﻿ 声速误差对圆走航水下控制点定位影响分析
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 大地测量与地球动力学  2020, Vol. 40 Issue (12): 1299-1302  DOI: 10.14075/j.jgg.2020.12.018

### 引用本文

SUN Wenzhou, YIN Xiaodong, XIA Wenjie. Analysis of the Influence of Sound Velocity Error on Sailing-Circle Positioning Method for Seafloor Control Points[J]. Journal of Geodesy and Geodynamics, 2020, 40(12): 1299-1302.

### Foundation support

National Natural Science Foundation of China, No. 41876103, 41874016.

### Corresponding author

YIN Xiaodong, PhD, professor, majors in marine geodesy, E-mail:triest@163.com.

### 第一作者简介

SUN Wenzhou, PhD candidate, majors in marine geodesy, E-mail:1519374228@qq.com.

### 文章历史

1. 国家海洋技术中心漳州基地筹建办公室，厦门市西林东路149号，361000;
2. 海军大连舰艇学院军事海洋与测绘系，大连市解放路667号，116018

1 声速测距误差对圆走航定位坐标解算的影响

 $\delta {\rho _v} = - \frac{{{c_b}}}{{S \times {c_h}}} \times \rho \times \Delta S$ (1)

 $\begin{array}{l} {\rm{d}}\mathit{\boldsymbol{x}} = {({\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{B}})^{ - 1}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{L}} = {({\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{B}})^{ - 1}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\delta {\rho _v} = \\ \left[ {\begin{array}{*{20}{c}} {\frac{{\sin \gamma \cdot \cos {\beta _1}}}{{\sum\limits_{i = 1}^n {{{(\sin \gamma \cdot \cos {\beta _i})}^2}} }}}&{\frac{{\sin \gamma \cdot \cos {\beta _2}}}{{\sum\limits_{i = 1}^n {{{(\sin \gamma \cdot \cos {\beta _i})}^2}} }}}& \cdots &{\frac{{\sin \gamma \cdot \cos {\beta _n}}}{{\sum\limits_{i = 1}^n {{{(\sin \gamma \cdot \cos {\beta _i})}^2}} }}}\\ {\frac{{\sin \gamma \cdot \sin {\beta _1}}}{{\sum\limits_{i = 1}^n {(\sin \gamma \cdot \sin {\beta _i})} }}}&{\frac{{\sin \gamma \cdot \sin {\beta _2}}}{{\sum\limits_{i = 1}^n {(\sin \gamma \cdot \sin {\beta _i})} }}}& \cdots &{\frac{{\sin \gamma \cdot \sin {\beta _n}}}{{\sum\limits_{i = 1}^n {(\sin \gamma \cdot \sin {\beta _i})} }}}\\ {\frac{{\cos \gamma }}{{\sum\limits_{i = 1}^n {{{(\cos \gamma )}^2}} }}}&{\frac{{\cos \gamma }}{{\sum\limits_{i = 1}^n {{{(\cos \gamma )}^2}} }}}& \cdots &{\frac{{\cos \gamma }}{{\sum\limits_{i = 1}^n {{{(\cos \gamma )}^2}} }}} \end{array}} \right] \cdot \\ \left[ {\begin{array}{*{20}{c}} {\delta {\rho _{v1}}}\\ {\delta {\rho _{v2}}}\\ \vdots \\ {\delta {\rho _{vn}}} \end{array}} \right] = \left[ {\frac{{\sum\limits_{i = 1}^n {\left( {\cos {\beta _i} \cdot \delta {\rho _{vi}}} \right)} }}{{\sum\limits_{i = 1}^n {{{\left( {\sin \gamma \cdot \cos {\beta _i}} \right)}^2}} }}\frac{{\sum\limits_{i = 1}^n {\left( {\sin {\beta _i} \cdot \delta {\rho _{vi}}} \right)} }}{{\sum\limits_{i = 1}^n {{{\left( {\sin \gamma \cdot \sin {\beta _i}} \right)}^2}} }}\frac{{\cos \gamma \cdot \sum\limits_{i = 1}^n {\left( {\delta {\rho _{vi}}} \right)} }}{{\sum\limits_{i = 1}^n {{{(\cos \gamma )}^2}} }}} \right] \end{array}$ (2)

 ${\rm{d}}\mathit{\boldsymbol{x}} = {\left[ {\frac{{\delta {\rho _{vb}} \cdot \sin \gamma \cdot \sum\limits_{i = 1}^n {\cos } {\beta _i}}}{{\sum\limits_{i = 1}^n {{{(\sin \gamma \cdot \cos {\beta _i})}^2}} }}\frac{{\delta {\rho _{vb}} \cdot \sin \gamma \cdot \sum\limits_{i = 1}^n {\sin } {\beta _i}}}{{\sum\limits_{i = 1}^n {{{(\sin \gamma \cdot \cos {\beta _i})}^2}} }}\quad \frac{{n \cdot \delta {\rho _{vb}} \cdot \cos \gamma }}{{\sum\limits_{i = 1}^n {{{(\cos \gamma )}^2}} }}} \right]^{\rm{T}}}$ (3)

 ${\delta {\rho _{vp}} = {k_p} \cdot \Delta {S_0} \cdot \sin ({\omega _v} \cdot t + {\varphi _v})}$ (4)
 ${{k_p} = \frac{{{c_b} \cdot \rho }}{{S \cdot {c_h}}}}$ (5)

