A methodology for combination of GPS/Leveling geoid as boundary data with the gravity boundary data within a gravimetric boundary value problem


Nowadays, combination of GPS heights with orthometric heights, derived from precise leveling, is broadly used to obtain point-wise solutions of the geoid, which is called “GPS/Leveling geoid”. The “GPS/Leveling geoid” is commonly used to constrain the gravimetric geoid solutions in a least squares surface fitting process. In this paper, unlike the usual application, the “GPS/Leveling geoid” is used as a boundary data. More specifically, in this paper we have developed a methodology for combination of “GPS/Leveling geoid”, as the boundary data, with other geodetic boundary data within the Fixed-Free Two-Boundary Value Problem (FFTBVP) for geoid computations. The proposed methodology can be explained algorithmically as follows:
1. Removal of the global topography and terrain effects via ellipsoidal harmonic expansion to degree and order 360 plus the centrifugal effect from the gravity boundary data at the surface of the Earth using the known GPS coordinates of the boundary points.
2. Removal of the local terrain masses using analytical solution of the Newton integral in the “cylindrical equiareal map projection” of the reference ellipsoid.
3. Formation of integral equations of the Abel-Poissn type for the harmonic residual gravity boundary data at the surface of the Earth, derived from the aforementioned remove steps.
4. Linearization and discretization of the formulated integral equations.
5. Application of the “GPS/Leveling geoid” within the ellipsoidal Bruns formula as the constraints to the system of equations developed in step (4) for the residual gravity data.
6. Least squares solution of the developed constraint problem of step (5), to estimate incremental gravitational potential values over the solution grid used for linearization in step (4) on the surface of the reference ellipsoid.
7. Restoration of the removed effects of steps (1) and (2) over the grid points on the reference ellipsoid.
8. Application of the Bruns formula to compute point-wise geoid over the grid points on the reference ellipsoid.
As a case study the proposed method is used for the geoid computation within the geographical region of Iran based on gravity and GPS/Leveling geoid as boundary data. The numerical results show the success of the methodology.
Finally the advantages of the proposal methodology can be summarized as follows:
1. Strictly following the principle of Gravimetric Boundary Value Problems (GBVP) for the geoid computation
2. Increasing the degree of freedom of the GBVP from a statistical point of view.
3. Making the downward continuation step of the GBVP solution more stable.