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Volume 44 Issue 10
Oct.  2019
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Article Contents

CUI Jiawu, ZHANG Xingfu, ZHOU Boyang, DU Xiangfeng, WEI Dehong. Improved step-by-step elimination method for optimal selection of GNSS/leveling points[J]. Geomatics and Information Science of Wuhan University, 2019, 44(10): 1505-1510. doi: 10.13203/j.whugis20180074
Citation: CUI Jiawu, ZHANG Xingfu, ZHOU Boyang, DU Xiangfeng, WEI Dehong. Improved step-by-step elimination method for optimal selection of GNSS/leveling points[J]. Geomatics and Information Science of Wuhan University, 2019, 44(10): 1505-1510. doi: 10.13203/j.whugis20180074

Improved step-by-step elimination method for optimal selection of GNSS/leveling points

doi: 10.13203/j.whugis20180074
Funds:

The National Natural Science Foundation of China 41674006

The National Natural Science Foundation of China 41504013

More Information
  • Author Bio:

    CUI Jiawu, postgraduate, specializes in mearsurement data processing. E-mail: 864885814@qq.com

  • Corresponding author: ZHANG Xingfu, PhD, associate professor. E-mail: xfzhang77@163.com
  • Received Date: 2018-08-29
  • Publish Date: 2019-10-05
  • The reasonable selection of GNSS/leveling points is very important to GNSS height fitting, step-by-step elimination method is a good method for the optimization selection of joint-observation points. The traditional elimination method selects the GNSS/leveling points based on the minimum fitting error of height anomaly, which will easily lead to uneven joint-observation points. According to this view, this paper proposes to optimize the joint-observation point based on the area size of Thiessen polygons generated by the GNSS/leveling points, and on this basis to improve the traditional method, namely taking into account both the size of the height anomaly fitting error and the area size of polygons generated by Thiessen method (referred to as the synthesis method). The experimental results show that the synthesis method can improve the distribution of joint-observation points, and get the fitting results of height anomaly with high stability and accuracy.
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Improved step-by-step elimination method for optimal selection of GNSS/leveling points

doi: 10.13203/j.whugis20180074
Funds:

The National Natural Science Foundation of China 41674006

The National Natural Science Foundation of China 41504013

  • Author Bio:

  • Corresponding author: ZHANG Xingfu, PhD, associate professor. E-mail: xfzhang77@163.com

Abstract: The reasonable selection of GNSS/leveling points is very important to GNSS height fitting, step-by-step elimination method is a good method for the optimization selection of joint-observation points. The traditional elimination method selects the GNSS/leveling points based on the minimum fitting error of height anomaly, which will easily lead to uneven joint-observation points. According to this view, this paper proposes to optimize the joint-observation point based on the area size of Thiessen polygons generated by the GNSS/leveling points, and on this basis to improve the traditional method, namely taking into account both the size of the height anomaly fitting error and the area size of polygons generated by Thiessen method (referred to as the synthesis method). The experimental results show that the synthesis method can improve the distribution of joint-observation points, and get the fitting results of height anomaly with high stability and accuracy.

