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.
- easting
- 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.nanif the observation point falls in a singular point: prism vertices or prism edges perpendicular to the upward direction.
- g_uu
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