Advance theory

Kriging system expansion:

Gradient Covariance-Matrix \({\bf{C_{\partial {\bf{Z}}/ \partial u}}}\)

The following equations have been derived from the work in :cite:`d-Aug.2004,Lajaunie.1997,chiles2009geostatistics

The gradient covariance-matrix, \({\bf{C_{\partial {\bf{Z}}/ \partial u}}}\), is made up of as many variables as gradient directions that are taken into consideration. In 3-D, we would have the Cartesian coordinates dimensions—\({\bf{Z}}/ \partial x\), \({\bf{Z}}/ \partial y\), and \({\bf{Z}}/ \partial z\)—and therefore, they will derive from the partial differentiation of the covariance function \(\sigma(x_i,x_j)\) of \(\bf{Z}\).

2-D representation of the decomposition of the orientation vectors into Cartesian axis. Each Cartesian axis represent a variable of a sub CoKriging system. The dotted green line represent the covariance distance, \(r\).

As in our case the directional derivatives used are the 3 Cartesian directions we can rewrite gradients covariance, \({\bf{C_{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial v}}}\) for our specific case as:

\[\begin{split}{\bf{C_{\partial {\bf{Z}}/ \partial x, \, \partial {\bf{Z}}/ \partial y \, \partial {\bf{Z}}/ \partial z}}} = \left[ \begin{array}{ccc} {\bf{C_{\partial {\bf{Z}}/ \partial x, \, \partial {\bf{Z}}/ \partial x}}} & {\bf{C_{\partial {\bf{Z}}/ \partial x, \, \partial {\bf{Z}}/ \partial y}}} & {\bf{C_{\partial {\bf{Z}}/ \partial x, \, \partial {\bf{Z}}/ \partial z}}} \\ {\bf{C_{\partial {\bf{Z}}/ \partial y, \, \partial {\bf{Z}}/ \partial x}}} & {\bf{C_{\partial {\bf{Z}}/ \partial y, \, \partial {\bf{Z}}/ \partial y}}} & {\bf{C_{\partial {\bf{Z}}/ \partial y, \, \partial {\bf{Z}}/ \partial z}}}\\ {\bf{C_{\partial {\bf{Z}}/ \partial z, \, \partial {\bf{Z}}/ \partial x}}} & {\bf{C_{\partial {\bf{Z}}/ \partial z, \, \partial {\bf{Z}}/ \partial y}}} & {\bf{C_{\partial {\bf{Z}}/ \partial z, \, \partial {\bf{Z}}/ \partial z}}} \end{array} \right] \label{C_g}\end{split}\]

Notice, however, that covariance functions by definition are described in a polar coordinate system, and therefore it will be necessary to apply the chain rule for directional derivatives. Considering an isotropic and stationary covariance we can express the covariance function as:

\[\sigma(x_i,x_j) = C(r)\]

with:

\[r = \sqrt{h^2_x+h^2_y}\]

therefore we need to apply the chain rule in partial differentiation. For the case of the covariance in a single direction:

\[\begin{split}{\bf{C_{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial u}}} = \frac{\partial^2 C_{Z} (r)}{\partial h^2_u} = \frac{\partial C_{Z} (r)}{\partial r} \frac{\partial}{\partial h_u} \left( \frac{\partial r}{\partial h_u} \right) + \frac{\partial}{\partial h_u} \left( \frac{\partial C_Z (r)}{\partial r}\right) \frac{\partial r}{\partial h_u} \\\end{split}\]

where:

\[\frac{\partial C_{Z} (r)}{\partial r} = \frac{\partial C_{Z} (r)}{\partial r} = C'_Z(r)\]
\[\frac{\partial r}{\partial h_u} = \frac{h_u}{\sqrt{h^2_u+h^2_v}} = -\frac{h_u}{r} \label{eq:}\]
\[\frac{\partial}{\partial h_u} \left( \frac{\partial r}{\partial h_u} \right) = \frac{\partial}{\partial h_u} \left(\frac{h_u}{\sqrt{h^2_u+h^2_v}}\right) = -\frac{2 h_u^2}{2\sqrt{h^2_u+h^2_v}} + \frac{1}{\sqrt{h^2_u+h^2_v}} = -\frac{h_u^2}{r^3} + \frac{1}{r}\]
\[\frac{\partial}{\partial h_u} \left( \frac{\partial C_Z (r)}{\partial r}\right) = \frac{\partial C'_Z(r)}{\partial h_u} = \frac{\partial C'_{Z} (r)}{\partial r} \frac{\partial r}{\partial h_u} = - \frac{h_u}{r}C''_Z\]