CUI Jiawu, ZHANG Xingfu, ZHOU Boyang, DU Xiangfeng, WEI Dehong. Improved step-by-step elimination method for optimal selection of GNSS/leveling points[J]. Geomatics and Information Science of Wuhan University, 2019, 44(10): 1505-1510. doi: 10.13203/j.whugis20180074
Citation: CUI Jiawu, ZHANG Xingfu, ZHOU Boyang, DU Xiangfeng, WEI Dehong. Improved step-by-step elimination method for optimal selection of GNSS/leveling points[J]. Geomatics and Information Science of Wuhan University, 2019, 44(10): 1505-1510. doi: 10.13203/j.whugis20180074
  • GNSS(global navigation satellite system)技术以其高精度、低成本等优势被广泛应用于获取地面点的平面坐标,但由于GNSS测量高为大地高,无法直接用于实际生产,通过GNSS高程转换可以将GNSS获取的大地高转换为正常高,其精度可以替代四等甚至更高等级水准测量[1-5]。与传统的水准测量劳动强度大、工作效率低相比,GNSS高程测量在劳动强度及效率等方面具有明显优势。GNSS高程转换一般采用区域似大地水准面模型直接转换,或者利用一组GNSS水准联测点,采用一定的拟合(如平面拟合等)方法获得[6-10]。基于区域似大地水准面模型法的GNSS高程转换精度相对稳定,而拟合法的GNSS高程转换精度易受测区范围大小、高程异常起伏大小、GNSS水准点的分布、密度、拟合方法等多种因素的综合影响[8-19]。在选定拟合方法的前提下,合理确定GNSS点水准高的联测方案具有非常重要的现实意义[10-13]。为此,沈云中等[10]提出以高程异常拟合误差最小作为目标函数,对GNSS水准点的联测方案进行优化,通过实验分析了全组合法和剔除法的计算效率和效果,得出在确保精度的前提下,逐步剔除法大大提高了计算效率。逐步剔除法仅仅是一种GNSS/水准点的优化选取方法,它适用于任何高程拟合模型,只要选定拟合模型,均可以获得该模型的优化选点方案。

    当采用低阶多项式(如1~2次)拟合函数时,逐步剔除法易出现选点不均匀情况。为此,本文对GNSS/水准点优化选择的逐步剔除法进行改进,在分析依据泰森法生成的多边形面积大小进行选点的基础上,同时考虑高程异常拟合误差大小及由泰森法生成的多边形面积大小进行优化选点。本文仅以1~3次多项式拟合模型为研究对象,实测数据结果表明,GNSS/水准点优化选择的综合法可在改善点分布的同时,获得稳定性好、精度较高的高程异常拟合结果。

  • 2004年,沈云中等[10]提出以高程异常拟合误差最小为目标函数,从而实现GNSS/水准点的优化选择(方案1),以下简称传统法。该方法以拟剔除某个点后所有点的方差和大小作为该点对拟合精度贡献大小的评判标准,通过逐步剔除对拟合精度贡献小的点,达到以少量的GNSS水准联测点获得较高拟合精度的目的。

    若高程异常的拟合函数为多项式,则高程异常可利用一组线性无关的基函数fk(x, y)表示如下:

    式中,αk为拟合参数;s为拟合参数的个数。其中1、2、3次多项式拟合模型可表示为:

    假定全网共布设n个控制点,选取其中m个点进行GNSS水准联测(ms),则误差方程式为:

    其中,$ \mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {{f_1}({x_1}, {y_1})}& \cdots &{{f_s}({x_1}, {y_1})}\\ \vdots & \vdots & \vdots \\ {{f_1}({x_m}, {y_m})}& \cdots &{{f_s}({x_m}, {y_m})} \end{array}} \right]$, $\mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} {{a_1}}\\ {{a_2}}\\ \vdots \\ {{a_s}} \end{array}} \right], \mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} {{\zeta _1}}\\ {{\zeta _2}}\\ \vdots \\ {{a_m}} \end{array}} \right] $。

    给定单位权方差σ0值,根据站点坐标及误差传播定律可计算每个站点的高程异常拟合误差mζ1:

    式中,Q=(ATPA)-1为协因数阵;Bi=$[{f_1}({x_i}, {y_i}) \cdots {f_s}({x_i}, {y_i})] $;qi为第i点的协因数。

    假设GNSS网中共有n个点,现采用逐步剔除法剔除对精度贡献小的点。在第t轮剔除中拟剔除联测点j(j∈[1, n]),利用剩余的n-t个点组成误差方程系数阵Aj(t),并计算Qj(t),由式(4)可算得剔除j点后i点(i∈[1, n])的拟合误差$ {\sigma _0}\sqrt {q_{j, i}^{(t)}} $,从而可以计算所有点的拟合误差平方和Mj(t)

