Built-in Variogram Models
Intro
\[
d \, - \, \text{distance}
\]
\[
r \, - \, \text{range}
\]
\[
c \, - \, \text{sill}
\]
\[
V(d) \, - \, \text{variogram function}
\]
\[
C(d) \, - \, \text{covariance function}
\]
In general, the covariance function is defined as follows:
\[
C(d) = c - V(d)
\]
For more information about variogram model parameters, please refer to this tutorial or see this section of the documentation.
Gaussian
\[
V(d) = c \cdot \left(1 - \exp\left(-\frac{d^2}{2 \cdot r^2}\right)\right)
\]
\[
C(d) = c \cdot \exp\left(-\frac{d^2}{2 \cdot r^2}\right)
\]
Exponential
\[
V(d) = c \cdot \left(1 - \exp\left(-\frac{|d|}{r}\right)\right)
\]
\[
C(d) = c \cdot \exp\left(-\frac{|d|}{r}\right)
\]
Spherical
\[
V(d) = \begin{cases}
c \cdot \left(1.5 \frac{|d|}{r} - 0.5 \left(\frac{|d|}{r}\right)^3\right) & \text{for } 0 \leq |d| \leq r \\
c & \text{for } |d| > r
\end{cases}
\]
\[
C(d) = \begin{cases}
c \cdot \left(1 - 1.5 \frac{|d|}{r} + 0.5 \left(\frac{|d|}{r}\right)^3\right) & \text{for } 0 \leq |d| \leq r \\
0 & \text{for } |d| > r
\end{cases}
\]
Linear
\[
V(d) = \begin{cases}
c \cdot \frac{|d|}{r} & \text{for } 0 \leq |d| \leq r \\
c & \text{for } |d| > r
\end{cases}
\]
\[
C(d) = \begin{cases}
c \cdot \left(1 - \frac{|d|}{r}\right) & \text{for } 0 \leq |d| \leq r \\
0 & \text{for } |d| > r
\end{cases}
\]