Comparison of the geodetic height correcting surface determination methods: A case study for Tehran

During the last decades, GPS has been used in many applications of geodesy, geophysics and surveying. The ellipsoidal height which is provided by GPS is a geometric height and lacks a physical meaning. So it cannot be used in most of engineering applications. Therefore, we need another type of height known as orthometric height which is one of the physical heights. The main advantage of leveling is its high accuracy but on the other hand, leveling measurements involve large amounts of time, cost and labor work. So another way to reach orthometric heights from GPS measurements is to determine geodetic height correcting surface of the area. Depending on the data availability and accuracy requirements, there are two basic approaches to transform the ellipsoidal heights to orthometric heights which include the gravimetric approach and geometric approach. The gravimetric approach uses the gravity data to determine the correcting surface. The geometric approach is to use the GPS/leveling data and interpolation methods in order to determine the correcting surface. In the present research, the geometric approach is applied to determine the geodetic height correcting surface from GPS/Leveling data. To do so, some reference points with known ellipsoidal and orthometric heights are used. Since there is a linear relation between the triplet of ellipsoidal, orthometric and geoid heights, one can calculate the geoid undulation by means of that relation and finally calculate the geoid undulation in every point of the area by the use of interpolation methods. First, basic definitions and equations of the radial basis function are explained. The method of radial basis functions is a global interpolation method used for interpolation of scattered data. This method was developed by Richard L. Hardy. Different kernels can be used as the kernel of radial basis function like multiquadric, Gaussian, thin plate spline and linear. In this paper, multiquadric and thin plate spline kernels are used to determine the correcting surface. The multiquadric method contains a parameter called shape parameter which is defined by the user and in actual applications, it affects the accuracy of the method. There are different ways to determine the shape parameter, among which the cross-validation procedure is explained. Then, the artificial neural network is defined as the other method to determine the correcting surface. Neural networks have different types but the multilayer perceptron neural network is the method commonly used in interpolation applications. Therefore, in this research, a three-layer perceptron neural network is used. Finally, as the case study, the GPS/leveling data of Tehran is analyzed. The ellipsoidal heights and orthometric heights of 147 benchmark points distributed all over the area are used in order to calculate the geoid undulations. And then the geodetic height correcting surface is determined using the multiquadric and thin plate spline and the artificial neural network methods. At last, their root-mean-square (RMS) values are compared with each other and the method with the lowest RMS is chosen as the most accurate method. The results show that the thin plate spline method leads to better results in Tehran.