interpolation of surfaces spatial data analysis. spatial interpolation spatial interpolation is the...

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Interpolation of Surfaces Spatial Data Analysis

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Page 1: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Interpolation of Surfaces

Spatial Data Analysis

Page 2: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spatial Interpolation

Spatial interpolation is the prediction of exact values of attributes at un-sampled locations from measurements made at control points within the same area.

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Page 3: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spatial Interpolation

Page 4: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spatial Interpolation

Automating interpolation is performed with GIS:

1. Proximity Polygons2. Inverse Distance Weighted Spatial Average3. Spline Smoothing4. Krigging

Page 5: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Interpolation of SurfacesInverse Distance Weighted

Page 6: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spatial Interpolation

Proximity PolygonsThis technique was introduced a century ago by

Thiessen (1911)Abrupt change at edges is the main pitfall of this

method

Page 7: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Inverse Distance Weighted Spatial Average

Inverse distance weighting models work on the premise that observations further away should have their contributions diminished according to how far away they are.

The simplest model involves calculating the weighted mean for all of the observations

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i ij

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i ijij

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1

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Page 8: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Inverse Distance Weighted Spatial Average

While the weight is the inverse of the distance it is from the target point raised to a power α

ijij dw /1

By defining the higher {power} option, more emphasis can be put on the nearest points. Thus, nearby data will have the most influence, and the surface will have more detail (be less smooth). As the power increases, the interpolated values begin to approach the value of the nearest sample point. Specifying a lower value for power will provide a bit more influence to surrounding points a little farther away.

Page 9: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Inverse Distance Weighted Spatial Average

Effect of α:

Source Data

Page 10: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Inverse Distance Weighted Spatial Average

Effect of α:

α = 1

Page 11: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Inverse Distance Weighted Spatial Average

Effect of α:

α = 3

Page 12: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Inverse Distance Weighted Spatial Average

Many GIS packages provide this kind of inverse distance model for interpolation, as it is simple to implement and to understand. Often the model is generalised in a number of ways:

1.a faster rate of distance decay may be provided, by including a power function of distance, α >1, rather than simple linear distance. While any α value convenient for a given application may be used, common practice is to use distance (α = 1) or distance squared (α = 2).

Page 13: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Example

Page 14: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Example

α 1

Control Point zij Distance

Inverse Distance wij wijzij

1 104 2.00 0.50 52.002 100 1.41 0.71 70.713 96 1.00 1.00 96.004 88 1.00 1.00 88.00

Total 3.21 306.71

Zj 95.63

Page 15: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Interpolation of SurfacesSpline Interpolation

Page 16: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spline Interpolation Method

Definition: The Spline method is an interpolation method that estimates values using a mathematical function that minimizes overall surface curvature, resulting in a smooth surface that passes exactly through the input points.

Page 17: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spline Interpolation Method

The basic form of the minimum curvature Spline interpolation imposes the following two conditions on the interpolant:

1. The surface must pass exactly through the data points.

2. The surface must have minimum curvature—the cumulative sum of the squares of the second derivative terms of the surface taken over each point on the surface must be a minimum.

Page 18: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spline Interpolation Method

Spline bends a sheet of rubber that passes through the input points while minimizing the total curvature of the surface.

It fits a mathematical function to a specified number of nearest input points while passing through the sample points.

This method is best for generating gently varying surfaces such as elevation, water table heights, or pollution concentrations.

Page 19: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spline Interpolation Method

Page 20: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Spline Interpolation Method

There are two Spline methods: Regularized and Tension.

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

The Tension method controls the stiffness of the surface according to the character of the modeled phenomenon. It creates a less smooth surface with values more closely constrained by the sample data range.

Page 21: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Interpolation of SurfacesKrigging

Page 22: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Krigging

Krigging is a statistical interpolation method that is optimal in the sense that it makes best use of what can be inferred about the spatial structure in the surface to be interpolated from an analysis of control point data.

In IDW we calculate the weights from the inverse of distance raised to some power.

Krigging finds the optimum weights for the data values in the interpolation at each unknown location.

Page 23: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Krigging

To interpolate in this way, three distinct steps are involved:

1. Producing a description of the spatial vario-gram in the sample control data.

2. Summarizing this spatial variation by a regular mathematical function

3. Using this model to determine the interpolation weights

Page 24: Interpolation of Surfaces Spatial Data Analysis. Spatial Interpolation Spatial interpolation is the prediction of exact values of attributes at un-sampled

Mathematical Function summarizing semivariogram