Geoid - Spherical Harmonics Representation

Spherical Harmonics Representation

Spherical harmonics are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is EGM96 (Earth Gravity Model 1996), determined in an international collaborative project led by NIMA. The mathematical description of the non-rotating part of the potential function in this model is

$V=\frac{GM}{r}\left(1+{\sum_{n=2}^{n_{max}}}\left(\frac{a}{r}\right)^n{\sum_{m=0}^n} \overline{P}_{nm}(\sin\phi)\left\right),$

where and are geocentric (spherical) latitude and longitude respectively, are the fully normalized associated Legendre polynomials of degree and order, and and are the numerical coefficients of the model based on measured data. Note that the above equation describes the Earth's gravitational potential, not the geoid itself, at location the co-ordinate being the geocentric radius, i.e., distance from the Earth's centre. The geoid is a particular equipotential surface, and is somewhat involved to compute. The gradient of this potential also provides a model of the gravitational acceleration. EGM96 contains a full set of coefficients to degree and order 360 (i.e. ), describing details in the global geoid as small as 55 km (or 110 km, depending on your definition of resolution). The number of coefficients, and, can be determined by first observing in the equation for V that for a specific value of n there are two coefficients for every value of m except for m = 0. There is only one coefficient when m=0 since . There are thus (2n+1) coefficients for every value of n. Using these facts and the formula, it follows that the total number of coefficients is given by

$\sum_{n=2}^{n_{max}}(2n+1) = n_{max}(n_{max}+1) + n_{max} - 3 = 130317$ using the EGM96 value of $\ n_{max} = 360$ .

For many applications the complete series is unnecessarily complex and is truncated after a few (perhaps several dozen) terms.

New even higher resolution models are currently under development. For example, many of the authors of EGM96 are working on an updated model that should incorporate much of the new satellite gravity data (see, e.g., GRACE), and should support up to degree and order 2160 (1/6 of a degree, requiring over 4 million coefficients). NGA has announced the availability of EGM2008, complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree 2190 and order 2159. Software and data is on the Earth Gravitational Model 2008 (EGM2008) - WGS 84 Version page.