Geoid Determination through Ellipsoidal Stokes Boundary-Value Problem

Martinec and Grafarend (1997) have shown how the construction of Green's function in the Stokes boundary-value problem with gravity data distributed on an ellipsoid of revolution is approached in the O(e ₀ ² )-approximation. They have also expressed the ellipsoidal Stokes function describing the effect of ellipticity of the boundary as a finite sum of elementary functions. We present an effective method of avoiding the singularity of spherical and the ellipsoidal Stokes functions, and also an analytical expression for the ellipsoidal Stokes integral around the computational point suitable for numerical solution. We give the numerical results of solving the ellipsoidal Stokes boundary-value problem and their difference with respect to the spherical Stoke boundary-value problem.     

Far-Zone Contribution in Ellipsoidal Stokes Boundary-Value Problem

In the first attempt to solve the Stokes boundary-value problem in ellipsoidal coordinates numerically (Ardestani and Martinec, 2000), we focused on the near-zone contribution since the effect of the ellipsoidal Stokes function in the far-zone contribution is not considered. We present a method for solving the ellipsoidal Stokes integral in far-zone contribution. The numerical results of computing the magnitude of this term for an area in north of Canada are presented.     

Ellipsoidal Stokes Boundary-Value Problem with Ellipsoidal Corrections in the Boundary Condition

We would like to solve the Stokes boundary-value problem taking into consideration the ellipsoidal corrections in the boundary condition in ellipsoidal coordinates The original problem, i.e., the ellipsoidal Stokes boundary-value problem has been solved by Martinec and Grafarend (1997) We use the same philosophy expressed by Martinec (1998) to solve the spherical Stokes boundary-value problem with ellipsoidal corrections in the boundary condition We wish to show the magnitude of the integration kernel describing the effect of the ellipsoidal corrections in the boundary condition in a cap around the computational point.     

Solution to the Stokes Boundary-Value Problem on an Ellipsoid of Revolution

We have constructed Green's function to Stokes's boundary-value problem with the gravity data distributed over an ellipsoid of revolution. We show that the problem has a unique solution provided that the first eccentricity e₀ of the ellipsoid of revolution is less than 0·65041. The ellipsoidal Stokes function describing the effect of ellipticity of the boundary is expressed in the E-approximation as a finite sum of elementary functions which describe analytically the behaviour of the ellipsoidal Stokes function at the singular point = 0. We prove that the degree of singularity of the ellipsoidal Stokes function in the vicinity of its singular point is the same as that of the spherical Stokes function.