Linear mixing in thermal infrared temperature retrieval

Virtually all remotely sensed thermal infrared (IR) pixels are, to some degree, mixtures of different materials or temperatures: real pixels are rarely thermally homogeneous. As sensors improve and spectral thermal IR remote sensing becomes more quantitative, the concept of homogeneous pixels becomes inadequate. Quantitative thermal IR remote sensors measure radiance. Planck's Law defines a relationship between temperature and radiance that is more complex than linear proportionality and is strongly wavelength-dependent. As a result, the area-averaged temperature of a pixel is not the same as the temperature derived from the radiance averaged over the pixel footprint, even for blackbodies. This paper uses simple linear mixing of pixel elements (subpixels) to examine the impacts of pixel mixtures on temperature retrieval and ground leaving radiance. The results show that for a single material with one temperature distribution and with a subpixel temperature standard deviation of 6K (daytime images), the effects of subpixel temperature variability are small but can exceed 0.5K in the 3-5μm band and about a third of that in the 8-12μm band. For pixels with a 50 : 50 mixture of materials (two temperature distributions with different means) the impact of subpixel radiance variability on temperature retrieval can exceed 6K in the 3-5μm band and 2K in the 8-12μm band. Sub-pixel temperatures determined from Gaussian distributions and also from high-resolution thermal images are used as inputs to our linear mixing model. Model results are compared directly to these broadband thermal images of plowed soil and senesced barley. Finally, a theoretical framework for quantifying the effect of non-homogeneous temperature distributions for the case of a binary combination of mixed pixels is derived, with results shown to be valid for the range of standard deviations and temperature differences examined herein.