choclo.point.kernel_uu#

choclo.point.kernel_uu(easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance)[source]#

Second derivative of the inverse of the distance along upward-upward.

Important

The coordinates of the two points must be in Cartesian coordinates and have the same units.

Parameters:

easting_p, northing_p, upward_pfloat

Easting, northing and upward coordinates of point $\mathbf{p}$.

easting_q, northing_q, upward_qfloat

Easting, northing and upward coordinates of point $\mathbf{q}$.

distancefloat

Euclidean distance between points $\mathbf{p}$ and $\mathbf{q}$.

Returns:

kernelfloat

Value of the kernel function.

Notes

Given two points $\mathbf{p}=(x_p,y_p,z_p)$ and $\mathbf{q}=(x_q,y_q,z_q)$ defined in a Cartesian coordinate system, compute the following kernel function:

$$k_{zz}(\mathbf{p},\mathbf{q})=\frac{{\partial }^2}{\partial z^2}\left(\frac{1}{\Vert \mathbf{p}-\mathbf{q}{\Vert }_2}\right)=\frac{3(z_p-z_q{)}^2}{\Vert \mathbf{p}-\mathbf{q}{\Vert }_2^5}-\frac{1}{\Vert \mathbf{p}-\mathbf{q}{\Vert }_2^3}$$

where $\Vert \cdot {\Vert }_2$ refer to the $L_2$ norm (the Euclidean distance between $\mathbf{p}$ and $\mathbf{q}$).