Friday, June 26, 2015

Spatial Interpolation

Spatial  Interpolation

What is a spatial interpolation?
Interpolation predicts values for cells in a raster from a limited number of sample data points. It can be used to predict unknown values for any geographic point data: elevation, rainfall, chemical concentrations, noise levels, and so on.
On the left is a point dataset of known values. On the right is a raster interpolated from these points. Unknown values are predicted with a mathematical formula that uses the values of nearby known points.

Interpolation is based on the assumption that spatially distributed objects are spatially correlated; in other words, things that are close together tend to have similar characteristics.
It is important to understand that the interpolated values are approximations only of the real values of the surface and that the interpolated values differ depending upon the interpolation method used.

Why interpolate?
Visiting every location in a study area to measure the height, magnitude, or concentration of a phenomenon is usually difficult or expensive. Instead, dispersed sample input point locations can be selected and a predicted value can be assigned to all other locations. Input points can be either randomly, strategically, or regularly spaced points containing height, concentration, or magnitude measurements.
A typical use for point interpolation is to create an elevation surface from a set of sample measurements. Each point represents a location where the elevation has been measured. The values between these input points are predicted by interpolation.

There are effectively two types of techniques for generating raster surfaces
Deterministic Models use a mathematical function to predict unknown values and result in hard classification of the value of features.
GeoStatistical Techniques produce confidence limits to the accuracy of a prediction but are more difficult to execute since more parameters need to be set.

Deterministic Models
Deterministic models include Inverse Distance Weighted (IDW), Rectangular, Natural Neighbours, and Spline. You can also develop a trend surface using polynomial functions to create a customized and highly accurate surface.

1Inverse Distance Weighting (IDW)


The IDW technique calculates a value for each grid node by examining surrounding data points that lie within a user-defined search radius. The node value is calculated by averaging the weighted sum of all the points. A radius is generated around each grid node from which data points are selected to be used in the calculation.
Options to control the use of IDW include
Ø Power a high power more emphasis is placed on the nearest points and the resulting surface will have more detail and be less smoothed. Its values range between one and ten.
Ø  Search Radius defines the maximum size, in map units, of a circular zone centered on each grid node within which point values from the original data set are averaged and weighted according to their distance from the node.

The IDW is usually applied to highly variable data not desirable to local high/low values but rather to look at a moving average of nearby data points and estimate the local trends.

2Natural Neighbourhood Interpolation

Natural Neighbourhood Interpolation
It is like IDW interpolation, except that the data points used to interpolate the surface values for each cell are identified and weighted using a Delaunay triangulation.The method thereby allows the creation of accurate surface models from data sets that are very sparsely distributed or very linear in spatial distribution.
Spline estimates values using a mathematical function that minimizes overall surface curvature, resulting in a smooth surface that passes exactly through the input points.This method is best for gently varying surfaces, such as elevation, water table heights, or pollution concentrations. There are two spline methods…

3Spline Interpolation
Spline Interpolation

Spline estimates values using a mathematical function that minimizes overall surface curvature, resulting in a smooth surface that passes exactly through the input points.

This method is best for gently varying surfaces, such as elevation, water table heights, or pollution concentrations. There are two spline methods…

Spline the Regularized Method
The regularized method creates a smooth, gradually changing surface with values that may lie outside the sample data range.
Spline the Tension Method

It creates a less-smooth surface with values more closely constrained by the sample data range. For Tension, the higher the weight the coarser the generated surface. The values entered have to equal or greater than zero. The typical values are 0, 1, 5, and 10.

4Trend Interpolation

Trend surfaces are good for identifying coarse scale patterns in data; the interpolated surface rarely passes through the sample points.
Modelers often work to the "fifth order" polynomial analysis.
Trend Interpolation
A Trend surface for a set of points, in transparent grey, and the IDW interpolated surface for the same points. Spline and Trend interpolation interpolate best-fit surfaces to the sample points using polynomial and least-squares methods. 

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