Applications of numerical integration in geodesy and geophysics

Analysis of one-dimensional methods and presenting two-dimensional spherical splines numerical integrators using Bernstein polynomials


In this paper, two applications of numerical integration in geodesy and geophysics are presented. In the first application, the Molodenskij truncation coefficients for the Abel-Poisson kernel are computed using eleven different numerical integration procedures, namely two-, three-, four-, and five-point Gaussian, Gauss–Kronrod, trapezoidal rule, Simpson and its adaptive mode, Romberg, Lobatto, and Sard’s approximating functional numerical integration methods. The coefficients are computed for truncation degree 90, and truncation radius \(6^\circ\). The results are then compared with an independent method for calculating these coefficients. 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 different 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 coefficients—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 coefficients 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.

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Availability of data and material

The observation data for this study are available for public use.

Code availability

MATLAB and C++ codes for the applications presented in this paper are available.


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Correspondence to Mostafa Kiani Shahvandi.

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Communicated by Michal Malinowski (CO-EDITOR-IN-Chief)/Ramon Zuñiga, Ph.D. (CO-EDITOR-IN-CHIEF).

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Kiani Shahvandi, M. Applications of numerical integration in geodesy and geophysics. Acta Geophys. (2021).

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  • Numerical integration
  • Spline interpolant
  • Bernstein polynomials
  • Gravity acceleration at sea surface
  • Shipborne gravimetry
  • Stokes’ formula
  • Inverse gravimetric problem