Kriging equations
Intrinsic model
\[
\mathbb{E} \left[ ( z( \boldsymbol{ \mathrm{ x } } ) - m )( z( \boldsymbol{ \mathrm{ x } } + \boldsymbol{ \mathrm{ h } } ) - m ) \right] = C(h)
\]
\[
h = \lVert \boldsymbol{ \mathrm{ x } } - ( \boldsymbol{ \mathrm{ x } } + \boldsymbol{ \mathrm{ h } } ) \rVert =
\sqrt{
( x_{1} - x'_{1} ) ^ 2 +
( x_{2} - x'_{2} ) ^ 2 +
( x_{3} - x'_{3} ) ^ 2
}
\]
Expression of the Intrisic model via semi-variogram:
\[
\gamma(h) = \frac{1}{2} \mathbb{E}\left[ z(\boldsymbol{ \mathrm{ x } } ) - z( \boldsymbol{ \mathrm{ x } } + \boldsymbol{ \mathrm{ h } } ) ^ 2 \right]
= C(0) - C(h)
\]
Weighted averaging:
\[
\hat{ z }( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) =
\sum\limits_{\alpha}^{N} \lambda_{\alpha} z(\boldsymbol{ \mathrm{ x } }_\alpha)
\]
where \(\lambda_{\alpha}\) is the weighting coefficient. This quantity can be estimated from the variance (\(\mathbb{V}\)):
\[\begin{split}
\begin{array}{l}
\mathbb{V}( z( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) ) =
\mathbb{E} \{ \left[ z( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) - \hat{ z }( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) \right ] ^ 2 \} \\
\phantom{\mathrm{ Var( z( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) ) }} =
C(0) - 2 \sum\limits_{\alpha} \lambda_{\alpha}
C( \lVert \boldsymbol{ \mathrm{ x } }_{ \alpha } \boldsymbol{ \mathrm{ x } }_{ \kappa } \rVert )
+ \sum\limits_{ \alpha } \sum\limits_{ \beta } \lambda_{ \alpha } \lambda_{ \beta }
C( \lVert \boldsymbol{ \mathrm{ x } }_{ \alpha } \boldsymbol{ \mathrm{ x } }_{ \beta } \rVert )
\end{array}
\end{split}\]
\[
\sum\limits_{\alpha}^{N} \lambda_{\alpha} = 1
\]
Differentiation with respect to \(\lambda_\alpha\) to find the predicted value while also minimizing the variance:
\[
\sum\limits_{\beta}^{ N } \lambda_{ \beta }
C_{n}( \lVert \boldsymbol{ \mathrm{ x } }_{ \alpha } \boldsymbol{ \mathrm{ x } }_{ \beta } \rVert )
- \mu =
C_{n}( \lVert \boldsymbol{ \mathrm{ x } }_{ \alpha } \boldsymbol{ \mathrm{ x } }_{ \kappa } \rVert )
\]
\[
\sum\limits_{\beta}^{N} \lambda_{\beta} = 1
\]
This can also be expressed as a system of linear equations that can be solved to determine \(\lambda_{\alpha}\) and
\(\lambda_{\beta}:\)
\[\begin{split}
\begin{bmatrix}
\gamma(d(u_1,u_1)) & \gamma(d(u_1,u_2)) & \cdots & \gamma(d(u_1,u_n)) & 1 \\
\gamma(d(u_2,u_1)) & \gamma(d(u_2,u_2)) & \cdots & \gamma(d(u_2,u_n)) & 1 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
\gamma(d(u_n,u_1)) & \gamma(d(u_n,u_2)) & \cdots & \gamma(d(u_n,u_n)) & 1 \\
1 & 1 & \cdots & 1 & 0
\end{bmatrix}
\begin{bmatrix}
\lambda_1 \\
\lambda_2 \\
\vdots \\
\lambda_n \\
\mu
\end{bmatrix}
=
\begin{bmatrix}
\gamma(d(u,u_1)) \\
\gamma(d(u,u_2)) \\
\vdots \\
\gamma(d(u,u_n)) \\
1
\end{bmatrix}
\end{split}\]
where \(\mu\) is the Lagrangian coefficient:
\[
\mu = \frac{
1 - 1^{T} \lambda
}{
\mathbf{1}^{T} K^{-1} \boldsymbol{ \mathrm{ k } }
}
\]
where \(K\) is the \(n~\mathrm{x}~n\) covariance matrix, \(\boldsymbol{ \mathrm{ k } }\) is the
\(n~\mathrm{x}~1\) vector of covariance estimates between sampled and unsampled locations,
\(1\) corresponds to a vector of ones, and \(\mathbf{1}^{T}\) is the transpose of the vector
of ones. Thus, the above system of linear equations for \(\lambda\) can be solved via:
\[
\lambda = K^{-1} (\mathbf{k} - \mu)
\]
This can also be expressed directly in terms of the semivariogram:
\[
\sum\limits_{\beta}^{ N } \lambda_{ \beta }
\gamma_{n}( \lVert \boldsymbol{ \mathrm{ x } }_{ \alpha } \boldsymbol{ \mathrm{ x } }_{ \beta } \rVert )
+ \mu =
\gamma_{n}( \lVert \boldsymbol{ \mathrm{ x } }_{ \alpha } \boldsymbol{ \mathrm{ x } }_{ \kappa } \rVert )
\]
\[
\sum\limits_{\beta}^{N} \lambda_{\beta} = 1
\]
Once the \(\lambda_{\beta}\) and \(\mu\) estimates have been obtained, the kriged prediction variance can be
estimated via:
\[\begin{split}
\begin{array}{l}
\mathbb{V}( z( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) ) =
\mathbb{E} \{ \left[ z( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) - \hat{ z }( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) \right ] ^ 2 \} \\
\phantom{\mathrm{ Var( z( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) ) }} =
C(0) + \mu
- \sum\limits_{\alpha} \lambda_{\alpha}
C( \lVert \boldsymbol{ \mathrm{ x } }_{ \alpha } - \boldsymbol{ \mathrm{ x } }_{ \kappa } \rVert ) \\
\phantom{\mathrm{ Var( z( \boldsymbol{ \mathrm{ x } }_{ \kappa } ) ) }} =
\mu + \sum\limits_{\alpha} \lambda_{\alpha}
\gamma( \lVert \boldsymbol{ \mathrm{ x } }_{ \alpha } - \boldsymbol{ \mathrm{ x } }_{ \kappa } \rVert )
-\gamma(0)
\end{array}
\end{split}\]
Ordinary kriging estimation equation:
\[
\hat{z}(u) = \sum_{i=1}^{n} \lambda_i z(u_i)
\]
\[
\sum\limits_{i=1}^{n} \lambda_{i} = 1
\]