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.     

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.     

Time‐space weak‐constraint data assimilation for nonlinear models

A general perturbation–linearization scheme is proposed for the problem of data assimilation with an imperfect and nonlinear model, allowing for the application of the weak constraint representer method. The scheme is shown in discrete formalism for a generic model. An application example is given with computer‐generated data in the case of the Burgers equation. Discussion in reference to the assimilation example concerns: the rôle of the model error, seen as a forcing term in the dynamics; the rôle of representers as a posteriori error covariances; a comparison among different choices for a priori dynamic error variance and strong constraint assimilation. Weak and strong constraint methods are also compared in a forecasting experiment.     

Open‐ocean convection and polynya formation in a large‐scale ice‐ocean model

The physical processes responsible for the formation in a large‐scale ice–ocean model of an offshore polynya near the Greenwich meridian in the Southern Ocean are analysed. In this area, the brine release during ice formation in autumn is sufficient to destabilise the water column and trigger convection. This incorporates relatively warm water into the surface layer which, in a first step, slows down ice formation. In a second step, it gives rise to ice melting until the total disappearance of the ice at the end of September. Two elements are crucial for the polynya opening. The first one is a strong ice‐transport divergence in fall induced by south‐easterly winds, which enhances the amount of local ice formation and thus of brine release. The second is an inflow of relatively warm water at depth originating from the Antarctic Circumpolar Current, that sustains the intense vertical heat flux in the ocean during convection. The simulated polynya occurs in a region where such features have been frequently observed. Nevertheless, the model polynya is too wide and persistent. In addition, it develops each year, contrary to observations. The use of a climatological forcing with no interannual variability is the major cause of these deficiencies, the simulated too low density in the deep Southern Ocean and the coarse resolution of the model playing also a role. A passive tracer released in the polynya area indicates that the water mass produced there contributes significantly to the renewal of deep water in the Weddell Gyre and that it is a major component of the Antarctic Bottom Water (AABW) inflow into the model Atlantic.     

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.     

Changes in average and extreme precipitation in two regional climate model experiments

Two regional climate model experiments for northern and central Europe are studied focussing on greenhouse gas‐induced changes in heavy precipitation. The average yearly maximum one‐day precipitation P _max shows a general increase in the whole model domain in both experiments, although the mean precipitation P _mean decreases in the southern part of the area, especially in one of the experiments. The average yearly maximum six‐hour precipitation increases even more than the one‐day P _max, suggesting a decrease in the timescale of heavy precipitation. The contrast between the P _mean and P _max changes in the southern part of the domain and the lack of such a contrast further north are affected by changes in wet‐day frequency that stem, at least in part, from changes in atmospheric circulation. However, the yearly extremes of precipitation exhibit a larger percentage increase than the average wet‐day precipitation. The signal‐to‐noise aspects of the model results are also studied in some detail. The 44 km grid‐box‐scale changes in P _max are very heavily affected by inter‐annual variability, with an estimated standard error of about 20% for the 10‐year mean changes. However, the noise in P _max decreases sharply toward larger horizontal scales, and large‐area mean changes in P _max can be estimated with similar accuracy to those in P _mean. Although a horizontal averaging of model results smooths out the small‐scale details in the true climate change signal as well, this disadvantage is, in the case of P _max changes, much smaller than the advantage of reduced noise.     

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.     

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.     

Some remarks on the westward propagation of the monsoon depression

Several westward propagation properties of the Indian monsoon depression were neglected by previous studies. They include:(1) the slower propagation speed of the depression depicted by a quasi‐geostrophic model, (2) the initiation of the asymmetric secondary circulation with respect to the depression center, and (3) the absence of the depression perturbation in the upper troposphere. Some further insights into these neglected propagation properties of the depression are obtained from the streamfunction budget analysis with the ECMWF (European Centre for Medium Range Weather Forecasts) reanalysis data. (1) The inclusion of relative vorticity stretching, which is neglected in a quasi‐geostrophic model, increases the depression's westward propagation speed. (2) Within the large‐scale environment of the summer monsoon, the coupling of the east‐west differentiation of the meridional absolute vorticity advection with the CISK mechanism is conducive to the initiation and development of the asymmetric secondary circulation associated with the depression. (3) The Tibetan high is formed by summertime global‐scale stationary waves which are maintained by a Sverdrup balance. The positive streamfunction tendency induced by the upper‐tropospheric vortex stretching over the monsoon region suppresses the development of the monsoon depression in the upper troposphere.     

