Ellipsoid Interface Under The Third Kind Boundary Integral Solution

This paper makes a comprehensive and systematic study of the establishment of the earth's ellipsoidal gravity model and the solution of the boundary value problem under ellipsoidal boundary.Firstly, according to the Laplace equation of ellipsoidal coordinates and the ellipsoidal harmonic functions in common use, the ellipsoidal series expansion of harmonic functions outside the referential ellipsoidal is introduced. However, the second associated Legendre function is too complicated to compute, so a simple and practical ellipsoidal harmonic series expansion is introduced, and the series solution and integral solution of the Dirichlet boundary value problem and the Neumann boundary value problem under ellipsoidal boundary are given. For the third boundary value problem under ellipsoidal boundary, firstly we simplify the form of the third boundary value problem. Then, the solution of the the third boundary value problem under ellipsoidal boundary is discussed in detail, along with its series solution, the integral expression is mainly studied.As all the solutions reserve an accuracy of ε~2, the O(T ε~4) can thus be acquired. That is. the relative accuracy could approximate 10~-9 to 10~10. Once the accuracy of model linearization is reserved, a cm accuracy of geoid is guaranteed theoretically.At the end of this paper, the transformation between spherical coefficients and ellipsoidal coefficients is introduced, from which the series solution of the boundary value problem under ellipsoidal boundary can be derived easily.