﻿ 中国似大地水准面模型CQG2000的梯度分析
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 大地测量与地球动力学  2023, Vol. 43 Issue (1): 61-64, 99  DOI: 10.14075/j.jgg.2023.01.012

### 引用本文

NIE Jianliang, ZHANG Xueping, GUO Xinwei, et al. Analysis of Gradient of Chinese Quasi-Geoid CQG2000[J]. Journal of Geodesy and Geodynamics, 2023, 43(1): 61-64, 99.

### Foundation support

National Natural Science Foundation of China, No.41774004, 41904040.

### Corresponding author

ZHANG Xueping, senior engineer, majors in key technologies of multi-source surveying and mapping data fusion, E-mail: 250163892@qq.com.

### 第一作者简介

NIE Jianliang, senior engineer, majors in geodetic data processing, E-mail: niejianliang@163.com.

### 文章历史

1. 陕西测绘地理信息局，西安市友谊东路334号，710054;
2. 自然资源部大地测量数据处理中心，西安市友谊东路334号，710054;
3. 自然资源部第二地形测量队，西安市测绘路6号，710054

1 梯度模型及计算方法 1.1 梯度模型

 \begin{aligned} & \operatorname{grad} f(B, L)=\frac{\partial f}{\partial B} \boldsymbol{i}+\frac{\partial f}{\partial L} \boldsymbol{j}= \\ & \frac{\zeta_{m+1, n}-\zeta_{m, n}}{\Delta B} \boldsymbol{i}+\frac{\zeta_{m, n+1}-\zeta_{m, n}}{\Delta L} \boldsymbol{j} \end{aligned} (1)

1.2 三阶反距离平方权差分方法

 图 1 梯度计算示意图 Fig. 1 Schematic diagram of gradient calculation
 $f_B=\frac{\partial f}{\partial B}=\frac{\zeta_7-\zeta_1+2\left(\zeta_8-\zeta_2\right)+\zeta_9-\zeta_3}{8 \Delta B}$ (2)
 $f_L=\frac{\partial f}{\partial L}=\frac{\zeta_3-\zeta_1+2\left(\zeta_6-\zeta_4\right)+\zeta_9-\zeta_7}{8 \Delta L}$ (3)

 $\alpha = \frac{3}{2}{\rm{ \mathit{ π} }} + {\rm{arctan}}\left( {\frac{{{f_L}}}{{{f_B}}}} \right) - \frac{1}{2}{\rm{ \mathit{ π} }}\frac{{{f_B}}}{{\left| {{f_B}} \right|}}$ (4)
1.3 梯度精度推导

 $\begin{gathered} f(B+\Delta B, L)=f(B, L)+f_B(B, L) \Delta B+ \\ f^{\prime \prime}{ }_B(B, L) \frac{(\Delta B)^2}{2}+f^{\prime \prime \prime}{ }_B\left(\zeta_B, L\right) \frac{(\Delta B)^3}{3 !} \end{gathered}$ (5)
 $\begin{gathered} f(B-\Delta B, L)=f(B, L)-f_B(B, L) \Delta B+ \\ f^{\prime \prime}{ }_B(B, L) \frac{(\Delta B)^2}{2}-f^{\prime \prime \prime}{ }_B\left(\gamma_B, L\right) \frac{(\Delta B)^3}{3 !} \end{gathered}$ (6)

 $\begin{array}{c} {f_B} = \left\{ {\frac{{2\left( {{\zeta _8} - {\zeta _2}} \right) + \left( {{\zeta _7} - {\zeta _1}} \right) + \left( {{\zeta _9} - {\zeta _3}} \right)}}{{8\Delta B}} + } \right.\\ \frac{{2\left( {{\rm{d}}{\zeta _8} - {\rm{d}}{\zeta _2}} \right) + \left( {{\rm{d}}{\zeta _7} - {\rm{d}}{\zeta _1}} \right) + \left( {{\rm{d}}{\zeta _9} - {\rm{d}}{\zeta _3}} \right)}}{{8\Delta B}} - \\ \frac{{{{(\Delta B)}^2}}}{{8 \times 3!}}\left[ {2\left( {{f^{\prime \prime \prime }}_B\left( {{\zeta _B}, L} \right) - {f^{\prime \prime \prime }}_B\left( {{\gamma _B}, L} \right)} \right) + } \right.\\ \left( {{f^{\prime \prime \prime }}_B\left( {{\zeta _B}, L - \Delta L} \right) - {f^{\prime \prime \prime }}_B\left( {{\gamma _B}, L - \Delta L} \right)} \right) + \\ \left. {\left. {\left( {{f^{\prime \prime \prime }}_B\left( {{\zeta _B}, L + \Delta L} \right) - {f^{\prime \prime \prime }}_B\left( {{\gamma _B}, L + \Delta L} \right)} \right)} \right]} \right\} \end{array}$ (7)