 ${\rm{d}}\mathit{\boldsymbol{x}} = \left[ {\begin{array}{*{20}{l}} {{k_p} \cdot \sin \gamma \cdot \Delta {S_0} \cdot \sum\limits_{i = 1}^n {(\cos {\beta _i} \cdot \sin (} {\omega _v} \cdot t + {\varphi _v}))/\sum\limits_{i = 1}^n {{{(\sin \gamma \cdot \cos {\beta _i})}^2}} }\\ {{k_p} \cdot \sin \gamma \cdot \Delta {S_0} \cdot \sum\limits_{i = 1}^n {(\sin {\beta _i} \cdot \sin (} {\omega _v} \cdot t + {\varphi _v}))/\sum\limits_{i = 1}^n {{{(\sin \gamma \cdot \cos {\beta _i})}^2}} }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_p} \cdot \sin \gamma \cdot \Delta {S_0} \cdot \sum\limits_{i = 1}^n {(\cos \gamma \cdot \sin (} {\omega _v} \cdot t + {\varphi _v}))/\sum\limits_{i = 1}^n {{{(\cos \gamma )}^2}} } \end{array}} \right]$ (6)

 $f(t) = \int_0^{{T_s}} {\sin } \left( {\frac{{2{\rm{ \mathsf{ π} }}}}{{{T_c}}} \cdot t} \right) \cdot \sin \left( {\frac{{2{\rm{ \mathsf{ π} }}}}{{{T_v}}} \cdot t + {\varphi _v}} \right){\rm{d}}t$ (7)

 $g(t) = \sin \left( {\frac{{2{\rm{ \mathsf{ π} }}}}{{{T_c}}} \cdot t} \right) \cdot \sin \left( {\frac{{2{\rm{ \mathsf{ π} }}}}{{{T_v}}} \cdot t + {\varphi _v}} \right)$ (8)

 $\begin{array}{*{20}{l}} {2g(t) = \cos \left( {2{\rm{ \mathsf{ π} }} \cdot t \cdot \left( {\frac{1}{{{T_c}}} + \frac{1}{{{T_v}}}} \right) + {\varphi _v}} \right) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos \left( {2{\rm{ \mathsf{ π} }} \cdot t \cdot \left( {\frac{1}{{{T_v}}} - \frac{1}{{{T_c}}}} \right) + {\varphi _v}} \right)} \end{array}$ (9)

 $\begin{array}{*{20}{l}} {2f(t) = \int_0^{{T_s}} {\cos } \left( {\frac{{2{\rm{ \mathsf{ π} }}}}{{\left( {\frac{{{T_c} \cdot {T_v}}}{{{T_c} + {T_v}}}} \right)}} \cdot t} \right){\rm{d}}t - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_0^{{T_s}} {\cos } \left( {\frac{{2{\rm{ \mathsf{ π} }}}}{{\left( {\frac{{{T_c} \cdot {T_v}}}{{{T_c} - {T_v}}}} \right)}} \cdot t} \right){\rm{d}}t} \end{array}$ (10)

${T_m} = \frac{{{T_c} \cdot {T_v}}}{{{T_c} + {T_v}}}, {T_n} = \frac{{{T_c} \cdot {T_v}}}{{{T_c} - {T_v}}}$。若Tv为测距误差周期项的周期，为使f(t)的积分为0，Ts应为TmTn的整数倍。设Ts=kc·TcTc=kv·TvTsTmTn的比值分别为ηmηn，则：

 ${\eta _m} = \frac{{{T_s}}}{{{T_m}}} = \frac{{{k_c} \cdot {k_s} \cdot {T_v}}}{{\frac{{{k_v}}}{{{k_v} + 1}} \cdot {T_v}}} = {k_c}{k_v} + {k_c}$ (11)
 ${\eta _n} = \frac{{{T_s}}}{{{T_s}}} = \frac{{{k_c} \cdot {k_v} \cdot {T_v}}}{{\frac{{{k_v}}}{{{k_v} - 1}} \cdot {T_v}}} = {k_c}{k_v} - {k_c}$ (12)

kc的定义可知，其必为整数，因此ηmηn为整数的条件是kckv为整数。当ηmηn同时为整数时，函数f(t)为0，即水平坐标y不受周期性误差的影响。但当kc=kv=1时，f(t)函数前一项积分为0，后一项不为0，此时水平坐标y的解算结果不为0。

 $g(t) = \sin \left( {\frac{{2{\rm{ \mathsf{ π} }}}}{{{T_v}}} \cdot t + {\varphi _v}} \right)$ (13)

 ${{T_s} = {\eta _s} \cdot {T_v}}$ (14)
 ${{\eta _s} = {k_c} \cdot {k_v}}$ (15)

2 仿真实验分析

3 结语

1) 圆走航单点定位中测距误差常数项不会对水平方向的解算结果产生影响，但会影响垂直方向坐标；

2) 当满足式(11)、(12)和式(14)、(15)时，测距误差周期项不会对定位计算结果的水平方向和垂直方向坐标产生影响，但当kc=kv=1时，测距误差周期项会对水平方向坐标产生影响。

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Analysis of the Influence of Sound Velocity Error on Sailing-Circle Positioning Method for Seafloor Control Points
SUN Wenzhou1     YIN Xiaodong2     XIA Wenjie1
1. Office for Establishing Zhangzhou Base, National Ocean Technology Center, 149 East-Xilin Road, Xiamen 361000, China;
2. Department of Military Oceanography and Hydrography and Cartography, Dalian Naval Academy, 667 Jiefang Road, Dalian 116018, China
Abstract: This paper focuses on the problem of deviation of seafloor control point coordinate solution caused by inaccurate measurement of sound velocity profile when using the sailing-circle positioning method. First, the effect of uncertainty in sound velocity on the ranging error are introduced. Then, the ranging error is divided into four parts: background sound velocity error, ranging random error, long-period term and short-period term of ranging error. The influence of these four parts on the control point coordinates are shown through analyzing the coordinate correction equation. The theoretical results are consistent with the simulation experiments.
Key words: seafloor control point; ranging error; ST law; least square method; sound velocity profile