顺序数据同化的Bayes滤波框架

数据同化是在动力学模型的运行过程中不断融合新的观测信息的方法论,Bayes理论是数据同化的基石.从原理、方法和符号系统为Bayes滤波在数据同化中的应用勾勒一个统一的框架.首先对连续数据同化和顺序数据同化的各种方法做了分类,然后给出了非线性系统顺序数据同化的Bayes递推滤波形式,并在此基础上介绍了典型的顺序数据同化方法--粒子滤波和集合Kalman滤波.粒子滤波实质上是一种基于递推Bayes估计和Monte Carlo模拟的滤波方法,而集合Kalman滤波相当于一种权值相等的粒子滤波.Bayes滤波理论为顺序数据同化提供了更广义的理论框架,从基础的数学理论上揭示了数据同化的基本原理.     

评估集合卡曼滤波反演土壤湿度廓线的性能

集合卡曼滤波由于易于使用而被广泛地应用到陆面数据同化研究中,它是建立在模型为线性、误差为正态分布的假设上,而实际土壤湿度方程是高度非线性的,并且当土壤过干或过湿时会发生样本偏斜.为了全面评估它在同化表层土壤湿度观测来反演土壤湿度廓线的性能,特引入不需要上述假设的采样重要性重采样粒子滤波,比较非线性和偏斜性对同化算法的影响.结果显示:不管是小样本还是大样本,集合卡曼滤波都能快速、准确地逼近样本均值,而粒子滤波只有在大样本时才能缓慢地趋近;此外,集合卡曼滤波的粒子边缘概率密度及其偏度和峰度与粒子滤波完全不同,前者粒子虽不完全满足正态分布,但始终为单峰状态,而后者粒子随同化推进经历了单峰到双峰再到单峰的变化.     

EnKF同化的背景误差协方差矩阵局地化对比研究

在集合数据同化中,背景场误差的协方差估计特别重要。通常有限个成员的集合在估计背景误差协方差矩阵时会引入伪相关,从而造成协方差被低估、滤波发散。虽然协方差膨胀的经验性方法能一定程度缓解协方差被低估的问题,但不能消除协方差的伪相关问题。因此,结合EnKF方案探讨2种消除伪相关的局地化方法(协方差局地化方法和局地分析方法),分析这2种局地化方法对背景误差协方差矩阵、增益矩阵、集合转换矩阵以及同化结果的影响。实验结果表明:局地化方法不仅能消除背景误差协方差矩阵的伪相关,还可以增加背景误差协方差矩阵的秩;在"弱"同化强度下,2种局地化方法的增益矩阵和集合转换矩阵相等;随着同化强度的增大,增益矩阵和集合转换矩阵的差异会变大;在不同的同化强度下,2种局地化方法各具特色,相对而言,协方差局地化方法在更新集合均值和集合扰动上具有较强的鲁棒性。研究结论有助于背景场误差协方差的精细分析和估计。     

不同滤波算法在土壤湿度同化中的应用

为研究不同滤波算法在土壤湿度同化中的有效性,以及土壤湿度模拟结果对模型参数的敏感性,结合简单生物圈模型SiB2,设置敏感性实验,探求土壤饱和水力传导度对土壤湿度模拟结果的影响;并在此基础上,采用集合卡尔曼滤波（EnKF）、无迹卡尔曼滤波（UKF）和无迹粒子滤波（UPF）开展土壤湿度实时同化实验。结果表明：土壤饱和水力传导度能显著影响土壤湿度模拟精度;利用EnKF、UKF、UPF同化站点观测数据,均能改善土壤湿度模拟结果;3种同化方法在不同土壤层的同化效果不同,在土壤表层,EnKF的有效性优于UKF和UPF,在根域层和土壤深层,3种滤波方法有效性在降雨前后相差较大。因此,针对性地选择同化方法,是提高土壤湿度模拟精度的有效手段。     