Substituting:

\[{\bf{C_{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial u}}} = C'_Z(r) \left( -\frac{h_u^2}{r^3} + \frac{1}{r} \right) - \frac{h_u}{r}C''_Z \frac{h_u}{r} = C'_Z(r) \left( -\frac{h_u^2}{r^3} + \frac{1}{r} \right) + \frac{h_u^2}{r^2} C''_Z \label{huhu}\]

While in case of two different directions the covariance will be:

\[{\bf{C_{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial v}}} = \frac{\partial^2 C_{Z}(r)}{\partial h_u h_v} = \frac{\partial C_{Z} (r)}{\partial r} \frac{\partial}{\partial h_v} \left( \frac{\partial r}{\partial h_u} \right) + \frac{\partial}{\partial h_v} \left( \frac{\partial C_Z (r)}{\partial r}\right) \frac{\partial r}{\partial h_u}\]

with:

\[\frac{\partial}{\partial h_v} \left( \frac{\partial r}{\partial h_u} \right) = \frac{\partial}{\partial h_v} \left(\frac{h_u}{\sqrt{h^2_u+h^2_v}}\right) = -\frac{h_u h_v}{r^3}\]
\[\frac{\partial}{\partial h_v} \left( \frac{\partial C_Z (r)}{\partial r}\right) = \frac{\partial C'_Z(r)}{\partial h_v} = - C''_Z(r) \frac{h_v}{r}\]

we have:

\[{\bf{C_{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial v}}} = C'_Z(r) \left(-\frac{h_u h_v}{r^3} \right) + C''_Z(r) \frac{h_u h_v}{r^2}= \frac{h_u h_v}{r^2} \left( C''_Z (r) - \frac{C'_Z (r)}{r} \right) \label{huhv}\]

This derivation is independent to the covariance function choice, however, some covariances may lead to mathematical indeterminations.

Interface Covariance-Matrix

Distances \(r\) involved in the computation of the interface subsystem of the interpolation. Because all covariances are relative to a reference point \(x_{\alpha, \, 0}^i\), all four covariances must be taken into account (equation [eq:int_cov])

In a practical sense, keeping the value of the scalar field at every interface unfixed forces us to consider the covariance between the points within an interface as well as the covariance between different layers following equation,

\[{{C}}_{{\bf{x}}_{\alpha \, i}^r, \, {\bf{x}}_{\alpha \,j}^s} = C_{x^r_{\alpha, \,i} \, x^s_{\alpha, \,j}} - C_{x^r_{\alpha, \,0} \, x^s_{\alpha, \,j}} - C_{x^r_{\alpha, \,i} \, x^s_{\alpha, \,0}} + C_{x^r_{\alpha, \,0} \, x^s_{\alpha, \,0}} \label{eq:int_cov}\]

This lead to the subdivision of the CoKriging system respecting the interfaces:

\[\begin{split}{\bf{C_{\bf{Z}, \, \bf{Z}}}} = \begin{bmatrix} {\bf{C}}_{{\bf{x}}_{\alpha}^1, \, {\bf{x}}_{\alpha}^1}& {\bf{C}}_{{\bf{x}}_{\alpha}^1, \, {\bf{x}}_{\alpha}^2}& ... & {\bf{C}}_{{\bf{x}}_{\alpha}^1, \, {\bf{x}}_{\alpha}^s}\\ {\bf{C}}_{{\bf{x}}_{\alpha}^2, \, {\bf{x}}_{\alpha}^1}& {\bf{C}}_{{\bf{x}}_{\alpha}^2, \, {\bf{x}}_{\alpha}^2}& ... & {\bf{C}}_{{\bf{x}}_{\alpha}^2, \, {\bf{x}}_{\alpha}^s}\\ \vdots & \vdots & \ddots & \vdots \\ {\bf{C}}_{{\bf{x}}_{\alpha}^r, \, {\bf{x}}_{\alpha}^1}& {\bf{C}}_{{\bf{x}}_{\alpha}^r, \, {\bf{x}}_{\alpha}^2}& ... & {\bf{C}}_{{\bf{x}}_{\alpha}^r, \, {\bf{x}}_{\alpha}^s} \end{bmatrix} \label{C_i}\end{split}\]

