Land Surface Temperature (LST) Retrieval from GOES Satellite Observations - Satellite-based Applications on Climate Change - page 296

19.3.2.3 Triple-Window LST Algorithm

Starting from the radiative transfer equation, the radiance measured by channel i of
a satellite sensor can be written as

τ i þ R i "

B i TðÞ¼ε i B i TðÞþρ i R i #

(19.16)

where B i is the Planck function weighted for channel i ; T i is the brightness tempera-
ture measured at satellite level in the channel i ; τ i is the atmospheric transmittance for

channel i ; R i # is the hemispheric downward atmospheric radiance for the waveband

ρ i R i # is referred
to term R r in Eq. 19.2 ;and R i " is the upward radiance emitted by the atmosphere in the
waveband of channel i ; it corresponds to the thermal path radiance term R a in
Eq. 19.2 . Equation 19.16 is a simplification of Eq. 19.2 , considering channel values
instead of spectral values and accounting for part of the atmospheric downward
radiation reflected by the surface. For simplicity, we assume Lambertian reflection
ρ i ¼

of channel i ; ρ i is the channel bidirectional reflectivity of the surface;

(1 ε i ) and define brightness temperature at surface level T i *:

B i T i ðÞ¼ε i B i TðÞþ Þ R i # ð 1 ε i (19.17)

McMillin ( 1975 ) used the mean value theorem to define the mean radiative
temperature of the atmosphere in the upward direction T a " :

B i T a " ¼

R i "

(19.18a)

τ i

1

We can introduce a similar mean radiative temperature of the atmosphere in the
downward direction according to McMillin ( 1975 ) approach:

B i T a # ¼

R i #

(19.18b)

1 τ i

By inserting Eqs. 19.17 , 19.18a , and 19.18b into Eq. 19.16 ,

Þ B i T " a B i TðÞ¼τ i B i T i ðÞþ

ð 1 τ i (19.19)

Linearizing the Planck function in ( 19.19 ) around T i yields

@ B

@T j T i LTðÞ¼τ i @ B @T j T i T i T i þLTðÞ

ð Þ

Þ @ B

@T j T i T a " T i þLTðÞ

þ ð 1 τ i (19.20a)