基于Sigma点的非线性平差精度评定理论与方法研究

How to improve the theory on precision estimation of nonlinear adjustment is a very worthy of study. From the perspective of sigma point, the second and higher order information of precision estimation is expected to obtain based on processes of bias calculation and variance or covariance calculation. The aim of this project is a try to break through limitations existing in theory on precision estimation of depending on derivatives and ignoring biases of adjustment estimations. Methods of unscented transformation and Sterling interpolation based on determined sigma point, and adaptive Monte Carlo based on random sigma point are proposed for precision estimation related to general, constrained and ill-posed nonlinear adjustment models. And methods of Jackknife and Bootstrap based on resampling sigma point are applied and improved for nonlinear adjustment model with systematic error or gross error. On that basis, the effect of parameter estimation on nonlinear model and assessing nonlinearity are analyzed and conducted by proposed methods. The theory on precision estimation is applied for fields of surveying data processing, such as data fitting, navigation and position, and geodetic inversion. The research of this project is significant in aspects of theory and application, which also further improves and develops nonlinear adjustment theory.

如何完善非线性平差的精度评定理论是一个值得研究的问题。本项目从Sigma点的角度，通过偏差计算和方差或协方差计算，有望给出参数估值二阶精度及以上的精度评定信息，旨在突破现有精度评定理论依赖导数，没有顾及平差估值有偏性的局限性。本项目提出基于确定Sigma点的unscented transformation法和Sterling插值法及基于随机Sigma点的自适应Monte Carlo法处理一般的、附有约束和病态的非线性平差模型；提出基于重采样Sigma点的Jackknife法和Bootstrap法处理含有系统误差或粗差的非线性平差模型。在此基础上，本项目利用提出的方法进一步分析非线性平差参数估值对非线性模型的影响，进行非线性评价，把研究的精度评定理论应用于数据拟合、导航定位、大地测量反演等测量数据处理领域。本研究在理论与应用方面都具有重要的意义，是对非线性平差理论的进一步完善。

本项目系统研究了基于Sigma点的非线性平差精度评定理论与方法，超额完成了项目既定的全部研究内容，并达成了预期研究目标。取得的主要成果有：从近似函数概率分布的角度，提出了非线性平差精度评定的基于确定Sigma点的unscented transformation法、基于确定Sigma点的Sterling插值法、基于随机Sigma点的自适应Monte Carlo法、基于重采样Sigma点的Jackknife法、基于重采样Sigma点的Bootstrap法。构建的新型非线性平差精度评定理论与方法，解决了传统近似函数表达式法依赖于复杂求导运算的问题。本项目将整个精度评定理论体系分为非线性平差估值偏差计算和参数估值方差或协方差计算两个过程，给出了参数估值二阶精度以上且更为充分的精度评定信息，完善了非线性平差模型的精度评定理论框架。在非线性平差的精度评定理论的基础上，本项目利用精度评定方法进一步分析非线性平差参数估值对非线性模型的影响，并进行非线性评价。本项目的理论和方法在地震断层参数反演、火山Mogi模型参数反演、火山CDM模型参数反演、GNSS基线向量解算、数字高程DEM、坐标转换、前方交会、高斯投影坐标正算、卫星钟差预报、空间直线拟合、椭圆拟合等测量数据处理领域得到应用和证明。本项目的研究在理论与应用方面都具有重要的意义，进一步完善和发展了大地测量数据处理理论。项目研究成果发表论文62篇，其中SCI/EI检索共49篇；参加学术会议交流报告12人次；培养硕士研究生14名，获得江西省优秀硕士学位论文2篇，东华理工大学优秀硕士学位论文8篇。