Combining Eq [one_val] and Eq [C_i] the covariance for the property potential field will look like:

\[\begin{split}{\bf{C}}_{{\bf{x}}_{\alpha}^r, \, {\bf{x}}_{\alpha}^s} = \begin{bmatrix} C_{x^1_1 \, x^1_1} - C_{x^1_0 \, x^1_1} - C_{x^1_1 \, x^1_0} + C_{x^1_0 \, x^1_0} & C_{x^1_1 \, x^1_2} - C_{x^1_0 \, x^1_2} - C_{x^1_1 \, x^1_0} + C_{x^1_0 \, x^1_0} & ... & C_{x^1_1 \, x^s_j} - C_{x^1_0 \, x^s_j} - C_{x^1_1 \, x^s_0} + C_{x^1_0 \, x^s_0}\\ C_{x^1_2 \, x^1_1} - C_{x^1_0 \, x^1_1} - C_{x^1_2 \, x^1_0} + C_{x^1_0 \, x^1_0} & C_{x^1_2 \, x^1_2} - C_{x^1_0 \, x^1_2} - C_{x^1_2 \, x^1_0} + C_{x^1_0 \, x^1_0} & ... & C_{x^1_2 \, x^s_j} - C_{x^1_0 \, x^s_j} - C_{x^1_j \, x^s_0} + C_{x^1_0 \, x^s_0}\\ \vdots & \vdots & \ddots & \vdots \\ C_{x^r_i \, x^s_1} - C_{x^r_0 \, x^s_1} - C_{x^r_i \, x^s_0} + C_{x^r_0 \, x^s_0} & C_{x^r_i \, x^s_2} - C_{x^r_0 \, x^s_2} - C_{x^r_i \, x^s_0} + C_{x^r_0 \, x^s_0} & ... & C_{x^r_i \, x^s_j} - C_{x^r_0 \, x^s_j} - C_{x^r_i \, x^s_0} + C_{x^r_0 \, x^s_0} \end{bmatrix}\end{split}\]

Cross-Covariance

In a CoKriging system, the relation between the interpolated parameters is given by a cross-covariance function. As we saw above, the gradient covariance is subdivided into covariances with respect to the three Cartesian directions (Eq [C_g]), while the interface covariance is detached from the covariances matrices with respect to each individual interface (Eq [C_i]). In the same manner, the cross-covariance will reflect the relation of every interface to each gradient direction,

Distances \(r\) involved in the computation of the cross-covariance function. In a similar fashion as before, all interface covariance are computed relative to a reference point in each layer \(x_{\alpha, \, 0}^i\)

\[\begin{split}{\bf{C_{Z, \,\partial {\bf{Z}}/ \partial u}}} = \begin{bmatrix} {\bf{C}}_{{\bf{x}}_{\alpha \, 1}^1, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial x}& {\bf{C}}_{{\bf{x}}_{\alpha \, 2}^1, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial x}& ... & {\bf{C}}_{{\bf{x}}_{\alpha \, 1}^1, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial y}& ... & {\bf{C}}_{{\bf{x}}_{\alpha \, i}^1, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, j})/ \partial z}& \\ {\bf{C}}_{{\bf{x}}_{\alpha \, 1}^2, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial x}& {\bf{C}}_{{\bf{x}}_{\alpha \, 2}^2, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial x}& ... & {\bf{C}}_{{\bf{x}}_{\alpha \, 1}^2, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial y}& ... & {\bf{C}}_{{\bf{x}}_{\alpha \, i}^2, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, j})/ \partial z}& \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {\bf{C}}_{{\bf{x}}_{\alpha \, 1}^r, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial x}& {\bf{C}}_{{\bf{x}}_{\alpha \, 2}^r, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial x}& ... & {\bf{C}}_{{\bf{x}}_{\alpha \, 2}^r, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, 1})/ \partial y}& ... & {\bf{C}}_{{\bf{x}}_{\alpha \, i}^r, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, j})/ \partial z}& \\ \end{bmatrix}\end{split}\]