集合卡尔曼滤波估计水文地质参数的局域化修正

集合卡尔曼滤波(Ensemble Kalman Filter,EnKF)作为一种有效的数据同化方法,在众多数值实验中体现优势的同时,也暴露了它使用小集合估计协方差情况下精度较低的缺陷。为了降低取样噪声对协方差估计的干扰并提高滤波精度,应用局域化函数对小集合估计的协方差进行修正,即在协方差矩阵中以舒尔积的形式增加空间距离权重以限制远距离相关。在一个二维理想孔隙承压含水层模型中的运行结果表明,局域化对集合卡尔曼滤波估计地下水参数的修正十分有效,局域化可以很好地过滤小集合估计中噪声的影响,节省计算量的同时又可以防止滤波发散。相关长度较小的水文地质参数(如对数渗透系数)更容易受到噪声的干扰,更有必要进行局域化修正。     

两种耦合模糊控制的局地化方法研究

在集合数据同化过程中,由于远距离的观测与同化状态之间存在着虚假相关,局地化方法受到广泛关注.此外,由于集合数的限制,容易引起欠采样和协方差被低估等现象,使得滤波效果欠佳.因此,提出模糊控制算法,模糊控制算法主要用于判断观测点与状态更新点之间的距离来匹配相应的观测权重,进而调整局地化系数来更新背景误差协方差和观测误差协方差矩阵,从而得到有效的状态估计.基于背景误差协方差局地化方法和观测误差协方差局地化方法,耦合模糊控制,形成了新的算法—模糊控制的背景误差协方差局地化方法和模糊控制的观测误差协方差局地化方法.利用Lorenz-96模型,在小集合数和局地化半径下,得出模糊控制的背景误差协方差局地化方法和模糊控制的观测误差协方差局地化方法有较好的同化性能.通过分析泰勒图谱甄别出新算法与观测点具有高度的相关性以及较小的空间变异性.最后,在不同维数的模糊控制器下,新算法的有效性进一步得到验证.为今后数据同化误差处理方面提供了良好的研究平台.     

一个基于模拟退火法的陆面数据同化算法

陆面数据同化系统是近年来兴起的新领域。我们发展了一个实验型的陆面数据同化方案,它使用一种启发式优化算法——模拟退火法极小化目标泛函。与变分法和Kalman滤波方法比较,这一算法具有独立于目标泛函的优点,可处理模型和观测算子的非线性和不连续性。使用GAME—Tibet实验中的土壤水分观测值进行单点数值实验,成功地将土壤水分观测同化到陆面过程模型SiB2中。结果表明,与不进行同化相比,土壤水分的估计值有较大改善。     

集合—变分数据同化方法的发展与应用

近年来,集合—变分数据同化方法已成为大气数据同化领域研究的热点问题.该方法能够综合利用集合卡尔曼滤波和变分同化的优势,是实现“集合预报和数据同化一体化”的有效途径.在分析变分同化和集合卡尔曼滤波优缺点的基础上引出集合—变分数据同化的概念;按照不同实现方式,将集合—变分同化分为协方差线性组合和增加控制变量2类,介绍了相应的研究进展,并将集合—变分同化概念拓展;然后介绍了集合—变分同化在英美两国的应用;最后回顾了集合—变分同化研究的主要问题,展望了未来的发展趋势.     

一种新的耦合模糊控制局地化的同化方法

由于在数据同化过程中远距离的观测与同化状态之间存在着虚假相关,局地化方法受到广泛关注。同时,在集合数目较少的同化情况下,观测数据难以得到有效利用,使得同化效果欠佳。因此,提出了一种新的模糊控制局地化同化方法,通过模糊控制算法判断观测点与状态更新点之间的距离,构造观测位置模糊权重。利用非线性Lorenz-96模型,比较分析模糊控制局地化同化(FLETKF)算法与模糊控制同化(FETKF)方法、局地化分析同化(LETKF)算法和集合转换卡尔曼滤波(ETKF)算法在非线性强迫参数变化时的性能,同时探讨了4种算法在不同强度下的优劣。研究结果表明,新方法能够获得更有效的观测权重,避免了远距离观测与状态变量之间的虚假相关,减小由于观测数据难以得到有效利用而带来的误差,在不同观测误差协方差情况下,随着集合数的增加,4种算法中FLETKF能够保持较好的鲁棒性,在观测误差协方差较大时,FLETKF方法的均方根误差(RMSE)相对FETKF方法的RMSE值减小98.2%,提高了同化精度,但在同化所需时间上,由于模糊控制局地化同化方法在判断观测点与状态更新点之间的距离,构造观测位置等价权重需要较长的额外时间,因此,并行计算的性能需进一步研究。     

