3.3 Coordinates: geocentric versus geographic

Spherical harmonic models are formulated in geocentric (or ellipsoidal) coordinates as opposed to geographic (or geodetic) coordinates. The longitude is the same for both geographic and geocentric coordinates. But you have to convert your geographic (or geodetic) latitude $\varphi$ into geocentric latitude $\bar{\varphi}$. The formula for this is (Torge, 1991)

$$\bar{\varphi} = \arctan\frac{z}{\sqrt{x^{2}+y^{2}}} = \arctan\left[ \left(\frac{b}{a}\right)^2\tan\varphi\right].$$ (14)