The variogram of a spectral band ratio with a considerable correlation between the bands

The subject of this paper is to show how the variogram of a spectral band ratio u = arctan(x/y) is controlled by the statistical parameters and spatial variation of the individual bands x and y, as well as the correlation coefficient ρ between the bands. Based on certain assumptions about the histograms and variograms of x and y, an approximate mathematical expression for the variogram of the band ratio is derived, which shows that the sill of the variogram of u depends on the ratio of the variance of x to the variance of y. As long as the correlation coefficient ρ increases, the sill decreases. The spatial variation of u for various distances h between pixels depends on the ranges of the variograms of bands x and y, as well as on the correlation coefficient between the bands. Experimentation with satellite images shows a good agreement between theoretically calculated and actual variograms of u, in qualitative terms and, in a considerable extent, in quantitative terms. The proposed methodology may be useful in assessing the efficiency of spectral bands and vegetation indices in environmental and geological remote sensing.