Linear–Quadratic Mixing Model for Reflectances in Urban Environments

In the field of remote sensing, the unmixing of hyperspectral images is usually based on the use of a mixing model. Most existing spectral unmixing methods, used in the reflective range (0.4–2.5
$\mu\hbox{m}$
), rely on a linear model of endmember reflectances. Nevertheless, such a model supposes the pixels at the ground level to be uniformly irradiated and the scene to be flat. When considering a 3-D landscape, such a model is no longer valid as irradiated and shadowed areas are present, as well as radiative interactions between facing surfaces. This paper introduces a new mixing model adapted to urban environments and which aims to overcome these limitations. This model is derived from physical equations based on radiative transfer theory, and its analytic expression is linear–quadratic. Similar models have already been used in the literature for unmixing purposes but without being justified by physical analysis. Our proposed model is validated using a synthetic but realistic European 3-D urban scene. Then, simplifications are introduced, based on a study of the different radiative components contributing to the signal in a way to make the model easy to use for spectral unmixing. This paper also shows that the quadratic term cannot be neglected in many cases in urban environments since it can, e.g., range from 15% to 20% of the reflectances in canyons.