**Abstract**

Precise geoid determination on islands suffers from the lack of accurate gravity data on the open seas. Nowadays, sub-microgal accuracy for the land gravity observations is obtainable. But the sea gravity data which are collected via shipborne techniques, due to the measuring environment at sea area, are usually highly noisy and are contaminated with various systematic errors.

On the other hand, satellite altimetry has provided a new source of information for geoid determination at sea area. It should be noted that satellite altimetry has accuracy at centimeter level which reaches decimeters at coastal. Such accuracy in geometric space is equivalent to microgal in gravity space, which is equivalent to microgal in gravity space. Therefore, one can see the altimetry data as a relatively accurate source of information for gravity applications.

With satellite altimetry observations at the sea area and accurate gravity data on the islands, we can define a gravimetric-altimery boundary value problem. Geometry of the oceanic part of the Earth’surface is given by the altimetric data. Ergo the problem at the oceanic part is a fixed boundary value problem. At the continental part, now, GPS is operable. The availability of the GPS coordinates means the geometry of the continental part can be considered as known. Ergo, we deal with a fixed gravimetric-altimetry boundary value problem.

By applying variational techniques to the fixed gravimetric-altimetry boundary

value problem the existence and uniqueness of its weak solution can be proved (Keller, 1996).

In this paper, using satellite altimetry observations on the open sea and gravity

from gravimetry on the island, a fixed gravimetric-altimetry boundary value problem

for geoid computations at islands has been developed. The problem is defined as

follows:

Where is the gravity potential of the Earth, the norm of the gravity vector on the island, geoid from satellite altimetry observations, mass density, the angular velocity and geoid potential.

The first step towards the solution of the proposed fixed-free two-boundary value problem is the linearization of the problem. After linearization we obtained an oblique boundary condition on the island and a Dirichlet condition on the sea area.

The algorithmic steps of the solution of the fixed garvimetric-altimetry boundary value problem for geoid computations at islands are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for removal of the effect of the global gravity from gravity intensity at the surface of the island.

- The removal from the gravity intensity at the surface of the Earth the effect of residual masses at the radius of up to 55 km from the computational point.

- Derivation marine geoid from satellite altimetry data.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of ellipsoidal centrifugal field for removal of from the geoidal undulations derived from satellite altimetry the effect of the global gravity.

- The removal from geoidal undulations derived from satellite altimetry of the effect of water masses at the radius of up to 55 km from the computational point.

- Application of Koch and Kusche algorithm (Koch and Kusche, 2002) for derivation of disturbing gravity potential at the surface of the reference ellipsoid from residual gravity intensity and residual gravity potential of satellite altimetry data.

- Restoration of the removed effects on the surface of the reference ellipsoid.

- Application of ellipsoidal Bruns formula in order to compute geoid undulations.

- Computation of the geoid of Qeshm Island of Iran has successfully tested this methodology.

On the other hand, satellite altimetry has provided a new source of information for geoid determination at sea area. It should be noted that satellite altimetry has accuracy at centimeter level which reaches decimeters at coastal. Such accuracy in geometric space is equivalent to microgal in gravity space, which is equivalent to microgal in gravity space. Therefore, one can see the altimetry data as a relatively accurate source of information for gravity applications.

With satellite altimetry observations at the sea area and accurate gravity data on the islands, we can define a gravimetric-altimery boundary value problem. Geometry of the oceanic part of the Earth’surface is given by the altimetric data. Ergo the problem at the oceanic part is a fixed boundary value problem. At the continental part, now, GPS is operable. The availability of the GPS coordinates means the geometry of the continental part can be considered as known. Ergo, we deal with a fixed gravimetric-altimetry boundary value problem.

By applying variational techniques to the fixed gravimetric-altimetry boundary

value problem the existence and uniqueness of its weak solution can be proved (Keller, 1996).

In this paper, using satellite altimetry observations on the open sea and gravity

from gravimetry on the island, a fixed gravimetric-altimetry boundary value problem

for geoid computations at islands has been developed. The problem is defined as

follows:

Where is the gravity potential of the Earth, the norm of the gravity vector on the island, geoid from satellite altimetry observations, mass density, the angular velocity and geoid potential.

The first step towards the solution of the proposed fixed-free two-boundary value problem is the linearization of the problem. After linearization we obtained an oblique boundary condition on the island and a Dirichlet condition on the sea area.

The algorithmic steps of the solution of the fixed garvimetric-altimetry boundary value problem for geoid computations at islands are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for removal of the effect of the global gravity from gravity intensity at the surface of the island.

- The removal from the gravity intensity at the surface of the Earth the effect of residual masses at the radius of up to 55 km from the computational point.

- Derivation marine geoid from satellite altimetry data.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of ellipsoidal centrifugal field for removal of from the geoidal undulations derived from satellite altimetry the effect of the global gravity.

- The removal from geoidal undulations derived from satellite altimetry of the effect of water masses at the radius of up to 55 km from the computational point.

- Application of Koch and Kusche algorithm (Koch and Kusche, 2002) for derivation of disturbing gravity potential at the surface of the reference ellipsoid from residual gravity intensity and residual gravity potential of satellite altimetry data.

- Restoration of the removed effects on the surface of the reference ellipsoid.

- Application of ellipsoidal Bruns formula in order to compute geoid undulations.

- Computation of the geoid of Qeshm Island of Iran has successfully tested this methodology.

**Keywords**