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3.6 Reference ellipsoid

An important part of the computation is the subtraction of the reference ellipsoid, defined by the semi-major axis of the Earth $ a$ and the flattening parameter $ f$. To do so, the zonal coefficients of the spherical harmonic gravity model $ \bar{C}_{2n,0}$, $ n=1,2,3,4,5$ are corrected by

$\displaystyle \bar{C}_{2n,0} \longrightarrow \bar{C}_{2n,0}+ \frac{GM}{GM_{\mathrm{g}}}\left(\frac{a}{a_{\mathrm{g}}}\right)^{n} \cdot\frac{J_{2n}}{\sqrt{4n+1}}.$ (20)

The factor $ (GM/GM_{\mathrm{g}})\cdot(a/a_{\mathrm{g}})^{n}$ accounts for the fact that neither the gravity mass constants nor the semi-major axes (just a scaling factor for the geopotential model) have to be the same for geopotential model and reference ellipsoid. In fact, very often they are not the same.3 In general, it is sufficient to correct the zonal coefficients $ \bar{C}_{2,0}$ through $ \bar{C}_{10,0}$. The coefficients $ J_{2n}$ are determined from the parameters $ a$, $ f$, and the angular velocity $ \omega$ of the Earth by an expansion (Heiskanen and Moritz, 1967, p.73, eq.2-92):

$\displaystyle J_{2n} = (-1)^{n+1}\frac{3(E/a)^{2n}}{(2n+1)(2n+3)} \left(1-n+5n\frac{C-A}{ME^{2}}\right).$ (21)

This equation involves new parameters: the moments of inertia with respect to any axis in the equatorial plane $ A$ and with respect to the axis of rotation $ C$ (not to be confused with the spherical harmonic coefficients $ \bar{C}_{n,m}$), the total mass $ M$, and the linear eccentricity $ E=\sqrt{a^{2}-b^{2}}$. They can be computed by (Heiskanen and Moritz, 1967)

$\displaystyle \frac{C-A}{ME^{2}}$ $\displaystyle =\frac{1}{3}\left[1- \frac{2}{15}\left(\frac{me'}{q_{0}}\right)\right]$ (22)
$\displaystyle m$ $\displaystyle =\frac{\omega^{2}a^{2}b}{GM}$ (23)
$\displaystyle q_{0}$ $\displaystyle =\frac{1}{2}\left[\left(1 +\frac{3}{e'^{2}}\right)\arctan{e'} -\frac{3}{e'}\right].$ (24)

$ e'=E/b$ is again the second numerical eccentricity. Subtracting the reference ellipsoid is a very sensitive operation, because two large numbers are subtracted from each other and this is always prone to errors. So if your coefficients $ J_{2n}$ are only slightly wrong, this will have a big effect on your solution. If you are lucky, the precomputed values of the coefficients $ J_{2n}$, which are in principle defined by $ a$, $ f$, $ \omega$, and $ GM$, are provided with the reference ellipsoid.


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Next: 3.7 Fully normalized associated Up: 3 Geoid Height, Step Previous: 3.5 Permanent tide system   Contents
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