On the residual term in the linear mixture model and its dependence on the point spread function

The linear mixture model has been extensively used in the analysis of satellite data, especially for characterization of surface cover at subpixel scales. In the model, a multispectral signal is assumed to comprise a weighted sum of characteristic spectra, the weights corresponding to fractional coverage. The residual is almost invariably assumed to be independent of the weights, and usually taken to arise from Gaussian noise; the maximum-likelihood estimate of the abundances is then found by minimizing a quadratic objective function. Nonuniform sampling of the radiance distribution within the field of view means, however, that there is some dependence of the residual on the true surface abundances, when it is understood that these are simple area averages. We account for this signal-dependent noise by incorporating the modified variance-covariance matrix of the residual into the quadratic objective function. It is shown that, despite the increased complexity of the new objective function, it is minimized by the traditional estimator. This is true whether or not we require the estimated abundances to sum to unity, i.e., whether the minimization is constrained or unconstrained. The constrained and unconstrained estimators are here treated together by deploying a "sum-to-one" parameter.