Appendix B - Linear mixing model

Summary

An imaging system will necessarily have limited field of view and spatial resolution. This limitation is imposed by such factors as pixel size, detector-array format, the number of data collected, etc. The corresponding instantaneous field of view (IFOV) is therefore likely to encompass several “patches” of materials that possess different reflectance and/or emissivity properties. If we are lucky, the combined signal from each IFOV is a linear mixture of weighted radiances from each “pure” material within the IFOV.

Given sufficient signal to noise ratio (SNR), sub-pixel traces of a particular material may be detected based on the presence of distinctive spectral features in the combined signature. A simpler technique relies on a single spectral channel within which the target and the background exhibit different radiance. For example, the 3–5 µm window that may be used to detect smoldering fires in a natural background and the detection of narrow roads of concrete surrounded by vegetation is accomplished in the 0.6–0.7 µm band. Compare the spectra of healthy vegetation and soil to concrete or asphalt in this spectral region to see why.

Take a look at Figures B.1(a) and B.1(b) for some examples of a geometric interpretation of linearly mixed pixels.

A data cube from a space-borne spectrometer provides spectral data on spatial locations of interest, as shown in Figure B.2. The signal measured in each IFOV is radiometrically made up of the constituents, i.e. [A(x) + B(x) + C(x) + D(x)], along with their fractional area.