基于集合卡尔曼滤波数据同化方法的岩土力学参数时空变异性研究
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摘要
岩土工程问题中,施工扰动引起的岩土体位移变化过程实际上是一个非确定性过程,很难用数学方法准确地描述清楚,它客观地处于一个动态的随机过程。岩土力学参数并非某一确定量,在时间上是处于某种随机过程的状态估计量,需要采用随机系统的方法来解决。同时,岩土力学参数还在空间域上呈现出高度的非均质性、各向异性,需要引入随机场的理论和方法来表征。因而,将随机系统分析方法和岩土工程数值模拟相结合,充分考虑岩土力学参数在时间域和空间域上的变异性,并建立一种适应于岩土工程实际动态的随机反分析理论及相应的计算方法,已成为客观需要和发展趋势。论文引入来源于现代大气和海洋科学的集合卡尔曼滤波(Ensemble Kalman Filter)顺序数据同化(Sequential Data Assimilation)的思想和方法,在考虑观测数据时空分布的基础上,在数值模型的动态运行过程中不断融合新的观测数据,逐步估计力学参数的数值大小和空间分布。论文研究主要内容包括:
1.视岩土变形体为一个随机动态系统,将位移观测值作为系统的输出,用集合卡尔曼滤波模型来描述系统的状态。基于随机过程理论,利用时间序列的位移观测数据,考虑岩土力学参数的时变性,提出了集合卡尔曼滤波耦合数值模拟的岩土力学参数动态随机估计方法。在估值过程中,岩土力学参数每一步都经过滤波器的滤波计算,并产生新息和滤波值,反映了岩土力学参数变化的历时性和岩土体变形过程的真实性。
2.针对标准Kalman滤波在岩土力学参数估计中预测误差协方差矩阵计算和存储困难以及扩展Kalman滤波应用于非线性系统的近似问题,采用了Monte Carlo集合预测来估计预测误差协方差,把误差的统计量隐含在一组预测变量中,根据预测值的差异进行统计,得到新的误差协方差,避免了协方差演变方程预测过程中出现的计算不准确和关于协方差矩阵的大量数据的存储问题。在增益矩阵的计算过程中采用了奇异值分解方法,避免了矩阵的不满秩问题,并将集合卡尔曼滤波算法与现有大型数值模拟软件ANSYS、FLAC、FLAC3D等相耦合,通过数值模拟分析将模型预测和现场观测有机结合,利用观测数据来调整模型的运行轨迹,使积累的误差得到“释放”,充分发挥了监测和模拟各自的优势。
3.基于开挖扰动引起的岩土体位移变化过程是一个随机动态过程,测量位移是含误差的不定值,提出动态观测扰动的概念,采用Monte-Carlo模拟对观测值施加高斯白噪声,并在参数随机动态估计过程中使用了先验知识,保证了计算的稳定性和降低了结果的不确定性。分析结果给出了状态变量的一个集合,同时估计了岩土力学模型的状态值和参数值,不仅可以为集合预测提供初始值,而且在给出岩土力学参数集合平均的最佳估计之外,同时还提供了估计值的不确定性。
4.基于地质统计学理论,将分布于研究区的岩土力学参数视为区域化变量,通过变异函数描述岩土力学参数整体的空间结构性变化及局部的随机性变化,以变异函数理论模型作为刻画岩土力学参数空间变异规律的数学模型。采用LU协方差分解和高斯顺序模拟(SGS)方法构造岩土力学参数空间分布的随机场并对单元体力学参数进行赋值,真实地再现了岩土力学参数随机场具有的离散性和波动性。提出了岩土力学参数空间变异性研究的集合卡尔曼滤波数据同化方法,通过逐步融合时空分布的动态观测数据,使各个不同的参数集合成员之间的变异性逐渐减小,逐步逼近真实场的空间分布模式,把观测数据与模型模拟结果集成为具有时间一致性、空间一致性和物理一致性的各种状态的数据集。
5.给出了数据同化结果评价的根均方误差标准和集合散度标准,分析和讨论了采样集合大小、采样集合初始均值、动态噪声比例、相关步距、同化步数、测点数目等影响因素的敏感性,论证了集合卡尔曼滤波耦合岩土力学数值分析方法在参数空间变异性研究中的实用性和有效性。
It is difficult to describe clearly with mathematical method for displacement variation under excavation disturbance in geotechnical engineering, which actual is an uncertainty process and exist objectively in a dynamical stochastic process. Geomechanical parameter is not a certainty value, but a state estimation value in time existing in a stochastic process. So there is a demand for method of stochastic system to solve it. In addition, geomechanical parameter presents great heterogeneity and anisotropy in space, and there is a demand for random field theory to represent it. Therefore, inconsideration of variability of geomechancial parameter in time and space, it is an objective demand and a development trend to incorporate analysis of stochastic system with numerical modeling of geotechnical engineering, and established theory of stochastic back-analysis conforming with practical dynamics in geotechnical engineering, as well as corresponding computational procedure. The ideal and method of sequential data assimilation with ensemble Kalman filter originated from meteorology and oceanography 7is introduced in dissertation, and numerical magnitude and spatial distribution of geomechanical parameter are estimated on basis of observation data distributed in time and space after model incorporated new observation data in process of running. Main research content in dissertation includes the following:
