choclo.prism.gravity_u#

choclo.prism.gravity_u(easting, northing, upward, prism, density)[source]#

Upward component of the gravitational acceleration due to a prism

Returns the upward component of the gravitational acceleration 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 (1d-array) – One dimensional array containing the coordinates of the prism in the following order: west, east, south, north, bottom, top in a Cartesian coordinate system. All coordinates should be in meters.

Returns

g_u (float) – Upward component of the gravitational acceleration generated by the rectangular prism on the observation point in \(\text{m}/\text{s}^2\).

Notes

Returns the upward component \(g_z(\mathbf{p})\) of the gravitational acceleration \(\mathbf{g}\) 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 \(\rho\):

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

where

\[k_z(x, y, z) = - \left[ x \, \text{ln2} (y + r) + y \, \text{ln2} (x + r) - z \, \text{arctan2} \left( \frac{xy}{zr} \right) \right]\]
\[r = \sqrt{x^2 + y^2 + z^2},\]

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 \(\text{ln2}\) and \(\text{arctan2}\) functions are defined as follows:

\[\begin{split}\text{ln2}(x) = \begin{cases} 0 & |x| < 10^{-10} \\ \ln (x) \end{cases}\end{split}\]
\[\begin{split}\text{arctan2} \left( \frac{y}{x} \right) = \begin{cases} \text{arctan}\left( \frac{y}{x} \right) & x \ne 0 \\ \frac{\pi}{2} & x = 0 \quad \text{and} \quad y > 0 \\ -\frac{\pi}{2} & x = 0 \quad \text{and} \quad y < 0 \\ 0 & x = 0 \quad \text{and} \quad y = 0 \\ \end{cases}\end{split}\]

These were defined after [Fukushima2020] and guarantee a good accuracy on any observation point.

References