Interpolation

Often called gridding, interpolation creates images by estimating values for pixel centres (nodes) on a regular network of rows and columns from regularly or irregularly scattered data points.

Why?

Potential Problems

 

Interpolation Parameters

  1. Diameter of search area (tolerance circle)
  2. Method for calculating pixel values from points within search area
  3. Number of points to use in calculation
  4. Pixel size

 

1. SEARCH AREA

Geostatistics (autocorrelation and semivariance) provide insight into defining a reasonable search area.

The two most common methods for selecting points, within a search area, to calculate the value of a node are:

  1. Nearest neighbour - nearest points to node
  2. Radius - all points within a given radius

 

2. GRIDDING ALGORITHMS

  1. Local - node values computed from an equation with coefficients determined using a subset of scattered data points (e.g. IDW, splines [Minimum curvature], kriging)
  2. Global - node values computed from an equation with coefficients determined using all scattered data points (e.g. trend surface, double fourier)

 

3. NUMBER OF DATA POINTS

The number of data points used in computing the value of a node.

 

4. PIXEL SIZE

Selecting a pixel size: Nyquist Rule states that there should be 2-3 pixels between average spacing of data points. A good practical example of this concept can be found in the gridding of geophysical data.

A good test for pixel size is by image subtraction. If the difference between two image "volumes" of different pixel size is less than 5%, then a smaller pixel size will not increase the accuracy of the model.

 

 

 

1. SEARCH AREA (Defining a reasonable search area)

Geostatistics (regionalized variable theory) provides a very useful tool for helping define a reasonable search area for interpolation. This tool is called a semivariogram, a graph of semivariance vs. lag, and provides insight into the degree of autocorrelation in a dataset. We need a few definitions:

Term Defintion
Autocorrelation/ autocovariance statistical concepts expressing the degree to which the value of an attribute at spatially adjacent points varies with the distance or time separating the observations.
Regionalized variable a single-values function defined over a metric space (a set of coordinates) that represent the variation of natural phenomena that are too irregular at the scale of interest to be modeled analytically
Lag a user-specified distance class within which semivariance is computed for a set of data points.
Semivariance Given two locations x and (x + h), a measure of one-half of the mean square differences (the semivariance) produced by assigning the value z(x + h) to the value z(x), where h (known as thelag) is the inter-sample distance, i.e.
where N refers to the number of data pairs that are separated by the same distance h (Source)
Semivariogram A graph of semivariance versus lag h.
Kriging Kriging is a weighted average method of gridding which determines weights based on the location of the data and the degree of spatial continuity present in the data as expressed by a semi-variogram. The weights are determined so that the average error of estimation is zero and the variance of estimation minimized.


Lag

 

2. GRIDDING ALGORITHMS

3. NUMBER OF DATA POINTS

4. PIXEL SIZE

Selecting a pixel size: Nyquist Rule states that there should be 2-3 pixels between average spacing of data points. A good practical example of this concept can be found in the gridding of geophysical data.

A good test for pixel size is by image subtraction. If the difference between two image "volumes" of different pixel size is less than 5%, then a smaller pixel size will not increase the accuracy of the model.

 

Good introduction to interpolation, semivariance, kriging, etc.

The following is an excellent reference for gridding methods:
http://kopernik.whoi.edu/12.747/notes/lect05/lectno05.html

This Web page is one of many useful pages linked from:
http://kopernik.whoi.edu/12.747/lectures.html

Krajewski, Dr. Stephen A. & Gibbs, Betty L. (1994) Understanding Contouring: A practical guide to spatial estimation and contouring using a computer AND Basics of Using Variograms. Gibbs Associates, Boulder, CO. ISBN 0-943909-16-3