On optimal solution error covariances in variational data assimilation problems

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the optimal solution error covariance operator through the input data error covariance operators, respectively. The quasi-Newton BFGS algorithm is adapted to construct the covariance matrix of the optimal solution error using the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. Preconditioning is applied to reduce the number of iterations required by the BFGS algorithm to build a quasi-Newton approximation of the inverse Hessian. Numerical examples are presented for the one-dimensional convection–diffusion model.     

以全球谱模式为约束的四维变分资料同化系统YH4DVAR的设计和实现

四维变分资料同化仍将是未来相当长时间内业务数值天气预报中所使用的主流同化方法．针对全球数值天气预报业务系统对气象资料同化技术的需求,在WRFDA软件框架结构的基础上,发展了一个与全球谱模式配套使用的四维变分资料同化系统YH4DVAR．系统将背景场、观测处理、重力波控制和偏差订正进行综合考虑,设计了一体化目标函数,引入了小波背景场误差协方差模型,实现了增量方法以及卫星遥感资料的直接同化．单点试验表明YH4DVAR的背景场误差模型具有各向异性、垂直相关和水平相关不可分离性、以及与位置的相关等特性．从2009年7月到2010年6月的分析预报试验结果表明,由YH4DVAR和全球谱模式组成的分析预报系统在北半球和亚洲地区的可用预报时效可以达到8天以上．     

Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.     

变分同化结合广义变分最佳分析对微波散射计资料进行海面风场反演

针对微波散射计资料的海面风场反演,本文首先利用QuikSCAT卫星散射计的近实时资料,借助荷兰皇家气象学会的SeaWinds散射计数据处理模式,在不同风速和海域的情况下,采用多解方案结合二维变分同化方法进行风向的模糊去除,达到消除解的不唯一性的目的,且对4种反演结果（低速风、中速风、高速风和近海岸风）与美国国家海洋和大气管理局卫星资料中心的近实时风场数据的对比分析,说明了该技术的有效性.在此基础上,对反演得到的中速风场利用广义变分最佳分析方法进行了调整,调整后的风场结构有了进一步的改善.此方法可为我国散射计资料反演海面风场的研究提供新的思路和方法.     

变分同化结合广义变分最佳分析对微波散射计资料进行海面风场反演

针对微波散射计资料的海面风场反演,本文首先利用QuikSCAT卫星散射计的近实时资料,借助荷兰皇家气象学会的SeaWinds散射计数据处理模式,在不同风速和海域的情况下,采用多解方案结合二维变分同化方法进行风向的模糊去除,达到消除解的不唯一性的目的,且对4种反演结果（低速风、中速风、高速风和近海岸风）与美国国家海洋和大气管理局卫星资料中心的近实时风场数据的对比分析,说明了该技术的有效性.在此基础上,对反演得到的中速风场利用广义变分最佳分析方法进行了调整,调整后的风场结构有了进一步的改善.此方法可为我国散射计资料反演海面风场的研究提供新的思路和方法. 关键词： 多解方案 二维变分同化 广义变分最佳分析 正则化方法     

GPS掩星一维弯曲角算子在四维变分资料同化系统中的实现方法研究

GPS无线电掩星资料的良好特性使其作为十分重要的观测资料在欧洲 及美国数值天气预报中效果明显. 本文设计了一维弯曲角算子在四维变分资料同化系统中的实现方法和质量控制策略. 详细阐述了一维弯曲角正演算子及其切线性/伴随算子在全球四维变分资料同化系统中的实现步骤和过程, 其中折射率计算中采用Rüeger的折射率最佳平均因子的三项表达式, 计算出更加精确的折射率值. 对2012年8月份的GPS无线电掩星资料进行了观测误差统计, 引入了更加符合四维变分资料同化系统实际情况的一维弯曲角观测误差模型和质量控制策略. 同时, 对比了观测误差统计试验中同化GPS无线电掩星资料的模式背景场和未同化GPS无线电掩星资料的背景场的观测误差统计结果, 发现同化GPS无线电掩星资料之后的观测误差统计结果在全球区域正作用十分明显, 尤其是在中低纬度地区, 高度在10 km 以上的区域观测误差减小十分明显. 关键词： GPS无线电掩星 一维弯曲角算子 四维变分同化系统 质量控制     

Regeneration of small eddies by data assimilation in turbulence

The effect of data assimilation of large-scale eddies on small-scale eddies in turbulence is studied by direct numerical simulations (DNSs) of Navier-Stokes turbulence with Taylor microscale Reynolds numbers up to 179. The DNSs show that even if the data of small-scale eddies are lost at some initial instant, they can be regenerated from the data of large-scale eddies under the condition that Fourier modes with wave number less than a critical wave number k(*) are continuously assimilated, where k(*) approximately 0.2eta(-1) with eta identical with(nu(3)/epsilon)(1/4), epsilon the mean energy dissipation rate, and nu the viscosity.     

