A new method for height determinations based on solving the geodetic fixed boundary value problem



The simple method for height determination is spirit leveling. The leveled height differences are path-dependent. Ergo height from spirit leveling is not unique. The problem can be solved by converting the path-dependent leveled height differences into unique path-independent height differences (Vanicek and Krakiwsky, 1986).
The difference between the potentials of two close equipotential surface can be written as:
Practically, instead of potential, it is better to use geopotential numbers :
Having derived geopotential numbers, we can compute various height systems such as dynamic height, orthometric height and normal height. According to (1), with gravity measured along the leveling line, the potential difference as a path-independent quantity can be determined. In this method we can only compute geopotential numbers for points where we can establish a leveling path.
On the other hand with geoid height at hand and the availability of geodetic height from GPS, we can compute orthometric height. With computation of mean gravity, orthometric height can be converted to geopotential number. But there are still some open problems with computation of geoid and mean gravity. The aim of this paper is the presentation of a new method for computation of geopotentail numbers based on solving the fixed geodetic boundary value problem.
With the availability of GPS coordinates for the point on the Earth’s surface, the outer boundary of the Earth can be regarded as a known and fixed boundary. With boundary observations on the Earth’s surface at hand, we deal with a fixed geodetic boundary value problem. The problem is defined as follows:

where is the gravity potential of the Earth, the norm of the gravity vector on the Earth’s surface, mass density, the angular velocity and geoid potential.
The first step towards the solution of the proposed fixed geodetic boundary value problem is the linearization of the problem. After linearization we obtained a linear boundary value problem as follows:

In this paper we propose a new method for solving the linear boundary value problem based on harmonic splines. The main steps of the new method are as follows:
- Application of the ellipsoidal harmonic expansion complete up to degree and order of 360 and of the ellipsoidal centrifugal field for removal of the effect of the global gravity from gravity intensity at the surface of the Earth.
- The removal from the gravity intensity at the surface of the Earth of the effect of residual masses at a radius of up to 55 km from the computational point.
-Solution of the linear boundary value problem based on harmonic splines.
-Restoration of the removed effects in order to compute potential on the surface of the Earth.
-Subtraction of the geoid potential from the computed potential on the Earth's surface in order to obtain geopotential numbers.
-Computation of various height systems.
Computation of various height systems in the central part of Iran has successfully tested this methodology.