choclo.point.kernel_uu

choclo.point.kernel_uu(easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance)

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_pfloat

Easting coordinate of point \(\mathbf{p}\).

northing_pfloat

Northing coordinate of point \(\mathbf{p}\).

upward_pfloat

Upward coordinate of point \(\mathbf{p}\).

easting_qfloat

Easting coordinate of point \(\mathbf{q}\).

northing_qfloat

Northing coordinate of point \(\mathbf{q}\).

upward_qfloat

Upward coordinate 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:

\[k_{zz}(\mathbf{p}, \mathbf{q}) = \frac{\partial^2}{\partial z^2} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2} \right) = \frac{ 3 (z_p - z_q)^2 }{ \lVert \mathbf{p} - \mathbf{q} \rVert_2^5 } - \frac{ 1 }{ \lVert \mathbf{p} - \mathbf{q} \rVert_2^3 }\]

where \(\lVert \cdot \rVert_2\) refer to the \(L_2\) norm (the Euclidean distance between \(\mathbf{p}\) and \(\mathbf{q}\)).