Kriging from Raster

Functionality

Kriging from Raster is similar to the Ordinary Kriging interpolation c.q. prediction method but requires a raster map as input, instead of a point map. The operation can be seen as a raster interpolation and returns a raster map with estimations and optionally an error map. The estimations are weighted averaged input pixel values, similar to the Moving Average operation. The weight factors in Kriging from Raster are determined by using a user-specified semi-variogram model (based on the output of the Autocorrelation - Semivariance operation), the distribution of input pixels, and are calculated in such a way that they minimize the estimation error in each output pixel. The estimated or predicted values are thus a linear combination of the input values and have a minimum estimation error. The optional error map contains the standard errors of the estimates.

In Kriging from Raster the interpolation is exact in all input data pixels whose value is defined. This means that the output pixels which where already defined as input pixels need not be kriged but their value can just be copied from the input map. In case of a non zero nugget and isolated data input pixels, the effect is a sharp discontinuity. At the place of an undefined input pixel, an output pixel is computed if its surrounding limiting circle contains enough defined input pixels. In this way one can fill up the undefined gaps of a raster map on the basis of the model of spatial continuity.

Point kriging interpolation on the other hand is seldom exact because the input data points normally don not coincide exactly with any output pixel midpoint. Besides filling up gaps of a raster map, Raster Kriging may also be a useful operation to get a much denser network of sample points. When point samples are measured on a regularly grid, it is much quicker to rasterize the point map and using Kriging from Raster than to use any other point interpolation operation.

Preparation:

Besides an input raster map, Kriging from Raster requires a semi-variogram model including the type of the model and values for the parameters nugget, sill and range; this can be obtained from the Autocorrelation - Semivariance operation.

Display the output table of Autocorrelation - Semivariance, and create a graph (i.e. a semi-variogram) in which you plot the horizontal or vertical semi-variance values against the distance classes.
Subsequently, model the semi-variogram: select a model like Spherical, Exponential, Gaussian, etc., and choose values for sill, nugget and range. This shows as a line in the graph through the points.
Superimpose trials of various models with various defining parameters in order to find the best approximation.

For more information, see Spatial correlation : functionality, section on Semi-variograms, or Graph window : Add semi-variogram model.

General Kriging tips:

When a global trend is apparent, it should be removed before Kriging. You could use the Trend surface operation and appropriate TabCalc subtractions.
To take a local trend into account, use Spatial Correlation and Universal Kriging.
To take anisotropy into account, use Variogram Surface, Spatial Correlation (bidirectional), and Anisotropic Kriging.
When you like to Krige a sparsely sampled variable with help of a well sampled variable, use Cross Variogram and CoKriging.

For more information, see How to use Kriging.

Limiting distance and number of input pixels:

In Kriging from Raster, you can influence the number of valid input pixels that should be taken into account in the calculation of an output pixel value by specifying a limiting distance and a minimum and maximum number of valid input pixel values:

only the values of input pixels that fall within the limiting distance or search radius to an output pixel will be used in the calculation for that output pixel value; the limiting distance can be specified in meters or in pixels,
within the limiting distance towards each output pixel, at least the specified minimum number of pixels should be found, otherwise the pixel will be assigned the undefined value,
within the limiting distance towards each output pixel, only the specified maximum number of pixels will be taken into account; when more than the specified maximum number of pixels are found within the limiting distance, only the pixels which are nearest to the output pixel will be taken into account.

For each output pixel, a set of simultaneous equations needs to be solved to find the weight values for those pixels that contribute to the output value of the pixel.

Limitations:

the maximum limiting distance which can be used is 40*pixelsize meters or 40 pixels (this will result in a matrix of 81*81).
the maximum number of input pixels which can be used within each limiting distance is 100.

These limitations are implemented both for the dialog box and the command line.

In general, it can be said that when more input pixel values are used (maximum limiting distance, maximum nr of pixels), the output Kriging estimates will be more reliable, but the calculation will take more time.

Error map:

Optionally, an output error map can be created which will contain the standard error of the estimate, i.e. the square root of the error variance.

The error variance in each estimated output pixel depends on:

the semi-variogram model including its parameters,
the spatial distribution of the input pixels,
the position of an output pixel with respect to the position of the input pixels.

A standard error which is larger than the original sample standard deviation denotes a rather unreliable prediction.

Input map requirements:

The input raster map should be a value map. You can also use a raster map with an ID domain which has a suitable value column in its attribute table.

Domain and georeference of output raster maps:

The output raster map containing the Kriging estimates or predictions will use the same value domain as the input raster map or the attribute column. The value range and precision can be adjusted for the output map; it is advised to choose a wider value range for the output map than the input value range.

The output raster map will also use the same georeference as the input raster map.

The optional error map will obtain the same name as specified for the output Kriging map, followed by the string _Error. The output error map will use the same domain and the same georeference as the output Kriging map with the predictions.

Confidence interval maps:

From the combination of a Kriged output map containing the estimates and its output error map, you can create confidence interval maps by using some MapCalc statements. For more information, see How to calculate confidence interval maps.

Tip 1:

When the output raster map shows undefined pixels, this can be due to several factors:

The value range of the output raster map is too narrow: because of extrapolation, certain output values may fall beyond the range limits and ILWIS converts them to undefined.
Remedy: extend the value range in the dialog box and compare the results.
The minimum number of pixels to be taken into account is too high in relation to the limiting distance, i.e. the minimum number of pixels specified are not found within the specified limiting distance.
Remedy: lower the minimum number of pixels and/or increase the limiting distance.

Tip 2:

The output can become erratic when:

Input pixels are too sparse in certain areas to ensure an estimate with small variance.
Remedy: increase limiting distance, find supplementary data, investigate possible anisotropy.
The semi-variogram Range parameter is set incorrectly due to poor interpretation of a correct output of the Autocorrelation - Semivariance
Remedy: Try another pixel shift.
The semi-variogram model is incorrect: the geometric distribution of the sample points is unbalanced or the user is unaware of an existing anisotropy. There is perhaps an error in the range setting (horizontal scale in graph) or sill and/or nugget (vertical scale).