﻿ ﻿ ﻿ ﻿ ﻿ 连续观测系统平差模型与有色噪声补偿
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1. 长安大学 地测学院，陕西 西安 710054；
2. 中国测绘科学研究院，北京 100830；
3. 西安测绘研究所 地理空间信息工程国家重点实验室，陕西 西安 710054

Adjustment Model and Colored Noise Compensation of Continuous Observation System
XUE Shuqiang1,2,YANG Yuanxi3
1. School of Geological and Surveying Engineering,Chang’an University,Xi’an 710054,China;
2. Chinese Academy of Surveying and Mapping,Beijing 100830,China;
3. National Key Laboratory of Geo-spatial Information,Xi’an Research Institute of Surveying and Mapping,Xi’an 710054,China
First author: XUE Shuqiang (1980—),male,PhD candidate,majors in error theory and adjustment.E-mail： xuesq@casm.ac.cn
Abstract:The affection caused by the colored noises should be taken into account to the adjustment model. As useful signals,these colored noises should be accurately identified and extracted by Fourier analysis. A continuous adjustment model is introduced with respect to the colored noises,and then it can be generalized that 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 or compensated by optimally designing the observational system.
Key words: adjustment     continuous observation     least squares     colored noise     Hilbert space

1 引 言

2 连续函数空间和连续平差模型 2.1 函数空间的基函数

Hilbert空间是具有几何结构(度量或内积)的函数空间。在平方可积函数空间L2(a,b)={f(t)|abf2(t)dt＜∞}中，任意连续函数f(t),g(t)∈L2(a,b)的内积可定义为[4, 16]

2.2 连续观测系统的平差模型

3 矛盾连续观测方程的最小二乘解

4 有色噪声对平差解的影响分析

5 算例分析

 图 1 连续定位构型[21] Fig. 1 Continuous configuration of positioning[21]

5.1 方案1：连续平差模型

(1) 当t∈[0,π/2]时，将关系式

(2) 当t∈[0,π]时，可得

(3) 当t∈[0,2π]时，可得

5.2 方案2：离散平差模型

6 结 论

(1) 本文提出的连续平差模型是经典测量平差模型向无穷维空间的一种推广。连续平差模型的最小二乘解是经典离散最小二乘解的极限。经典测量平差的许多结论对于连续平差模型也是成立的。

(2) 当连续平差模型的基函数为一组标准正交基时，连续观测方程平差退化为观测信号的傅里叶分析。

(3) 通过合理的观测系统设计和采样设计，可消除有色噪声，从而实现平差解的无偏估计。有色噪声函数正交于连续平差模型的基函数，是消除有色噪声的前提条件。

(4) 提高系统的采样频率只能消除观测高频噪声，有色噪声的消除取决于连续观测系统的重复观测周期和观测时长。最佳观测时长应覆盖有色噪声的整数周期，而并非采样时间越长越好。

(5) 在测量中，许多观测模型为连续数学模型，对这些模型进行采样不但增加计算成本，而且会损失精度。若使用傅里叶分析将离散实测数据消噪后建立连续观测函数，并建立连续观测平差模型，则有望简化计算。

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http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0054

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#### 文章信息

XUE Shuqiang,YANG Yuanxi.

Adjustment Model and Colored Noise Compensation of Continuous Observation System

Acta Geodaetica et Cartographica Sinica,2014,43(4):360-365,371.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0054