30942_Estimation_of_the

In geostatistics the spatial correlation is analyzed by the variogram instead of a correlogram or covariogram (Bivand et al. 2008). If the random function $Z(s)$ , is intrinsic stationarity, which means it has a constant mean $(E(Z(s)) = \mu)$ , then the variogram is defined as

$2\gamma(\mathbf h) = \mathbf E[(Z(\mathbf s_i)-Z(\mathbf s_i+h))^2]\, ,$

where $Z(\mathbf s_i)$ is the value of a target variable at some sampled location and $Z(\mathbf s_i+h))$ is the value of the neighbor at distance $\mathbf s_i+h$ . In other words, the covariance between values of $Z(s)$ at any two locations depends on only their relative locations or, equivalently, on their spatial separation distance, $h$ , also denoted as spatial lag. The quantity $2\gamma(s)$ is known as the variogram and the function $\gamma(h)$ is called the semivariogram which is simply written as

$\gamma(\mathbf h) = \frac{1}{2}\mathbf E[(Z(\mathbf s_i)-Z(\mathbf s_i+h))^2]\, .$

Suppose that there are $n$ point observations, this yields $n \cdot (n-1)/2$ pairs for which a semivariance can be calculated (Hengl 2007).

When data locations are irregularly spaced, there is generally little to no replication of lags among the data locations. To obtain quasi-replication of lags, we partition the lag space into lag classes or bins. Then, $N_h$ is the number of lags that fall into the bin $h_j$ . The more bins that are used, the smaller they are and the better the lags are approximated by $h_j$ , but the fewer the number of observed lags belonging to $h_j$ . One popular rule of thumb is to require $N_h$ to be at least 30 and to require the length of $h_j$ to be less than half the maximum lag length among data locations (Gelfand at al. 2010).

If we assume isotropy, which is direction independence of semivariance, the variogram can be estimated from $N_h$ sample data pairs $z(s_i), z(s_i + h)$ for a number of distances (or distance intervals) $\hat h_j$ by

$\hat \gamma(\hat h_j) = \frac{1}{2N_h}\sum_{i=1}^{N_h}(Z(s_i)-Z(s_i+h))^2, \quad \forall h \in \hat h_j\, .$

This estimate is called the sample variogram or experimental variogram.

Citation

The E-Learning project SOGA-Py was developed at the Department of Earth Sciences by Annette Rudolph, Joachim Krois and Kai Hartmann. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Rudolph, A., Krois, J., Hartmann, K. (2023): Statistics and Geodata Analysis using Python (SOGA-Py). Department of Earth Sciences, Freie Universitaet Berlin.