The problem of the spatial variation of a spectral band ratio: a first order approach based on probability theory

Abstract

Numerous efforts have been made to study how the spatial distribution of ground surface objects controls the image semivariogram. The present paper is centered on how the histograms and semivariograms of the individual bands x and y influence the spatial variation of a simple spectral ratio u = arctan(x/y). The image histogram of each separate band is described by a proper distribution. The exponential model is used to describe the semivariograms of x and y. Taking the first derivatives of the spectral ratio u for x and y and taking into account the mathematical behavior of the histograms of bands x and y, an approximate expression for the semivariogram γ u of the spectral ratio is derived. This mathematical expression shows how the spatial variation of the spectral ratio depends on the standard deviations of the histograms, as well as the ranges of the semivariograms of x and y. Experimentation with multispectral images is then carried out and it shows that theoretical predictions agree, in qualitative terms, with real data. The results and conclusions of this paper may be useful in assessing the efficiency of various spectral band ratios and vegetation indices, which are often used in geological and environmental research (mapping of hydrothermal zones and land cover types).