SecondOrderStationarity

In the field of stochastic processes, a process is said to be second-order stationary (or weakly stationary, or wide-sense stationary) if it satisfies certain statistical properties that are invariant over time. For a second-order stationary process $Z(x)$, the following two conditions must hold:

1. The mean function, $μ(x) = E[Z(x)]$, is constant. This means that the expected value or average value of the process does not depend on the location $x$. Mathematically, this is expressed as:

$$μ(x) = E[Z(x)] = μ$$

for all $x$, where $E[]$ denotes the expectation operator, and $μ$ is the constant mean of the process.

2. The autocovariance function, $C(h) = Cov[Z(x + h), Z(x)]$, depends only on the lag $h$ and not the actual spatial location $x$. This means that the covariance between the process at different locations depends only on the distance and direction between the locations, not the actual locations themselves. Mathematically, this is expressed as:

$$C(h) = Cov[Z(x + h), Z(x)]$$

for all $x$ and for all $h$, where $Cov[]$ denotes the covariance operator, and $C(h)$ is the covariance function.

In the context of variograms, if the process $Z(x)$ is second-order stationary, the variogram can be expressed in terms of the covariance function:

$$γ(h) = C(0) - C(h)$$

Where: $C(0)$ is the covariance at distance zero (which is just the variance of the process), $C(h)$ is the covariance function evaluated at distance $h$, and $γ(h)$ is the variogram function.