Inverse Distance to a Power
The Inverse Distance to a Power gridding method is a weighted average interpolator, and can be either an exact or a smoothing interpolator. Grapher automatically uses the Inverse Distance to a Power gridding method whenever a contour data map or surface data map is created.
With Inverse Distance to a Power, data are weighted during interpolation such that the influence of one point relative to another declines with distance from the grid node. Weighting is assigned to data through the use of a weighting power that controls how the weighting factors drop off as distance from a grid node increases. The greater the weighting power, the less effect points far from the grid node have during interpolation. As the power increases, the grid node value approaches the value of the nearest point. For a smaller power, the weights are more evenly distributed among the neighboring data points.
Normally, Inverse Distance to a Power behaves as an exact interpolator. When calculating a grid node, the weights assigned to the data points are fractions, and the sum of all the weights are equal to 1.0. When a particular observation is coincident with a grid node, the distance between that observation and the grid node is 0.0, and that observation is given a weight of 1.0, while all other observations are given weights of 0.0. Thus, the grid node is assigned the value of the coincident observation. The Smoothing parameter is a mechanism for buffering this behavior. When you assign a non-zero Smoothing parameter, no point is given an overwhelming weight so that no point is given a weighting factor equal to 1.0. You can adjust the smoothing used by Inverse Distance to a Power in the Gridding section of contour data map or surface data map properties. See Grid Properties - Contour Maps or Grid Properties - Surface Maps for more information.
One of the characteristics of Inverse Distance to a Power is the generation of "bull's-eyes" surrounding the position of observations within the gridded area. Adjusting the power used by the Inverse Distance to a Power feature reduces the "bull's-eye" effect by smoothing the interpolated grid.
Inverse Distance to a Power Math
The equation used for Inverse Distance to a Power is:
where:
|
|
is the effective separation difference between grid node "j" and neighboring point "i"; |
|
|
is the interpolated value for grid node "j"; |
|
|
are the neighboring points; |
|
|
is the difference between grid node "j" and neighboring point "i"; |
|
|
is the weighting power (the Power parameter); and |
|
|
is the Smoothing parameter. |
Inverse Distance to a Power References
Davis, John C. (1986), Statistics and Data Analysis in Geology, John Wiley and Sons, New York.
Franke, R. (1982), Scattered Data Interpolation: Test of Some Methods, Mathematics of Computations, v. 33, n. 157, p. 181-200.