Application of the nonlinear optimisation in regional gravity field modelling using spherical radial base functions

The gravity field is a signature of the mass distribution and interior structure of the Earth, in addition to all its geodetic applications especially geoid determination and vertical datum unification. Determination of a regional gravity field model is an important subject and needs to be investigated and developed. Here, the spherical radial basis functions (SBFs) are applied in two scenarios for this purpose: interpolating the gravity anomalies and solving the fundamental equation of physical geodesy for geoid or disturbing potential determination, which has the possibility of being verified by the Global Navigation Satellite Systems (GNSS)/levelling data. Proper selections of the number of SBFs and optimal location of the applied SBFs are important factors to increase the accuracy of estimation. In this study, the gravity anomaly interpolation based on the SBFs is performed by Gauss-Newton optimisation with truncated singular value decomposition, and a Quasi-Newton method based on line search to solve the minimisation problems with a small number of iterations is developed. In order to solve the fundamental equation of physical geodesy by the SBFs, the truncated Newton optimisation is applied as the Hessian matrix of the objective function is not always positive definite. These two scenarios are applied on the terrestrial free-air gravity anomalies over the topographically rough area of Auvergne. The obtained accuracy for the interpolated gravity anomaly model is 1.7 mGal with the number of point-masses about 30% of the number of observations, and 1.5 mGal in the second scenario where the number of used kernels is also 30%. These accuracies are root mean square errors (RMSE) of the differences between predicted and observed gravity anomalies at check points. Moreover, utilising the optimal constructed model from the second scenario, the RMSE of 9 cm is achieved for the differences between the gravimetric height anomalies derived from the model and the geometric height anomalies from GNSS/levelling points.