MATLAB实现高斯-克吕格投影反算

高斯投影(高斯-克吕格投影)的反算

更新2020-06，将坐标系统统一换为WGS-84坐标系，整理一下脚本函数

高斯投影的反算是指由当地的局部坐标系(x,y)转换为当地的地理坐标系(B: 纬度, L: 经度)。由于之前的博文MATLAB实现高斯-克吕格投影正算已经对高斯投影进行过简要的说明，故本博文不再对高斯-克吕格投影的原理进行介绍，只给出高斯投影反算的算法流程和实现的MATLAB脚本。本博文参考文献资料如下：

[1] 孔祥元, 郭际明, 刘宗泉, 等. 大地测量学基础(第二版)[M]. 武汉: 武汉大学出版社, 2009.
[2] 程鹏飞,成英燕,文汉江,等.2000国家大地坐标系实用宝典[M]. 北京:测绘出版社, 2008.
[3] 牛丽娟. 测量坐标转换模型研究与转换系统实现[D]. 长安大学, 2010.
[4] 李厚朴,边少锋.高斯投影的复变函数表示[J].测绘学报,2008(01):5-9

高斯投影反算公式

当地纬度$B$的计算公式如下：

$$\begin{array}{rl}B=& B_f-\frac{\rho t_f}{2M_f}y\left(\frac{y}{N_f}\right)[1-\frac{1}{12}\left(5+3t_f^2+{\eta }_f^2-9{\eta }_f^2{t}_f^2\right){\left(\frac{y}{N_f}\right)}^2\\ & +\frac{1}{360}\left(61+90t_f^2+45t_f^4\right){\left(\frac{y}{N_f}\right)}^4]\end{array}$$

当$y=0$时的纬度称为底点纬度$B_f$，底点纬度的迭代计算公式为：

$$X=a_{0}B-\frac{a_2}{2}\sin 2B+\frac{a_4}{4}\sin 4B-\frac{a_6}{6}\sin 6B+\frac{a_8}{8}\sin 8B$$其中，$$\{\begin{array}{r}{a}_0=m_0+\frac{1}{2}{m}_2+\frac{3}{8}{m}_4+\frac{5}{16}{m}_6+\frac{35}{128}{m}_8\\ a_2=\frac{1}{2}{m}_2+\frac{1}{2}{m}_4+\frac{15}{32}{m}_6+\frac{7}{16}{m}_8\\ a_4=\frac{1}{8}{m}_4+\frac{3}{16}{m}_6+\frac{7}{32}{m}_8\\ a_6=\frac{1}{32}{m}_6+\frac{1}{16}{m}_8\\ a_8=\frac{1}{128}{m}_8\end{array};\{\begin{array}{r}{m}_0=a(1-e^2)\\ m_2=\frac{3}{2}{e}^2{m}_0\\ m_4=\frac{5}{4}{e}^2{m}_2\\ m_6=\frac{7}{6}{e}^2{m}_4\\ m_8=\frac{9}{8}{e}^2{m}_6\end{array}$$
$B_f$迭代计算过程为：
迭代初始: 令$B_f^0=\frac{x}{a_0}$
迭代过程: 令$F^i=-\frac{a_2}{2}\sin 2B_f^i+\frac{a_4}{4}\sin 4B_f^i-\frac{a_6}{6}\sin 6B_f^i+\frac{a_8}{8}\sin 8B_f^i$; 令$B_f^{i+1}=\frac{X-F^i}{a_0}$
迭代停止: 当$|B_f^{i+1}-B_f^i|<ϵ$时则停止迭代.
注：$ϵ$为给定的阈值，阈值通常小于1E-8.

当地经度$l$的计算公式如下:

$$\begin{array}{rl}l=& \frac{\rho }{\cos B_f}\left(\frac{y}{N_f}\right)[1-\frac{1}{6}\left(1+2t_f^2+{\eta }_f^2\right){\left(\frac{y}{N_f}\right)}^2+\frac{1}{120}(5+28t_f^2\\ & +24t_f^4+6{\eta }_f^2+8{\eta }_f^2{t}_f^2){\left(\frac{y}{N_f}\right)}^4]\end{array}$$

上述式中:

