总体最小二乘精度评定方法研究

发布时间：2018-03-05 02:10

本文选题：总体最小二乘　切入点：精度评定　出处：《东华理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：如何依据大地测量学科的发展来进一步完善非线性平差理论是一个值得研究的重要问题。总体最小二乘(TLS,total least squares)方法是一类同时顾及观测向量误差和系数矩阵误差的非线性平差方法。相比于总体最小二乘丰富的参数估计算法,总体最小二乘精度评定理论却没有引起足够的重视,尚需要进一步发展。本文依据非线性函数的误差传播理论,基于近似函数表达式和近似函数概率分布两种思想,研究总体最小二乘精度评定的改进方法或新方法,旨在获得更为合理和精度更高的精度评定信息。本文的具体研究如下:研究了精度评定的基于二阶导数的近似函数法。基于高斯-赫尔默特(GH,Gauss-Helmert)模型,本文推导了总体最小二乘参数估值、改正数、观测量及观测量平差值之间一阶近似的协因数阵和互协因数阵计算公式。基于非线性高斯-马尔科夫模型(GM,Gauss-Markov),本文推导了总体最小二乘参数估值和改正数对于观测误差的二阶近似泰勒展开式,依据非线性函数的误差传播公式,进一步给出了适用范围更广的参数估值和改正数偏差以及参数估值二阶精度的协方差阵及均方误差矩阵计算公式。研究了精度评定的sigma点法,包括SUT(scaled unscented transformation)法和Sterling插值法。为了避免复杂的求导运算以及处理难以获取导数的精度评定问题,本文把采用sigma点这种确定样本点的SUT法和Sterling插值法融入到总体最小二乘精度评定中。本文把精度评定分为偏差计算和近似协方差阵或均方误差矩阵计算两个过程,并设计了对sigma点进行非线性变换的两种方案,方案一为把总体最小二乘迭代过程表示成非线性函数,方案二为直接进行总体最小二乘迭代解算。算例结果表明,SUT法和Sterling插值法的精度评定结果能够达到二阶近似精度,采用方案二的SUT法和Sterling插值法适用性更强,SUT法的精度稍优于Sterling插值法,Sterling插值法在实施上比SUT法更简单。研究了精度评定的自适应Monte Carlo法。针对Monte Carlo法模拟次数的选择具有主观性,无法对结果进行直接控制,以及没有同时考虑到总体最小二乘参数估值、改正数和单位权方差估值有偏性等问题,本文把自适应Monte Carlo法融入到总体最小二乘精度评定理论中,并明确了数值容差的含义和选择方法。通过基于自适应Monte Carlo法的偏差计算和近似协方差阵计算,本文设计了总体最小二乘精度评定的一套算法流程。基于对偶变量思想,提出了参数估值偏差计算的对偶自适应MonteCarlo法。算例结果表明,自适应Monte Carlo法能够自主选择模拟次数,同时兼顾计算结果的精度和计算量,获得稳定且合理的精度评定结果;对偶自适应Monte Carlo法计算估值偏差的收敛速度更快,效率更高。把近似函数法、sigma点法和对偶自适应Monte Carlo法应用到震源参数估值对格林函数系数矩阵的影响分析中。考虑到滑动分布反演中格林函数矩阵元素是震源参数估值的非线性函数,震源参数估值的随机性使得滑动分布反演成为一类总体最小二乘参数估计问题。本文通过依据矩形位错模型计算位移偏差来分析不同精度的断层长度、宽度、深度和倾角对断层单位走滑位错、单位倾滑位错和单位张裂位错对应的位移产生的影响,以期为总体最小二乘法的使用和格林函数矩阵定权提供一定的依据。模拟断层结果表明,sigma点法的计算效率最高;矩形位错模型的非线性主要体现在二阶项;位移偏差大约集中在以断层中心为中心的5km范围内,主要的位移偏差位于断层附近;单位张裂位错对应的位移受震源参数估值的影响最大,单位倾滑位错次之,单位走滑位错最小;三种单位位错对应的垂直位移比平面位移更易受震源参数估值的影响;当位移偏差接近毫米级时,可以考虑总体最小二乘方法。
[Abstract]:According to the development of Geodesy to further improve the nonlinear adjustment theory is an important issue worthy of study. The total least squares (TLS total, least squares) is a class of nonlinear vector error coefficient matrix and taking into account the observation error adjustment method. Compared to the total least squares parameter estimation algorithm is rich, the accuracy of total least squares the evaluation theory has not caused enough attention, it still needs further development. Based on the error propagation theory of nonlinear function, approximate function and approximate probability distribution function of two kinds of thoughts based on the improved method of assessing the accuracy of total least squares or new methods, in order to obtain more information and reasonable accuracy evaluation of higher accuracy. The research of this paper is as follows: Based on the precision of the approximation method for two order derivative evaluation. Based on the Gauss Hull silent Special (GH, Gauss-Helmert) model, in this paper the total least squares parameter estimation and correction, the covariance matrix and cross covariance matrix calculation formula between the measured values and measured values of level difference of one order approximation. The nonlinear Gauss Markoff model based on (GM, Gauss-Markov), this paper deduces the general least squares parameter estimation and correction the observation error of the two order approximation of Taylor expansion, according to the error propagation formula of nonlinear function, further is given a wider range of parameter estimation and correction of deviation and covariance matrix to estimate the parameters of two order accuracy and mean square error matrix. The calculation formula of assessing the accuracy of Sigma method, including SUT (scaled unscented transformation) method and Sterling interpolation method. In order to avoid the complicated derivation of accuracy evaluation and treatment is difficult to obtain the derivative of the