A diagnostic stabilized finite-element ocean circulation model

A stabilized finite-element (FE) algorithm for the solution of oceanic large scale circulation equations and optimization of the solutions is presented. Pseudo-residual-free bubble function (RFBF) stabilization technique is utilized to enforce robustness of the numerics and override limitations imposed by the Babuška–Brezzi condition on the choice of functional spaces. The numerical scheme is formulated on an unstructured tetrahedral 3d grid in velocity–pressure variables defined as piecewise linear continuous functions. The model is equipped with a standard variational data assimilation scheme, capable to perform optimization of the solutions with respect to open lateral boundary conditions and external forcing imposed at the ocean surface. We demonstrate the model performance in applications to idealized and realistic basin-scale flows. Using the adjoint method, the code is tested against a synthetic climatological data set for the South Atlantic ocean which includes hydrology, fluxes at the ocean surface and satellite altimetry. The optimized solution proves to be consistent with all these data sets, fitting them within the error bars.The presented diagnostic tool retains the advantages of existing FE ocean circulation models and in addition (1) improves resolution of the bottom boundary layer due to employment of the 3d tetrahedral elements; (2) enforces numerical robustness through utilization of the RFBF stabilization, and (3) provides an opportunity to optimize the solutions by means of 3d variational data assimilation. Numerical efficiency of the code makes this a desirable tool for dynamically constrained analyses of large datasets.     

Impact on oceanic dynamics from assimilation of satellite surface height anomaly data into the Hybrid Coordinate Ocean Model Ocean Model (HYCOM) over the Atlantic Ocean

Observed along-track data of sea surface height anomalies (SSHAs) over the Atlantic Ocean from the Jason-1 and Jason-2 satellites were assimilated into the Hybrid-Coordinate Ocean Model (HYCOM) with the Ensemble Optimal Interpolation scheme (EnOI). The impact of assimilation of SSHA with focus on oceanic dynamics was investigated. Time series of analyzed and forecasted values were compared with a model free run with the same forcing but without assimilation. In addition, the results were compared with an independent run, the so-called HYCOM + NCODA analysis from the US Navy. The study shows that the assimilation technique with some modifications allowed substantial improvement in the 24 h ocean prediction by reducing the forecast errors in comparison with the free run. It is also shown that the analyzed sea surface fields contain mesoscale and synoptic variability, which are poorly seen in the free run.     

热带大西洋年际和年代际变率的时空结构模拟

使用美国夏威夷大学发展的中等复杂程度海洋模式(IOM)在给定表面强迫条件下模拟了热带大西洋上层海洋年际和年代际变率的时空结构.利用NCEP的41a(1958～1998年)逐月平均表面资料作为强迫场,积分海洋模式41a作为控制试验,并利用模式分别做动量(风应力)通量和热量通量无异常变化的平行试验,与控制试验作比较.对3组试验模拟上层海洋变率状况的比较,并按年际和年代际时间尺度分别分析,揭示表面风应力和热通量异常对海表面温度和温跃层深度变化的影响,并比较了其影响的相对重要性.结果表明模式成功地模拟出了热带大西洋上层海洋的变率.模式模拟的海表面温度年际变化主要表现为弱ENSO型,年代际变化表现为南、北大西洋变化相反的偶极子型.在年际时间尺度上,热力强迫和动力强迫对海表温度变化都有贡献,其中赤道外海表面温度异常(SSTA)变化主要由热通量异常引起,而近赤道SSTA的变化主要由动量异常强迫引起.在年代际时间尺度上,热通量强迫的作用远比动量强迫重要.模式不仅能够模拟SST在年际和年代际时间尺度上的变率,还能够模拟温跃层深度在年际和年代际时间尺度上的变率.年际和年代际时间尺度上,温跃层深度的变率主要由动量异常决定,热通量异常强迫的贡献很小.     

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     

Freshwater forcing as a booster of thermohaline circulation

Making use of a simple two‐layer model, we analyze the impact of freshwater forcing on the thermohaline circulation. We consider the forward‐type circulation dominated by thermal forcing, implying that the freshwater forcing acts to reduce the density contrast associated with the equator‐to‐pole temperature contrast (prescribed in the model). The system is described by two variables: the depth of the upper layer ( H ) and the density contrast between the upper and lower layer (Δρ), which decreases with salinity contrast. The rate of poleward flow of light surface water and the diapycnal flow (i.e., upwelling) driven by widespread small‐scale mixing are both modeled in terms of H and Δρ. Steady states of thermohaline circulation are found when these two flows are equal. The representation of the diapycnal flow ( M_D ) is instrumental for the dynamics of the system. We present equally plausible examples of a physically based representation of M_D for which the thermohaline circulation either decreases or increases with density contrast. In the latter case, contrary to the traditional wisdom, the freshwater forcing amplifies the circulation and there exists a thermally dominated equilibrium for arbitrary intensity of freshwater forcing. Here, Stommel's famous feedback between circulation and salinity contrast is changed from a positive to a negative feedback. The interaction of such a freshwater boosted thermohaline circulation with the climate system is fundamentally different from what is commonly assumed, an issue which is briefly addressed.     