1. Geomechanical deformation is treated as a dynamic stochastic system, and displacement observation is looked as the output. Furthermore, ensemble Kalman filter is used as model to describe the system state. The dynamical estimation method coupled ensemble Kalman filter with numerical modeling is presented on basis of stochastic process, which considers time variation property of geomechanical parameter and utilizes displacement observation in time series. Geomechanical parameter is filtered every step during estimation process, then innovation and adjusted value are generated, which reflects the diachronism of geomechanical parameter the truth of deformation in geomechanical media.
2. Aim at computing and storing trouble of forecast error covariance matrix of Kalman filter applied in geomechanical parameter estimation, as well as approximating problem of extended Kalman filter applied in nonlinear system, ensemble forecast with Monte Carlo is adopted to estimate forecast error covariance, which implies the statistical error in a set of forecast variable. Thus, new error covariance can be obtained according to difference statistical of forecast value, so it is avoided for computing inaccuracy in process of covariance evolution equation forecasting, as well as storing difficulty of massive data in covariance matrix. Furthermore, singular value decomposition is adopted to compute the Kalman gain matrix. What’s more, the ensemble Kalman filter algorithm is incorporated with available numerical simulating software, such as ANSYS, FLAC, FLAC3D, which fuses in situ observation into model forecast. By adjust the model running with observation data, the accumulative error is released. Therefore, the advantages of monitoring and modeling are adequately exerted.
3. It is under a dynamical stochastic process for displacement variation under excavation disturbance in geomechanical media, and the measuring displacement is an uncertainty value with error. The dynamical disturbed observation concept is posed, then Monte Carlo simulation is adopted to add Gaussian white noise in observation. At the same time, a priori knowledge is utilized to ensure computational stability and reduce resultant uncertainty in process of dynamical stochastic estimation. Analysis result gives a ensemble of state variable, and also estimates simultaneously the state and parameter value of geomechanical model, which not only can afford the initial value for ensemble prediction, but also can offer the estimated uncertainty, besides the optimal estimation of geomechanical parameter ensemble mean.
4. Geomechanical parameter is viewed as zonal variable in terms of geostatistics. Sequentially, variation function is given to describe integral spatial structural variation and local stochastic variation, and theoretical model of variation function is served as mathematical model to depict spatial variation law of geomechanical parameter. Moreover, LU decomposition and Gaussian Sequential Simulation are adopted to construct the random field representing spatial distribution of geomechanical parameter, and then assign to element, which recur veritably the discreteness and fluctuation in random field of geomechanical parameter. The sequential data assimilation method with ensemble Kalman filter is presented to research spatial variation of geomechanical parameter. After gradually fusing dynamical observation data distributed in time and space, the variability in various ensemble realizations decrease step by step, and all approach to the distribution pattern of true field finally. The data assimilation with ensemble Kalman filter integrates observation data and simulation result into various state data sets with consistency of time, space and physics.
