choclo.prism.gravity_nu#
- choclo.prism.gravity_nu(easting, northing, upward, prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top, density)[source]#
Northing-upward component of the gravitational tensor due to a prism
Returns the northing-upward component of the gravitational tensor produced by a single rectangular prism on a single computation point.
- Parameters:
- easting, northing, upward
float Easting, northing and upward coordinates of the observation point. Must be in meters.
- prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top
float The boundaries of the prism. Must be in meters.
- density
float Density of the rectangular prism in kilograms per cubic meter.
- easting, northing, upward
- Returns:
- g_nu
float Northing-upward component of the gravitational tensor generated by the rectangular prism on the observation point in \(\text{m}/\text{s}^2\). Return
numpy.nanif the observation point falls in a singular point: prism vertices or prism edges parallel to the easting direction.
- g_nu
Notes
Returns the northing-upward component \(g_{yz}(\mathbf{p})\) 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 \(\rho\):
\[g_{yz}(\mathbf{p}) = G \rho \,\, \Bigg\lvert \Bigg\lvert \Bigg\lvert k_{yz}(x, y, z) \Bigg\rvert_{X_1}^{X_2} \Bigg\rvert_{Y_1}^{Y_2} \Bigg\rvert_{Z_1}^{Z_2}\]where
\[k_{yz}(x, y, z) = \operatorname{safe-ln} \left( x, r \right),\]\[r = \sqrt{x^2 + y^2 + z^2},\]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 \(\operatorname{safe-ln}\) function is defined as follows:
\[\begin{split}\operatorname{safe-ln}(x, r) = \begin{cases} 0 & x = 0, r = 0 \\ \ln(x + r) & x \ge 0 \\ \ln((r^2 - x^2) / (r - x)) & x < 0, r \ne -x \\ -\ln(-2 x) & x < 0, r = -x \end{cases}\end{split}\]It was defined after [Fukushima2020] and guarantee a good accuracy on any observation point.
References