Land Surface Temperature (LST) Retrieval from GOES Satellite Observations - Satellite-based Applications on Climate Change - page 297

with LTðÞ¼B i Tð = @B

@T j T i

(19.20b)

The Planck function can be well approximated using a simple power function
(Price 1989):

B i TðÞα i T n i

(19.21)

Parameters α i and n i are constants obtained by a least-square regression fitting. In

order to have the best approximation of the Planck function, we divide the temper-
ature range into two parts, (a) less than 285 K, and (b) more than 285 K. The
parameter n i is given in Table 19.3 for each case.
The power law approximation is very useful for analyses involving the Planck
function, with this approximation:

α i T i n i
α i n i T i n i 1 ¼

LTðÞ¼B i Tð = @B

T i
n i @T j T i

(19.22)

Inserting Eqs. 19.21 and 19.22 into Eqs. 19.20a and 19.20b , the atmospheric
correction for brightness temperature can be written as

1 τ i

τ i

T i T i ¼ T i T a " (19.23)

We linearize Planck function in ( 19.17 ) around T i * and obtain the emissivity
correction:

T i
n i þ

ð 1 ε i Þ ð n i 1 Þ

T s T i ¼ Þ T i ÞT a # ð 1 τ i ð 1 τ i (19.24)

ε i n i

Inserting ( 19.23 ) into ( 19.24 ), we get

T s ¼ C 1 i T i C 2 i T a " C 3 i T a #

where

1
τ i

ð 1 Þ

n i ε i þ

ε i ð 1 ε i Þ ð 1 τ i Þ n i ð 1 Þ

C 1 i ¼ þ

1

n i ε i

ð 1 τ i Þ ð 1 Þ

n i ε i þ

ε i ð 1 ε i Þ ð 1 τ i Þ n i ð 1 Þ

C 2 i ¼ 1 þ (19.25)

τ i n i ε i

ð 1 ε i Þ

C 3 i ¼ ð 1 τ i Þ

ε i