremental Strong constraint version of ROMS-4DVAR (ROMS-IS4DVAR). Results from both twin and real data experiments are presented where it is demonstrated that ROMS-IS4DVAR always reduces the difference between the model and the observations that are assimilated. However, without corrections to the surface forcing, the assimilation of surface data can degrade the temperature structure at depth. When using surface forcing adjustment in ROMS-IS4DVAR the system does not degrade the temperature structure at depth, because differences between the model and surface observations can be reduced through corrections to surface forcing rather than to temperature at depth. However, corrections to surface forcing can generate abnormal spatial and temporal variability in the structure of the wind stress or surface heat flux fields if not properly constrained. This behavior can be partially controlled via the choice of decorrelation length scales that are assumed for the forcing errors. Abnormal forcing corrections may also arise due to the effects of model error which are not accounted for in IS4DVAR. In particular, data assimilation tends to weaken the alongshore wind stress in an attempt to reduce the rate of coastal upwelling, which seems to be too strong due to other sources of error. However, corrections to wind stress and surface heat flux improve systematically the ocean state analyses. Trends in the correction of surface heat fluxes indicate that, given the ocean model used and its potential limitations, the heat flux data from the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) used to impose surface conditions in the model are generally too low except in spring-summer, in the upwelling region, where they are too high. Comparisons with independent data provide confidence in the resulting forecast ocean circulation on timescales ～14 days, with less than 1.5 °C, 0.3 psu, and 9 cm RMS error in temperature, salinity and sea surface height anomaly, respectively, compared to observations.     

海洋环流模式中卫星遥感资料同化的应用进展

物理海洋研究长久以来一直受到观测资料不足的制约,然而这一状况随着现代观测技术的迅猛发展得到了很大的改善。卫星遥感技术的发展提供了覆盖全球的、连续、实时的卫星观测数据,这是其他任何资料都无法比拟的。这些数据大部分难以直接运用来改善气候预测或数值模拟分析,然而资料同化技术的出现和发展改善了这一情况。     

Numerical solution to the problem of variational assimilation of operational observational data on the ocean surface temperature

The problem of variational assimilation of satellite observational data on the ocean surface temperature is formulated and numerically investigated in order to reconstruct surface heat fluxes with the use of the global three-dimensional model of ocean hydrothermodynamics developed at the Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS), and observational data close to the data actually observed in specified time intervals. The algorithms of the numerical solution to the problem are elaborated and substantiated, and the data assimilation block is developed and incorporated into the global three-dimensional model. Numerical experiments are carried out with the use of the Indian Ocean water area as an example. The data on the ocean surface temperature over the year 2000 are used as observational data. Numerical experiments confirm the theoretical conclusions obtained and demonstrate the expediency of combining the model with a block of assimilating operational observational data on the surface temperature.     

Data assimilation with a barotropically unstable shallow water system using representer algorithms

The cycling representer algorithm of Xu and Daley (2000) is a weak constraint four-dimensional variational data assimilation algorithm. It was successfully applied to a one-dimensional transport problem and was able to successfully extract the signal from noisy and sparse observations. The algorithm, however, has not previously been applied to a multivariate, multidimensional system with dynamic instability. The algorithm is also very computationally demanding and awaits considerable enhancement in computer power before being practical for operational forecast models. We have two objectives in this paper. The first is to apply the cycling representer algorithm to a two-dimensional, multivariate barotropically unstable linear shallow water system. The second objective is to formulate and test an accelerated representer algorithm that is much more computationally tractable than the cycling representer algorithm itself. A linear shallow water system with a barotropically unstable basic state was used as a test bed to conduct data assimilation experiments. The evolution of a 'neutral' eastward-propagating singular vector was selected as the 'truth', against which all data assimilation experiments were to be evaluated. The results indicated that the cycling representer algorithm was capable of providing satisfying state estimates for a multivariate, multidimensional system. The results from the accelerated representer algorithm were very encouraging because it is sufficiently computationally tractable to be used on present day multi-processor machines for operational applications.     

气候模式中海洋数据同化对热带降水偏差的影响

本文采用海洋卫星观测海表温度(SST)和海面高度异常(SLA)数据,对国家海洋局第一海洋研究所地球系统模式FIO-ESM(First Institute of Oceanography Earth System Model version 1.0)中海洋模式分量进行了集合调整卡尔曼滤波(EAKF)同化,对比分析了大气环流、湿度和云量对海洋数据同化的响应,探讨了海洋同化对热带降水模拟偏差的影响。结果表明:海洋数据同化能有效改善海表温度和上层海洋热含量的模拟,30°S~30°N纬度带内年平均SST的绝均差降低60%。同化后大气模式模拟的赤道两侧信风得到明显改善,上升气流在赤道以北热带地区增强而在赤道以南热带地区减弱,热带降水模拟的动力结构更为合理,水汽和云量分布也更切合实际。热带年平均降水的空间分布和强度在同化后均得到改善,赤道以南的纬向年平均降水峰值显著降低,降水偏差明显减小,同化后30°S~30°N纬度带内年平均降水绝均差降低35%。     

A theoretical study of the multigrid three-dimensional variational data assimilation scheme using a simple bilinear interpolation algorithm

In order to solve the so-called "bull-eye" problem caused by using a simple bilinear interpolation as an observational mapping operator in the cost function in the multigrid three-dimensional variational(3DVAR) data assimilation scheme,a smoothing term,equivalent to a penalty term,is introduced into the cost function to serve as a means of troubleshooting.A theoretical analysis is first performed to figure out what on earth results in the issue of "bull-eye",and then the meaning of such smoothing term is elucidated and the uniqueness of solution of the multigrid 3DVAR with the smoothing term added is discussed through the theoretical deduction for one-dimensional(1D) case,and two idealized data assimilation experiments(one-and two-dimensional(2D) cases).By exploring the relationship between the smoothing term and the recursive filter theoretically and practically,it is revealed why satisfied analysis results can be achieved by using such proposed solution for the issue of the multigrid 3DVAR.     

海洋温盐度资料多变量同化研究进展

早期海洋资料同化仅考虑温度的调整而忽略盐度的变化,这往往会带来虚假信息,可能导致密度场被严重恶化,同化后的结果甚至比没有同化任何观测资料时还要差。为了解决这个问题,海洋资料同化中的一些温、盐度多变量调整方案便被提出来了。本文对广泛应用于多变量分析的资料同化方法及不同温、盐度多变量调整方案进行了系统的回顾,对它们的优缺点进行了分析与讨论,并指出了不同调整方案的适用条件及应用现状,最后对Argo资料在海洋资料同化中的重要性及今后的研究重点进行了探讨。