基于最小二乘原理及間接平差的變形監測控制網數據處理分析

摘 要：在某幾何模型中，總的觀測量為，必要觀測量，選擇t個必要觀測量為參數，將每一個觀測值表示成所選參數的函數，列立n個誤差方程，這種以誤差方程為函數模型的平差方法，稱為間接平差法。在變形監測控制網數據處理中，通常可以采用基于最小二乘原理的間接平差法對網中的觀測數據進行平差和點位精度分析，求出最弱點中誤差、最弱邊中誤差、繪制誤差橢圓。其中處理的關鍵是誤差方程式的列立、非線性條件方程式線性化、誤差方程式的求解及未知點精度評定。本文以在益陽市資江三橋布設的橋梁控制網為例，推導了全部的橋梁控制網解算過程，并得出了結論。

關鍵詞：間接平差；法方程；最小二乘原理；線性方程組

Analysis of Data Processing for Deformation Monitoring Control Network Based on Least Squares Principle and Indirect Adjustment

Cao Yuanzhi1 Nie Zhiping1 Peng Zhihong1 Peng Zheng2

1.Hunan City University Institute of Municipal and Mapping Engineering Hunan Yiyang 413000 ；

2. Hunan Tunnel Engineering Corporation Hunan Yiyang 413000

Abstract： I In a certain geometric model， the total observation quantity is， necessary observations， and t necessary observations are selected as parameters. Each observation value is represented as a function of the selected parameter， and n error equations are listed. This adjustment method using the error equation as the function model is called indirect adjustment method. In the data processing of deformation monitoring and control networks， the indirect adjustment method based on the least squares principle can usually be used to adjust and analyze the point accuracy of the observed data in the network， calculate the mean square error of the weakest point， the mean square error of the weakest edge， and draw the error ellipse. The key to processing is the formulation of error equations， linearization of nonlinear conditional equations， solution of error equations， and accuracy evaluation of unknown points. This article takes the bridge control network deployed at Zijiang Third Bridge in Yiyang City as an example to derive the entire calculation process of the bridge control network and draw conclusions.

Keywords： Indirect adjustment;normal equation;Least squares principle;System of linear equations

在現代橋梁控制網布設中，通常布設成GPS網或邊角網[1]。布設為GPS網的優點是不受地形和通視條件的限制，布設為邊角網的優點是既觀測角度又觀測距離，精度相對較高[2]。本課題結合邊角網的布設優點對益陽市資江三橋進行控制測量。2021年以來，益陽市資江三橋由于車流量較大，投入使用時間較長，在運營過程中出現了橋墩傾斜下沉、橋面平整度發生變化的情況。為了了解資江三橋的真實變形速率及沉降規律，筆者布置了一測角測距的邊角控制網。所用儀器為徠卡TC702型全站儀，測角中誤差為±2″，測距精度為1mm+2ppm×D（公里）。如下圖所示，A，B，C是已知點，P1，P2為待定點，網中觀測了12個角度，2條邊長（在條件平差中，不用觀測邊長；在間接平差中，用16，17作為觀測邊長）。……