A methodology for mean gravity value computation based on harmonic splines and their application to boundary value problem

Height is amongst the most delicate subjects of geodesy. Thanks to the Global Navigation Satellite Systems (GNSS) like GPS or GLONASS, 3D point positioning of points, by geometrical positioning, since years ago has become a common practice. The height derived from these ways has a geometrical concept. In civil projects the physical concept of the height is more demanded. Orthometric height, , is one kind of the physical concepts of the height. The Orthometric height of point i, , can be calculated by
where is the mean value of the gravity along the plumb line between the geoid and the surface point i and is the geo-potential number of point i, which is calculated using

One of the problems in orthometric height calculation is computation of .The value of the gravity at the point with mean height is calculated by

Where is gravity observation value at point i.
The orthometric height computed by this mean value of the gravity is called Helmert orthometric height, according to Sanso and Sona (1993) the idea for earth gravity determination.
In this paper a methodology to calculate value of the gravity at the point with mean height from the geoid has been supposed. Derived gravity from this method is composed from three parts, (1) global and regional gravity computed by ellipsoidal harmonic expansion to degree and order 360 plus the centrifugal acceleration (2) gravitational due to terrain masses within the radius of 55km around the computational point (3) incremental gravity intensity at the computational point. The first and second parts are computed by global geopotential models and digital terrain models.
Computation of the third part is possible by solving a boundary value problem. In this paper for computing the incremental gravity intensity at the point with mean height, a method by solving a fixed-free two-boundary nonlinear value problem is addressed. This boundary value problem constructed for observables of the type (i) modulus of gravity (ii) gravity potential (iii) satellite altimetry data (iv) astronomical latitude (v) astronomical longitude.
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 that its solution gives us the incremental gravity potential at the surface of the reference ellipsoid. Out of the reference ellipsoid surface, this answer could be obtained by solving the following Dirichlet boundary value problem:

In this paper harmonic splines supposed by Freeden (1987) are used to solve the Dirichlet problem. By applying the gradient operator on the incremental gravity potential, due to solving Dirichlet problem, incremental gravity at every point out of the reference ellipsoid can be calculated (Jekeli, 2005).
Second section of this paper is an introduction on harmonic splines analysis. The construction of self productive Hilbert space and optimum interpolation answer is presented in the third section. In the final section the application of harmonic splines for solving the Dirichlet boundary value problem is discussed and by the proposed methodology the mean value of gravity in the first order leveling of Iran is calculated.