﻿ 基于蚁群算法的多面函数在GPS高程拟合中的应用
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 大地测量与地球动力学  2019, Vol. 39 Issue (1): 31-35  DOI: 10.14075/j.jgg.2019.01.006

### 引用本文

PU Lun, TANG Shihua, ZHANG Ziping, et al. Application of Multi-Quadric Function Based on Ant Colony Algorithm in GPS Elevation Fitting[J]. Journal of Geodesy and Geodynamics, 2019, 39(1): 31-35.

### Foundation support

Open Fund of Guangxi Key Laboratory of Spatial Information and Geomatics, No.15-140-07-05, 16-380-25-13, 16-380-25-25; Natural Science Foundation of Guangxi Province, No.2018JJA150047.

### Corresponding author

TANG Shihua, PhD, professor, majors in the automatic collection and processing of geomatics data and application development of measuring robot, E-mail:58650875@qq.com.

### 第一作者简介

PU Lun, postgraduate, majors in GNSS data processing and application, E-mail:pulun16@163.com.

### 文章历史

1. 桂林理工大学测绘地理信息学院，桂林市雁山街319号，541006;
2. 广西空间信息与测绘重点实验室，桂林市雁山街319号，541006;
3. 青海省生态环境遥感监测中心，西宁市南山东路116号，810007

1 多面函数 1.1 多面函数基本原理

 $\sum {{{\left( {f\left( {{x_i},{y_i}} \right) - \varphi \left( {{x_i},{y_i}} \right)} \right)}^2}} = \min$ (1)

 $\varphi \left( {x,y} \right) = \sum\limits_{j = 1}^u {{\beta _j}{Q_j}\left( {\left( {x,y} \right),\left( {{x_j},{y_j}} \right)} \right)}$ (2)

 $Q = {\left( {{{\left( {x - {x_j}} \right)}^2} + {{\left( {y - {y_j}} \right)}^2} + \delta } \right)^{ - 1/2}}$ (3)

 $Q = {\left( {{{\left( {x - {x_j}} \right)}^2} + {{\left( {y - {y_j}} \right)}^2} + \delta } \right)^{1/2}}$ (4)

 $Q = {\left( {{{\left( {x - {x_j}} \right)}^2} + {{\left( {y - {y_j}} \right)}^2}} \right)^{3/2}} + \delta$ (5)

 $\begin{array}{l} {v_1} = \sum\limits_{j = 1}^u {{{\hat \beta }_j}{Q_j}\left( {\left( {{x_1},{y_1}} \right),\left( {{x_{0j}},{y_{0j}}} \right)} \right)} - {\zeta _1}\\ {v_2} = \sum\limits_{j = 1}^u {{{\hat \beta }_j}{Q_j}\left( {\left( {{x_2},{y_2}} \right),\left( {{x_{0j}},{y_{0j}}} \right)} \right)} - {\zeta _2}\\ \;\;\;\;\;\; \cdots \\ {v_n} = \sum\limits_{j = 1}^u {{{\hat \beta }_j}{Q_j}\left( {\left( {{x_n},{y_n}} \right),\left( {{x_{0j}},{y_{0j}}} \right)} \right)} - {\zeta _n} \end{array}$ (6)

 $\mathit{\boldsymbol{V}} = \mathit{\boldsymbol{A\hat \beta }} - \mathit{\boldsymbol{\zeta }}$ (7)

 $\begin{array}{*{20}{c}} {\mathop {\mathit{\boldsymbol{A}}}\limits_{n \times u} = }\\ {\left[ \begin{array}{l} {Q_1}\left( {\left( {{x_1},{y_1}} \right),\left( {{x_{01}},{y_{01}}} \right)} \right), \cdots ,{Q_u}\left( {\left( {{x_1},{y_1}} \right),\left( {{x_{0u}},{y_{0u}}} \right)} \right)\\ {Q_1}\left( {\left( {{x_2},{y_2}} \right),\left( {{x_{01}},{y_{01}}} \right)} \right), \cdots ,{Q_u}\left( {\left( {{x_2},{y_2}} \right),\left( {{x_{0u}},{y_{0u}}} \right)} \right)\\ \;\;\;\;\;\; \cdots \\ {Q_1}\left( {\left( {{x_n},{y_n}} \right),\left( {{x_{01}},{y_{01}}} \right)} \right), \cdots ,{Q_u}\left( {\left( {{x_n},{y_n}} \right),\left( {{x_{0u}},{y_{0u}}} \right)} \right) \end{array} \right]} \end{array}$ (8)

