Finite Element Method for Precise Geoid Modeling for GNSS Positioning in Palestine

Abstract

Nowadays, the use of modern and precise GNSS technologies for precise positioning is the most common tool for field surveyors. The output coordinate of GNSS are divided into geometric horizontal coordinates (latitude, longitude), or equivalently the mathematically transformed and projected coordinates (Easting, Northing), and the ellipsoidal Normal heights (h). The ellipsoidal heights (h) need to be transformed to match with properties and values of the engineering used physical/orthometric heights (H) that are typically produced using precise leveling. The transformation between both types of heights requires the availability of precise geoid model as a height reference surface (HRS). Typically, the modeling process of a Geoid requires dense networks of precise leveling, gravity and astronomical deflections of vertical. Here, the requirements of the availability of dense leveling and gravity networks for classical geoid modeling methods are overridden by the integration of the limited number of benchmarks and the freely available global geoid models (EGM2008, Eigen05c, EGM96 … etc.) is applied using finite elements method. Conceptually, the modeling area is divided into patches with dimensions (50-70km) to transform the global models' reference datum to fit to the local vertical datum. Afterward, each patch is then divided into smaller elements/meshes with the size of (5x5km) that are represented by 2 nd /3 rd order polynomial. To apply the least squares solution for the parameters of the polynomials, a combined system observation equations is applied using GNSS/Leveling and additionally Geoid heights and deflections of vertical by the global models for further observations and densification of the solution. To guarantee the continuity and the smoothness of the modeled surface, one least squares solution is applied for all element using zero, first and second-order continuity conditions. Finally, statistical analysis of the least squares solution and test points were used for the validation and accuracy assessment of the model. Residuals less than 3cm were obtained by the solution. Consistently, the accuracy of 1-3cm could be achieved using the test points.