Abstract: In space geodesy time, GNSS is capable of determining geodetic heights for ground points, enabling gravity disturbances to be direct observables. As a result, the second geodetic boundary value problem(GBVP)with gravity disturbances as the boundary condition can be applied in geodesy. The combination of its solution with GNSS is becoming a new approach to measuring the elevation above sea level, which is expected to have a bright future in application. The paper briefly discusses in principle the spherical approximation the 2nd GBVPs with two different boundary surfaces. The first kind uses the topographic surface as the boundary, and gives the analytical continuation solutions of surface height anomalies and deflections of the vertical. The second kind, with the surface of the reference-ellipsoid as the boundary, first moves topographic masses outside the reference ellipsoid onto it according to the Helmert's second condensation method, and then convolutes the Hotine function with Helmert gravity disturbances at the boundary, which are analytically downward-continued from the earth's surface, and after considering the indirect effects of topography gives finally the Helmert solutions of geoidal heights, ellipsoid vertical deflections, height anomalies and surface deflections of the vertical. In the discussion part a comparison between the 2nd GBVP and the 3rd GBVP is made, and an approach is presented to converting the orthometric or normal height into the geodetic height for existing gravity points, and the application prospect of the 2nd GBVP is looked into the future.

Key words: the second geodetic boundary value problem, geoidal heights, height anomalies, deflections of the vertical, analytical continuation solutions, Helmert solutions