# Local Methods or Per-Cell Functions to Compute a Raster Output Dataset (Thiessen Polygons, Density Estimation, Inverse Distance Weighted-IDW, and Spline) (Especially for GATE-Geospatial 2022)

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## 1. Thiessen Polygons

It assumes that any point within a polygon is closer to the polygon՚s known point than any other known points.

## 2. Density Estimation

Density estimation measures cell densities in a raster by using a sample of known points. There are simple and kernel density estimation methods.

Figure shows the input and output of an example of simple density estimation method. Density values in the raster are expected values rather than probabilities. Kernel density estimation usually produces a smoother surface than the simple estimation method does. As a surface interpolation method, kernel density estimation has been applied to a wide variety of fields such as forest resources.

## 3. Inverse Distance Weighted (IDW)

It is an exact method that enforces the condition that the estimated value of a point is influenced more by nearby known points than by those farther away. The general equation for the IDW method is:

Where is the estimated value at point is the Z value at known points is the distance between points I and point 0, s is the number of known points used in estimation and k is the specified power.

## 4. Spline

Spline creates a surface that passes through the control points and has the least possible changes in slope at all the points.

Where x and y are the coordinates of the point to be interpolated,

And and are the coordinates of control point.

Unlike the IDW method, the predicted values from Spline are not limited within the range of maximum and minimum values of the known points.

There are two types of Splines:

### I. Regularized

Yields a smooth surface and smooth first derivatives

With the Regularized option, higher values used for the [weight] parameter produce smoother surfaces. The values entered for this parameter must be equal to or less than zero. Typical values that may be used are .

### II. Tension

A Tension Spline is flatter than a Regularized Spline of same sample points, forcing the estimate to stay closer to the sample data. It produces a surface more rigid according to the character of the modeled phenomenon.

With the Tension option, higher values entered for the [weight] parameter result in somewhat coarser surfaces, but surfaces that closely conform to the control points. The values entered must be equal to or greater than zero. The typical values are .