choclo.prism.kernel_pot#
- choclo.prism.kernel_pot(easting, northing, upward, radius)[source]#
Kernel for the potential field due to a rectangular prism
Evaluates the integration kernel for the potential field 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 natural logarithmic function and 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,northingandupwardshifted coordinates.
- easting
- Returns:
- kernel
float Value of the numerical kernel function for the potential field due to a rectangular prism evaluated on a single vertex.
- kernel
Notes
Computes the following numerical kernel on the passed shifted coordinates:
\[\begin{split}k_V(x, y, z) &= x y \, \operatorname{safe\_ln} (z, r) + y z \, \operatorname{safe\_ln} (x, r) + z x \, \operatorname{safe\_ln} (y, r) \\ & - \frac{x^2}{2} \operatorname{safe-arctan} \left( yz, xr \right) - \frac{y^2}{2} \operatorname{safe-arctan} \left( zx, yr \right) - \frac{z^2}{2} \operatorname{safe-arctan} \left( xy, zr \right)\end{split}\]where
\[\begin{split}\operatorname{safe\_ln}(x, r) = \begin{cases} 0 & r = 0 \\ \ln(x + r) & x \ge 0 \\ \ln((y^2 + z^2) / (r - x)) & x < 0, r \ne |x| \\ -\ln(-2 x) & x < 0, r = |x| \end{cases}\end{split}\]and
\[\begin{split}\operatorname{safe-arctan} \left( y, x \right) = \begin{cases} \text{arctan}\left( \frac{y}{x} \right) & x \ne 0 \\ \frac{\pi}{2} & x = 0 \quad \text{and} \quad y > 0 \\ -\frac{\pi}{2} & x = 0 \quad \text{and} \quad y < 0 \\ 0 & x = 0 \quad \text{and} \quad y = 0 \\ \end{cases}\end{split}\]References