 $\mathit{\boldsymbol{\hat \beta }} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{\zeta }}$ (9)

$\mathit{\boldsymbol{\hat \beta }}$代入式(2)求得多面函数拟合模型，再根据拟合模型代入已知点位数据，即可完成高程异常拟合。

1.2 稳健权改进的多面函数

 $\rho \left( v \right) = \left\{ \begin{array}{l} {v^2}/2,\left| v \right| < 1.5\sigma \\ \left| v \right|,1.5\sigma < \left| v \right| < 2.5\sigma \\ d,\left| v \right| > 2.5\sigma \end{array} \right.$ (10)

 $w\left( v \right) = \left\{ \begin{array}{l} 1,\left| v \right| < 1.5\sigma \\ 1/\left( {\left| v \right| + {k_0}} \right),1.5\sigma < \left| v \right| < 2.5\sigma \\ 0,\left| v \right| > 2.5\sigma \end{array} \right.$ (11)

 ${{\mathit{\boldsymbol{\hat \beta }}}^{\left( 1 \right)}} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PA}}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{P\zeta }}$ (12)
 ${\mathit{\boldsymbol{V}}^{\left( 1 \right)}} = \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat \beta }}}^{\left( 1 \right)}} - \mathit{\boldsymbol{\zeta }}$ (13)

 $\left| {{{\mathit{\boldsymbol{\hat \beta }}}^{\left( k \right)}} - {{\mathit{\boldsymbol{\hat \beta }}}^{\left( {k - 1} \right)}}} \right| \le \varepsilon$ (14)

 ${{\mathit{\boldsymbol{\hat \beta }}}^{\left( k \right)}} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}{{\mathit{\boldsymbol{\bar P}}}^{\left( {k - 1} \right)}}\mathit{\boldsymbol{A}}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}{{\mathit{\boldsymbol{\bar P}}}^{\left( {k - 1} \right)}}\mathit{\boldsymbol{\zeta }}$ (15)
 ${\mathit{\boldsymbol{V}}^{\left( k \right)}} = \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat \beta }}}^{\left( k \right)}} - \mathit{\boldsymbol{\zeta }}$ (16)

2 基于蚁群算法的多面函数

3 算例分析

3.1 蚁群多面函数法拟合GPS高程

 图 1 蚂蚁寻找最优路径对比 Fig. 1 Ant looking for optimal path comparison chart

 图 2 蚁群+稳健估计拟合模型三维效果 Fig. 2 Ant colony and robust estimation fitting model 3D effect

3.2 结果分析

 图 3 蚁群算法与均匀格网法拟合残差对比 Fig. 3 Comparison of ant colony algorithm and uniform grid method

4 结语

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Application of Multi-Quadric Function Based on Ant Colony Algorithm in GPS Elevation Fitting
PU Lun1,2     TANG Shihua1,2     ZHANG Ziping3     HU Xinkai1,2     XIAO Yan1,2
1. College of Geomatics and Geoinformation, Guilin University of Technology, 319 Yanshan Street, Guilin 541006, China;
2. Guangxi Key Laboratory of Spatial Information and Geomatics, 319 Yanshan Street, Guilin 541006, China;
3. Qinghai Ecological Environment Remote Sensing Monitoring Center, 116 East-Nanshan Road, Xining 810007, China
Abstract: Aiming at the problem that the central node of the multi-quadric function fitting method is difficult to select, we introduce the method of the ant colony algorithm into the multi-quadric function and construct the high-precision fitting model with robust estimation.The ant colony algorithm is used to find feature points in complex terrain, and combines several non-feature points together as a central node in the construction model. Robustness is added to the multi-faceted function, and the effect of gross error on the fitting model is eliminated using the iterative method.GPS elevation fitting data processing example shows that the fitting method of multi-quadric function and robust estimation based on ant colony algorithm effectively eliminates the influence of gross error, fitting precision is improved by 26%.
Key words: elevation anomaly; ant colony algorithm; robust estimate; multi-quadric function; center node