*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?**

**extrapolate data beyond point locations****have regularly spaced data for contouring or raster calculations****visualize trends in point data****smooth or enhance estimated surface variability**

**Potential Problems**

**values incorrectly extrapolated into areas with sparse data leading to misguided interpretations****discontinuities are difficult to model****algorithmic artifacts may produce phantom features, noise, or unrealistic surface undulation****may require many iterations to optimize model**

**Interpolation Parameters**

**Diameter of search area (tolerance circle)****Method for calculating pixel values from points within search area****Number of points to use in calculation****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:

- Nearest neighbour - nearest points to node

- Radius - all points within a given radius

**2. GRIDDING ALGORITHMS**

**Local**- node values computed from an equation with coefficients determined using a subset of scattered data points (e.g. IDW, splines [Minimum curvature], kriging)

**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.

- Few points --> enhance local anomalies
- Many points --> subdue local anomalies

**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