# choclo.prism.kernel_uu#

choclo.prism.kernel_uu(easting, northing, upward, radius)[source]#

Kernel for upward-upward component of the tensor due to a prism

Evaluates the integration kernel for the upward-upward component of the tensor generated by a prism [Nagy2000] on a single vertex of the prism. The coordinates that must be passed are shifted coordinates: the coordinates of the vertex from a Cartesian coordinate system whose origin is located in the observation point.

This function makes use of a safe arctangent function [Fukushima2020] that guarantee a good accuracy on every observation point.

Parameters:
• easting (float) – Shifted easting coordinate of the vertex of the prism. Must be in meters.

• northing (float) – Shifted northing coordinate of the vertex of the prism. Must be in meters.

• upward (float) – Shifted upward coordinate of the vertex of the prism. Must be in meters.

• radius (float) – Square root of the sum of the squares of the easting, northing and upward shifted coordinates.

Returns:

kernel (float) – Value of the kernel function for the upward-upward component of the tensor due to a rectangular prism evaluated on a single vertex.

Notes

Computes the following numerical kernel on the passed shifted coordinates:

$k_{zz}(x, y, z) = - \operatorname{safe-arctan} \left( xy, zr \right)$

where

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

References