5. The root mean square error and ensemble spread criteria are put forward to evaluate the data assimilation result. Then, sensitivity is discussed for some influencing factors, such as sampling ensemble size, initial mean of sampling ensemble, dynamical noise ratio, correlative distance, assimilation step, observational number. It is demonstrated that the method of ensemble Kalman filter coupled with numerical modeling is practical and valid.
1.视岩土变形体为一个随机动态系统,将位移观测值作为系统的输出,用集合卡尔曼滤波模型来描述系统的状态。基于随机过程理论,利用时间序列的位移观测数据,考虑岩土力学参数的时变性,提出了集合卡尔曼滤波耦合数值模拟的岩土力学参数动态随机估计方法。在估值过程中,岩土力学参数每一步都经过滤波器的滤波计算,并产生新息和滤波值,反映了岩土力学参数变化的历时性和岩土体变形过程的真实性。
2.针对标准Kalman滤波在岩土力学参数估计中预测误差协方差矩阵计算和存储困难以及扩展Kalman滤波应用于非线性系统的近似问题,采用了Monte Carlo集合预测来估计预测误差协方差,把误差的统计量隐含在一组预测变量中,根据预测值的差异进行统计,得到新的误差协方差,避免了协方差演变方程预测过程中出现的计算不准确和关于协方差矩阵的大量数据的存储问题。在增益矩阵的计算过程中采用了奇异值分解方法,避免了矩阵的不满秩问题,并将集合卡尔曼滤波算法与现有大型数值模拟软件ANSYS、FLAC、FLAC3D等相耦合,通过数值模拟分析将模型预测和现场观测有机结合,利用观测数据来调整模型的运行轨迹,使积累的误差得到“释放”,充分发挥了监测和模拟各自的优势。
3.基于开挖扰动引起的岩土体位移变化过程是一个随机动态过程,测量位移是含误差的不定值,提出动态观测扰动的概念,采用Monte-Carlo模拟对观测值施加高斯白噪声,并在参数随机动态估计过程中使用了先验知识,保证了计算的稳定性和降低了结果的不确定性。分析结果给出了状态变量的一个集合,同时估计了岩土力学模型的状态值和参数值,不仅可以为集合预测提供初始值,而且在给出岩土力学参数集合平均的最佳估计之外,同时还提供了估计值的不确定性。
4.基于地质统计学理论,将分布于研究区的岩土力学参数视为区域化变量,通过变异函数描述岩土力学参数整体的空间结构性变化及局部的随机性变化,以变异函数理论模型作为刻画岩土力学参数空间变异规律的数学模型。采用LU协方差分解和高斯顺序模拟(SGS)方法构造岩土力学参数空间分布的随机场并对单元体力学参数进行赋值,真实地再现了岩土力学参数随机场具有的离散性和波动性。提出了岩土力学参数空间变异性研究的集合卡尔曼滤波数据同化方法,通过逐步融合时空分布的动态观测数据,使各个不同的参数集合成员之间的变异性逐渐减小,逐步逼近真实场的空间分布模式,把观测数据与模型模拟结果集成为具有时间一致性、空间一致性和物理一致性的各种状态的数据集。
5.给出了数据同化结果评价的根均方误差标准和集合散度标准,分析和讨论了采样集合大小、采样集合初始均值、动态噪声比例、相关步距、同化步数、测点数目等影响因素的敏感性,论证了集合卡尔曼滤波耦合岩土力学数值分析方法在参数空间变异性研究中的实用性和有效性。
It is difficult to describe clearly with mathematical method for displacement variation under excavation disturbance in geotechnical engineering, which actual is an uncertainty process and exist objectively in a dynamical stochastic process. Geomechanical parameter is not a certainty value, but a state estimation value in time existing in a stochastic process. So there is a demand for method of stochastic system to solve it. In addition, geomechanical parameter presents great heterogeneity and anisotropy in space, and there is a demand for random field theory to represent it. Therefore, inconsideration of variability of geomechancial parameter in time and space, it is an objective demand and a development trend to incorporate analysis of stochastic system with numerical modeling of geotechnical engineering, and established theory of stochastic back-analysis conforming with practical dynamics in geotechnical engineering, as well as corresponding computational procedure. The ideal and method of sequential data assimilation with ensemble Kalman filter originated from meteorology and oceanography 7is introduced in dissertation, and numerical magnitude and spatial distribution of geomechanical parameter are estimated on basis of observation data distributed in time and space after model incorporated new observation data in process of running. Main research content in dissertation includes the following:
1. Geomechanical deformation is treated as a dynamic stochastic system, and displacement observation is looked as the output. Furthermore, ensemble Kalman filter is used as model to describe the system state. The dynamical estimation method coupled ensemble Kalman filter with numerical modeling is presented on basis of stochastic process, which considers time variation property of geomechanical parameter and utilizes displacement observation in time series. Geomechanical parameter is filtered every step during estimation process, then innovation and adjusted value are generated, which reflects the diachronism of geomechanical parameter the truth of deformation in geomechanical media.
