Logo Tesseroids 1.0: User Manual and API Documentation

Theoretical background

About Coordinate Systems

The two coordinate systems involved in the computations are the Global and Local coordinate systems.

The Global system has origin on the center of the Earth and Z axis aligned with the Earth's mean rotation axis. The X and Y axis are contained on the equatorial parallel with X intercepting the mean Greenwich meridian and Y completing a right-handed system.

The Local system has origin on the computation point. It's z is oriented along the radial direction and points away from the center of the Earth. The x and y axis are contained on a plane normal to the z axis and x points North and y East.

The tesseroids are defined using the Global Coordinate system with spherical coordinates, while the gravitational fields are calculated on the Local Coordinate system of the computation point.

WARNING: The $ g_z $ component is an exception to this. In order to conform with the regular convention of z-axis pointing toward the center of the Earth, this component ONLY is calculated with an inverted z axis.

tesseroid_coordsys.png

Figure1: View of a tesseroid, the integration point Q, the global coordinate system, the computation P and it's local coordinate system.

Gravitational Fields of a Tesseroid

The gravitational attraction of a tesseroid can be calculated using the formula (Grombein et al., 2010):

\[ g_{\alpha}(r_p,\phi_p,\lambda_p) = G \rho \displaystyle\int_{\lambda_1}^{\lambda_2} \displaystyle\int_{\phi_1}^{\phi_2} \displaystyle\int_{r_1}^{r_2} \frac{\Delta x_{\alpha}}{\ell^3} \kappa \ d r' d \phi' d \lambda' \ \ \alpha \in \{1,2,3\} \]

The gravity gradients can be calculated using the general formula (Grombein et al., 2010):

\[ g_{\alpha\beta}(r_p,\phi_p,\lambda_p) = G \rho \displaystyle\int_{\lambda_1}^{\lambda_2} \displaystyle\int_{\phi_1}^{\phi_2} \displaystyle\int_{r_1}^{r_2} I_{\alpha\beta}\ d r' d \phi' d \lambda' \ \ \alpha,\beta \in \{1,2,3\} \]

\[ I_{\alpha\beta} = \left(\frac{3\Delta x_{\alpha} \Delta x_{\beta}}{\ell^5} - \frac{\delta_{\alpha\beta}}{\ell^3} \right)\kappa\ \ \ \alpha,\beta \in \{1,2,3\} \]

where $ \rho $ is density, the subscripts 1, 2, and 3 should be interpreted as the x, y, and z axis, $ \delta_{\alpha\beta} $ is the Kronecker delta function, and

\begin{eqnarray*} \Delta x_1 &=& r' K_{\phi} \\ \Delta x_2 &=& r' \cos \phi' \sin(\lambda' - \lambda_p) \\ \Delta x_3 &=& r' \cos \psi - r_p\\ \ell &=& \sqrt{r'^2 + r_p^2 - 2 r' r_p \cos \psi} \\ \cos\psi &=& \sin\phi_p\sin\phi' + \cos\phi_p\cos\phi' \cos(\lambda' - \lambda_p) \\ K_{\phi} &=& \cos\phi_p\sin\phi' - \sin\phi_p\cos\phi' \cos(\lambda' - \lambda_p)\\ \kappa &=& {r'}^2 \cos \phi' \end{eqnarray*}

$ \phi $ is latitude, $ \lambda $ is longitude, $ r $ is radius. The subscript $ p $ is for the computation point.

Numerical Integration

The above integrals are solved using the Gauss-Legendre Quadrature rule (Asgharzadeh et al., 2007):

\[ g_{\alpha\beta}(r_p,\phi_p,\lambda_p) \approx G \rho \frac{(\lambda_2 - \lambda_1) (\phi_2 - \phi_1)(r_2 - r_1)}{8} \displaystyle\sum_{k=0}^{N^{\lambda} - 1} \displaystyle\sum_{j=0}^{N^{\phi} - 1} \displaystyle\sum_{i=0}^{N^r - 1} W^r_i W^{\phi}_j W^{\lambda}_k I_{\alpha\beta}({r'}_i, {\phi'}_j, {\lambda'}_k )\kappa\ \ \alpha,\beta \in \{1,2,3\} \]

where $ W^r $,$ W^{\phi} $, and $ W^{\lambda} $ are weighting coefficients and $ N^r $,$ N^{\phi} $, and $ N^{\lambda} $ are the number of quadrature nodes, ie the order of the quadrature.

References

  • Asgharzadeh, M.F., von Frese, R.R.B., Kim, H.R., Leftwich, T.E. & Kim, J.W. (2007): Spherical prism gravity effects by Gauss-Legendre quadrature integration. Geophysical Journal International, 169, 1-11.
  • Grombein, T.; Seitz, K.; Heck, B. (2010): Untersuchungen zur effizienten Berechnung topographischer Effekte auf den Gradiententensor am Fallbeispiel der Satellitengradiometriemission GOCE. KIT Scientific Reports 7547, ISBN 978-3-86644-510-9, KIT Scientific Publishing, Karlsruhe, Germany. (http://digbib.ubka.uni-karlsruhe.de/volltexte/documents/1336300).
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