LST from satellites

As mentioned before, if there was no atmosphere interposing between the ground (assumed as a Lambertian surface) and the sensor, and assuming the emissivity was known, the surface temperature could be obtained from the measured spectral radiance by inverting Planck's function.

However, we cannot forget the layers composed of clouds, trace gases and aerosols that interfere with the propagation of radiation from the surface towards the top of the Atmosphere (TOA) by attenuating and re-emitting thermal infrared radiation. Figure 4 illustrates the several contributions to the OLR (Outgoing Longwave Radiation). OLR is the total energy emitted from the Earth and its atmosphere out to space in the form of thermal radiation. Most of it is emitted by the surface (Planck's law), followed by upward radiance emitted by atmospheric constituents (mainly water vapor) in the several layers of the atmosphere, Latm . This term varies strongly with the vertical structure of the atmosphere - warm/moist layers increase its contribution.

The atmosphere also emits longwave radiation downwards, which is then reflected back from the surface to the TOA (1 - ε) Lλ atm (this term is very small, nearly zero for a spectral emissivity close to one).

Fig. 4: Contributions to the total outgoing longwave radiance reaching the top of the atmosphere (TOA): surface emission (Planck's law ελBλ(LST)) + upward radiance emitted by atmospheric constituents (mainly water vapor) in the different layers of the atmosphere, L↑atm + downward longwave radiation which is then reflected back from the surface to the TOA.

The actual longwave radiation reaching the TOA corresponds to the sum of all these terms multiplied by atmospheric transmittance (τλ) (aerosol absorption and scattering are considered to be negligible and are generally ignored (Prata et al 1995)):

Lλ = ελBλ(LST)τλ + Lλatmτλ + (1-ελ)Lλatmτλ  (eq.11)

which is known as the Radiative Transfer Equation (RTE).

To summarize: Lλ is the total radiance we can measure by a satellite sensor at TOA and what we want to retrieve is LST. So in principle, if we had the information about the spectral surface emissivity and atmospheric composition, we could solve the RTE to get LST. This is assuming we have solved the issue of cloud masking, since LST from thermal infrared measurements can only be retrieved for clear sky pixels.