# 精度衰减因子

## 数学定义

### 函数模型

${\displaystyle p_{\text{r}}^{\text{s}}=\rho _{\text{r}}^{\text{s}}+(dt_{\text{r}}-dt_{\text{s}})+I_{\text{r}}^{\text{s}}+T_{\text{r}}^{\text{s}}+\varepsilon _{\text{r}}^{\text{s}}}$

${\displaystyle {\begin{pmatrix}{\hat {p}}_{\text{r}}^{\text{1}}\\{\hat {p}}_{\text{r}}^{\text{2}}\\\vdots \\{\hat {p}}_{\text{r}}^{\text{n}}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial \rho _{\text{r}}^{1}}{\partial {e}_{\text{r}}}}&{\frac {\partial \rho _{\text{r}}^{1}}{\partial {n}_{\text{r}}}}&{\frac {\partial \rho _{\text{r}}^{1}}{\partial {u}_{\text{r}}}}&1\\{\frac {\partial \rho _{\text{r}}^{2}}{\partial {e}_{\text{r}}}}&{\frac {\partial \rho _{\text{r}}^{2}}{\partial {n}_{\text{r}}}}&{\frac {\partial \rho _{\text{r}}^{2}}{\partial {u}_{\text{r}}}}&1\\\vdots &\vdots &\vdots &\vdots \\{\frac {\partial \rho _{\text{r}}^{n}}{\partial {e}_{\text{r}}}}&{\frac {\partial \rho _{\text{r}}^{n}}{\partial {n}_{\text{r}}}}&{\frac {\partial \rho _{\text{r}}^{n}}{\partial {u}_{\text{r}}}}&1\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {e}}_{\text{r}}\\{\hat {n}}_{\text{r}}\\{\hat {u}}_{\text{r}}\\{\hat {dt}}_{\text{r}}\end{pmatrix}}+{\begin{pmatrix}-dt_{\text{1}}+I_{\text{r}}^{\text{1}}+T_{\text{r}}^{\text{1}}\\-dt_{\text{2}}+I_{\text{r}}^{\text{2}}+T_{\text{r}}^{\text{2}}\\\vdots \\-dt_{\text{n}}+I_{\text{r}}^{\text{n}}+T_{\text{r}}^{\text{n}}\end{pmatrix}}}$

${\displaystyle {\hat {\mathbf {p} }}=\mathbf {H} {\hat {\mathbf {x} }}+\mathbf {l} }$

${\displaystyle d\mathbf {p} =\mathbf {H} d\mathbf {x} }$
${\displaystyle d\mathbf {p} ={\hat {\mathbf {p} }}-\mathbf {p} _{0}}$
${\displaystyle d\mathbf {x} ={\hat {\mathbf {x} }}-\mathbf {x} _{0}}$

${\displaystyle d\mathbf {x} =\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)^{-1}\mathbf {H} ^{\text{T}}d\mathbf {p} }$

### 随机模型

${\displaystyle \operatorname {cov} (d\mathbf {x} )={\begin{pmatrix}\sigma _{e_{\text{r}}}^{2}&\sigma _{e_{\text{r}}n_{\text{r}}}^{2}&\sigma _{e_{\text{r}}u_{\text{r}}}^{2}&\sigma _{e_{\text{r}}dt_{\text{r}}}^{2}\\\sigma _{e_{\text{r}}n_{\text{r}}}^{2}&\sigma _{n_{\text{r}}}^{2}&\sigma _{n_{\text{r}}u_{\text{r}}}^{2}&\sigma _{n_{\text{r}}dt_{\text{r}}}^{2}\\\sigma _{e_{\text{r}}u_{\text{r}}}^{2}&\sigma _{n_{\text{r}}u_{\text{r}}}^{2}&\sigma _{u_{\text{r}}}^{2}&\sigma _{u_{\text{r}}dt_{\text{r}}}^{2}\\\sigma _{e_{\text{r}}dt_{\text{r}}}^{2}&\sigma _{n_{\text{r}}dt_{\text{r}}}^{2}&\sigma _{u_{\text{r}}dt_{\text{r}}}^{2}&\sigma _{dt_{\text{r}}dt_{\text{r}}}^{2}\\\end{pmatrix}}}$