sigma, using the To determine the total least squares into the accuracy assessment of sample points in the SUT method and Sterling interpolation method. The accuracy of calculation and the approximate covariance matrix is divided into deviation or mean square error matrix of two processes, and two design scheme of sigma nonlinear transform, a scheme for the general least squares iterative process representation a nonlinear function scheme for direct total least squares iterative solution. Numerical results show that the evaluation results of SUT method and Sterling interpolation accuracy can reach two order approximation accuracy, the scheme of two SUT method and Sterling interpolation method is more applicable, the precision of SUT method is slightly better than Sterling interpolation, Sterling interpolation in the implementation method is more simple than SUT. The study of adaptive Monte Carlo method of precision evaluation is subjective. For Monte Carlo simulation of the choice of the number, can not be straight to the results Take control, and not taking into account the total least squares parameter estimation and correction of unit weight variance and bias of the problem, the adaptive Monte Carlo method into the total least squares accuracy evaluation theory, and makes clear the definition and selection method. Through the numerical tolerance deviation adaptive Monte Carlo method and approximate calculation of covariance based on matrix calculation, this paper designs the overall accuracy evaluation of a set of least squares algorithm process. The dual variables based on the proposed dual adaptive MonteCarlo method parameter estimation deviation calculation. Numerical results show that the adaptive Monte Carlo method can choose the number of simulations, both the precision and computation results, to obtain stable accuracy evaluation results and the reasonable convergence rate; valuation deviation calculation method of dual adaptive Monte Carlo faster and more efficient. The approximate function method, s Igma method and dual adaptive Monte Carlo method is applied to analyze the influence of source parameters valuation of Green function coefficient matrix. Considering the Green function matrix element inversion of slip distribution is a nonlinear function of source parameters estimation, stochastic parameter estimation source makes the slip distribution inversion become a total least squares parameter estimation problem. This paper through the rectangular according to the calculated displacement deviation of dislocation model to analyze the different accuracy of fault length, width, depth and dip angle of fault unit slip dislocations, dislocation and dip slip displacement unit unit corresponding to the tensile fracture dislocation, in order to provide a basis for the use of Green function and total least squares weighting matrix. The simulation results of fault show that the sigma method to calculate the highest efficiency; nonlinear rectangular dislocation model is mainly embodied in two order displacement deviation about set; In the center of the fault in the range of 5km, mainly located near the fault displacement deviation; displacement unit corresponding to dislocation rifting influence of source parameter estimation of the maximum dip slip dislocation of the unit, the unit of strike slip dislocation minimum; vertical displacement of three kinds of units corresponding to the dislocation influence of source parameter estimation more easily than the plane displacement; when the displacement error is close to the mm level, can consider the total least squares method.

【学位授予单位】：东华理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：P207

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