ON ACCURATE GEOID MODELING: DERIVATION OF DIRICHLET PROBLEMS THAT GOVERN GEOIDAL UNDULATIONS AND GEOID MODELING BY MEANS OF THE FINITE DIFFERENCE METHOD AND A HYBRID METHOD

Abstract

The geoid is the reference surface used to measure heights (orthometric). These are
used to study any mass variability in the Earth system. As the Earth is represented
by an oblate spheroid (Ellipsoid), the geoid is determined by geoidal undulations
(N) which are the separation between these surfaces. N is determined from gravity
data by Stokes's Integral. However, this approach takes a Spherical rather than an
Ellipsoidal Earth. Here it is derived a Partial Differential Equation (PDE) that
governs N over the Earth by means of a Dirichlet problem and show a method to
solve it which precludes the need for a Spherical Earth. Moreover, Stokes's Integral
solves a boundary value problem defined over the whole Earth. It was found that the
Dirichlet problem derived here is defined only over the region where a geoid model
is to be computed, which is advantageous for local geoid modeling. Moreover, the
method eliminates several of the sources of uncertainty in Stokes's Integral.
However, estimates indicate that the errors due to discretization are very large in
this new method which calls for its modification. So, here it is also proposed an optimal combination of techniques by means of a Hybrid method and shown that it
alleviates the uncertainty in Finite Difference Method. Moreover, a rigorous error
analysis indicates that the Hybrid method proposed here may well outperform
Stokes's Integral.