2. Aim at computing and storing trouble of forecast error covariance matrix of Kalman filter applied in geomechanical parameter estimation, as well as approximating problem of extended Kalman filter applied in nonlinear system, ensemble forecast with Monte Carlo is adopted to estimate forecast error covariance, which implies the statistical error in a set of forecast variable. Thus, new error covariance can be obtained according to difference statistical of forecast value, so it is avoided for computing inaccuracy in process of covariance evolution equation forecasting, as well as storing difficulty of massive data in covariance matrix. Furthermore, singular value decomposition is adopted to compute the Kalman gain matrix. What’s more, the ensemble Kalman filter algorithm is incorporated with available numerical simulating software, such as ANSYS, FLAC, FLAC3D, which fuses in situ observation into model forecast. By adjust the model running with observation data, the accumulative error is released. Therefore, the advantages of monitoring and modeling are adequately exerted.
3. It is under a dynamical stochastic process for displacement variation under excavation disturbance in geomechanical media, and the measuring displacement is an uncertainty value with error. The dynamical disturbed observation concept is posed, then Monte Carlo simulation is adopted to add Gaussian white noise in observation. At the same time, a priori knowledge is utilized to ensure computational stability and reduce resultant uncertainty in process of dynamical stochastic estimation. Analysis result gives a ensemble of state variable, and also estimates simultaneously the state and parameter value of geomechanical model, which not only can afford the initial value for ensemble prediction, but also can offer the estimated uncertainty, besides the optimal estimation of geomechanical parameter ensemble mean.
4. Geomechanical parameter is viewed as zonal variable in terms of geostatistics. Sequentially, variation function is given to describe integral spatial structural variation and local stochastic variation, and theoretical model of variation function is served as mathematical model to depict spatial variation law of geomechanical parameter. Moreover, LU decomposition and Gaussian Sequential Simulation are adopted to construct the random field representing spatial distribution of geomechanical parameter, and then assign to element, which recur veritably the discreteness and fluctuation in random field of geomechanical parameter. The sequential data assimilation method with ensemble Kalman filter is presented to research spatial variation of geomechanical parameter. After gradually fusing dynamical observation data distributed in time and space, the variability in various ensemble realizations decrease step by step, and all approach to the distribution pattern of true field finally. The data assimilation with ensemble Kalman filter integrates observation data and simulation result into various state data sets with consistency of time, space and physics.
5. The root mean square error and ensemble spread criteria are put forward to evaluate the data assimilation result. Then, sensitivity is discussed for some influencing factors, such as sampling ensemble size, initial mean of sampling ensemble, dynamical noise ratio, correlative distance, assimilation step, observational number. It is demonstrated that the method of ensemble Kalman filter coupled with numerical modeling is practical and valid.
引文
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