The spherical horizontal and spherical vertical boundary value problem - vertical deflections and geoidal undulations - the completed Meissl diagram

In a comparison of the solution of the spherical horizontal and vertical bo undary value problems of physical geodesy it is aimed to construct downward continuation operators for vertical deflections (surface gradient of the i ncremental gravitational potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the E arth's topographic surface or of the three-dimensional (3-D) Euclidean spac e nearby down to the international reference sphere (IRS). First the horizo ntal and vertical components of the gravity vector, namely spherical vertic al deflections and spherical gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geo metry space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of horizontal spherical boundary problem in terms of the horizontal vector-valued Green function co nverts vertical deflections given on the IRS to the incremental gravitation al potential external in the 3-D Euclidean space. The horizontal Green func tions specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the vertical spherical b oundary value problem is solved in terms of the vertical scalar-valued Gree n function. Fourth, the operators for upward continuation of vertical defle ctions given on the IRS to vertical deflections in its external 3-D Euclide an space are constructed. Fifth, the operators for upward continuation of i ncremental gravity given on the IRS to incremental gravity to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward continuation of horizontal and verti cal gravity data, namely vertical deflection and incremental gravity, are p roduced.