# Efficiency investigation of tesseroid based methods for computing gravimetric terrain correction

Author

Assistant Professor, Civil Engineering Faculty, Shahrood University of Technology, Iran

Abstract

The gravitational effect of topographical masses is one of the important component of the gravity field, which plays a key role in geophysical and geodetic studies. For geophysical interpretations, it is necessary to eliminate the effect of topography as a disturbing factor from the observed gravity data. In geodetic applications, the solution of geodetic boundary problem such as Stokes requires mass free space above the geoid. In present study efficiency of different tesseroid based methods are compared with well-known rectangular prism to evaluate the gravimetric terrain corrections up to distance of 1.5 arc-degree known as the Hayford-Bowie zone. For this purpose, the mathematical formula: the vertical derivative of Newton integral and the digital elevation model (DEM) are used as data. In computing the topographic effect, we are involved with the two factors: 1- the integral element (point, line, plane, rectangular prism, tesseroid, etc.) and 2- geometry of topography (planar, spherical and ellipsoidal), which causes some difficulties to understand the subject. Finite element method is a general and standard method for estimating the terrain correction. In this method, the gravitational topographic effect is evaluated as the total gravitational effect of the smaller elements.
Tesseroid is the geometrical body bounded by two concentric spheres. This element uses the spherical geometry of topography which introduces relative error of about 1% (Novak and Grafarend, 2005). By choosing this element, the Newton integral and its radial derivatives do not have an analytic solution, and numerical integration must be applied. The rectangular prism element, has been used frequently to compute terrain correction in various studies. It uses planar geometry and has an analytical solution for Newton's integral and its derivatives. Recently many studies investigated tesseroid based method to compute the potential and attraction of topographic masses, see, [Fukushima, 2017; Grombein et al., 2013; Heck and Seitz, 2007; Uieda et al., 2016]. Fukushima's method utilizes the 3D numerical double-exponential integration method, HS's method uses the Tylor series up to term 2 and the PM method is the zero term approximation of HS method. The simulation studies demonstrated the higher accuracy of tesseroid based methods compared to the method of prism in the literature. However, their performance is not tested for gravimetric terrain correction. The main goal of this study is the investigation of efficiency, in terms of speed and accuracy, of four tesseroid methods: Fukushima, Martinec-Vanicek (MV), Heck-Seitz (HS), point mass (PM) compared with prism in Hayford-Bowie zone.
To investigate the computation accuracy, we used bounded spherical shell with constant thinness and density for which the analytical exact solution exists. The thinness of the shell have been chosen 1000 meter and the computation point is located on the origin of bounded spherical shell on the equator in the spherical coordinate (0,0,1000). The computation of terrain correction are discretized in different zones: innermost, inner and outer correspond respectively to ,  and  and with different sizes. The contribution of innermost zone is over 75% of total effect. Numerical results indicate the success of the prism for topographic effect in all three zones, especially for masses in neighborhoods of computation points, than those methods based on tesseroid. To overcome the effect of Earth's curvature, the elevation of computation point is corrected using a simple formula. Also, our calculations show that, in innermost zone, the topography should be discretized in 30 meter elements to achieve 10 Gal level of accuracy.

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Main Subjects

#### References

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