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choclo.point.kernel_u

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choclo.point.kernel_u#

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

Upward component of the gradient of the inverse of the distance

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:

\[k_z(\mathbf{p}, \mathbf{q}) = \frac{\partial}{\partial z} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2} \right) = - \frac{ z_p - z_q }{ \lVert \mathbf{p} - \mathbf{q} \rVert_2 }\]

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

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