Downward continuation of Helmert gravity anomaly to precise determination of geoid in Iran

The use of Stokes boundary values problems for the determination of the geoid requires that gravity anomalies (as boundary values) are known on geoid (as boundary). While, the gravity observations are measured on the Earth's surface, to obtain boundary data, the surface gravity anomalies does are downward from the terrain onto geoid. For downward continuation (DWC) of surface gravity anomalies, the Poisson integral can be used if the disturbing potential corresponding to gravity anomalies have harmonics everywhere above the geoid. But free-air gravity anomalies are non harmonics due to presence of (topographical+athmospherical) masses above the geoid. In the geoid accounting for the topography was first suggested by Helmert put in practice for example by Martinec and Van??ek (1994), and ultimately applied in the Stokes-Helmert scheme (Van??ek et. al, 1999). The second condensation method proposed by Helmert, involves the condensation of the topographic and atmospheric masses outside the geoid onto the geoid in the form of a surface layer.
As the Helmert disturbing potential is harmonic above the geoid, we use the Poisson solution formulated for harmonic functions. A critical and unavoidable problem encountered in the DWC is the implementation of discretization of the Poisson integral. Several discretization schemes have been proposed by, e.g., Matrince (1996), Van??ek et al. (1996) and Sun and Van??ek (1998). In mean-mean model the mean values on the surface are transformed to the corresponding mean values on the geoid by doubly averaged Poisson kernel. The point-point model transformed point surface values to point geoid values. The in point-mean the mean values of geoid are obtained from the point values on the Earth's surface by averaged Poisson kernel.
Normally, mean anomalies are evaluated from several/many point values on the Earth's surface on the regular grid. Also the finite element method of evaluation of Stokes integral needs that mean anomaly to be given on the geoid. Therefore we have to use the mean-mean model of discretization of Poisson integral. But In Iran, the distribution of observed point gravity data are very spars and it involves the big gaps in Alborz and Zagros mountainous areas as well as in deserts and sea. The gaps usually are filled by the high resolution satellite geopotential model. Therefore in most cells, the mean gravity anomalies are predicted/computed from few point values or geopotential model. In this case, the mean values tend to point values.
We employed both point-mean and mean-mean models of DWC in Stokes-Helmert method to precise determination of geoid in Iran. The long wavelengths part of geoid up to degree and order 180/180 is determined using the EGM08 model. As external evidence, the two geoids were compared to the GPS solution at 213 points into the national height network. The RMS of differences between two geoids and GPS-levelling data are 46cm without any applying correction surface. The RMS of difference between two geoid models is about 4cm and it can be reached up to 35cm in mountainous area. As a result, we cannot able to decide to suitable discretization model of Poisson integral due to very poor distribution of gravity data in Iran.