Adjustment Model and Colored Noise Reduction of Continuous Observation System

Abstract

Abstract:

Nowadays, surveying technique mainly relies on continuous earth observation systems. In practice, while the precision and the reliability of estimated parameters are seriously affected by colored noises, these colored noises contain a great amount of useful information to the earth science. On the one hand, the affection caused by these colored noises should be taken into account to the adjustment model, on the other hand, as useful signals these colored noises can be accurately identified and extracted by Fourier analysis. In this paper, a continuous adjustment model is introduced with respect to the colored noises, and then we generalize the traditional adjustment theory from the finite space to the infinite space so called as Hilbert space. This extension is to provide a new technique to perform the continuous observational system design, Fourier analysis as well as the parameter estimation. It shows that the Gramer’s determinant provides maximization criteria in the system optimization design as well as a rule in diagnosing the adjustment model. Related with the definition of the integral, the least squares solution of the continuous adjustment model becomes the limit of the traditional least squares solution in finite space. Moreover, the influence caused by the colored noises is systematic, but it can be eliminated by optimally designing the observational system and the observational scheme.

Key words: adjustment, continuous observation, least squares, colored noise, Hilbert space

CLC Number:

p208

Xue shuqiang Yuanxi Yang. Adjustment Model and Colored Noise Reduction of Continuous Observation System[J]. Acta Geodaetica et Cartographica Sinica.

References

[1] 周江文. 系统误差的数学处理. 测绘工程 1999;8:1-4. (ZHOU Jiangwen, Mathematical Processing of Problems with Systematic Errors, ENGINEERING OF SURVEYING AND MAPPING, 1999(8)2:1-4.).
[2] 杨元喜, 崔先强. 动态定位有色噪声影响函数——以一阶AR模型为例. 测绘学报 2003;32:6-10. (YANG Yuanxi, CUI Xianqiang. Influence Functions of Colored Noises on Kinematic Positioning—Taking the AR Model of First Class As an Example[J].Acta Geodaetica et Cartographica Sinica, 2003(32)1:110-114).
[3] 杨元喜. 卫星导航的不确定性、不确定度与精度若干注记. 测绘学报 2012 41:646-50. (YANG Yuanxi. Some Notes on Uncertainty,Uncertainty Measure and Accuracy in Satellite Navigation[J].Acta Geodaetica et Cartographica Sinica, 2012(41)5:646-650 ).
[4] 黄振友, 杨建新, 华踏红, 刘景麟. 泛函分析. 北京: 科学出版社(HUANG Jianxin, YANG Jianxin, HUA Tahong, LIU Jinlin Functional Analysis, Science Press, 2003) 2003.
[5] 曾令候. 泛函方法在测量平差中的应用——基本理论. 测绘学报 1962:176-84. (ZENG Linghou. Applications of Functional Analysis Method in Adjustment—Basic Theory[J].Acta Geodaetica et Cartographica Sinica, 1962(5)3:176-183).
[6] 杨元喜. 泛函分析在最小二乘平差中的应用. 测绘学报 1986;15:110-4. (YANG Yuanxi. Funcutional Alalysis and Its Applications to Least Squares Adjustment[J].Acta Geodaetica et Cartographica Sinica, 1986(15)2:110-114).
[7] 党诵诗. 关于测量平差中的几个问题. 测绘学报 1983;12:208-13. (Dang Songshi. Some Problems on the Adjustment in Surveying[J].Acta Geodaetica et Cartographica Sinica, 1983(12)3:208-213).
[8] 祝永刚, 白亿同. 按泛函分析理论来剖析平差计算中的数学结构. 武汉测绘学院学报 1985:50-6. (ZHU Yonggang, BAI Yitong. Study of the Mathematical Structure of Adjustment Computation based upon the Theory of Functional Analysis, Geomatics and Information Science of Wuhan University,1985, 4:50-56.).
[9] Teunissen PJG, Technische Universiteit Delft. Faculty of Civil Engineering and Geosciences. Dept. of Mathematical Geodesy and Positioning. Adjustment theory : an introduction. 1. druk. ed. Delft: Delft University Press; 2000.
[10] Grafarend EW. Linear and nonlinear models : fixed effects, random effects, and mixed models. Berlin ; New York: Walter de Gruyter; 2006.
[11] 王珂, 肖鹏峰, 冯学智, 吴桂平, 李晖. 基于改进二维离散希尔伯特变换的图像边缘检测方法. 测绘学报 2012;41:421-7+33. (WANG Ke, XIAO Pengfeng, FENG Xuezhi, WU Guiping, LI Hui. The Modified Algorithm of Image Edege Features Detection from Phase Congruency Model Based on 2-D Hi lbert Transformt[J].Acta Geodaetica et Cartographica Sinica, 2012(41)3:421-427+433).
[12] 谭琨, 杜培军. 基于再生核Hilbert空间小波核函数支持向量机的高光谱遥感影像分类. 测绘学报 2011;40:142-7.(TAN Kun, DU Peijun, Dang Songshi. Wavelet Support Vector Machines Based on Reproducing Kernel Hilbert Space for Hyperspectral Remote Sensing Image Classification[J].Acta Geodaetica et Cartographica Sinica, 2011(40)2:142-147.).
[13] 李新武, 郭华东, 廖静娟, 王长林, 范典. 基于快速傅立叶变换的干涉SAR基线估计. 测绘学报 2003;32:70-2. (LI Xinwu, GUO Huadong, LIAO Jingjuan,WANG Changlin, FAN Dian. Baseline Estimation of Interferometric SAR Based on Fast Fourier Transform[J].Acta Geodaetica et Cartographica Sinica, 2003(32)1:70-72.).
[14] 刘大杰, 于正林. 动态测量系统与卡尔曼滤波. 测绘学报 1988;17:254-62. (LIU Dajie, YU Zhenglin. Dynamic Surveying System and Kalman Filter[J].Acta Geodaetica et Cartographica Sinica, 1988(17)4:254-262.).
[15] 汪嘉祯. 拟合自由项解算多普勒单点定位. 测绘学报 1988;01:9-20. (WANG Jiazhen. A Solution of Doppler Single Point Positioning by Fitting Free Terns[J].Acta Geodaetica et Cartographica Sinica, 1988,(17)1:9-20).
[16] Courant R, Hilbert D. Methods of Mathematical Physics. JOHN WILEY & SONS 1989.
[17] 陈永奇, 吴子安, 吴中如. 变形监测分析与预报. 测绘出版社 1998. (CHEN Yongqi, WU Zian, WU Zhongru, Deformation Monitoring Analysis and Forecasting, Beijing:Surveying and Mapping Press,1981).
[18] XUE S, DANG Y, CHEN W. Matrix Volume Method of the Mean Square Deviation Computation of Least Square Solution. Geomatics and Information Science of Wuhan University 2009;34:1106-9.(薛树强,党亚民,陈武. 最小二乘估值均方差计算的矩阵体积法[J]. 武汉大学学报(信息科学版),2009,v.3409:1106-9.).
[19] 薛树强. 矩阵体积及其在网形设计中的应用. 中国测绘科学研究院 2007.
[20] Serre D, SpringerLink (Online service). Matrices theory and applications. Graduate texts in mathematics v 216. 2nd ed. New York: Springer Science+Business Media, LLC; 2010.