CoKriging

Intro

The term kriging is traditionally reserved for liear regression using data on the same attribute as that being estimated. For example, an unsampled porosity value \(z(u)\) is estimated from neighboring porosity sample values defined on the same volume support.

The term cokriging is reserved for linear regression that also uses data defined on different attributes. For example, the porosity values \(z(u)\) may be estimated from combination of porosity samples and related acoustic data values.

In the case of a single secondary variable (\(Y\)), the ordinary cokriging estimator of \(Z(\mathbf{u})\) is written:

\[Z_{COK}^{*}(\mathbf{u}) =\sum_{{\alpha}_{1}=1}^{{n}_{1}}{{\lambda}_{{\alpha}_{1}}(\mathbf{u})Z({\mathbf{u}}_{{\alpha}_{1}})} +\sum_{{\alpha}_{2}=1}^{{n}_{2}}{{\lambda}_{{\alpha}_{2}}^{'}(\mathbf{u})Y({\mathbf{u}}_{{\alpha}_{2}}^{'})}\]

where the \({\lambda}_{{\alpha}_{1}}\) are the weights applied to the \({n}_{1}\) \(z\) samples and the \({\lambda}_{{\alpha}_{2}}^{'}\) are the weights applied to the \(n_2\) y samples.

Kriging requires a model for the \(Z\) covariance. Cokriging requires a joint model for the matrix of covariance functions including the \(Z\) covariance \(C_{Z}(\mathbf{h})\), the \(Y\) covariance \(C_{Y}(\mathbf{h})\), the cross \(Z-Y\) covariance \(C_{ZY}(\mathbf{h})=Cov\{Z(\mathbf{u}),Y(\mathbf{u+h})\}\), and the cross \(Y-Z\) covariance \(C_{YZ}(\mathbf{h})\)

The covariance matrix requires \(K^2\) covariance functions when \(K\) different variables are considered in a cokriging exercise. The inference becomes extremely demanding in terms of data and the subsequent joint modeling is particularly tedious. This is the main reason why cokriging has not been extensively used in practice. Algorithms such as kriging with an external drift and collocated cokriging have been developed to shortcut the tedious inference and modeling process required by cokriging.

Ordinary Cokriging

The sum of the weights applied to the primary variable is set to one, and the sum of the weigths applied to any other variable is set to zero. In the case of two variables, these two conditions are:

\[\begin{split}\begin{cases} \sum\limits_{{\alpha}_{1}}^{}{{\lambda}_{{\alpha}_{1}}(\mathbf{u})}=1\\ \sum\limits_{{\alpha}_{2}}^{}{{\lambda}_{{\alpha}_{2}}(\mathbf{u})}=0 \end{cases}\end{split}\]

The problem with this traditional formalism is that the second condition tends to limit severely the influence of the secondary variables.

Standardized Ordinary Cokriging

Often, a better approach consists of creating new secondary variables with the same mean as the primary variable. Then all the weights are constrained to sum to one.

In the case of two variables, the expression could be written as:

\[Z_{COK}^{*}(\mathbf{u}) =\sum_{{\alpha}_{1}=1}^{{n}_{1}}{{\lambda}_{{\alpha}_{1}}(\mathbf{u})Z({\mathbf{u}}_{{\alpha}_{1}})} +\sum_{{\alpha}_{2}=1}^{{n}_{2}}{{\lambda}_{{\alpha}_{2}}^{'}(\mathbf{u})[Y({\mathbf{u}}_{{\alpha}_{2}}^{'})+{m}_{Z}-{m}_{Y}]}\]

with a single condition:

\[\sum_{{\alpha}_{1}=1}^{{n}_{1}}{{\lambda}_{{\alpha}_{1}}}(\mathbf{u})+\sum_{{\alpha}_{2}=1}^{{n}_{2}}{{\lambda}_{{\alpha}_{2}}}(\mathbf{u})=1\]

where \(m_Z=E\{Z(u)\}\) and \(m_Y=E\{Y(u)\}\) are stationary means of \(Z\) and \(Y\).

Simple Cokriging

There is no constraint on the weights. Just like simple kriging, this version of cokriging requires working on data residuals or equivalently, on variables whose means have all been standardized to zero. This is the case when applying simple cokriging in an MG approach (the normal score transforms of each variable have a stationary mean of zero).

Collocated Cokriging

A reduced form of cokriging consists of retaining only the collocated variable \(y(\mathbf{u})\), provided that it is availible at all locations \(\mathbf{u}\) being estimated. The cokriging estimator is written as:

\[Z_{COK}^{*}(\mathbf{u}) =\sum_{{\alpha}_{1}=1}^{{n}_{1}}{{\lambda}_{{\alpha}_{1}}(\mathbf{u})Z({\mathbf{u}}_{{\alpha}_{1}})} +{\lambda}^{'}(\mathbf{u})Y(\mathbf{u})\]

The corresponding cokriging system requires knowledge of only the \(Z\) covariance \(C_{Z}(\mathbf{h})\) and the \(Z-Y\) cross-covariance \(C_{ZY}(\mathbf{h})\). The latter can be approximated through the following model:

\[C_{ZY}(\mathbf{h})=B\cdot C_{Z}(\mathbf{h}),\quad\forall \mathbf{h}\]

where \(B=\sqrt{C_Y(0)/C_Z(0)}\cdot{\rho}_{ZY}(0)\), \(C_Z(0)\), \(C_Y(0)\) are the variances of Z and Y, and \({\rho}_{ZY}(0)\) is the linear coefficient of correlation of collocated z-y data.