next up previous contents
Next: 4 Nota Bene Up: 3 Geoid Height, Step Previous: 3.7 Fully normalized associated   Contents

3.8 Putting it all together

Now you are have all the quantities you need to compute the geoid undulation $ N$ as a function of geographical longitude $ \lambda$ and latitude $ \varphi$ by the well-known formula (Heiskanen and Moritz, 1967)

\begin{displaymath}\begin{split}N(\lambda,\varphi) = \frac{GM_{\mathrm{g}}}{\gam...
...n(m\lambda)\right] \bar{P}_{nm}(\sin\bar{\varphi}). \end{split}\end{displaymath} (30)

Here, $ \bar{C}_{n,m}$ and $ \bar{S}_{n,m}$ are the spherical harmonic coefficients of degree $ n$ and order $ m$. Note that in this formula, you need to use the mass gravity constant $ GM_{\mathrm{g}}$ and the scale factor $ a_{\mathrm{g}}$ of the geopotential model. $ \bar{P}_{n,m}(\sin\bar{\varphi})$ are the fully normalized harmonics, or fully normalized associated Legendre functions. They are described in the previous section (Section 3.7). Note that the harmonics $ \bar{P}_{n,m}$ are evaluated at the geocentric latitude $ \bar{\varphi}$, and not at the geographical latitude $ \varphi$.



mlosch@awi-bremerhaven.de