本文選題：總體最小二乘　切入點(diǎn)：精度評(píng)定　出處：《東華理工大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：如何依據(jù)大地測(cè)量學(xué)科的發(fā)展來進(jìn)一步完善非線性平差理論是一個(gè)值得研究的重要問題�？傮w最小二乘(TLS,total least squares)方法是一類同時(shí)顧及觀測(cè)向量誤差和系數(shù)矩陣誤差的非線性平差方法。相比于總體最小二乘豐富的參數(shù)估計(jì)算法,總體最小二乘精度評(píng)定理論卻沒有引起足夠的重視,尚需要進(jìn)一步發(fā)展。本文依據(jù)非線性函數(shù)的誤差傳播理論,基于近似函數(shù)表達(dá)式和近似函數(shù)概率分布兩種思想,研究總體最小二乘精度評(píng)定的改進(jìn)方法或新方法,旨在獲得更為合理和精度更高的精度評(píng)定信息。本文的具體研究如下:研究了精度評(píng)定的基于二階導(dǎo)數(shù)的近似函數(shù)法。基于高斯-赫爾默特(GH,Gauss-Helmert)模型,本文推導(dǎo)了總體最小二乘參數(shù)估值、改正數(shù)、觀測(cè)量及觀測(cè)量平差值之間一階近似的協(xié)因數(shù)陣和互協(xié)因數(shù)陣計(jì)算公式。基于非線性高斯-馬爾科夫模型(GM,Gauss-Markov),本文推導(dǎo)了總體最小二乘參數(shù)估值和改正數(shù)對(duì)于觀測(cè)誤差的二階近似泰勒展開式,依據(jù)非線性函數(shù)的誤差傳播公式,進(jìn)一步給出了適用范圍更廣的參數(shù)估值和改正數(shù)偏差以及參數(shù)估值二階精度的協(xié)方差陣及均方誤差矩陣計(jì)算公式。研究了精度評(píng)定的sigma點(diǎn)法,包括SUT(scaled unscented transformation)法和Sterling插值法。為了避免復(fù)雜的求導(dǎo)運(yùn)算以及處理難以獲取導(dǎo)數(shù)的精度評(píng)定問題,本文把采用sigma點(diǎn)這種確定樣本點(diǎn)的SUT法和Sterling插值法融入到總體最小二乘精度評(píng)定中。本文把精度評(píng)定分為偏差計(jì)算和近似協(xié)方差陣或均方誤差矩陣計(jì)算兩個(gè)過程,并設(shè)計(jì)了對(duì)sigma點(diǎn)進(jìn)行非線性變換的兩種方案,方案一為把總體最小二乘迭代過程表示成非線性函數(shù),方案二為直接進(jìn)行總體最小二乘迭代解算。算例結(jié)果表明,SUT法和Sterling插值法的精度評(píng)定結(jié)果能夠達(dá)到二階近似精度,采用方案二的SUT法和Sterling插值法適用性更強(qiáng),SUT法的精度稍優(yōu)于Sterling插值法,Sterling插值法在實(shí)施上比SUT法更簡(jiǎn)單。研究了精度評(píng)定的自適應(yīng)Monte Carlo法。針對(duì)Monte Carlo法模擬次數(shù)的選擇具有主觀性,無法對(duì)結(jié)果進(jìn)行直接控制,以及沒有同時(shí)考慮到總體最小二乘參數(shù)估值、改正數(shù)和單位權(quán)方差估值有偏性等問題,本文把自適應(yīng)Monte Carlo法融入到總體最小二乘精度評(píng)定理論中,并明確了數(shù)值容差的含義和選擇方法。通過基于自適應(yīng)Monte Carlo法的偏差計(jì)算和近似協(xié)方差陣計(jì)算,本文設(shè)計(jì)了總體最小二乘精度評(píng)定的一套算法流程�；趯�(duì)偶變量思想,提出了參數(shù)估值偏差計(jì)算的對(duì)偶自適應(yīng)MonteCarlo法。算例結(jié)果表明,自適應(yīng)Monte Carlo法能夠自主選擇模擬次數(shù),同時(shí)兼顧計(jì)算結(jié)果的精度和計(jì)算量,獲得穩(wěn)定且合理的精度評(píng)定結(jié)果;對(duì)偶自適應(yīng)Monte Carlo法計(jì)算估值偏差的收斂速度更快,效率更高。把近似函數(shù)法、sigma點(diǎn)法和對(duì)偶自適應(yīng)Monte Carlo法應(yīng)用到震源參數(shù)估值對(duì)格林函數(shù)系數(shù)矩陣的影響分析中�？紤]到滑動(dòng)分布反演中格林函數(shù)矩陣元素是震源參數(shù)估值的非線性函數(shù),震源參數(shù)估值的隨機(jī)性使得滑動(dòng)分布反演成為一類總體最小二乘參數(shù)估計(jì)問題。本文通過依據(jù)矩形位錯(cuò)模型計(jì)算位移偏差來分析不同精度的斷層長度、寬度、深度和傾角對(duì)斷層單位走滑位錯(cuò)、單位傾滑位錯(cuò)和單位張裂位錯(cuò)對(duì)應(yīng)的位移產(chǎn)生的影響,以期為總體最小二乘法的使用和格林函數(shù)矩陣定權(quán)提供一定的依據(jù)。模擬斷層結(jié)果表明,sigma點(diǎn)法的計(jì)算效率最高;矩形位錯(cuò)模型的非線性主要體現(xiàn)在二階項(xiàng);位移偏差大約集中在以斷層中心為中心的5km范圍內(nèi),主要的位移偏差位于斷層附近;單位張裂位錯(cuò)對(duì)應(yīng)的位移受震源參數(shù)估值的影響最大,單位傾滑位錯(cuò)次之,單位走滑位錯(cuò)最小;三種單位位錯(cuò)對(duì)應(yīng)的垂直位移比平面位移更易受震源參數(shù)估值的影響;當(dāng)位移偏差接近毫米級(jí)時(shí),可以考慮總體最小二乘方法。
[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.

【學(xué)位授予單位】：東華理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：P207

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本文編號(hào)：1568284