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Applications of numerical integration in geodesy and geophysics Analysis of one-dimensional methods and presenting two-dimensional spherical splines numerical integrators using Bernstein polynomials

Kiani Shahvandi Mostafa

Acta Geophysica

2021

Vol. 69, no. 1

29--45

In this paper, two applications of numerical integration in geodesy and geophysics are presented. In the frst application, the Molodenskij truncation coefcients for the Abel-Poisson kernel are computed using eleven diferent numerical integration procedures, namely two-, three-, four-, and fve-point Gaussian, Gauss–Kronrod, trapezoidal rule, Simpson and its adaptive mode, Romberg, Lobatto, and Sard’s approximating functional numerical integration methods. The coefcients are computed for truncation degree 90, and truncation radius 6◦. The results are then compared with an independent method for calculating these coefcients. It is shown that numerical integration methods represent better accuracy. In the second application, the gravity accelerations at sea surface in Qeshm in southern Iran are calculated using the spherical spline numerical integration method. The formulae for spherical spline numerical integration in two diferent modes weighted and without weight are derived. The special case when the weight of the integral is the so-called Stokes’ kernel is thoroughly investigated. Then, the results are used to generate gravity accelerations. First, the geoid height from the sum of the mean sea level and sea surface topography is calculated. Then, a spherical spline analytical representation—with unknown coefcients—is considered for the gravity anomaly. In the next step, using the Stokes’ formula for the integral relation between geoid height and gravity anomaly, the unknown coefcients in the previous step are calculated and subsequently the gravity anomalies are derived. Adding the gravity of the reference ellipsoid to the gravity anomalies, the actual gravity accelerations at sea surface in Qeshm are calculated. To analyze the accuracy, the derived values are compared with the values observed by shipborne gravimetry. It is shown that using Bernstein polynomials as basis function for calculating numerical integration has a better accuracy than other numerical integration methods of the same degree.