On the effect of variable endmember spectra in the linear mixture model

The linear mixture model is frequently used to characterize surface cover over land, to model the reflectance of heterogeneous surfaces, and, by inversion, to estimate fractional cover from a multispectral satellite signal. It is usually assumed that certain parameters of this model, namely the so-called endmember spectra, are fixed, and that the model residual - the difference between a signal and its expected value in terms of the linear model - is systematically independent of all other parameters. In a small number of studies the endmember spectra have been allowed to have random fluctuations, giving rise to a covariance matrix for the residual that depends on the underlying proportions, and two distinct models exist for this mixed-pixel covariance matrix. In this note the linear model for mixed pixels is examined with varying endmember spectra, and it is shown that under a simple set of models for the variability of both endmembers and abundance, the covariance matrix for the residual is a weighted sum of the two previously considered cases. Generally, the balance between the two limiting cases is determined by the length scale for changes in the reflectance of any given cover type, and the length scale for changes in surface cover itself; one or other of the two limit models is preferred when these lengths are very different.