Extended Collocated Cokriging

Isatis window: Interpolate / Estimation / Bundled Collocated Cokriging.

This technique is used when trying to estimate a target variable Z, known on a sparse sampling, on a regular grid while a correlated variable Y is available at each node of this grid.

The original technique, strictly "Collocated Cokriging", has been extended in Isatis and is also referred to as "Multi Collocated Cokriging" in the literature.

The first task that must be performed by the user consists in writing the value of the variable Y at the points of the sparse sampling. Then he must perform the bivariate structural analysis using the variables Y and Z. This may lead to a severe problem due to the large heterotopy between these two variables: as a matter of fact, if the inference is carried out in terms of variograms, the two variables need to be defined at the same points. If the secondary variable Y is dense with regards to the primary variable Z, we can always interpolate Y at the points where Z is defined and therefore the influence (at least as far as the simple variogram and the cross-variogram are concerned) only considers those samples: all the remaining locations where Y only is defined are simply neglected.

In the literature, we also find another inference method. The variogram is constructed on the whole dense data set whereas the simple variogram and the cross variogram are set as being similar to up to the scaling of their sills and to the use of the nugget effect: the whole system must satisfy to the definite positiveness conditions. By definition, we are in the framework of the linear model of coregionalization. This corresponds to the procedure programmed in "Interpolate / Estimation / Collocated Cokriging (Bundled)".

The Cokriging step is almost similar to the one described in Paragraph "Kriging Two Variables in the Intrinsic Case", the only difference is the neighborhood search. Within the neighborhood (centered on the target grid node), any information concerning the Z variable must be used (because Z is the primary variable and because the variable is sparse). Regarding the Y variable (which is assumed to be dense with regards to Z), several possibilities are offered:

• not using any Y information: obviously this does not offer any interest,
• using all the Y information contained within the neighborhood: this may lead to an untractable solution because of too many information,
• the initial solution (as mentioned in Xu, W., Tran, T. T., Srivastava, R. M., and Journel, A. G. 1992, Integrating seismic data in reservoir modeling: The collocated cokriging alternative. SPE paper 24742, 67Th Annual Technical Conference and exhibition, p.833-842) consists in using the single value located at the target grid node location: hence the term collocated. Its contribution to the kriging estimate relies on the cross-correlation between the two variables at zero distance. But, in the Intrinsic case, the weights attached to the secondary variable must add up to zero and therefore, if only one data value is used, its single weight (or influence) will be zero.
• the solution used in Isatis is to use the Y variable at the target location and at all the locations where the Z variable is defined (Multi Collocated Cokriging). This neighborhood search has given the more reliable and stable results so far.

In general collocated cokriging is less precise than a full cokriging - making use of the auxiliary variable at all target points when estimating each of these.

Exception are models where the cross variogram (or covariance) between the two variables is proportional to the variogram (or covariance) of the auxiliary variable.

In this case collocated cokriging coincides with full cokriging, but is also strictly equivalent to the simple method consisting in kriging the residual of the linear regression of the target variable on the auxiliary variable.

The user interested by the different approaches to Collocated Cokriging can refer to Rivoirard J., Which Models for Collocated Cokriging?, In Math. Geology, Vol. 33, No 2, 2001, pp. 117-131.