Low‐dimensional representation of error covariance

Ensemble and reduced‐rank approaches to prediction and assimilation rely on low‐dimensional approximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time‐independent systems are used to identify factors that cause the steady‐state analysis error covariance to admit a low‐dimensional representation. A useful measure of forecast/analysis cycle stability is the bound matrix , a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady‐state analysis error covariance matrix eigenvalues are derived from the bound matrix. The estimates generalize to time‐dependent systems. If much of the steady‐state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix approximate those of the steady‐state analysis error covariance matrix. The analytical results are illustrated in two numerical examples where the Kalman filter is carried to steady state. The first example uses the dynamics of a generalized advection equation exhibiting non‐modal transient growth. Failure to observe growing modes leads to increased steady‐state analysis error variances. Leading eigenvectors of the steady‐state analysis error covariance matrix are well approximated by leading eigenvectors of the bound matrix. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest‐order approximation of the bound matrix are shown to approximate well the leading eigenvectors of the steady‐state analysis error covariance matrix.     

各向异性背景场误差协方差构建及在“凡亚比”台风的应用

利用相临过去时段预报结果中同一时刻不同时效的模式预报场差异,计算预报误差协方差,并基于集合-变分混合同化系统将其与静态背景场误差协方差结合,从而在同化系统中构建了具有各向异性和一定流依赖特征的背景场误差协方差。单点观测理想试验显示本方案改善了静态模型化背景场误差协方差的各向同性和流依赖性问题。“凡亚比”台风的一系列同化及模拟试验表明,从台风路径、强度等方面本文方案的效果都要优于三维变分法。本文方案在不需要集合预报,计算量与三维变分法相当的情况下,给同化系统引入了各向异性、一定流依赖特征的背景误差协方差,因此本方案适于在计算资源较为紧缺情况下,对时效要求较高的预报业务中应用。     

Data assimilation considerations for improved ocean predictability during the Gulf of Mexico Grand Lagrangian Deployment (GLAD)

Ocean prediction systems rely on an array of assumptions to optimize their data assimilation schemes. Many of these remain untested, especially at smaller scales, because sufficiently dense observations are very rare. A set of 295 drifters deployed in July 2012 in the north-eastern Gulf of Mexico provides a unique opportunity to test these systems down to scales previously unobtainable. In this study, background error covariance assumptions in the 3DVar assimilation process are perturbed to understand the effect on the solution relative to the withheld dense drifter data. Results show that the amplitude of the background error covariance is an important factor as expected, and a proposed new formulation provides added skill. In addition, the background error covariance time correlation is important to allow satellite observations to affect the results over a period longer than one daily assimilation cycle. The results show the new background error covariance formulations provide more accurate placement of frontal positions, directions of currents and velocity magnitudes. These conclusions have implications for the implementation of 3DVar systems as well as the analysis interval of 4DVar systems.     

Reducing systematic errors by empirically correcting model errors

A methodology for the correction of systematic errors in a simplified atmospheric general‐circulation model is proposed. First, a method for estimating initial tendency model errors is developed, based on a 4‐dimensional variational assimilation of a long‐analysed dataset of observations in a simple quasi‐geostrophic baroclinic model. Then, a time variable potential vorticity source term is added as a forcing to the same model, in order to parameterize subgrid‐scale processes and unrepresented physical phenomena. This forcing term consists in a (large‐scale) flow dependent parametrization of the initial tendency model error computed by the variational assimilation. The flow dependency is given by an analogues technique which relies on the analysis dataset. Such empirical driving causes a substantial improvement of the model climatology, reducing its systematic error and improving its high frequency variability. Low‐frequency variability is also more realistic and the model shows a better reproduction of Euro‐Atlantic weather regimes. A link between the large‐scale flow and the model error is found only in the Euro‐Atlantic sector, other mechanisms being probably the origin of model error in other areas of the globe.     

变分资料同化中不同的变分求解方法

在应用变分资料同化方法时面临着两方面的难题:一是背景场误差协方差矩阵的求逆问题;二是与背景场误差协方差矩阵相关的计算与存储问题。为了解决这两方面的问题,不同的求解方法便被提出来了。对主要的变分求解方法,包括增量法、运用空间滤波算子的变分分析法、预处理化法、物理空间统计分析法、谱统计插值法等进行了系统的回顾,对它们的优缺点进行了分析与讨论,并指出了变分资料同化中各种求解方法的适用条件。     

Preconditioned Optimizing Utility for Large-dimensional analyses (POpULar)

