Within dr, L changes (dL) from

1
Spectral Radiance, L(l,q,f) - W m-2 sr-1 mm -1
Within dr, L changes (dL) from sources
due to scattering emission losses
due to scattering absorption
2
mdLl/ddl -Ll(Q,?) alBl(T) wl/4? ?0 2p?? -1
1? Ll(m,?)P(Ys)dmd?
ddl ???se(l)dz vertical optical depth
?l sa(l)/ se(l) absorption number
alBl(T) emitted energy
wl/4? ?0 2p?? -1 1? Ll(m,?)P(Ys)dmd?
scattering term
-Ll(Q,?) radiance
Now we want to simplify equation .
3
Beer-Bouguer-Lambert Law
s1
Assume that no sources of radiance are possible
along a path
s
dL(s)/ L(s) -se(s) ds
4
Schwartzchilds Equation
no scattering (ss0) but include a source
function from emission B(l,T)
dL(l,s) -se(l,s) L(l,s)ds se(l,s) B(l,T(s))ds
multiply by e-d dd, and integrate from s to s1
(prime means along the path)

x
sum of
x
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normal or vertical path optical depth d(l,z)
This differs from the path optical depth by cos
q d(l,s) d (l,z)/m where m
cos q From now on d d (l,z) is our vertical
coordinate
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As an example
0
mcos q
q
direct transmittance e-d(z)/m
Jscat
Jth
direct transmittance e-dt/m
d(z)
dd
L(dtm,f)
dt
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Special Solutions 1, 2 and 3
We will develop special solutions of the
radiative transfer equation for radiance at the
top of the atmosphere. These solutions will
apply to the major applications of
satellite-based remote sensing. Appropriate
assumptions will be employed to simplify the
solutions to illustrate conceptual
principles. These include Solution 1 - No
path radiance Solution 2 - Path radiance from
emission only Solution 3 - Path radiance from
single-scattering only (Microwave solutions
will be presented later)
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Special Case 1 No Path Radiance We are at a
wavelength where B(l,T) 0 and there is no
scattering solution
Also known as Beer's Law if q 0 d 0.01
0.1 1.0 7 e-d 99 90.5 36.8 0.1
d ?
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The SAGE II (Stratospheric Aerosol and Gas
Experiment II) sensor was launched into a 57
degree inclination orbit aboard the Earth
Radiation Budget Satellite (ERBS) in October
1984. During each sunrise and sunset encountered
by the orbiting spacecraft, the instrument uses
the solar occultation technique to measure
attenuated solar radiation through the Earth's
limb in seven channels centered at wavelengths
ranging from 0.385 to 1.02 micrometers. The exo
atmospheric solar irradiance is also measured in
each channel during each event for use as a
reference in determining limb transmittances.
Optical Depths Examples
Unlike the lower atmosphere (or troposphere,
which extends from the surface to roughly 10 km),
the stratosphere does not have rain clouds as a
mechanism to quickly wash out pollutants.
Therefore, a heavy influx of aerosol pollutants,
like the plume from Mount Pinatubo, will remain
in the stratosphere for years until the processes
of chemical reactions and atmospheric circulation
can filter them out. In the case of Mount
Pinatubo, the result was a measurable cooling of
the Earth's surface for a period of almost two
years. Because they scatter and absorb
incoming sunlight, aerosol particles exert a
cooling effect on the Earth's surface. The
Pinatubo eruption increased aerosol optical depth
in the stratosphere by a factor of 10 to 100
times normal levels measured prior to the
eruption. ("Aerosol optical depth" is a measure
of how much light airborne particles prevent from
passing through a column of atmosphere.)
Consequently, over the next 15 months, scientists
measured a drop in the average global temperature
of about 10F (0.60C)
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This 10km resolution image of Water Cloud Optical
Thickness MODIS. Uses bands 1 (648nm) and band
2(858nm)
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Special Case 2 Emitted Path Radiance Only Here
emission is the only source of photons and there
is no scattering, so... J (l,z) sa(l,z)
B(l,T(z)) and se(l,z) sa(l,z) Also,
Lo(l,q,j) es(l,q) B(l,Ts) The solution
becomes (known as Schwartzchilds Equation)
d(l)
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For which wavelengths does this solution apply?
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Special Case 3 Source due to single-scattered
path radiance only(note in general multiple
scattering is required) J Jscat only Single
scattering implies each photon is scattered only
once along the path from the source to the
satellite. Therefore, the only source of photons
L(r,l,X) at some place (X) in the atmosphere is
the radiance from the source L(r,l,X)
L(q0,f0,l,X) L(q0,f0,l) e-d(l,z)/mo The path
radiance is then
L(q0,f0,l)
q
q0
What conditions are required for single scatter
to dominate?
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At the top of the atmosphere the result is
If we apply this to a single homogeneous layer
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Other Approximations
Substituting J Jscat
Some approaches simplify sources as in Special
Solution 3 above. Some approaches focus on
simplifying the scattering phase function p(ys).
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0.39-0.76 ?m --gt visible window
8.5-12.5 ?m --gt IR window
2-4 6cm --gt ?wave window
Radar bands Ku 2.1cm X
3.6 cm C 6 cm