Brightness temperature in remote sensing

[jones123]

Hi all,

I don't know if I'm on the right forum to ask this, but maybe somebody knows anything about brightness temperatures measured by remote sensing devices.

In a paper that I read "Atmospheric corrections for retrieving ground brightness temperature at commonly-used passive microwave frequencies" by Han et al. (2017), I found the following formula:

Tb = Tg.τ + Tba

where: Tb is the brightness temperature at the TOA, Tg the brightness temperature at the ground, τ the atmospheric transmittance and Tba the brightness temperature of the atmospheric layers emitting into the direction of the TOA.

This would mean that:
Tg = (Tb - Tba) / τ

Now: can anyone explain me why the brightness temperature at the TOA Tb is positively proportional to the atmospheric transmittance τ, whereas the ground brightness temperature is inversly proportional to it?

Thanks already!

 

[olivermsun]

One way to think about it is this:

Tb is inversely proportional to $\tau$ because atmospheric transmittance decreases Tb for a given Tg.

Tb is positively proportional to atmospheric transmittance because the observed brightness temperature increases if there is more transmittance, which allows more radiation from the surface to make it to the remote sensing device.