I present the derivation of the Preconditioned Optimizing Utility for Large-dimensional analyses (POpULar), which is developed for adopting a non-diagonal background error covariance matrix in nonlinear variational analyses (i.e., analyses employing a non-quadratic cost function). POpULar is based on the idea of a linear preconditioned conjugate gradient method widely adopted in ocean data assimilation systems. POpULar uses the background error covariance matrix as a preconditioner without any decomposition of the matrix. This preconditioning accelerates the convergence. Moreover, the inverse of the matrix is not required. POpULar therefore allows us easily to handle the correlations among deviations of control variables (i.e., the variables which will be analyzed) from their background in nonlinear problems. In order to demonstrate the usefulness of POpULar, we illustrate two effects which are often neglected in studies of ocean data assimilation before. One is the effect of correlations among the deviations of control variables in an adjoint analysis. The other is the nonlinear effect of sea surface dynamic height calculation required when sea surface height observation is employed in a three-dimensional ocean analysis. As the results, these effects are not so small to neglect.     

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.     

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     

Variational data analysis with control of the forecast bias

We propose a methodology for the treatment of the systematic model error in variational data assimilation. The principle of the method is to add a systematic error correction term in the model equations and to include it in the variational assimilation control vector.
This method is applied to a simplified ocean circulation model in an identical twin experiment framework. It shows a noticeable improvement compared to the result of a classical variational assimilation scheme in which the systematic error is not corrected. The estimated systematic error correction term is sufficiently consistent with that needed by the model that it allows improvements not just to the analysis, but also during the forecast phase.     

Inferred variables in data assimilation: quantifying sensitivity to inaccurate error statistics

The ability of data assimilation systems to infer unobserved variables has brought major benefits to atmospheric and oceanographic sciences. Information is transferred from observations to unobserved variables in two ways: through the temporal evolution of the predictive equations (either a forecast model or its adjoint) or through an error covariance matrix (or a parametrized approximation to the error covariance). Here, it is found that high frequency information tends to flow through the former route, low frequency through the latter. It is also noted that using the Kalman Filter analysis to estimate the correlation between the observed and unobserved variables can lead to a biased result because of an error correlation: this error correlation is absent when the Kalman Smoother is used.     

Moisture retrievals from simulated zenith delay "observations" and their impact on short‐range precipitation forecasts

The feasibility of assimilating the GPS total zenith delay into atmospheric models is investigated within the framework of the "Observing System Simulation Experiment." The total zenith delay is made up of two terms: one is proportional to the pressure at the site of the GPS ground‐based receiver and the other to the overlying amount of water vapor. Using the MM5 mesoscale model and its adjoint, a set of 4‐dimensional variational (4DVAR) experiments is performed. Results from the assimilation of simulated precipitable water observations are used as the benchmark. The model domain covers Southern California. The observations are simulated with a 10 km horizontal resolution model that includes full physics, while a 20‐km resolution and a less comprehensive physics package are used in the 4DVAR experiments. Both, the 10‐km and 20‐km models employ the same set of 15 vertical levels. Moisture fields retrieved from the total zenith delay are found to compare very well with those retrieved from the precipitable water. Verified against the observations, the vertically integrated moisture is found to be very accurate. An overall improvement is also achieved in the vertical profiles of the moisture fields. The use of the so‐called background term and model initialization are shown to greatly reduce the negative impact that the sole assimilation of the total zenith delay can have on the pressure field and integrated water vapor. The adverse effect stems from the poor resolution of the topography needed to evaluate the model pressure at the GPS sites. The analysis increments of all model fields are found to be similar to the counterparts obtained from the assimilation of the precipitable water. The same is true for the short‐range precipitation forecasts initiated from the 4DVAR‐optimal initial conditions.     

A 4D-variational approach applied to an eddy-permitting North Atlantic configuration: Synthetic and real data assimilation of altimeter observations

The increasing number of oceanic observations calls for the use of synthetic methods to provide consistent analyses of the oceanic variability that will support a better understanding of the underlying mechanisms. In this study, a 1/3° eddy-permitting model of the North Atlantic (from 20°S to 70°N) is combined with a 4D-variational method to estimate the oceanic state from altimeter observations. This resolution allows a better extraction of the physical content of altimeter data since the model spatial scales are more consistent with the data than coarser assimilation exercises because of a lower error in model representativity. Several strategies for the assimilation window are tested through twin experiments carried out under the following conditions: different window lengths and either a quasi-static (also known as progressive) variational assimilation with progressive extension of the window, or a simpler direct method without prior assimilation. From our set of experiments, the most efficient strategy is the use of both a simple direct assimilation method and a 90-day window. The assimilation of synthetic altimeter data constrains the model-temperature, -salinity and -velocity fields mainly over the first 1300 m where the error is the largest. Improvements occur not only in quiescent regions, but also in more energetic meso-scale regimes. Despite the existence of model- and surface forcing-errors as well as large errors in the first guess, the assimilation of real altimeter data proves to be consistent with our twin experiments. Indeed, the analyses show a better detachment of the Gulf Stream, weaker regional biases and more accurate positions for meso-scale structures. Independent hydrographic data (Argo floats and CTD cruises) and transports estimates along the OVIDE 2002 cruise show an improvement of the analysed oceanic state with respect to the assimilation-free case though water mass properties are still incorrectly represented. After assimilation, the North Atlantic heat transport in the model is in good agreement with independent estimates based on hydrographic data.     