1. $x$为北方向的坐标，
2. $y$为东方向的坐标(中国地区内需减去500000m)，
3. $\rho =180\times 3600/\pi$为弧度秒，
4. $a$为地球椭球长半径，
5. $e$为地球椭球第一偏心率，
6. $e^{\prime }$为地球椭球第二偏心率，
7. $N_f$为底点纬度$B_f$处地球的卯酉圈曲率半径，
8. $M_f$为底点纬度$B_f$处地球的子午圈曲率半径，
9. ${\eta }_f^2=e^{\prime 2}co{s}^2{B}_f$，
10. $t_f=tan{B}_f$。

高斯投影反算的MATLAB函数

function Coord = GaussProInverse(Pos)
% INPUT // Units of longitude and latitude is RAD (°)
% REF[1]// 程鹏飞,等.2000国家大地坐标系实用宝典[M].北京:测绘出版社,2008.
% REF[2]// 孔祥元,郭际明.大地测量学基础(第二版)[M].武汉:武汉大学出版社,2010.
% Longitude of the Earth central meridian
MerLon = 114;          % Wuhan is 114 deg
% Extract the local projected coordinate (x & y)
Coord = Pos;
Eth.D2R = 0.0174532925199433;  % pi/180
x = Pos(1);
y = Pos(2) - 500000;
%% Earth orientation parameters of WGS 84 Coordinate System
Eth.R0 = 6378137.0;
Eth.f = 1/298.257223563;
Eth.Rp = 6356752.3142452;      % R0*(1 - f);
% First Eccentricity and its Squared
Eth.e12 = 0.006694379990141;   % (2f - f*f);
Eth.e11 = 0.081819190842622;   % sqrt(2f - f*f);
% Second Eccentricity and its Squared
Eth.e22 = 0.00673949674227643; % (2f - f*f)/(1 + f*f - 2*f);
Eth.e21 = 0.08209443794969570; % sqrt((2f - f*f)/(1 + f*f - 2*f));
Bf = Meridian2Latitude(x, Eth.R0, Eth.e12);
tnBf = tan(Bf); tn2Bf = tnBf*tnBf; tn4Bf = tn2Bf*tn2Bf;
csBf = cos(Bf); cs2Bf = csBf*csBf; Eta2 = Eth.e22*cs2Bf;
COE = sqrt(1 - Eth.e12*sin(Bf)*sin(Bf));
Nf = Eth.R0/COE;
Mf = Eth.R0*(1 - Eth.e12)/COE^3;
%% Calculate Latitude(B)
YNf = y/Nf;
YNf2 = YNf*YNf;
YNf4 = YNf2*YNf2;
TYMN = 0.5*tnBf*y*YNf/Mf;
COE1 = (5 + 3*tn2Bf + Eta2 -9*Eta2*tn2Bf)*YNf2/12;
COE2 = (61 + 90*tn2Bf + 45*tn4Bf)*YNf4/360;
Lat = Bf - TYMN*(1 - COE1 + COE2);
%% Calculate Longitude(L)
YBNf = YNf/csBf;
COE3 = (1 + 2*tn2Bf + Eta2)*YNf2/6;
COE4 = (5 + 28*tn2Bf + 24*tn4Bf + 6*Eta2 + 8*Eta2*tn2Bf)*YNf4/120;
Lon = MerLon*Eth.D2R + YBNf*(1 - COE3 + COE4);
Coord(1) = Lat/Eth.D2R;
Coord(2) = Lon/Eth.D2R;
end

function Bf = Meridian2Latitude(x,a,e12)
m0 = a*(1 - e12);    m2 = 3*e12*m0/2;
m4 = 5*e12*m2/4;     m6 = 7*e12*m4/6;
m8 = 9*e12*m6/8;     a8 = m8/128;
a6 = m6/32 + m8/16;  a4 = m4/8 + 3*m6/16 + 7*m8/32;
a0 = m0 + m2/2 + 3*m4/8 + 5*m6/16 + 35*m8/128;
a2 = m2/2 + m4/2 + 15*m6/32 + 7*m8/16;
B0 = x/a0;
while 1
    F = -a2*sin(2*B0)/2 + a4*sin(4*B0)/4 - a6*sin(6*B0)/6 + a8*sin(8*B0)/8;
    Bf = (x - F)/a0;
    if abs(B0 - Bf)<1e-10
        break;
    end
    B0 = Bf;
end
end

该方法为经典的高斯反算的算法，对于高斯反算的详细推导可以参考大地测量的教材或者最新的文献。