Radiant and kinetic temperatures

The actual measurements by a TIR sensor will relate to the “spectral radiance” (measured in W m⁻² sr⁻¹ μm⁻¹) that reaches the sensor for a certain wavelength band. We know that the amount of radiation from an object depends on its temperature T and emissivity \epsilon. That means that a cold object with high emissivity can radiate just as much radiation as a considerably hotter one with low emissivity. Often the emissivity of the object is unknown. If we assume that the emissivity of the object is equal to 1.0, then with the help of Planck’s law we can calculate directly the ground temperature  that is needed to create this amount of radiance in the specified wavelength band of the sensor for the object with a perfect emissivity. The temperature calculated in this way is the radiant temperature or T_rad. The terms brightness or “top-of-the-atmosphere” temperature are also frequently used.

The radiant temperature calculated from the emitted radiation is in most cases lower than the true, kinetic temperature (T_kin) that we could measure on the ground with a contact thermometer. The reason for this is that most objects have an emissivity lower kinetic temperature than 1.0 and radiate incompletely. To calculate the true T_kin from the T_rad, we need to know or estimate the emissivity. The relationship between Tkin and Trad is:

With a single thermal band (e.g. Landsat-7 ETM+), \epsilon<!!!no support for symbol!!!> has to be estimated from other sources. One way of doing this is to do a land cover classification with all available bands and then assign an  value for each class from an emissivity table (e.g. 0.99 for water, 0.85 for granite).