Forecasting ocean wave energy: Tests of time-series models

This paper evaluates the ability of time-series models to predict the energy from ocean waves. Data sets from four Pacific Ocean sites are analyzed. The energy flux is found to exhibit nonlinear variability. The probability distribution has heavy tails, while the fractal dimension is non-integer. This argues for using nonlinear models. The primary technique used here is a time-varying parameter regression in logs. The time-varying regression is estimated using both a Kalman filter and a sliding window, with various window widths. The sliding window method is found to be preferable. A second approach is to combine neural networks with time-varying regressions, in a hybrid model. Both of these methods are tested on the flux itself. Time-varying regressions are also used to forecast the wave height and wave period separately, and combine the forecasts to predict the flux. Forecasting experiments are run at an hourly frequency over horizons of 1-4 h, and at a daily frequency over 1-3 days. All the models are found to improve relative to a random walk. In the hourly data sets, forecasting the components separately achieves the best results in three out of four cases. In daily data sets, the hybrid and regression models yield similar outcomes. Because of the intrinsic variability of the data, the forecast error is fairly high, comparable to the errors found in other forms of alternative energy, such as wind and solar.     

现代海洋/大气资料同化方法的统一性及其应用进展

海洋／大气资料同化的理论基础是用数值模式作为动力学强迫对观测信息进行提炼，或者说，从包含观测误差(噪声)的空间分布不均匀的实测资料中依据动力系统自身的演化规律(动力学方程或模式)来确定海洋／大气系统状态的最优估计。本文对主要的现代海洋／大气资料同化方法，包括最优插值(()ptimal Interpolation，简称()Ⅰ)、变分方法(3—Dimensional Variational和4—Dimensional Variational，分别简称3DVAR和4DVAR)和滤波方法(Filtering)的原理、算法设计和实际应用进行系统地回顾，并对这些资料同化方法的优缺点进行分析和讨论。在滤波框架下，所有的现代资料同化方法都被统一了：()Ⅰ和3DVAR是不随时间变化的滤波器，4DVAR和卡曼滤波是线性滤波器，即非线性滤波的退化情形；而集合滤波能构建非线性的滤波器，因为集合在某种程度上体现了系统的非高斯信息。一个非线性滤波器的主要优点是能计算和应用随时间变化的各阶误差统计距，如误差协方差矩阵。将非线性滤波器计算的随时间变化的误差协方差矩阵引入到（）Ⅰ或4DVAR中，也许能实质性地改进这些传统方法。在实际应用中，方法的优劣可能取决于所选用的数值模式和可获得的计算资源，因此需针对不同的问题选取不同的资料同化方法。由于各种资料同化方法具有统一性，因此可建立测试系统来评价这些方法，从而对各种方法获得更深入的理解，改进现有的资料同化技术，并提高人们对海洋／大气环境的预测能力。     

Examination of targeting methods in a simplified setting

The effectiveness of 2 methods for targeting observations is examined using a T21 L3 QG model in a perfect model context. Target gridpoints are chosen using the pseudo‐inverse (the inverse composed of the first three singular vectors only) and the quasi‐inverse or backward integration (running the tangent equations with a negative time‐step). The effectiveness of a target is measured by setting the analysis error to zero in a region surrounding the target and noting the impact on the forecast error in the verification region. In a post‐time setting, when the targets are based on forecast errors that are known exactly, both methods provide targets that are significantly better than targets chosen at random within a broad region upstream of the verification region. When uncertainty is added to the verifying analysis such that the forecast error is known inexactly, the pseudo‐inverse targets still perform very well, while the backward integration targets are degraded. This degradation due to forecast uncertainty is especially significant when the targets are a function of height as well as horizontal position. When an ensemble‐forecast difference is used in place of the inexact forecast error, the backward integration targets may be improved considerably. However, this significant improvement depends on the characteristics of the initial‐time ensemble perturbation. Pseudo‐inverse targets based on ensemble forecast differences are comparable to pseudo‐inverse targets based on exact forecast errors. Targets based on the largest analysis error are also found to be considerably more effective than random targets. The collocation of the backward integration and pseudo‐inverse targets appears to be a good indicator of target skill.     

