﻿ 最小二乘定权方法对流动重力计算结果的影响
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 大地测量与地球动力学  2020, Vol. 40 Issue (12): 1233-1236  DOI: 10.14075/j.jgg.2020.12.005

### 引用本文

LIU Dong, CUI Xingping, WANG Qinghua, et al. Influence of Least Squares Weighting Method on Gravity Calculation Results[J]. Journal of Geodesy and Geodynamics, 2020, 40(12): 1233-1236.

### Foundation support

Innovative Team for Seismic Geomagnetic Analysis and Earthquake Prediction; Special Fund for Earthquake Science and Technology of Yunnan Earthquake Agency, No. 2017ZX03.

### Corresponding author

CUI Xingping, engineer, majors in faults, E-mail: 813347090@qq.com.

### 第一作者简介

LIU Dong, PhD candidate, majors in gravity measurement, E-mail:doumo@whu.edu.cn.

### 文章历史

1. 武汉大学测绘学院，武汉市珞喻路129号，430071;
2. 云南省地震局，昆明市知春街249号，650093;
3. 河南测绘职业学院，郑州市工贸路，451464

1 测点情况及计算公式

 图 1 云南省重力观测点分布 Fig. 1 Distribution of gravity points in Yunnan province

 ${a_1}{p_1} + {a_2}{p_2} = b$ (1)

2 不等权和等权计算结果

3 最小二乘计算结果 3.1 最小二乘原理

m×n(m>n)的线性方程组有：

 $\left\{ \begin{array}{l} {a_{11}}{P_1} + {a_{12}}{P_2} + \ldots + {a_{1n}}{P_n} = {b_1}\\ {a_{21}}{P_1} + {a_{22}}{P_2} + \ldots + {a_{2n}}{P_n} = {b_2}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots \\ {a_{m1}}{P_1} + {a_{m2}}{P_2} + \ldots + {a_{mn}}{P_n} = {b_m} \end{array} \right.$ (2)

 $\mathit{\boldsymbol{AP}} = \mathit{\boldsymbol{b}}$ (3)

 ${\delta _i} = \sum\limits_{j = 1}^n {{a_{ij}} - {b_i}} , i = 1, 2, \ldots , m$ (4)

 $\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi ({P_1}, {P_2}, \ldots , {P_n}) = \\ \sum\limits_{i = 1}^m {\delta _i^2} {\rm{ = }}\sum\limits_{i = 1}^m {{{\left( {\sum\limits_{j = 1}^n {{a_{ij}}{P_j} - {b_i}} } \right)}^2}} \end{array}$ (5)

 $\frac{{\partial \varphi }}{{\partial {x_k}}} = 2\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {({a_{ij}}{P_j} - {b_i}){a_{jk}}} } = 0$ (6)

 $\sum\limits_{i = 1}^m {\left( {\sum\limits_{j = 1}^n {{a_{ij}}{a_{jk}}} } \right){P_j}} = \sum\limits_{j = 1}^n {{a_{ij}}{b_i}}$ (7)

 ${\mathit{\boldsymbol{A}}^{\text{T}}}\mathit{\boldsymbol{AP}} = {\mathit{\boldsymbol{A}}^{\text{T}}}\mathit{\boldsymbol{b}}$ (8)

3.2 最小二乘计算结果

3.3 重力异常与地震的关系

 图 2 3种定权方式计算云南省2018~2019年重力异常与MS＞4.0地震的关系 Fig. 2 Gravity anomalies obtained by 3 weights and MS > 4.0 earthquakes in Yunnan Province during 2018 to 2019

4 结语

1) 仪器的权值确定侧面反映了仪器本身的性能，与仪器各自的观测精度一致;

2) 不等权的定权方式虽然是根据仪器各自观测精度来定权，但在对其中一台仪器取较大权值时削弱了另一台仪器的观测作用，虽然得到的观测值精度更高，但也削弱了反映MS4.0~5.0地震的有用的重力异常信息;

3) 采用最小二乘定权方式的计算结果与绝对观测点的计算结果一致，从地震监测预报上来说是一种比较好的方法，相比不等权的计算结果，重力异常与MS4.0以上地震的发生有很好的对应关系，相比等权的计算结果，其精度更好，与绝对观测点得到的重力值更接近。

 [1] 肖云, 王云鹏, 刘晓刚, 等. 空域最小二乘法用于重力卫星误差分析[J]. 武汉大学学报:信息科学版, 2019, 44(3): 340-346 (Xiao Yun, Wang Yunpeng, Liu Xiaogang, et al. Application of Space-Wise Least Square Method to Error Analysis for Satellite Gravimetry[J]. Geomatics and Information Science of Wuhan University, 2019, 44(3): 340-346) (0) [2] 汪健, 孙少安, 邢乐林, 等. CG-5重力仪的漂移特征[J]. 大地测量与地球动力学, 2016, 36(6): 556-560 (Wang Jian, Sun Shao'an, Xing Lelin, et al. Drift Characteristics of CG-5 Gravimeter[J]. Journal of Geodesy and Geodynamics, 2016, 36(6): 556-560) (0) [3] 蒋萍, 于宏亮, 王司光. 不等精度测量中权的确定方法[J]. 济南大学学报:自然科学版, 2017, 30(3): 229-232 (Jiang Ping, Yu Hongliang, Wang Siguang. Weights Determining Method of Unequal Precision Measurement[J]. Journal of University of Jinan: Science and Technology, 2017, 30(3): 229-232) (0) [4] 高冰. 不等精度测量数据处理中的加权原则[J]. 数学理论与应用, 2002, 22(1): 119-122 (Gao Bing. The Principle of Adding Weight in Unequal Precision Measured Data Processing[J]. Mathematical Theory and Application, 2002, 22(1): 119-122) (0)
Influence of Least Squares Weighting Method on Gravity Calculation Results
LIU Dong1,2     CUI Xingping2     WANG Qinghua2     MU Baosheng3     ZHANG Yong2
1. School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430071, China;
2. Yunnan Earthquake Agency, 249 Zhichun Street, Kunming 650093, China;
3. Henan College of Surveying and Mapping, Gongmao Road, Zhengzhou 451464, China
Abstract: We research how to obtain the optimal weight value in flow gravity observation data processing, which can be used to calculate gravity value. In the processing, we use the least-square method to solve the overdetermined equations consisting of the weight coefficient and the observations. The results show that the accuracy of the point value derived from the equal-weight method is not good enough. However, it contains more information for gravity anomaly seismic analysis and prediction. Although the accuracy of the point value solved by the unequal-weight method is better than the equal-weight method, it contains less information which may inhibit gravity anomaly seismic analysis and prediction. Finally, the results obtained from the least-square method not only have higher accuracy of the point value, but also is the best among the three methods, which can be used to analyze and predict the earthquakes with magnitude large than MS4.0.
Key words: equal weight; unequal weight; overdetermined equations; least square method