Efficient calculation of a Tesseroid's potential, and its first and second derivatives

K. Seitz, F. Wild, B. Heck

Geodetic Institute, University of Karlsruhe (TH), Englerstr. 7, D-76128 Karlsruhe

Topographic reductions in Physical Geodesy

 

Theory from Stokes (Stokes’ formula)

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Objective: Geoid determination

Boundary value problem (BVP) by Stokes:
• • Border area = (co-) geoid
? Topographic (+ isostatic) reductions

 

Theory from Molodensky

Objective: Determination of the surface S and the gravity potential W in the outer space

Boundary value problem by Molodensky:

  • In principle, no topographic reductions are required
  • Be used for smoothing

? RTM (Residual Terrain Modelling)
? RRT (Remove-Restore Technique)

 BILD

 

Approximation of the (residual) topography using a homogeneous vertical prism

  • Coordinate system defined as the cuboid’s system of the edges
  • Transformation of the effect of the vertical prism on dg and M in the topocentric horizon system of the calculation point P
  • Distance between the calculation point Pand the variable integration point Q:

 

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BILD

Effect on the potential in P
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Effect on the gravity in P
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Effect on the second radial derivative in P
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For the vertical prism with constant density, analytical solutions exist in terms of their potential and first derivatives, and the elements of the Marussi Tensor
  • High numerical cost
  • Intensive computing time and evaluation of the functions atan and ln

 

Approximation of the (residual) topography using Tesseroids

  • Limited by geographical grid lines and surfaces with a constant height
  • Division of the reference ellipsoid’s surface with respect to the geographic grid lines
  • Spherical approximation of Tesseroids
  • Distance between the calculation point P and the variable integration point Q:
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Effect on the potential in P
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Effect on the gravity in P
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Effect on the second radial derivative in P
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There are no elementary solutions: Elliptical integrals!
  • Approximative solution by the Taylor-Series expansion of the integrand
  • Taylor point = geometric centre of Tesseroids
  • Series expansion of the first order? corresponds to the approximation by a point mass
  • Second-order terms are omitted due to symmetry, if the Taylor point is chosen at the centre of Tesseroids
  • In our third order theory there are four coefficients

? Easy to use, strong reduction of the required computation time

 

Approximation error of potentials, their first and second radial derivative

Tesseroid dimensions: 5’ x 5’ x 2 km
Calculation point: P(r,f,?) at pole

r = R + h;       h(V) = h(g*z) = 2 km;       h(Mrr) = 260 km

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Comparison of the computation time Tesseroid versus point mass versus prism
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  • Comparison of the computation time
  • Calculated with the global DGM JGP95E (5’ x 5’)

? The computation time is reduced by ten-fold when Tesseroids are used instead of prisms