Cokriging

Cokriging is an interpolation technique that allows one to better estimate map values by kriging if the distribution of a secondary variate sampled more intensely than the primary variate is known. If the primary variate is difficult or expensive to measure, then cokriging can greatly improve interpolation estimates without having to more intensely sample the primary variate.

Consider the following example. After an accident, plutonium was sampled across an 80 x 80 m area at a sample density indicated by the quartile plot below. Block kriging (following variography) resulted in the adjacent map:

images\ur232_sparse_quantiles.gif images\ur232_sparse_kriged.gif

Soil carbon, easier to measure than Uranium, was sampled at the same locations as Uranium and additionally at another 60 locations as noted in the quartile map below left. Regression of carbon against plutonium showed that the variates were highly correlated (right), suggesting that cokriging might improve the map of plutonium.

images\ur232_covariate_c_not_sparse_quantiles.gif images\ur232_vs_c_regression.gif

Using carbon as a covariate to produce a cokriged map of plutonium results in the below-right map, plotted next to the original kriged map. Note the substantial improvement in the definition of contour (isoline) differences, especially in the upper right quadrant of the map where the uranium was sampled most sparsely:

images\ur232_sparse_kriged.gif images\ur232_sparse_cokriged.gif

The kriging estimate is based not only on distance to nearby sample locations for Z and the variogram for Z, but also distance to nearby sample locations for Z2, the variogram for Z2, and the cross-variogram for Z x Z2. This can provide a more robust estimate of Z at unsampled locations if Z and Z2 are sufficiently correlated.

Prior to cokriging you must a) define a covariate in the Data Worksheet Field Assignment dialog, b) perform semivariance analysis (including variogram modeling) for the primary variate Z, for the covariate Z2, and c) for the cross variate Z x Z2. You will also want to check that the covariate is in fact correlated with the primary variate in the Data Summary – Covariate window.

Once the three variograms are modeled, you can choose the Cokrig tab in the Interpolation window:

images\interpolation_cokrig_tab.gif

Cokriging Type

GS+ provides three types of cokriging: Simple, Ordinary, and Standardized Ordinary.

Simple cokriging places no constraints on the weights applied to the measured values during interpolation.

Ordinary cokriging sets the sum of weights applied to the primary variate to one, and the sum of weights applied to the covariate to zero. This limits the influence of the covariate relative to other cokriging types, so may not be preferred.

Standardized ordinary cokriging recalculates the covariate to have the same mean as the primary variate, and constrains all weights to sum to one. This is often the preferred cokriging method and is the default cokriging type in GS+.

Variogram Model Type

Variogram models for isotropic and anisotropic variograms are defined and chosen using the Model command in the Autocorrelation Analyses windows. There is a separate autocorrelation window for the primary variate Z, for the secondary variate Z2, and for the cross variate Z x Z2. Here in the Cokriging window, you can specify whether to use the isotropic or anisotropic model for each of these variograms. You can also choose to use a relative variogram, in which the nugget and structural components are rescaled from 0 to 1.0 for each of the three models.

Discretization grid

Choose either Point or Block cokriging in this section. The choices in this section are the same as for Kriging,

Search Neighborhood – Covariate

The search strategy for covariate Z2 values in cokriging can be different from that for the primary variate Z values. In the Search Neighborhood Covariate tab you can specify whether search parameters should be the same as for the primary variate Z or different.

images\interpolation_search_covariate_tab.gif

See Search Neighborhood in the Interpolation Window for further information about the options available