Macroscopic cell representation of density matrix equation and optimal choice of projectors by variational principle

We introduce a complete set of projectors on Liouvillespace and a “cell representation” of the equation of motion of the density matrixρ adjusted to the macroscopic observables of the system. Using a variational principle, the projector on the relevant density matrixρ _rel is determined by the postulate that the subdynamics ofρ _rel should include an optimal part of the Liouvillean?. The relation of the reduced subdynamics equation to wellknown master equations is investigated.     

变分同化中水平误差函数的正交小波模拟新方法

背景误差协方差是变分资料同化系统中的一个重要组成部分,能将观测信息从观测点传播到周围的模式格点和垂直层上.为了模拟背景误差协方差中水平误差函数的非均匀性和各向异性,提出了一种用正交小波变换模拟水平误差函数的新方法.试验结果表明：新方法能模拟出水平误差函数中固有的非均匀性和各向异性,极好地表示了背景误差协方差中固有的结构和特征. 关键词： 变分资料同化 背景误差协方差 二维正交小波 水平误差函数     

Growth optimal investment and pricing of derivatives

We introduce a criterion how to price derivatives in incomplete markets, based on the theory of growth optimal strategy in repeated multiplicative games. We present reasons why these growth-optimal strategies should be particularly relevant to the problem of pricing derivatives. Under the assumptions of no trading costs, and no restrictions on lending, we find an appropriate equivalent martingale measure that prices the underlying and the derivative security. We compare our result with other alternative pricing procedures in the literature, and discuss the limits of validity of the lognormal approximation. We also generalize the pricing method to a market with correlated stocks. The expected estimation error of the optimal investment fraction is derived in a closed form, and its validity is checked with a small-scale empirical test.     

Diagnosis and impacts of non-Gaussianity of innovations in data assimilation

Most of the atmospheric and oceanic data assimilation (DA) schemes rely on the Best Linear Unbiased Estimator (BLUE), which is sub-optimal if errors of assimilated data are non-Gaussian, thus calling for a full Bayesian data assimilation. This paper contributes to the study of the non-Gaussianity of errors in the observational space. Possible sources of non-Gaussianity range from the inherent statistical skewness and positiveness of some physical observables (e.g. moisture, chemical species), the nonlinearity, both of the data assimilation models and of the observation operators among others. Deviations from Gaussianity can be justified from a priori hypotheses or inferred from statistical diagnostics of innovations (observation minus background), leading to consistency relationships between the error statistics. From samples of observations and backgrounds as well as their specified error variances, we evaluate some measures of the innovation non-Gaussianity, such as the skewness, kurtosis and negentropy. Under the assumption of additive errors and by relating statistical moments from both data errors and innovations, we identify potential sources of the innovation non-Gaussianity. These sources range from: (1) univariate error non-Gaussianity, (2), nonlinear correlations between errors, (3) spatio-temporal variability of error variances (heteroscedasticity) and (4) multiplicative noise. Observational and background errors are often assumed independent. This leads to variance-dependent bounds for the skewness and the kurtosis of errors. From innovation statistics, we assess the potential DA impact of some scenarios of non-Gaussian errors. This impact is measured through the mean square difference between the BLUE and the Minimum Variance Unbiased Estimator (MVUE), obtained with univariate observations and background estimates. In order to accomplish this, we compute maximum entropy probability density functions (pdfs) of the errors, constrained by the first four order moments. These pdfs are then used to compute the Bayesian posterior pdf and the MVUE. The referred impact is studied for a large range of statistical moments, being higher for skewed innovations and growing in average with the skewness of data errors, specially if the skewnesses have the same sign. An application has been performed to the quality-accepted ECMWF innovations of brightness temperatures of a set of High Resolution Infrared Sounder (HIRS) channels. In this context, the MVUE has led in some extreme cases to a potential reduction of 20%-60% of the posterior error variance as compared to the BLUE, specially for extreme values of the innovations.     

