Formulation of Stokes-Helmert boundary value problem using no topography space



To solve the geodetic boundary-value problem the gravity anomalies have to be continued from the Earth's surface down onto the geoid surface. Helmert’s gravity anomalies (multiplied by the geocentric radius) are harmonic above the geoid. It is thus possible to use them in the Poisson solution of the inverse Dirichlet boundary value problem to get their values on the geoid. The downward continuation (DWC) of Helmert gravity anomalies has raised some criticism, however, originating from the suspicion that the anomalies, even though they are smoother than free-air and Faye anomalies, are still too rough to yield a reasonably good solution particularly for denser grid of gravity anomalies.
No topography (NT) space is the gravity Earth's field after the removal of the gravitational effect of the all masses above geoid. Van??ek et al., 2003 proposed the Spherical Complete Bouguer Anomaly (SCBA) as a 3D solid gravity anomaly that is suitable for DWC. The SCBA is a harmonic and smooth function that conforms to an accurate and stable downward continuation. The topography above the geoid contributes toward the high frequency portion of the free-air gravity anomaly on the Earth surface. By contrast, the SCBA is a smooth function because it is already corrected for the topographical effect. Hence, the Bouguer anomaly, especially the SCBA, is much smoother with attenuated high frequency content, compared to the Helmert anomaly.
By the nature of the Downward Continuation (DWC) process, which amplifies high frequencies, the SCBA is better suited (Heck, 2003). However, in contrast to the Helmert anomaly (a free-air type), the SCBA involves the disturbed isostasic equilibrium of the crust on the mantle because of the removal of the topographical effect. This phenomenon results in a powerful low frequency content in the SCBA and a large indirect effect (PITE), ultimately amounting to a few hundred meters which is detrimental to the accuracy of the geoid solution. To reduce the large effect to a few decimeters, the SCBA on the geoid is transformed back to the Helmert space.
In this paper we formulate the Stokes-Helmert BVP using No topographical space and SCBA in a three space scenario.
1. transforming the observed gravity anomaly on the terrain to the NT space on the same position,
2. downward continuation of the SCBA anomalies from the observation positions to the geoid level,
3. transforming the SCBA on the geoid to the Helmert space,
4. evaluation of the co-geoid using the generalized Stokes integral,
5. transforming the co-geoid back to the geoid. i.e., from Helmert’s space to the real space by applying the PITE.
In theory, all different reductions yield the same result in geoid determination. We compared the geoid by NT deduced Stokes-Helmert method and geoid by Stokes-Helmert method 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 gravimetric geoids were compared to the geometric GPS-levelling solution at 213 points. The RMS of differences between two geoids and GPS-levelling data are 46cm without any applying correction surface. However, the RMS of difference between two geoid models is about 4cm and it can be reached up to 1.67meter in mountainous area.