Land Surface Temperature (LST) Retrieval from GOES Satellite Observations - Satellite-based Applications on Climate Change - page 294

atmospheric layer modulated by the transmittance of the air above that emitting
layer:

Z

1

d

Lðλ; μÞ¼ε 0 ðλ; μÞB λ; T s ð Þτ 0 ðλ; μÞþ B λ; T p τðλ; μ; pÞ (19.8)

τ 0

where T p is the air temperature at vertical layer p and p is the pressure of the vertical
emitting layer.
For a specific land surface type with surface emissivity close to unity, the
radiance error introduced by the atmosphere

ΔL can be represented as

Z

1

d

ΔL ¼ B λ; T s ð ÞLðλ; μÞ¼B λ; T s ð ÞB λ; T s ð Þ τ 0 ðλ; μÞ B λ; T p τðλ; μ; pÞ

τ 0

Z Z

1 1

d

¼ B λ; T s ð Þ d τðλ; μ; pÞ B λ; T p τðλ; μ; pÞ

τ 0 τ 0

Z

1

¼ ðB λ; T s ð ÞB λ; T p Þ d τðλ; μ; pÞ

τ 0

(19.9)

From the Planck function, we find

T s T s T λ

Þ @ B
@T

ΔL ¼ B λ; T s ð ÞLðλ; μÞ¼B λ; T s ð ÞB λ; T λ ð ð Þ (19.10)

where T λ is brightness temperature at wavelength .

For an optically thin gas, the following approximations can be made:

λ

d τ ¼ d exp f ð k λ l Þ g d1 ð k λ l Þ¼ k λ d l (19.11a)

where k λ is the absorption coefficient and l is the optical path length:

d l ¼ ρ d z ρ 0 exp ðz=HÞ d z (19.11b)

ρ ρ 0 is the density at 0 km, H is the atmospheric

scale height, and z is the height. If we assume that the Planck function is adequately
represented by a first-order Taylor series expansion in each window channel, then

is the density of the absorption gas,

T s

@ B
@T

B λ; T s ð ÞB λ; T p T s T p (19.12)