Transcript Radiation Transfer Equation

Illumination Dependent Aerosol
Optical Properties



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Volume scattering function
Path function
Path radiance
Atmospheric modulation transfer function
Radiation Transfer Equation

dN r  ,  , r 


 bext N r  ,  , r   N   ,  , r 
dr
loss
gain
Equilibrium Radiance

dN r  ,  , r 


 bext N r  ,  , r   N   ,  , r 
dr
When Nr has some special value, Nq, such
that bextNq= N  , then dNq/dr = 0. Nq is
independent of r and is commonly referred to
as the equilibrium radiance.
dN r
 bext N r  N q 
dr
Radiance Difference
If Nq is constant, the previous equation can be
integrated to yield
Nr  Nq
No  N q
r
where
Tr  e
 Tr
  bext r dr 
o
Path Radiance
Rearranging yields
N r  N oTr  N q 1  Tr 
The second term is the path radiance

(airlight), N r , which results from scattering
processes throughout the sight path. The

parameter N  is the sky radiance
N  N q 1  T   Ns


Path Radiance Approximation
If T is approximately zero, then Nq=
(sky radiance) and:
N
N  N s 1  Tr 

r

=Ns

In general:

r
r
N   N Tr dr
o
N   hs   Nd
4
bs   d
4
Radiance Difference and Real World
b
.
l
N r  Tr b N o  N

r
N r  Tr l N o  N

r
 l N r  b N r   Tr  l N o  b N o 
Apparent and Inherent Contrast
Define:
Apparent Contrast:
Cr   l N r  b N r / b N r
Inherent Contrast:
Co   l N o  b N o / b N o
Radiance Difference Transmittance
 l N r  b N r   Tr  l N o  b N o 
Cr   l N r  b N r / b N r
Contrast Transmittance
No
C r  Co
Tr
b Nr
b
Cr
r 

Co
1

Nr
1
b N oTr
If background is sky then:
b

r
N r  Tr b N o  N  Ns
l
and
N r  Tr l N o  N
No / bNr  1
b

r
Estimating Extinction
N o bext r
C r  Co
e
b Nr
b
No / bNr  1
If background is sky, Ns: b
and,
C r  Co e
bext
 bext r
which yields
1 Cr
  ln
r Co
The Visual Range Concept
The distance at which Cr approaches a
threshold contrast of between -0.02 or -0.05
defines the visual range, Vr. If Co= 1
(black object) and -0.02 is taken to be a
threshold contrast, then the previous
equation becomes
3.912
Vr 
bext
Visibility Metrics
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Apparent contrast: Cr = Coe-bextr
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Visual Range: vr = 3.912/ bext
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decView: dv = 10 ln(bext/.01)
Contrast Sensitivity Surface (back)
Contrast Sensitivity Surface (forward)
Forward Scattering
Back Scattering
Optics of Visibility
La Sals –Forward Scattering
La Sals – Backward Scattering
La Sals -- 30km
Mt Trumbull -- 130km
Mt Trumbull -- 130km
North Rim of Grand Canyon Through Smoke
Plume
Easterly View of Grand Canyon from Desert
View Watch Tower
Westerly View from Desert View Watch Tower
Clouds Evaporate Leaving Sulfate Haze
Clouds Evaporated Leaving Sulfate
Haze
Sulfate Haze Trapped Below Rim
Next Day After Haze is Blown Out
Path Radiance Absent Because of Clouds
Nitrogen Dioxide Haze
Nitrogen Dioxide Plume
Illumination
Effects