Sensitivity analysis of the climate of a chaotic system

This paper addresses some fundamental methodological issues concerning the sensitivity analysis of chaotic geophysical systems. We show, using the Lorenz system as an example, that a naïve approach to variational ("adjoint") sensitivity analysis is of limited utility. Applied to trajectories which are long relative to the predictability time scales of the system, cumulative error growth means that adjoint results diverge exponentially from the "macroscopic climate sensitivity"(that is, the sensitivity of time‐averaged properties of the system to finite‐amplitude perturbations). This problem occurs even for time‐averaged quantities and given infinite computing resources. Alternatively, applied to very short trajectories, the adjoint provides an incorrect estimate of the sensitivity, even if averaged over large numbers of initial conditions, because a finite time scale is required for the model climate to respond fully to certain perturbations. In the Lorenz (1963) system, an intermediate time scale is found on which an ensemble of adjoint gradients can give a reasonably accurate (O(10%)) estimate of the macroscopic climate sensitivity. While this ensemble‐adjoint approach is unlikely to be reliable for more complex systems, it may provide useful guidance in identifying important parameter‐combinations to be explored further through direct finite‐amplitude perturbations.     

不同 AMDAR资料同化时间窗对台风“摩羯”预报影响研究

2018年第14号台风“摩羯”对山东造成了大范围暴雨和大风天气,基于WRF（Weather Research and Forecasting）模式及其Hybrid-3DVAR混合同化预报系统,对Hybrid-3DVAR不同集合协方差比例和不同航空气象数据转发（aircraft meteorological data relay,以下简称AMDAR）资料同化时间窗对台风“摩羯”预报的影响进行了数值研究。结果表明：加大集合协方差比例对台风“摩羯”路径预报有较大影响和改进;当全部取来自集合体的流依赖误差协方差时,预报的台风路径最好,降水预报也最接近实况;AMDAR资料同化对于台风路径和降水预报也有正的改进作用,但加大集合协方差比例到100%时对台风路径预报影响更大;不同资料同化时间窗会影响同化的AMDAR资料数量,从而影响台风降水精细化预报;45 min同化时间窗的要素预报误差最小,对台风造成的强降水精细特征预报最接近实况;不同资料同化时间窗主要影响台风降水预报落区分布,对台风路径预报影响相对较小。     

Predicting the East Australian Current

Results are presented from an ensemble prediction study (EPS) of the East Australian Current (EAC) with a specific focus on the examination of the role of dynamical instabilities and flow dependent growing errors. The region where the EAC separates from the coast, is characterized by significant mesoscale eddy variability, meandering and is dominated by nonlinear dynamics thereby representing a severe challenge for operational forecasting. Using analyses from OceanMAPS, the Australian operational ocean forecast system, we explore the structures of flow dependent forecast errors over 7 days and examine the role of dynamical instabilities. Forecast ensemble perturbations are generated using the method of bred vectors allowing the identification of those perturbations to a given initial state that grow most rapidly. We consider a 6 month period spanning the Austral summer that corresponds to the season of maximum eddy variability. We find that the bred vector (BV) structures occur in areas of instability where forecast errors are large and in particular in regions associated with the Tasman Front and EAC extension. We also find that very few BVs are required to identify these regions of large forecast error and on that basis we expect that even a small BV ensemble would prove useful for adaptive sampling and targeted observations. The results presented also suggest that it may be beneficial to supplement the static background error covariances typically used in operational ocean data assimilation systems with flow dependent background errors calculated using a relatively cheap EPS.     

基于不同背景场条件的雷达资料同化在登陆台风“桑美”中的应用研究

本文针对2006年登陆我国的超强台风“桑美”,分别采用美国国家环境预报中心的全球预报系统(Global Forecasting System, GFS)再分析资料和日本气象厅(Japan Meteorological Agency, JMA)区域客观再分析资料作为背景场,利用中尺度数值模式WRF(Weather Research and Forecasting Model)及其三维变分同化系统进行多普勒雷达资料同化和数值模拟试验,考察不同的背景场条件下雷达资料同化对台风初始场、内部结构及其随后确定性预报的影响。结果表明:GFS试验和JMA试验在同化了雷达资料之后分析出的台风700 hPa风场和500 hPa高度场相比其初始场均有所增强,JMA试验在3 h同化窗内的均方根误差和最小海平面气压的改进效果均比GFS试验显著,同时对台风动力和热力结构的改进效果也优于GFS试验;JMA试验对台风降水、路径、强度的预报均优于GFS试验,且能预报出台风前沿的降水,更加接近观测实况。