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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.

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