Weight¶

Here is some details about how to calculate the weight

Distance Weight¶

Distance between points are used as measurements to measure the density of the points. The area with less points should be given larger weights as a balance

1. Sphere Relative Distance Weighting¶

The points here are restricted to be located at the surface of sphere. So position could be expressed as [latitude, longitude]. The distance of two points D_ij is the spheric distance, which is the great-circle distance.

Give a set of points, P_i, located at the surface of a sphere, and the distance between two points, P_i and P_j, the distance between two points, D_ij, is the great-circle distance. Thus, the weight of the points P_i could be expressed as:

\[W_i = \frac{1}{\sum_{j=1}^N exp(-(\frac{D_{ij}}{\Delta})^2)}\]

where \(\Delta\) is the reference distance. Here we keep the i=j term, using \(D_{ij}=0\), when i=j. Physically, it means the distance between one point and itself is 0.

However, you still have the freedom to choose \(\Delta\), and the value should be picked really carefully.

2. Sphere Voronoi Weighting¶

This method used the area of voronoi cell as the weighting term

Azimuth Weighting¶

If you want to use azimuth weighting, you need to set a central point to calculte the azimuth(the azimuth is from the central point to a given point).

1. Sphere Azimuth Bin Weighting¶

This weighting strategy sort the azimuth, and depending on the number of points falls into that bins, the weighting is determined by 1 over number of points in that bin

2. Sphere Relative Azimuth Weighting¶

This is very similar to Sphere Relative Distance Weighting