On the validity of the variational approximation in discrete nonlinear Schrödinger equations

The variational approximation is a well known tool to approximate localized states in nonlinear systems. In the context of a discrete nonlinear Schrödinger equation with a small coupling constant, we prove error estimates for the variational approximations of site-symmetric, bond-symmetric, and twisted discrete solitons. This is shown for various trial configurations, which become increasingly more accurate as more parameters are taken. It is also shown that the variational approximation yields the correct spectral stability result and controls the oscillatory dynamics of stable discrete solitons for long but finite time intervals.     

Validity of the time-dependent variational approximation to the Gaussian wavepacket method applied to double-well systems

We have examined the validity of the time-dependent variational approximation (TDVA) to the Gaussian wavepacket method (GWM) for quantum double-well (DW) systems, by using the quasi-exact spectral method (SM). Comparisons between results of wavefunctions, averages of position and momentum, the auto-correlation function, and an uncertainty product calculated by SM and TDVA have been made. It has been shown that a given initial Gaussian wavepacket in SM is quickly deformed at t>0$t>0$ where a wavepacket cannot be expressed by a single Gaussian, and that assumptions on averages of higher-order fluctuations in TDVA are not justified. These results cast some doubt on an application of TDVA to DW systems. Gaussian wavepacket dynamics in anharmonic potential systems is studied also.     

资料同化中的数字滤波弱约束试验及分析

气象数值预报中,由于分析过程引入初始非平衡,从而引起虚假快波振荡,重力波控制弱约束把资料分析过程和初始化过程结合在一起,通过数字滤波弱约束在极小化过程中实现对分析场的平衡约束,克服非平衡问题. 以2008年初的一次南方雨雪天气为研究个例,进行了数字滤波弱约束的同化试验和预报试验,结果表明,数字滤波弱约束4D-Var能充分控制快波振荡的出现和初始调整现象,使得到的分析场不仅能更好的逼近观测,而且能更好地与模式动力相协调. 预报检验的结果表明,在同化过程中施加数字滤波弱约束,能有效滤除由于地形或观测资料等因素 关键词： 变分同化 初始非平衡 数字滤波 弱约束     

On the reconstruction of an optical potential on the basis of experimental data in the WKB approximation

The inverse scattering problem at a fixed energy for a complex-valued potential is solved in the WKB approximation. The method is used to reconstruct the optical potential for elastic ¹⁶O + ¹⁶O scattering at E _lab = 350 MeV. The stability of the solution against small changes in the scattering matrix is studied.     

Statistical mechanics of learning: a variational approach for real data

Using a variational technique, we generalize the statistical physics approach of learning from random examples to make it applicable to real data. We demonstrate the validity and relevance of our method by computing approximate estimators for generalization errors that are based on training data alone.     

Analysis of the Fenton-Karma model through an approximation by a one-dimensional map

The Fenton-Karma model is a simplification of complex ionic models of cardiac membrane that reproduces quantitatively many of the characteristics of heart cells; its behavior is simple enough to be understood analytically. In this paper, a map is derived that approximates the response of the Fenton-Karma model to stimulation in zero spatial dimensions. This map contains some amount of memory, describing the action potential duration as a function of the previous diastolic interval and the previous action potential duration. Results obtained from iteration of the map and numerical simulations of the Fenton-Karma model are in good agreement. In particular, the iterated map admits different types of solutions corresponding to various dynamical behavior of the cardiac cell, such as 1:1 and 2:1 patterns. (c) 2002 American Institute of Physics.     

利用变分结合正则化方法对高度计风速资料调整海面风场的研究

利用变分结合正则化方法,对高度计风速资料在无辐散和有辐散两种情形下的背景风场进行调整,同时进行了数值试验.试验表明:高度计风速对背景风场的调整有积极作用,特别是高度计沿轨区域风场调整效果更为明显.对高度计后向散射截面进行了敏感性试验,当后向散射截面存在随机扰动时,利用高度计风速调整海面风场具有较强的抗噪性.最后进行实例试验,结果表明该方法是切实可行的.