{\displaystyle {\begin{aligned}\operatorname {cov} (d\mathbf {x} )&=\operatorname {E} \left[d\mathbf {x} d\mathbf {x} ^{\text{T}}\right]\\&=\operatorname {E} \left[{\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)}^{-1}\mathbf {H} ^{\text{T}}d\mathbf {p} d\mathbf {p} ^{\text{T}}\mathbf {H} {\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)}^{-1}\right]\\&={\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)}^{-1}\mathbf {H} ^{\text{T}}\operatorname {cov} (d\mathbf {p} )\mathbf {H} {\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)}^{-1}\end{aligned}}}

### DOP值的定义

${\displaystyle \operatorname {cov} (d\mathbf {p} )=I_{n{\times }n}\sigma _{\text{UERE}}^{2}}$

${\displaystyle \operatorname {cov} (d\mathbf {x} )=\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)^{-1}\sigma _{\text{UERE}}^{2}}$

${\displaystyle \sigma _{\text{UERE}}={\sqrt {\sigma _{e_{\text{r}}}^{2}+\sigma _{n_{\text{r}}}^{2}+\sigma _{u_{\text{r}}}^{2}+\sigma _{dt_{\text{r}}}^{2}}}={\sqrt {\operatorname {tr} \left[\operatorname {cov} (d\mathbf {x} )\right]}}}$

${\displaystyle {\text{(G)DOP}}={\frac {\sigma _{\text{G}}}{\sigma _{\text{UERE}}}}={\frac {\sqrt {\operatorname {tr} \left[\operatorname {cov} (d\mathbf {x} )\right]}}{\sigma _{\text{UERE}}}}={\sqrt {\operatorname {tr} \left[\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)^{-1}\right]}}}$

${\displaystyle {\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)}^{-1}={\begin{pmatrix}D_{11}&D_{12}&D_{13}&D_{14}\\D_{21}&D_{22}&D_{23}&D_{24}\\D_{31}&D_{32}&D_{33}&D_{34}\\D_{41}&D_{42}&D_{43}&D_{44}\\\end{pmatrix}}}$

GDOP值亦可表示为：

${\displaystyle {\text{GDOP}}={\sqrt {D_{11}+D_{22}+D_{33}+D_{44}}}={\sqrt {\operatorname {tr} \left[\left(\mathbf {H} ^{\text{T}}\mathbf {H} \right)^{-1}\right]}}}$

• ${\displaystyle {\text{PDOP}}={\frac {\sigma _{\text{P}}}{\sigma _{\text{UERE}}}}={\frac {\sqrt {\sigma _{e_{\text{r}}}^{2}+\sigma _{n_{\text{r}}}^{2}+\sigma _{u_{\text{r}}}^{2}}}{\sigma _{\text{UERE}}}}={\sqrt {D_{11}+D_{22}+D_{33}}}}$
• ${\displaystyle {\text{HDOP}}={\frac {\sigma _{\text{H}}}{\sigma _{\text{UERE}}}}={\frac {\sqrt {\sigma _{e_{\text{r}}}^{2}+\sigma _{n_{\text{r}}}^{2}}}{\sigma _{\text{UERE}}}}={\sqrt {D_{11}+D_{22}}}}$
• ${\displaystyle {\text{VDOP}}={\frac {\sigma _{\text{V}}}{\sigma _{\text{UERE}}}}={\frac {\sigma _{u_{\text{r}}}^{2}}{\sigma _{\text{UERE}}}}={\sqrt {D_{33}}}}$
• ${\displaystyle {\text{TDOP}}={\frac {\sigma _{\text{T}}}{\sigma _{\text{UERE}}}}={\frac {\sigma _{dt_{\text{r}}}^{2}}{\sigma _{\text{UERE}}}}={\sqrt {D_{44}}}}$

${\displaystyle {\text{GDOP}}={\sqrt {{\text{PDOP}}^{2}+{\text{TDOP}}^{2}}}={\sqrt {{\text{HDOP}}^{2}+{\text{VDOP}}^{2}+{\text{TDOP}}^{2}}}}$