MBML${f^{\prime \prime \prime }}_B, {f^{\prime \prime \prime }}_L$关于BL的三阶导数最大范围，其误差具有随机性，则有：

 $\begin{array}{c} {f_B} = \left\{ {\frac{{2\left( {{\zeta _8} - {\zeta _2}} \right) + \left( {{\zeta _7} - {\zeta _1}} \right) + \left( {{\zeta _9} - {\zeta _3}} \right)}}{{8\Delta B}} + } \right.\\ \frac{{2\left( {{\rm{d}}{\zeta _8} - {\rm{d}}{\zeta _2}} \right) + \left( {{\rm{d}}{\zeta _7} - {\rm{d}}{\zeta _1}} \right) + \left( {{\rm{d}}{\zeta _9} - {\rm{d}}{\zeta _3}} \right)}}{{8\Delta B}} - \\ \left. {\frac{{{{(\Delta B)}^2}}}{{8 \times 3!}}\left( {4{M_B} + 2{M_B} + 2{M_B}} \right)} \right\} = \\ \left\{ {\frac{{2\left( {{\zeta _8} - {\zeta _2}} \right) + \left( {{\zeta _7} - {\zeta _1}} \right) + \left( {{\zeta _9} - {\zeta _3}} \right)}}{{8\Delta B}} + } \right.\\ \frac{{2\left( {{\rm{d}}{\zeta _8} - {\rm{d}}{\zeta _2}} \right) + \left( {{\rm{d}}{\zeta _7} - {\rm{d}}{\zeta _1}} \right) + \left( {{\rm{d}}{\zeta _9} - {\rm{d}}{\zeta _3}} \right)}}{{8\Delta B}} - \\ \left. {\frac{{{{(\Delta B)}^2}}}{{3!}}{M_B}} \right\} \end{array}$ (8)

 $\begin{gathered} f_L=\left\{\frac{2\left(\zeta_6-\zeta_4\right)+\left(\zeta_3-\zeta_1\right)+\left(\zeta_9-\zeta_7\right)}{8 \Delta L}+\right. \\ \frac{2\left(\mathrm{~d} \zeta_6-\mathrm{d} \zeta_4\right)+\left(\mathrm{d} \zeta_3-\mathrm{d} \zeta_1\right)+\left(\mathrm{d} \zeta_9-\mathrm{d} \zeta_7\right)}{8 \Delta L}- \\ \left.\frac{(\Delta L)^2}{3 !} M_L\right\} \end{gathered}$ (9)

 $\begin{gathered} \sigma_{f_B}^2=\left(\frac{1}{8 \Delta B}\right)^2\left[4 \sigma_\zeta^2+4 \sigma_\zeta^2+\sigma_\zeta^2+\sigma_\zeta^2+\sigma_\zeta^2+\sigma_\zeta^2\right]+ \\ \left(\frac{(\Delta B)^2}{3 !}\right)^2 M^2=\frac{12 \sigma_\zeta^2}{64(\Delta B)^2}+\frac{(\Delta B)^4 M^2}{36}= \\ \frac{3 \sigma_\zeta^2}{16(\Delta B)^2}+\frac{(\Delta B)^4 M^2}{36} \end{gathered}$ (10)

 $\sigma_{f_L}^2=\frac{3 \sigma_{\xi}^2}{16(\Delta L)^2}+\frac{(\Delta L)^4 M^2}{36}$ (11)

 $\begin{gathered} \sigma_\alpha^2=\left(\frac{f_L}{f_B^2+f_L^2}\right)^2 \sigma_{f_B}^2+\left(\frac{f_B}{f_B^2+f_L^2}\right)^2 \sigma_{f_L}^2= \\ \frac{f_L^2 \sigma_{f_B}^2+f_B^2 \sigma_{f_L}^2}{\left(f_B^2+f_L^2\right)^2} \end{gathered}$ (12)

