[Proj] Analyzing the bumps in the EGM2008 geoid

[Charles Karney]

Partly in response to Mikael Rittri's questions about compressing the
data files for the EGM models, I looked into seeing how well EGM2008 is
approximated by a few terms in its spherical harmonic expansion.  The
quick answer is not very well.  The bumps in EGM2008 are at all scales
and the coefficients in the spherical harmonic expansion decay very
slowly.

However, I thought it worth reporting on the magnitudes of the low order
terms.  Recall that the range in geoid heights (in meters) relative to
WGS84 is about [-107, 86].

The lowest order terms, Y00, Y10, Y11, Y20, Y21 are all small (less than
0.5m), "by construction".  (The volume if WGS84 is about right, the COM
of the geoid nearly matches WGS84, the flattening nearly matches WGS84.)

The biggest spherical harmonic component is the Y22 term, which is the
component that makes WGS84 into a triaxial ellipsoid.  This makes the
equator an ellipse with major/minor equatorial axes

   6378137 +/- 35 meters

The major axis is lon = -15, 165; the minor axis is lon = -105, 75.  The
amount EGM2008 deviates from this triaxial shape (WGS84 + Y22 term) is
[-72, 70].  The reason that a triaxial model of the earth is not useful
is that you add a lot of mathematical complexity going from an oblate
ellipsoid to a triaxial ellipsoid and yet you have not gained much
(about 25%) in how well you approximate the geoid.

The next biggest contributions are the Y3m components which together
with their amplitudes (meters) above/below WGS84 are

     Y31:  +/- 29
     Y33:  +/- 21
     Y30:  +/- 16
     Y32:  +/- 14

These results were derived by taking spherical harmonic moments of the
gridded EGM2008 geoid numerically using the longitude and geographic
co-latitude as the independent variables.

I found it necessary to carry out these integrals directly rather than
using the spherical harmonic expansions provided by the NGA because the
NGA provides two expansions: one for the gravitational potential and
another for an undulation correction, and both of these use geocentric
co-latitude.  I found it simplest just to perform the analysis on the
geoid height directly.

-- 
Charles Karney <charles.karney at sri.com>
SRI International, Princeton, NJ 08543-5300
Tel: +1 609 734 2312