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more on geodetic coordinates

 1 \chapter{Geodetic Coordinates}\label{sec:geodetic} 2 3 The geodetic coordinates are more appropriate for describing near surface data of the Earth treating the 4 the Earth as an ellipsoid. The location of a point 5 is described by its geodetic latitude $\phi$, longitude $\lambda$ and geodetic height $h$ where we assume 6 \begin{equation} 7 -\frac{\pi}{2} \le \phi \le \frac{\pi}{2} \mbox{ and } -\pi \le \lambda \le \pi \;. 8 \end{equation} 9 In the following we refer to the $(\phi, \lambda, h)$ as the Geodetic Coordinate system\index{geodetic coordinates}. 10 The Cartesian coordinates $(x_0,x_1,x_2)$ of a point are given as 11 \begin{equation} 12 \begin{array}{rcll} 13 x_0 & = & (N + h) & \cdot \cos(\phi) \cdot \cos(\lambda) \\ 14 x_1 & = & (N + h) & \cdot \cos(\phi) \cdot \sin(\lambda) \\ 15 x_2 & = & (N \cdot (1-e^2) + h ) & \cdot \sin(\phi)\\ 16 \end{array} 17 \label{equ:geodetic:1} 18 \end{equation} 19 where $N$ is given as 20 \begin{equation} 21 N = \frac{a}{\sqrt{1- e^2 \cdot \sin^2(\phi) }} 22 \label{equ:geodetic:2} 23 \end{equation} 24 with the semi major axis length $a$ and $b$ ($a \ge b$), and the eccentricity 25 \begin{equation} 26 e = \sqrt{2f - f^2} \mbox{ with flattening } f = 1-\frac{b}{a} \ge 0 27 \label{equ:geodetic:3} 28 \end{equation} 29 Notice that the surface of the ellipsoid (Earth) is described by the case $h=0$. The following 30 table shows values for flattening semi-major axis for the major reference systems of the Earth: 31 \begin{center} 32 \begin{tabular}{c|lll} 33 Ellipsoid reference & Semi-major axis a & Semi-minor axis b & Inverse flattening (1/f)\\ 34 \hline 35 GRS 80 & 6 378 137.0 m & 6 356 752.314 140 m & 298.257 222 101\\ 36 WGS 84 & 6 378 137.0 m & 6 356 752.314 245 m & 298.257 223 563 37 \end{tabular} 38 \end{center} 39 We need to translate any object from the Cartesian coordinates $(x_i)$ in terms of Geodetic Coordinate system 40 $(\phi, \lambda, h)$. In the following indexes running through $(\phi, \lambda, h)$ are denoted by little Greek letters $\alpha$, 41 $\beta$ 42 to separate them from indexes running through the components of the Cartesian coordinates system in little Latin letters. 43 With this convection the derivative of a function $g$ with respect to the Geodetic coordinates which is given as 44 \begin{equation} 45 \begin{array}{rcl} 46 g_{,\phi} & = & g_{,0} \cdot x_{0,\phi} + g_{,1} \cdot x_{1,\phi} + g_{,2} \cdot x_{2,\phi} \\ 47 g_{,\lambda} & = & g_{,0} \cdot x_{0,\lambda} + g_{,1} \cdot x_{1,\lambda} + g_{,2} \cdot x_{2,\lambda} \\ 48 g_{,h} & = & g_{,0} \cdot x_{0,h} + g_{,1} \cdot x_{1,h} + g_{,2} \cdot x_{2,h} \\ 49 \end{array} 50 \end{equation} 51 via chain rule can be written in the compact form 52 \begin{equation} 53 g_{,\alpha} = g_{,i} \cdot x_{i,\alpha} 54 \end{equation} 55 56 \begin{equation} 57 x_{i,\alpha} 58 = 59 \left[ 60 \begin{array}{ccc} 61 -(M + h) \cdot \sin(\phi) \cdot \cos(\lambda) & -(M + h) \cdot \sin(\phi) \cdot \sin(\lambda) & (M + h ) \cdot \cos(\phi) \\ 62 - (N + h) \cdot \cos(\phi) \cdot \sin(\lambda) & (N + h) \cdot \cos(\phi) \cdot \cos(\lambda) & 0 \\ 63 \cos(\phi) \cdot \cos(\lambda) & \cos(\phi) \cdot \sin(\lambda) & \sin(\phi) \\ 64 \end{array} 65 \right] 66 \end{equation} 67 with 68 \begin{equation} 69 M = \frac{a \cdot (1-e^2) }{(1- e^2 \cdot \sin^2(\phi))^{\frac{3}{2}}} 70 \label{equ:geodetic:5} 71 \end{equation} 72 With the coordinate vectors $(u_{\alpha})$ defined as 73 \begin{equation} 74 u_{\alpha i} 75 = 76 \left[ 77 \begin{array}{ccc} 78 -\sin(\phi) \cdot \cos(\lambda) & - \sin(\lambda) & \cos(\phi) \cdot \cos(\lambda) \\ 79 - \sin(\phi) \cdot \sin(\lambda) & \cos(\lambda) & \cos(\phi) \cdot \sin(\lambda) \\ 80 \cos(\phi) & 0 & \sin(\phi) \\ 81 \end{array} 82 \right] 83 \end{equation} 84 and scaling factors 85 \begin{equation} 86 v_{\phi \phi} = (M + h) \; , 87 v_{\lambda \lambda} = (N + h) \cdot \cos(\phi) \mbox{ and } 88 v_{h h} = 1 89 \end{equation} 90 we get 91 \begin{equation} 92 x_{i,\alpha} = u_{\alpha i} v_{\alpha \alpha} \mbox{ and } g_{,\alpha} = g_{,i} u_{\alpha i} v_{\alpha \alpha} 93 \end{equation} 94 With the fact that 95 \begin{equation} 96 u_{\alpha i} u_{\alpha j} = \delta_{ij} \mbox{ and } u_{\alpha i} u_{\beta i} = \delta_{\alpha \beta} 97 \end{equation} 98 \begin{equation} 99 g_{,i} = \frac{1}{ v_{\alpha \alpha}} g_{,\alpha} u_{\alpha i} 100 \end{equation} 101 or 102 \begin{equation} 103 g_{,i} = \frac{1}{M + h} g_{,\phi} u_{\phi i} + 104 \frac{1}{(N + h) \cdot \cos(\phi) } g_{,\lambda} u_{\lambda i} + 105 g_{,h} u_{h i} 106 \end{equation} 107 Moreover for integrals we get by substitution rule 108 \begin{equation} 109 dx_0 \; dx_1 \; dx_2 =\det((x_{i,\alpha})) \; d \phi \; d\lambda \; dh = v \; d \phi \; d\lambda \; dh 110 \end{equation} 111 with $v= (M + h) \cdot (N + h) \cdot \cos(\phi)$. 112 113 Notice that for a spherical Earth $e=0$ for which $M=N$ is the radius of the Earth and $M+h$ is the distance from 114 the center.