陆面数据同化系统误差问题研究综述

同化系统中的误差问题一直被认为是制约数据同化性能的瓶颈问题。从分析陆面数据同化系统的误差问题研究现状出发,统一定义了同化系统的误差来源及误差表现,简要综述了顺序同化方法及连续同化方法中的误差定义和相关理论问题。从误差估计的角度,重点介绍了目前研究中各种误差估计的方法和面临的困难。针对误差处理方法的研究,介绍了在集合数据...     

A multiresolution ensemble Kalman filter using the wavelet decomposition

We present a method of using classical wavelet-based multiresolution analysis to separate scales in model and observations during data assimilation with the ensemble Kalman filter. In many applications, the underlying physics of a phenomena involve the interaction of features at multiple scales. Blending of observational and model error across scales can result in large forecast inaccuracies since large errors at one scale are interpreted as inexact data at all scales due to the misrepresentation of observational error. Our method uses a partitioning of the range of the observation operator into separate observation scales. This naturally induces a transformation of the observation covariance and we put forward several algorithms to efficiently compute the transformed covariance. Another advantage of our multiresolution ensemble Kalman filter is that scales can be weighted independently to adjust each scale’s affect on the forecast. To demonstrate feasibility, we present applications to a one-dimensional Kuramoto-Sivashinsky (K–S) model with scale-dependent observation noise and an application involving the forecasting of solar photospheric flux. The solar flux application uses the Air Force Data Assimilative Photospheric Transport (ADAPT) model which has model and observation error exhibiting strong scale dependence. Results using our multiresolution ensemble Kalman filter show significant improvement in solar forecast error compared to traditional ensemble Kalman filtering.     

Cross-covariances and localization for EnKF in multiphase flow data assimilation

The ensemble Kalman filter has been successfully applied for data assimilation in very large models, including those in reservoir simulation and weather. Two problems become critical in a standard implementation of the ensemble Kalman filter, however, when the ensemble size is small. The first is that the ensemble approximation to cross-covariances of model and state variables to data can indicate the presence of correlations that are not real. These spurious correlations give rise to model or state variable updates in regions that should not be updated. The second problem is that the number of degrees of freedom in the ensemble is only as large as the size of the ensemble, so the assimilation of large amounts of precise, independent data is impossible. Localization of the Kalman gain is almost universal in the weather community, but applications of localization for the ensemble Kalman filter in porous media flow have been somewhat rare. It has been shown, however, that localization of updates to regions of non-zero sensitivity or regions of non-zero cross-covariance improves the performance of the EnKF when the ensemble size is small. Localization is necessary for assimilation of large amounts of independent data. The problem is to define appropriate localization functions for different types of data and different types of variables. We show that the knowledge of sensitivity alone is not sufficient for determination of the region of localization. The region depends also on the prior covariance for model variables and on the past history of data assimilation. Although the goal is to choose localization functions that are large enough to include the true region of non-zero cross-covariance, for EnKF applications, the choice of localization function needs to balance the harm done by spurious covariance resulting from small ensembles and the harm done by excluding real correlations. In this paper, we focus on the distance-based localization and provide insights for choosing suitable localization functions for data assimilation in multiphase flow problems. In practice, we conclude that it is reasonable to choose localization functions based on well patterns, that localization function should be larger than regions of non-zero sensitivity and should extend beyond a single well pattern.     