2 CQG2000的梯度分析

 图 2 CQG2000的梯度分布 Fig. 2 The distribution of gradient for CQG2000

1) 102°E以东地区CQG2000的梯度方位角方向主要为东，量级小于0.2 m/(′)，且变化平缓。其中，东北地区、华南地区梯度方向主要为东南，角度约为东向南45°；30°~40°N区间内的梯度方位角方向为正东。102°E以西地区CQG2000梯度在不同地类区域的方向不同，且梯度数值变化幅度较大。

2) 我国东部、西北、西南地区的梯度分量精度σfBσfL随似大地水准面格网点精度σζ的增大而增大，随分辨率的增大而减小。西南地区σα误差最大，为50.1°；西北地区σα误差最小，为16.4°。造成上述差异的主要原因为梯度方位角精度σα主要受梯度分量及其精度的影响，较小的梯度数值变化会导致较大的梯度方位角精度变化。进一步说明，相比于梯度分量精度σfBσfL，梯度方位角误差σα对梯度方位角α更加敏感。

3) 新疆北部与南部、青海、成都平原等地存在多个梯度扩散中心，区域似大地水准面格网点高程异常值由中心向四周逐渐增大；喜马拉雅山脉、新疆中部地区出现区域似大地水准面梯度会聚现象；河西走廊也出现区域似大地水准面梯度会聚现象，甘南西部梯度方向为正南，同时顺时针旋转；宁夏南部、甘肃东北部地区区域似大地水准面梯度方向为正东，同时逆时针旋转。喜马拉雅山脉、新疆南部昆仑山脉等地的区域似大地水准面梯度数值较大，最大值位于西藏墨脱县(0.625 m/(′))；青藏高原绝大部分区域的区域似大地水准面梯度较小，变化较平缓。

4) CQG2000梯度变化剧烈区域与我国大山脉走向一致，说明区域似大地水准面与地球内部质量分布密切相关。在喜马拉雅山、昆仑山、天山、秦岭等山脉区域，区域似大地水准面变化速度较大，梯度数值变化为小-大-小。

5) CQG2000自身精度为dm级，实际分辨率为15′×15′。虽然区域似大地水准面梯度精度不高，但能整体反映我国范围内似大地水准面空间域的变化特征。此外，CQG2000梯度与全国范围内的垂线偏差分布[15]一致, 仅在西藏东南部地区存在一定差异, 可能是该区域缺乏重力等基础资料、似大地水准面精度较低所致。

6) 由于CQG2000是利用EGM96参考重力场模型获取高精度重力似大地水准面, 再通过GNSS水准点纠正融合得到, 因此重力似大地水准面梯度与似大地水准面梯度的总体趋势一致二者系统偏差较小, 梯度大小与方向差异也较小CQG2000梯度与高精度EGM2008重力场模型梯度在我国中东地区的差异较小, 在喜马拉雅山、昆仑山、天山、秦岭等山脉区域存在一定差异, 最大差值位于喜马拉雅山地区, 约为0.18m/('), 可能是CQG2000在该区域缺乏基础资料所致。

3 结语

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Analysis of Gradient of Chinese Quasi-Geoid CQG2000
NIE Jianliang1     ZHANG Xueping1     GUO Xinwei2     WANG Lili1     ZHAO Wenpu3
1. Shaanxi Bureau of Surveying, Mapping and Geoinformation, 334 East-Youyi Road, Xi'an 710054, China;
2. Geodetic Data Processing Center, MNR, 334 East-Youyi Road, Xi'an 710054, China;
3. The Second Topographic Surveying Brigade, MNR, 6 Cehui Road, Xi'an 710054, China
Abstract: We obtain the gradient for the changes of CQG2000 model, to meet the needs of height conversion between map sheets at different scales, analyze the change correlation of the variety of quasigeoid in the spatial domain. Firstly, we define the gradient model of CQG2000. Secondly, we calculate the gradient value of CQG2000 by the 3rd order finite difference weighted by reciprocal of squared distance to derive the error of the gradient and the azimuth. Finally, to analyze the change trend of national height anomaly in China, we use the gradient of CQG2000 model. The results show that the gradient value in the area east of 102° E is small, the change is soft, and the direction of azimuth is eastward; while in the area west of 102°E, the gradient value is large, the local change is abrupt, and the direction of azimuth has obvious trend changes locally. Some gradient diffusion and convergence centers exist in western regions, such as Xinjiang, Qinghai-Tibet, where there are great gradient varieties, and the maximum gradient value of 0.625 m/(′) is near Motuo.