## Variogram

A variogram is calculated from the variance of pairs of points at different separation. For several distance classes or lags, all point pairs are identified which matches that separation and the variance is calculated. Repeating this process for various distance classes yields a variogram. These functions can be used to measure spatial variability of point data but also of maps or images.

### Spatial Auto-correlation of Point Data

The statistical analysis referred to as spatial auto-correlation, examines the correlation of a random process with itself in space. Many variables that have discrete values measured at several specific geographic positions (i.e., individual observations can be approximated by dimensionless points) can be considered random processes and can thus be analyzed using spatial auto-correlation analysis. Examples of such phenomena are: Total amount of rainfall, toxic element concentration, grain size, elevation at triangulated points, etc.

The spatial auto-correlation function, shown in a graph is referred to as spatial auto-correlogram, showing the correlation between a series of points or a map and itself for different shifts in space or time. It visualizes the spatial variability of the phenomena under study. In general, large numbers of pairs of points that are close to each other on average have a lower variance (i.e., are better correlated), than pairs of points at larger separation. The auto-correlogram quantifies this relationship and allows gaining insight into the spatial behaviour of the phenomenon under study.

### Point Interpolation

A point interpolation performs an interpolation on randomly distributed point values and returns regularly distributed point values. The various interpolation methods are: Voronoi Tesselation, moving average, trend surface and moving surface.

Example: Nearest Neighbor (Voronoi Tessellation)-In this method the value, identifier, or class name of the nearest point is assigned to the pixels. It offers a quick way to obtain a Thiessen map from point data (Figure 9).

Figure 9: (a) An input point map, (b) The output map obtained as the result of the interpolation operation applying the Voronoi Tessellation method

In this section the basic concept of various vector operations are dealt in detail. There are multi layer operations, which allow combining features from different layers to form a new map and give new information and features that were not present in the individual maps.

Topological overlays: Selective overlay of polygons, lines and points enables the users to generate a map containing features and attributes of interest, extracted from different themes or layers. Overlay operations can be performed on both raster (or grid) and vector maps. In case of raster map calculation tool is used to perform overlay. In topological overlays polygon features of one layer can be combined with point, line and polygon features of a layer.

Polygon-in-polygon overlay:

Output is polygon coverage. Coverages are overlaid two at a time.

There is no limit on the number of coverages to be combined.

New File Attribute Table is created having information about each newly created feature.

Line-in-polygon overlay:

Output is line coverage with additional attribute. No polygon boundaries are copied. New arc-node topology is created.

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