Groundwater parameter estimation using the ensemble Kalman filter with localization

The ensemble Kalman filter (EnKF), an efficient data assimilation method showing advantages in many numerical experiments, is deficient when used in approximating covariance from an ensemble of small size. Implicit localization is used to add distance-related weight to covariance and filter spurious correlations which weaken the EnKF??s capability to estimate uncertainty correctly. The effect of this kind of localization is studied in two-dimensional (2D) and three-dimensional (3D) synthetic cases. It is found that EnKF with localization can capture reliably both the mean and variance of the hydraulic conductivity field with higher efficiency; it can also greatly stabilize the assimilation process as a small-size ensemble is used. Sensitivity experiments are conducted to explore the effect of localization function format and filter lengths. It is suggested that too long or too short filter lengths will prevent implicit localization from modifying the covariance appropriately. Steep localization functions will greatly disturb local dynamics like the 0-1 function even if the function is continuous; four relatively gentle localization functions succeed in avoiding obvious disturbance to the system and improve estimation. As the degree of localization of the L function increases, the parameter sensitivity becomes weak, making parameter selection easier, but more information may be lost in the assimilation process.     

An efficient matrix-free algorithm for the ensemble Kalman filter

In this work, we present an efficient matrix-free ensemble Kalman filter (EnKF) algorithm for the assimilation of large data sets. The EnKF has increasingly become an essential tool for data assimilation of numerical models. It is an attractive assimilation method because it can evolve the model covariance matrix for a non-linear model, through the use of an ensemble of model states, and it is easy to implement for any numerical model. Nevertheless, the computational cost of the EnKF can increase significantly for cases involving the assimilation of large data sets. As more data become available for assimilation, a potential bottleneck in most EnKF algorithms involves the operation of the Kalman gain matrix. To reduce the complexity and cost of assimilating large data sets, a matrix-free EnKF algorithm is proposed. The algorithm uses an efficient matrix-free linear solver, based on the Sherman–Morrison formulas, to solve the implicit linear system within the Kalman gain matrix and compute the analysis. Numerical experiments with a two-dimensional shallow water model on the sphere are presented, where results show the matrix-free implementation outperforming an singular value decomposition-based implementation in computational time.     

Ensemble-Based Seismic and Production Data Assimilation Using Selection Kalman Model

Data assimilation in reservoir modeling often involves model variables that are multimodal, such as porosity and permeability. Well established data assimilation methods such as ensemble Kalman filter and ensemble smoother approaches, are based on Gaussian assumptions that are not applicable to multimodal random variables. The selection ensemble smoother is introduced as an alternative to traditional ensemble methods. In the proposed method, the prior distribution of the model variables, for example the porosity field, is a selection-Gaussian distribution, which allows modeling of the multimodal behavior of the posterior ensemble. The proposed approach is applied for validation on a two-dimensional synthetic channelized reservoir. In the application, an unknown reservoir model of porosity and permeability is estimated from the measured data. Seismic and production data are assumed to be repeatedly measured in time and the reservoir model is updated every time new data are assimilated. The example shows that the selection ensemble Kalman model improves the characterisation of the bimodality of the model parameters compared to the results of the ensemble smoother.

     

Combining sensitivities and prior information for covariance localization in the ensemble Kalman filter for petroleum reservoir applications

Sampling errors can severely degrade the reliability of estimates of conditional means and uncertainty quantification obtained by the application of the ensemble Kalman filter (EnKF) for data assimilation. A standard recommendation for reducing the spurious correlations and loss of variance due to sampling errors is to use covariance localization. In distance-based localization, the prior (forecast) covariance matrix at each data assimilation step is replaced with the Schur product of a correlation matrix with compact support and the forecast covariance matrix. The most important decision to be made in this localization procedure is the choice of the critical length(s) used to generate this correlation matrix. Here, we give a simple argument that the appropriate choice of critical length(s) should be based both on the underlying principal correlation length(s) of the geological model and the range of the sensitivity matrices. Based on this result, we implement a procedure for covariance localization and demonstrate with a set of distinctive reservoir history-matching examples that this procedure yields improved results over the standard EnKF implementation and over covariance localization with other choices of critical length.