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I am trying to figure out how the Weight Field option of Geostatistical Analyst IDW in ArcMap 10.3 works.

On IDW - ArcGIS Desktop Help I can find just this simple explanation:

Used to emphasize an observation. The larger the weight, the more impact it has on the prediction. For coincident observations, assign the largest weight to the most reliable measurement.

What is the algorithm that creates the weight which emphasizes the observation?

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The explanation about the Weight Field parameter is subtle, but

Used to emphasize an observation

means that an additional set of weights (besides the ones based on distances) are applied directly to the observed data (the data to be interpolated). Those weights correspond to values from a Weight Field in the input vector layer (i.e. a field different from the one being interpolated).

Compare IDW formulas with and without the Weight Field parameter.

IDW without Weight Field:

Z(0) = {∑[zi/di²]/∑[1/di²]}

in which:

  • Z(0) is the value to be predicted at location 0;
  • zi are observed values at locations 1, ... , n.
  • di² are the distances (with a power of 2, could have been 1, 3, etc.) between observed points and the prediction point.

In this case the interpolation depends solely on the distances among points.

IDW with Weight Field:

Z(0) = {∑[(zi*qi)/di²]/∑[qi/di²]}

in which:

  • qi are values from the Weight Field corresponding to sample i.

Now, not only the prediction result is influenced by the inverse distance among points, but also by the Weight Field in a direct relation.


Example:

Suppose we want to predict a z_value for a prediction point Z(0) of coordinates (2,2) from the following set of points:

obs_point  x_coordinate  y_coordinate  d² to Z(0)  z_value  column1
1          2             1             1           4        1
2          0             2             4           3        2
3          3             3             2           1        4

IDW based only on the inverse distance²:

Z(0) = {[4/1 + 3/4 + 1/2]/[1/1 + 1/4 + 1/2]} = 3

IDW using the column1 as the Weight Field parameter:

Z(0) = {[(4*1)/1 + (3*2)/4 + (1*4)/2]/[1/1 + 2/4 + 4/2]} = 2.14  
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When an algorithm is not provided in the Help for ArcGIS Desktop, and you seek more details about it, I think you will need to contact the developers of that software (Esri) to see if they will share that information.

You could do this direct with them or via one of their international distributors as a support request.

Alternatively, or if that fails, you could post an ArcGIS Idea for a documentation enhancement to include more details about this algorithm.

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