From the Radiative Transfer Equation to

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From the Radiative Transfer Equation
to Single-Scattering Albedo
Ron Resmini
September 15, 2004
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A Thread Through
Hapke, B., (1993). Theory of Reflectance and
Emittance Spectroscopy. Cambridge Univ.
Press, 455 p.
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Eq. and pg. numbers refer to Hapke (1993).
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Motivation...
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Some basic questions to ask when considering
PDEs

• Whats the problem space/geometry?
• What are the ICs?
• What are the BCs?
• How do you solve the _at_!?ing thing?

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• Bidirectional reflectance configuration
• Vacuum (i.e., no atmosphere)
• Semi-infinite particular medium
• Homogeneous physically and chemically
• Proxy for soils, regoliths, atmospheres
• Other assumptions as we go along e.g
• Isotropic scattering
• Collimated source input

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Diffuse Illumination
Clouds
Atmosphere
Veg. Canopy
Material of interest (perhaps intimate mixture)
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Immerse a cylindrical control volume (cv) in a
radiance field the radiant power entering the
bottom of the cv is
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The radiant power leaving the top of the cv is
The difference in radiant power is
As an aside...
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Stuff happens to the radiance as it passes
through the cv

• Extinction
• Absorption
• Scattering out
• Scattering
• Scattering in
• Emission
• Single Scattering
• Thermal
• Fluorescence and Luminescence
• Stimulated Emission

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Extinction
Note that if extinction is the only process
acting in the control volume
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Scattering
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Emission
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Equate contributions to net I leaving the control
volume
This gives
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Dividing through by
gives
This is the general form of the radiative
transfer equation. It is an integro-partial
differential equation.
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Making the substitution and dividing by E(z)
gives
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Define optical depth, t, as follows
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The equation is now in terms of things we measure.
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Solving the RT equation The multistream
method The two-stream method
See pg. 177 of Hapke (1993).
The intensity in the jth directional region
becomes
Eq. 7.53, pg. 178 of Hapke (1993).
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8.G. The bidirectional reflectance 8.G.3. The
bidirectional reflectance of a medium of
isotropic scatters 8.G.3.a. The two-stream
solution with collimated source (pg. 201 of
Hapke, 1993).
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Were now down to one ODE
The general solution is
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The solution in terms of reflectance, r
Add a term for multiple scattering from isotropic
scatters
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Add a term for the opposition effect
Rearrange to obtain single-scattering albedo
(...actually the albedo factor)