Simple Kriging - Transform

%matplotlib widget
import matplotlib.pyplot as plt
plt.ioff()
# turn of warnings
import warnings
warnings.filterwarnings('ignore')

Simple kriging assumes a known mean of the data. For simplicity we assume a mean of 0, which can be achieved by subtracting the mean from the observed values and subsequently adding it to the resulting data.

The resulting equation system for $W$ is given by:

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

(1)

Here, $C(x_i,x_j)$ is the directional covariance of the given observations.

Example¶

Here we use simple kriging in 1D (for plotting reasons) with 5 given observations/conditions. The mean of the field has to be given beforehand.

import numpy as np
import gstools as gs

# 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 = gs.Gaussian(dim=1, var=0.5, len_scale=2)

krig = gs.krige.Simple(
    model,
    mean=1,
    cond_pos=cond_pos, 
    cond_val=cond_val,
)
field, var = krig(gridx)

ax = krig.plot()
ax.scatter(cond_pos, cond_val, color="k", zorder=10, label="Conditions")
ax.legend()

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