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)

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