Variogram

A variogram is a fundamental concept in the field of spatial statistics and is used to analyze the spatial continuity of stochastic processes.

A variogram refers to two related aspects depending on the context in which it is used: the function itself and its plot. The variogram function is the mathematical definition that quantifies the spatial dependence of a random field or stochastic process $Z(x)$, where $x$ represents location in space. Specifically, the variogram characterizes the variance of the difference between random variables at two locations across different distances.

Formally, the variogram $\gamma(h)$ for a stationary (in the strict sense) random field $Z(x)$ is defined as:

$$\gamma^{*}(h) = E[(Z(x + h) - Z(x))^2] = \frac{1}{N(h)} \sum\limits_{i=1}^{N(h)} (Z(x_i + h) - Z(x_i))^2$$

Here:

• $E[x]$ denotes the expected value operator, implying an average over all locations $x$ in the field,
• $Z(x + h)$ and $Z(x)$ are the values of the stochastic process at the positions $x + h$ and $x$, respectively,
• $h$ is the vector representing separation distance and direction between two locations,
• $\gamma^{*}(h)$ represents the variogram value, i.e., the variance of the difference between random field values separated by distance $h$.

If the process $Z(x)$ is second-order stationary (i.e., its mean and variance are constant over space and its autocovariance only depends on the distance $h$), the variogram can also be expressed in terms of the covariance function $C(h)$:

$$\gamma^{*}(h) = C(0) - C(h)$$

The variogram plot is also referred to as the experimental variogram and displays the variogram values as a function of distance $h$. This plot allows for visual examination and interpretation of the spatial dependence structure of the stochastic process.

Thus, depending on the context, the term "variogram" can refer to both the variogram as a mathematical function and its graphical representation as function values plotted against distance.