Spatial autocorrelation analysis in geomorphology:: Definitions and tests.

Spatial autocorrelation analysis in geomorphology: Definitions and tests. Spatial autocorrelation can be defined as the similarity of values of a given variable, in relation with their spatial location. Autocorrelation functions are used to quantify the spatial regularity of a phenomenon (a form of spatial complexity) and to assess the lag of spatial dependence in order to design a sampling procedure for which the data are independent, which permits the use of traditional statistical tests. Three points have been developed: i) Definition of autocorrelation functions used for categorical and quantitative variables: Geary's c, Moran's 1, semivariogram, non ergodic covariance and correlation, J statistics; ii) Definition of the randomization tests used to test the null hypothesis of no autocorrelation: iii) examples illustrating the two previous objectives. Three sets of simulated data were used to compare different autocorrelation functions (Geary's c, non ergodic covariance and correlation): the first one has no spatial structure, the second one has a periodic spatial structure whereas the third one is characterized by a linear gradient. Spatial autocorrelation has also been assessed on measured geomorphological data. Two sets were studied: i) a set of elementary channel segments of 500 m in length on which mean active channel width and degradation have been measured, ii) a set of pixels of an image of two basins representing the different types of hillslope erosion forms. In the first case, the nested structure of homogeneous geomorphological reaches is highlighted at different spatial scales. The second example, which illustrates autocorrelation assessment on a categorical variable, shows that omnidirectional analysis can underestimate the autocorrelation lag when the studied phenomenon is characterized by a preferential geographical orientation. In this particular case, it may not be not possible to define a sample of data which are spatially independent and on which it is possible to use classical statistical tests.