Correcting the errors in the initial conditions and wind stress in storm surge simulation using an adjoint optimal technique

An adjoint data assimilation methodology is applied to the Princeton Ocean Model and is evaluated by obtaining “optimal” initial conditions, sea surface forcing conditions, or both for coastal storm surge modelling. By prescribing different error sources and setting the corresponding control variables, we performed several sets of identical twin experiments by assimilating model-generated water levels. The experiment results show that, when the forecasting errors are caused by the initial (or surface boundary) conditions, adjusting initial (or surface boundary) conditions accordingly can significantly improve the storm surge simulation. However, when the forecasting errors are caused by surface boundary (or initial) conditions, data assimilation targeting improving the initial (or surface boundary) conditions is ineffective. When the forecasting errors are caused by both the initial and surface boundary conditions, adjusting both the initial and surface boundary conditions leads to the best results. In practice, we do not know whether the errors are caused by initial conditions or surface boundary conditions, therefore it is better to adjust both initial and surface boundary conditions in adjoint data assimilation.     

一种新的模式倾向误差估计算法及其在ENSO模拟中的应用

气候模式是我们理解、模拟和预报气候演变的重要工具。然而即使是目前最先进的耦合模式,其模拟和预报与大气/海洋的真实状态相比,仍存在较大偏差,这是由于在模式的倾向方程中不可避免地存在系统性的误差(倾向误差)。因此,减小模式倾向误差对改进模式的模拟和预报效果具有重要意义。该研究首先发展了一种新的计算模式倾向误差的估计算法——基于局地集合变换卡尔曼滤波器(local ensemble transform kalman filter, LETKF)同化技术的倾向误差估计算法。在此基础上,将新发展的算法应用到Zebiak-Cane (ZC)模式,通过同化海表面温度异常(sea surface temperature anomaly, SSTA)数据,估计随时空变化的倾向误差,并使用计算得到的倾向误差订正模式,进行积分模拟。结果表明: (1)倾向误差和ZC模式的模拟偏差具有高度相关性; (2)订正后的模式改善了对厄尔尼诺-南方涛动(El Niño-Southern Oscillation, ENSO)的一些重要特征的模拟。这说明新发展的模式倾向误差估计算法十分有效且在ENSO模拟中具有较好的应用价值,此外,这种新的模式倾向误差估计算法,计算高效简便,可便捷地应用于各模式中,利于推广。     

在Zebiak-Cane模式中观测MJO强迫对ENSO可预报性的影响, 第1部分:对最大预报误差的影响

With the observational wind data and the Zebiak-Cane model, the impact of Madden-Julian Oscillation (MJO) as external forcing on El Ni(n)o–Southern Oscillation (ENSO) predictability is studied. The obs...     

Different approaches to model error formulation in 4D-Var: a study with high-resolution advection schemes

All numerical models are imperfect. Weak constraint variational data assimilation ( VDA ), which provides a treatment of the modelling errors, is studied; building on the approach of Vidard et al. (Tellus, 56 A, pp. 177–188, 2004). The evolution of model error ( ME ) is modelled using ordinary differential equations, which involve a scalar parameter. These approaches were tested using different high-resolution advection schemes. The first set of experiments were constructed to see if it is possible to account for (numerical) discretization error within such a framework. In other set of experiments, a systematic source of modelling error was introduced by deliberately specifying an incorrect value for the Coriolis parameter in the model. Results with observational state at half of the model state resolution, are also presented. We also discuss a method of estimating the scalar parameter in the ME through VDA . In all cases, the inclusion of ME provides reduction in forecasting errors. Also, our experiments indicate that different settings of the model (e.g. using different high-resolution advection schemes) would need different ME formulation. Results presented in this paper could be used to formulate sophisticated ME forms to account for systematic errors in higher dimensional models with complex advection schemes.     

南大西洋副热带偶极子的年际变化

南大西洋副热带偶极子(South Atlantic Subtropical Dipole;SASD)为南大西洋海洋与大气相互作用的主要模态。它的空间型为海表面温度异常呈现东北-西南偶极子分布。当SASD指数大于1,为SASD正事件,小于-1,为负事件。根据1960-2016年HadISST(Hadley Center Global Sea Ice and Sea Surface Temperature)数据,本文鉴别出57年中共发生6次正事件和9次负事件。SASD存在显著的5~8年周期的年际变化特征。本文进一步利用1992-2016年ECCO2(Estimating the Circulation and Climate of the Ocean,PhaseⅡ)模式数据,根据温度倾向方程分别诊断了SASD西南极和东北极的混合层温度变化。诊断结果表明,SASD的年际变化主要来自于表面热力强迫项的年际变化。考虑到表面热力强迫项主要由短波辐射项控制,SASD的年际变化最终来源于短波辐射项的年际变化。     

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.