As the interfaces are relative to a point :math:` {bf{x}}_{alpha_, 0}^k` the value of the covariance function:

\[{\bf{C}}_{{\bf{x}}_{\alpha \, i}^r, \, \partial {\bf{Z}}({\bf{x}}_{\beta \, j})/ \partial x} = C_{Z({{\bf{x}}_{\alpha \,i}^r)}, \, \partial Z({\bf{x_{\beta \, j}}})/\partial x} - C_{Z({{\bf{x}}_{\alpha \,0}^r)}, \, \partial Z({\bf{x_{\beta \, j}}})/\partial x}\]

with the covariance of the scalar field being function the vector r, its directional derivative is analogous to the previous derivations:

\[{\bf{C}{_{\bf{Z}, \, \partial {\bf{Z}}/ \partial u}}} = \frac{\partial C_{\bf{Z}} (r)}{\partial r} \frac{\partial r}{\partial h_u} = - \frac{h_u}{r}C'_Z\]

Universal matrix

As the mean value of the scalar field is going to be always unknown, it needs to be estimated from data itself. The simplest approach is to consider the mean constant for the whole domain, i.e. ordinary Kriging. However, in the scalar field case we can assume certain drift towards the direction of the orientations. Therefore, the mean can be written as function of known basis functions:

\[\mu({\bf{x}}) = \sum^L_{l=0} a_lf^l({\bf{x}})\]

where l is the grade of the polynomials used to describe the drift. Because of the algebraic dependence of the variables, there is only one drift and therefore the unbiasedness can be expressed as:

\[{\bf{U}}_{\bf{Z}} \lambda_{1} + {\bf{U}}_{\partial {\bf{Z}}/ \partial u} \lambda_{2} = f_{10}\]

Consequently, the number of equations are determined according to the grade of the polynomial and the number of equations forming the properties matrices equations [C_i] and [C_g]:

