choclo.prism.gravity_uu#

choclo.prism.gravity_uu(easting, northing, upward, prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top, density)[source]#

Upward-upward component of the gravitational tensor due to a prism

Returns the northing-northing component of the gravitational tensor produced by a single rectangular prism on a single computation point.

Parameters

  • easting (float) – Easting coordinate of the observation point. Must be in meters.

  • northing (float) – Northing coordinate of the observation point. Must be in meters.

  • upward (float) – Upward coordinate of the observation point. Must be in meters.

  • prism_west (float) – The West boundary of the prism. Must be in meters.

  • prism_east (float) – The East boundary of the prism. Must be in meters.

  • prism_south (float) – The South boundary of the prism. Must be in meters.

  • prism_north (float) – The North boundary of the prism. Must be in meters.

  • prism_bottom (float) – The bottom boundary of the prism. Must be in meters.

  • prism_top (float) – The top boundary of the prism. Must be in meters.

  • density (float) – Density of the rectangular prism in kilograms per cubic meter.

Returns

g_uu (float) – Upward-upward component of the gravitational tensor generated by the rectangular prism on the observation point in \(\text{m}/\text{s}^2\). Return numpy.nan if the observation point falls in a singular point: prism vertices or prism edges perpendicular to the upward direction.

Notes

Returns the upward-upward component \(g_{zz}(\mathbf{p})\) of the gravitational tensor \(\mathbf{T}\) on the observation point \(\mathbf{p} = (x_p, y_p, z_p)\) generated by a single rectangular prism defined by its boundaries \(x_1, x_2, y_1, y_2, z_1, z_2\) and with a density \(\rho\):

\[g_{zz}(\mathbf{p}) = G \rho \,\, \Bigg\lvert \Bigg\lvert \Bigg\lvert k_{zz}(x, y, z) \Bigg\rvert_{X_1}^{X_2} \Bigg\rvert_{Y_1}^{Y_2} \Bigg\rvert_{Z_1}^{Z_2}\]

where

\[k_{zz}(x, y, z) = - \operatorname{safe-arctan} \left( xy, zr \right),\]
\[r = \sqrt{x^2 + y^2 + z^2},\]

and

\[\begin{split}X_1 = x_1 - x_p \\ X_2 = x_2 - x_p \\ Y_1 = y_1 - y_p \\ Y_2 = y_2 - y_p \\ Z_1 = z_1 - z_p \\ Z_2 = z_2 - z_p\end{split}\]

are the shifted coordinates of the prism boundaries and \(G\) is the Universal Gravitational Constant.

The \(\operatorname{safe-arctan}\) function is defined as follows:

\[\begin{split}\operatorname{safe-arctan} \left( y, x \right) = \begin{cases} \text{arctan}\left( \frac{y}{x} \right) & x \ne 0 \\ \frac{\pi}{2} & x = 0 \quad \text{and} \quad y > 0 \\ -\frac{\pi}{2} & x = 0 \quad \text{and} \quad y < 0 \\ 0 & x = 0 \quad \text{and} \quad y = 0 \\ \end{cases}\end{split}\]

It was defined after [Fukushima2020] and guarantee a good accuracy on any observation point.

References