Ordinary Kriging

Ordinary kriging will estimate an appropriate mean of the field, based on the given observations/conditions and the covariance model used.

The resulting system of equations for W is given by:

\begin{pmatrix}W\\\mu\end{pmatrix} = \begin{pmatrix}
\gamma(x_1,x_1) & \cdots & \gamma(x_1,x_n) &1 \\
\vdots & \ddots & \vdots  & \vdots \\
\gamma(x_n,x_1) & \cdots & \gamma(x_n,x_n) & 1 \\
1 &\cdots& 1 & 0
\end{pmatrix}^{-1}
\begin{pmatrix}\gamma(x_1,x_0) \\ \vdots \\ \gamma(x_n,x_0) \\ 1\end{pmatrix}

Thereby \gamma(x_i,x_j) is the semi-variogram of the given observations and \mu is a Lagrange multiplier to minimize the kriging error and estimate the mean.

Example

Here we use ordinary kriging in 1D (for plotting reasons) with 5 given observations/conditions. The estimated mean can be accessed by krig.mean.

import numpy as np
from gstools import Gaussian, krige

# condtions
cond_pos = [0.3, 1.9, 1.1, 3.3, 4.7]
cond_val = [0.47, 0.56, 0.74, 1.47, 1.74]
# resulting grid
gridx = np.linspace(0.0, 15.0, 151)
# spatial random field class
model = Gaussian(dim=1, var=0.5, len_scale=2)
krig = krige.Ordinary(model, cond_pos=cond_pos, cond_val=cond_val)
krig(gridx)
ax = krig.plot()
ax.scatter(cond_pos, cond_val, color="k", zorder=10, label="Conditions")
ax.legend()
../../_images/sphx_glr_01_ordinary_kriging_001.png

Total running time of the script: ( 0 minutes 0.118 seconds)

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