\[\begin{split}U_Z = \begin{bmatrix} x^1_1 - x^1_0 & x^1_2 - x^1_0 & ... & x^2_1 - x^2_0 & x^2_2 - x^2_0 & ... & x^r_{i-1} - x^r_0 & x^r_i - x^r_0 \\ y^1_1 - y^1_0 & y^1_2 - y^1_0 & ... & y^2_1 - y^2_0 & y^2_2 - y^2_0 & ... & y^r_{i-1} - y^r_0 & y^r_i - y^r_0 \\ z^1_1 - z^1_0 & z^1_2 - z^1_0 & ... & z^2_1 - z^2_0 & z^2_2 - z^2_0 & ... & z^r_{i-1} - z^r_0 & z^r_i - z^r_0 \\ x^1_1 x^1_1 - x^1_0x^1_0 & x^1_2 x^1_2 - x^1_0 x^1_0 & ... & x^2_1 x^2_1 - x^2_0x^2_0 & x^2_2 x^2_2 - x^2_0 x^2_0 & ... & x^r_{i-1}x^r_{i-1} - x^r_0x^r_0 & x^r_i x^r_i - x^r_0 x^r_0 \\ y^1_1 y^1_1 - y^1_0 y^1_0 & y^1_2 y^1_2 - y^1_0 y^1_0 & ... & y^2_1 y^2_1 - y^2_0 y^2_0 & y^2_2 y^2_2 - y^2_0 y^2_0 & ... & y^r_{i-1}y^r_{i-1} - y^r_0 y^r_0 & y^r_i y^r_i - y^r_0 y^r_0 \\ z^1_1 z^1_1 - z^1_0 z^1_0 & z^1_2 z^1_2 - z^1_0 z^1_0 & ... & z^2_1 z^2_1 - z^2_0 z^2_0 & z^2_2 z^2_2 - z^2_0 z^2_0 & ... & z^r_{i-1}z^r_{i-1} - z^r_0 z^r_0 & z^r_i z^r_i - z^r_0 z^r_0 \\ x^1_1 y^1_1 - x^1_0 y^1_0 & x^1_2 y^1_2 - x^1_0 y^1_0 & ... & x^2_1 y^2_1 - x^2_0 y^2_0 & x^2_2 y^2_2 - x^2_0 y^2_0 & ... & x^r_{i-1}y^r_{i-1} - x^r_0 y^r_0 & x^r_i y^r_i - x^r_0 y^r_0 \\ x^1_1 z^1_1 - x^1_0 z^1_0 & x^1_2 z^1_2 - x^1_0 z^1_0 & ... & x^2_1 z^2_1 - x^2_0 z^2_0 & x^2_2 z^2_2 - x^2_0 z^2_0 & ... & x^r_{i-1}z^r_{i-1} - x^r_0 z^r_0 & x^r_i z^r_i - x^r_0 z^r_0 \\ y^1_1 z^1_1 - y^1_0 z^1_0 & y^1_2 z^1_2 - y^1_0 z^1_0 & ... & y^2_1 z^2_1 - y^2_0 z^2_0 & y^2_2 z^2_2 - y^2_0 z^2_0 & ... & y^r_{i-1}z^r_{i-1} - y^r_0 z^r_0 & y^r_i z^r_i - y^r_0 z^r_0 \\ \end{bmatrix}\end{split}\]
\[\begin{split}\bf{U_{\partial {\bf{Z}}/ \partial u}} = \begin{matrix} \begin{matrix} {\bf{x}_{\beta 1}} & {\bf{x}_{\beta 2}} & ...& {\bf{x}_{\beta 1}} & {\bf{x}_{\beta 2}}& ...& {\bf{x}_{\beta i-1}} & {\bf{x}_{\beta i}} \end{matrix}\\ \begin{bmatrix} 1 & 1 & ... & 0 & 0 & ... & 0 & 0\\ 0 & 0 & ... & 1 & 1 & ... & 0 & 0\\ 0 & 0 & ... & 0 & 0 & ... & 1 & 1\\ 2x_1 & 2x_2 & ... & 0 & 0 & ... & 0 & 0\\ 0 & 0 & ... & 2y_1 & 2y_2 & ... & 0 & 0\\ 0 & 0 & ... & 0 & 0 & ... & 2z_{i-1} & 2z_i\\ y_1 & y_2 & ... & x_1 & x_2 & ... & 0 & 0\\ y_1 & y_2 & ... & 0 & 0 & ... & x_{i-1} & x_i\\ 0 & 0 & ... & z_1 & z_2 & ... & x_{i-1} & x_i\\ \end{bmatrix} & \begin{aligned}[l] &\left\}\begin{matrix} \partial {\bf{x}_{\beta i}}/\partial x \\ \partial {\bf{x}_{\beta i}}/\partial y \\ \partial {\bf{x}_{\beta i}}/\partial z \\ \partial^2 {\bf{x}_{\beta i}}/\partial x^2 \\ \partial^2 {\bf{x}_{\beta i}}/\partial y^2 \\ \partial^2 {\bf{x}_{\beta i}}/\partial z^2 \\ \partial^2 {\bf{x}_{\beta i}}/\partial x \partial y\\ \partial^2 {\bf{x}_{\beta i}}/\partial x \partial z\\ \partial^2 {\bf{x}_{\beta i}}/\partial y \partial z\\ \end{matrix}\right.\end{aligned} \end{matrix}\end{split}\]

Kriging Estimator

In normal Kriging the right hand term of the Kriging system (Eq. [krig_sys]) corresponds to covariances and drift matrices of dimensions \(m \times n\) where m is the number of elements of the data sets—either \({\bf{x}}_\alpha\) or \({\bf{x}}_\beta\)—and n the number of locations where the interpolation is performed, \({\bf{x}}_0\).