## 数值大小

### 四颗卫星的情况

${\displaystyle H={\begin{pmatrix}\cos {E}&0&\sin {E}&1\\-\displaystyle {\frac {1}{2}}\cos {E}&\displaystyle {\sqrt {\frac {3}{4}}}\cos {E}&\sin {E}&1\\-\displaystyle {\frac {1}{2}}\cos {E}&-\displaystyle {\sqrt {\frac {3}{4}}}\cos {E}&1&1\\0&0&\sin {E}&1\\\end{pmatrix}}}$

• ${\displaystyle D_{11}=D_{22}={\frac {2}{3}}\sec ^{2}{E}}$
• ${\displaystyle D_{33}={\frac {4}{3}}\left(\cos ^{2}{\frac {E}{2}}-\sin ^{2}{\frac {E}{2}}\right)^{-4}}$
• ${\displaystyle D_{44}={\frac {1}{6}}\left(\cos ^{2}{\frac {E}{2}}-\sin ^{2}{\frac {E}{2}}\right)^{-4}\left(5-3\cos {2E}\right)}$

## 应用情况

DOP值的等级及其含义[12][13]
DOP值 等级 含义
1 理想 置信度水平高
2－4 优秀 置信度水平满足所有的应用需求
4－6 良好 置信度水平满足高精度应用需求
6－8 中等 置信度水平满足大部分应用需求
8－20 一般 置信度水平较低，应评估应用风险
20－50 很差 置信度水平很差，基本无法满足应用需求

## 注释

1. ^ 由与接收机相关的三个坐标参数和一个钟差参数组成

## 參考文獻

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2. Langley, Richard B. Dilution of precision (PDF). GPS world. 1999, 10 (5): 52-59 （英语）.
3. 李征航，黄劲松编著．GPS测量与数据处理（第三版）．武汉：武汉大学出版社，2016．ISBN 978-7-307-17680-5
4. ^ HOROWITZ, LEO. Direct-Ranging LORAN. Navigation. 1970-06, 17 (2): 200–204. ISSN 0028-1522. doi:10.1002/j.2161-4296.1970.tb00039.x （英语）.
5. Kaplan, Elliott D.; Hegarty, C. (Christopher J.). Understanding GPS/GNSS : principles and applications Third edition. Boston. 2017. ISBN 978-1-63081-442-7. OCLC 1022790269 （英语）.
6. ^ GPS Accuracy: HDOP, PDOP, GDOP, Multipath & the Atmosphere. GIS Geography. 2017-03-13 [2020-11-10] （英语）.
7. J.G. Teunissen, Peter; Montenbruck, Oliver. Springer Handbook of Global Navigation Satellite Systems. Springer International Publishing. 2017. ISBN 978-3-319-42926-7 （英语）.
8. ^ Jijie Zhu. Calculation of geometric dilution of precision. IEEE Transactions on Aerospace and Electronic Systems. 1992-07, 28 (3): 893–895. doi:10.1109/7.256323 （英语）.
9. Zhang, Miaoyan; Zhang, Jun. A Fast Satellite Selection Algorithm: Beyond Four Satellites. IEEE Journal of Selected Topics in Signal Processing. 2009-10, 3 (5): 740–747. ISSN 1932-4553. doi:10.1109/JSTSP.2009.2028381 （英语）.
10. 李建文. 卫星导航中几何精度衰减因子最小值分析及应用. 测绘学报. 2011-05-05, 40 (Sup.): 83–85. ISSN 1001-1595.
11. ^ 中国有色金属工业协会．GB 50026—2007 工程测量规范[S]． 北京：中国计划出版社，2007．
12. ^ Tahsin, Mahdia; Sultana, Sunjida; Reza, Tasmia; Hossam-E-Haider, Md. Analysis of DOP and its preciseness in GNSS position estimation. 2015 International Conference on Electrical Engineering and Information Communication Technology (ICEEICT) (Savar, Dhaka, Bangladesh: IEEE). 2015-05: 1–6. ISBN 978-1-4673-6676-2. doi:10.1109/ICEEICT.2015.7307445 （英语）.
13. ^ 秦红磊; 从丽; 金天. 全球卫星导航系统原理、进展及应用. 高等教育出版社. 2019. ISBN 978-7-04-051733-0.