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

• northing_p (float) – Northing coordinate of point \(\mathbf{p}\).

• upward_p (float) – Upward coordinate of point \(\mathbf{p}\).

• easting_q (float) – Easting coordinate of point \(\mathbf{q}\).

• northing_q (float) – Northing coordinate of point \(\mathbf{q}\).

• upward_q (float) – Upward coordinate of point \(\mathbf{q}\).

• distance (float) – Euclidean distance between points \(\mathbf{p}\) and \(\mathbf{q}\).

Returns

kernel (float) – 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 \(\mathbf{q}\)).