# Analyzing the bumps in the EGM2008 geoid

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## Analyzing the bumps in the EGM2008 geoid

 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 <[hidden email]> SRI International, Princeton, NJ 08543-5300 Tel: +1 609 734 2312 _______________________________________________ Proj mailing list [hidden email] http://lists.maptools.org/mailman/listinfo/proj
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## Re: Analyzing the bumps in the EGM2008 geoid

 I should have included the definition I used for spherical harmonics (since different definitions are used in different fields).  I use the DLMF definition given in    http://dlmf.nist.gov/14.30.E1   http://dlmf.nist.gov/14.30.E3_______________________________________________ Proj mailing list [hidden email] http://lists.maptools.org/mailman/listinfo/proj
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## Re: Analyzing the bumps in the EGM2008 geoid

Thank you for this and the previous post. Very interesting.

 From: Charles Karney <[hidden email]> To: PROJ.4 and general Projections Discussions <[hidden email]> Date: 07/04/2011 01:59 PM Subject: Re: [Proj] Analyzing the bumps in the EGM2008 geoid Sent by: [hidden email]

I should have included the definition I used for spherical harmonics
(since different definitions are used in different fields).  I use the
DLMF definition given in

http://dlmf.nist.gov/14.30.E1

http://dlmf.nist.gov/14.30.E3
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