EGU2020-59, updated on 12 Jun 2020
EGU General Assembly 2020
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.

Subtleties in spherical harmonic synthesis of the gravity field

Ropesh Goyal1,2, Sten Claessens2, Will Featherstone1,2, and Onkar Dikshit1
Ropesh Goyal et al.
  • 1Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India (,
  • 2School of Earth and Planetary Sciences, Curtin University of Technology, GPO Box U1987, Perth WA 6845, Australia (,

Spherical harmonic synthesis (SHS) can be used to compute various gravity functions (e.g., geoid undulations, height anomalies, deflections of vertical, gravity disturbances, gravity anomalies, etc.) using the 4pi fully normalised Stokes coefficients from the many freely available Global Geopotential Models (GGMs).  This requires a normal ellipsoid and its gravity field, which are defined by four parameters comprising (i) the second-degree even zonal Stokes coefficient (J2) (aka dynamic form factor), (ii) the product of the mass of the Earth and universal gravitational constant (GM) (aka geocentric gravitational constant), (iii) the Earth’s angular rate of rotation (ω), and (iv) the length of the semi-major axis (a). GGMs are also accompanied by numerical values for GM and a, which are not necessarily identical to those of the normal ellipsoid.  In addition, the value of W0, the potential of the geoid from a GGM, needs to be defined for the SHS of many gravity functions. W0 may not be identical to U0, the potential on the surface of the normal ellipsoid, which follows from the four defining parameters of the normal ellipsoid.  If W0 and U0 are equal and if the normal ellipsoid and GGM use the same value for GM, then some terms cancel when computing the disturbing gravity potential.  However, this is not always the case, which results in a zero-degree term (bias) when the masses and potentials are different.  There is also a latitude-dependent term when the geometries of the GGM and normal ellipsoids differ.  We demonstrate these effects for some GGMs, some values of W0, and the GRS80, WGS84 and TOPEX/Poseidon ellipsoids and comment on its omission from some public domain codes and services (isGraflab.m, harmonic_synth.f and ICGEM).  In terms of geoid heights, the effect of neglecting these parameters can reach nearly one metre, which is significant when one goal of modern physical geodesy is to compute the geoid with centimetric accuracy.  It is also important to clarify these effects for all (non-specialist) users of GGMs.

How to cite: Goyal, R., Claessens, S., Featherstone, W., and Dikshit, O.: Subtleties in spherical harmonic synthesis of the gravity field , EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-59,, 2019