    Mj(t)值大,说明在第t轮剔除j点后整体拟合效果变差,即j点对拟合精度贡献大,应该保留;反之Mj(t)值越小,则说明j点对拟合精度贡献越小,因此找出该轮拟剔除点中M值最小的点进行剔除。依此循环,逐步剔除每轮中M值最小的点,直至剩余m个点。

  • 泰森多边形法是荷兰气象学家Thiessen提出的按欧氏距离最近原则划分区域的方法[20],可表示为:

    式中,Xi为点Pi的泰森多边形上的任意一点;|XiPi|表示Xi到点Pi的距离;|XiPj|表示Xi到其余点的距离。

    依据GNSS水准点均匀分布原则,采用泰森多边形法逐步剔除泰森多边形面积最小的待测点(方案2),以下简称为Thiessen法。具体过程为:首先划定高程异常拟合区域的边界,生成泰森多边形,计算每个点的泰森多边形面积,将面积最小的点剔除;利用剩余的点重新生成泰森多边形,继续剔除面积最小的点,依此循环,直到剩余预设置的m个GNSS水准联测点。

  • 为了避免传统优化选择方法可能出现的选点不均匀情况,本文将传统优化选择方法[10]与Thiessen多边形面积法相结合,称为GNSS/水准点优化选择的综合法(方案3),以下简称为综合法,其经验公式可表示为:

    式中,qi(0)是在不剔除任何点的情况下求得i点的协因数;Ssum为拟合区域的总面积;Sj(t)为第t轮点剔除中拟剔除点j的泰森多边形面积。剔除M值最小的点,剔除后将剩余的点重新生成泰森多边形并计算其M值,依此循环,逐步剔除至剩余m个联测点。

  • 本文挑选了由美国国家大地测量局(National Geodetic Survey,NGS)提供的密歇根湖以西20 km的杜佩奇郡(Du Page County, Illinois)及其周边区域分布较为均匀的33个GNSS水准点作为实验数据(https://www.ngs.noaa.gov/GEOID/GEOID12B/GPSonBM12B.shtml)。实验区域为高程起伏平缓的平原地区,海拔约为195~255 m,位于西经88°21.0′ ~ 87°52.8′、北纬41°43.4′ ~ 42°01.2′,南北方向约为33 km,东西方向约为39 km,面积约为1 287 km2

    实验分别采用传统法、Thiessen法、综合法3种方法按1次、2次、3次多项式拟合对联测点进行逐步剔除,利用保留点进行高程异常拟合,并将所有点的拟合高程异常与真实高程异常进行比较分析。另外,实验根据已知的GNSS水准数据设置了参照组,参照组利用已知高程异常数据,逐步剔除对高程异常拟合精度贡献最小的点。由于参照组以高程异常真实残差平方和为剔除依据,理论上可获得高程异常拟合的最优点组合或近似最优点组合。

    图 1为分别按上述3种方案及参照组进行逐点剔除后,利用保留点进行高程异常拟合时所有点(即33个点)的真实残差统计的均方根误差。图 2图 3为参照组及3种方案在剔除20个点后的选点分布图。Thiessen法在逐点剔除时仅依据了该点的泰森多边形面积大小,与采取的拟合方法无关,故其1 ~ 3次多项式拟合选点是一样的,如图 2所示,其余方案及参照组的选点均与拟合方法有关,如图 3所示。

    Figure 1.  Root Mean Square Errors of Height Anomaly Fitting

    Figure 2.  Selection of GNSS/Leveling Points by Thiessen Method

    Figure 3.  Selection of GNSS/Leveling Points by Reference Group, Traditional and Synthesis Methods

    图 1可知,参照组分别按3种拟合方法进行逐点剔除时,前期均能获得稳定的、较小的均方根误差,这说明逐点剔除法是一种有效的选点方法;随着保留点数接近必要观测数,均方根误差开始明显增大,因此建议联测的GNSS水准点个数要大于拟合模型的必要观测数。