Since, in this case, the parameters of the variogram functions are arbitrarily chosen, the Kriging variance does not hold any physical information of the domain. As a result of this, being interested only in the mean value, we can solve the Kriging system in the dual form [D1]:

\[\begin{split}Z({\bf{x}}_0)= \begin{bmatrix} a'_{{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial v}} & b'_{{\bf{Z}, \,\bf{Z}}} & c' \end{bmatrix} \left[ \begin{array}{cc} {\bf{c_{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial v}}} & {\bf{c_{\partial {\bf{Z}}/ \partial u, \, Z}}} \\ {\bf{c_{Z, \,\partial {\bf{Z}}/ \partial u}}} & {\bf{c_{\bf{Z}, \,\bf{Z}}}} \\ {\bf{f_{10}}} & {\bf{f_{20}}} \end{array} \right]\end{split}\]

where:

\[\begin{split}\begin{bmatrix} a_{{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial v}} \\ b_{{\bf{Z}, \,\bf{Z}}} \\ c \end{bmatrix} = \begin{bmatrix} \partial {\bf{Z}} \\ 0 \\ 0 \end{bmatrix} \left[ \begin{array}{ccc} {\bf{C_{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial v}}} & {\bf{C_{\partial {\bf{Z}}/ \partial u, \, Z}}} & \bf{U_{\partial {\bf{Z}}/ \partial u}} \\ {\bf{C_{Z, \, \partial {\bf{Z}}/ \partial u }}} & {\bf{C_{\bf{Z}, \, \bf{Z}}}} & {\bf{U_{Z}}} \\ \bf{U'_{\partial {\bf{Z}}/ \partial u}} & {\bf{U'_{Z}}} & {\bf{0}} \end{array} \right]^{-1}\end{split}\]

noticing that the 0 on the second row appears due to we are interpolation the difference of scalar fields instead the scalar field itself [eq_rel].

Example of covariance function: cubic

The choice of the covariance function will govern the shape of the iso-surfaces of the scalar field. As opposed to other Kriging uses, here the choice cannot be based on empirical measurements. Therefore, the choice of the covariance function is merely arbitrary trying to mimic as far as possible coherent geological structures.

Representation of a cubic variogram and covariance for an arbitrary range and nugget effect.

The main requirement to take into consideration when the time comes to choose a covariance function is that it has to be twice differentiable, \(h^2\) in origin to be able to calculate \({\bf{C_{\partial {\bf{Z}}/ \partial u, \, \partial {\bf{Z}}/ \partial v}}}\) as we saw in equation [huhv]. The use of a Gaussian model \(C(r) = \exp{-(r/a)^2}\) and the non-divergent spline \(C(r) = r^4 Log(r)\) and their correspondent flaws are explored in [D2].

The most widely used function in the potential field method is the cubic covariance due to mathematical robustness and its coherent geological description of the space.

\[\begin{split}C(r) = \begin{cases} C_0(1-7(\frac{r}{a})^2+ \frac{35}{4}(\frac{r}{a})^3 - \frac{7}{2}(\frac{r}{a})^5 +\frac{3}{4}(\frac{r}{a})^7) & \text{for } 0 \leq r \leq a \\ 0 & \text{for } r \geq a \end{cases}\end{split}\]

with \(a\) being the range and \(C_0\) the variance of the data. The value of \(a\) determine the maximum distance that a data point influence another. Since, we assume that all data belong to the same depositional phase it is recommended to choose values close to the maximum extent to interpolate in order to avoid mathematical artifacts. for the values of the covariance at 0 and nuggets effects so far only ad hoc values have been used so far. It is important to notice that the only effect that the values of the covariance in the potential-field method has it is to weight the relative influence of both CoKriging parameters (interfaces and orientations) since te absolut value of the field is meaningless. Regarding the nugget effect, the authors recommendation is to use fairly small nugget effects to give stability to the computation—since we normally use the kriging mean it should not have further impact to the result.

Example of Probabilistic Graphical Model

Here we can see the probabilistic graphical model of the Bayesian inference of Section [sec:geol-invers-prob]:

Probabilistic graphical model generated with pymc2. Ellipses represent stochastic parameters, while triangles are deterministic functions that return intermediated states of the probabilistic model such as the GemPy model

[D1]Georges Matheron. Splines and kriging: their formal equivalence. Down-to-earth-statistics: Solutions looking for geological problems, pages 77–95, 1981.
[D2]Christian Lajaunie, Gabriel Courrioux, and Laurent Manuel. Foliation fields and 3D cartography in geology: Principles of a method based on potential interpolation. Mathematical Geology, 29(4):571–584, 1997.