    图 1(a)可知,当采取1次多项式拟合选点时,Thiessen法和综合法均能获得较好的拟合精度,传统法的均方根误差随着剔除点数的增加而显著增大,其拟合精度明显差于另外两种方案。结合图 2图 3(b)3(c)可知,综合法与Thiessen法的选点吻合度非常高,相同率为92%,选点较为均匀;传统法选取的点皆为拟合区域边缘的点,中间大片区域无点,与综合法的选点相同率仅为38%。

    图 1(b)可知,当采取2次多项式拟合选点时,Thiessen法在点剔除前期能获得与参照组几乎一致的拟合精度,当剔除点数超过20后,其拟合误差急剧增大;传统法与综合法的拟合效果整体上较为稳定,其中传统法的拟合精度相对综合法波动大,但其在接近必要观测数时较为稳定。由图 2图 3(e)3(f)可知,综合法和Thiessen法选点较为均匀,选点吻合度较高,约为61.5%;传统法选取了拟合区域中心及边缘的点,出现了环状无点区,与综合法的选点相同率为46%。

    图 1(c)可知,当采取3次多项式拟合选点时,3种方案整体上均获得较好的拟合效果,其中综合法最优,传统法次之,Thiessen法在接近必要观测数时拟合误差急剧增大。由图 2图 3(h)3(i)可知,3种方案的选点均较为均匀,其中综合法和Thiessen法的选点相同率较1次、2次多项式拟合急剧下降,不足8%,同时综合法与传统法的选点相同率进一步增加至约61.5%,这说明了随着拟合多项式次数的增加,综合法中蕴含的传统法的权重逐步增大并起到主导作用。

    在实际生产测量中,希望通过联测少量的GNSS水准点而获得较优高程异常拟合精度。根据《工程测量规范》规定,GNSS/水准联测点的数量宜大于选用计算模型中未知参数个数的1.5倍[21],故本文针对1次、2次、3次多项式拟合模型选取联测点数分别为6、10、16进行进一步分析。由图 1(a)可知,在采取1次多项式拟合时,当剔除点数不超过27个点时(即保留6个联测点),参照组、Thiessen法、综合法的拟合均方根误差均不超过2.3 cm,传统法均方根误差约为3.2 cm,各点拟合残差如图 4(a)所示。由图 1(b)可知,在采取2次多项式拟合时,当剔除点数不超过23个点时(即保留10个联测点),参照组、综合法的均方根误差均不超过0.9 cm,传统法约为1.0 cm,Thiessen法约为1.2 cm,各点拟合残差如图 4(b)所示。由图 1(c)可知,在采取3次多项式拟合时,当剔除点数不超过17个点时(即保留16个联测点),参照组、Thiessen法、综合法、传统法的均方根误差均优于0.8 cm,各点拟合残差如图 4(c)所示。

    Figure 4.  Fitting Residual Error of Each Point

  • 本文对传统的GNSS/水准点优化选择法进行改进,提出同时考虑高程异常拟合误差大小及由泰森法生成的多边形面积大小进行优化选点,主要结论如下:

    1) 传统法在进行低阶次多项式拟合选点时,易出现选点不均匀情况。当采取1次多项式拟合选点时,其拟合精度较难控制,随着剔除点数的增加,容易出现拟合效果与真实高程异常偏差大的情况;在采取2次、3次多项式拟合选点时,可获得稳定性好、拟合精度高的点组合。Thiessen法是一种均匀分布的选点方法,更适用于1次多项式拟合选点。

    2) 综合法是一种融合传统法与Thiessen法的选点方法。在采取1次、2次多项式拟合选点时,起主导作用的主要是Thiessen法;随着拟合多项式次数的提高,传统法的权重逐渐增加,Thiessen法的权重逐渐降低,当采取3次多项式拟合选点时,传统法起主导作用。在1~3次多项式拟合选点中,均获得稳定性好、拟合精度高且分布较为均匀的点组合。

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