1. Tuesday, 20 January
Lecture 5: Radiative transfer theory where light comes from and how it gets to where it’s going
(scattering)
(refraction)
(Snell’s Law)
Review On Solid Angles, class website (Ancillary folder: Steradian.ppt)
Last lecture: color theory, data spaces,
color mixtures, absorption, photogrammetry

2. The Electromagnetic Spectrum (review)
Units:
Micrometer = 10-6 m
Nanometer = 10-9 m
Light emitted by the sun

3. Light from Sun – Light Reflected and Emitted by Earth
W m-2 μm -1
W m-2 μm-1 sr-1
Wavelength, μm

5. Atmospheric Constituents
Constant
Nitrogen (78.1%)
Oxygen (21%)
Argon (0.94%)
Carbon Dioxide (0.033%)
Neon
Helium
Krypton
Xenon
Hydrogen
Methane
Nitrous Oxide
Variable
Water Vapor ( %)
Ozone (0 – 12x10-4%)
Sulfur Dioxide
Nitrogen Dioxide
Ammonia
Nitric Oxide
All contribute to scattering
For absorption, O2, O3, and N2 are important in the UV
CO2 and H2O are important in the IR (NIR, MIR, TIR)

7. Atmospheric transmission

8. Modeling the atmosphere
To calculate t we need to know how k in the Beer-Lambert-Bouguer Law (called b here) varies with altitude. Modtran models the atmosphere as thin homogeneous layers.
Modtran calculates k or b for each layer using the vertical profile of temperature, pressure, and composition (like water vapor).
This profile can be measured made using a balloon, or a standard atmosphere can be assumed.
Fo is the incoming flux

9. Radiosonde data Altitude (km) Relative Humidity (%) Temperature (oC)20 15 10 5
Mt Everest
Mt Rainier

10. Terms and units used in radiative transfer calculations
Radiant energy – Q (J) - electromagnetic energy
Solar Irradiance – Itoa(W m-2) - Incoming radiation (quasi directional) from the sun at the top of the atmosphere.
Irradiance – Ig (W m-2) - Incoming hemispheric radiation at ground. Comes from: 1) direct sunlight and 2) diffuse skylight (scattered by atmosphere).
Downwelling sky irradiance – Is↓(W m-2) – hemispheric radiation at ground
Path Radiance - Ls↑ (W m-2 sr-1 ) (Lp in text) - directional radiation scattered into the camera from the atmosphere without touching the ground
Transmissivity – t - the % of incident energy that passes through the atmosphere
Radiance – L (W m-2 sr-1) – directional energy density from an object.
Reflectance – r -The % of irradiance reflected by a body in all directions (hemispheric: r·I) or in a given direction (directional: r·I·p-1)
Note: reflectance is sometimes considered to be the reflected radiance. In this class, its use is restricted to the % energy reflected.
Terms and units used in radiative transfer calculations
0.5º
Itoa L
Ls↑
Is↓ Ig

11. DN = a·Ig·r + b Radiative transfer equation Parameters that relate to
instrument and atmospheric
characteristics
DN = a·Ig·r + b
This is what we want
Ig is the irradiance on the ground
r is the surface reflectance
a & b are parameters that relate to
instrument and atmospheric
characteristics

12. Radiative transfer equation
DN = a·Ig·r + b
DN = g·(te·r · ti·Itoa·cos(i)/p + te· r·Is↓/p + Ls↑) + o
g amplifier gain
t atmospheric transmissivity
e emergent angle
i incident angle
r reflectance
Itoa solar irradiance at top of atmosphere
Ig solar irradiance at ground
Is↓ down-welling sky irradiance
Ls↑ up-welling sky (path) radiance
o amplifier bias or offset

13. The factor of p ∫ ∫ L sin  cos  d dw=pL ∫ ∫ sin  d dw=2p
Incoming directional radiance L at elevation angle  is isotropic
Reflected directional radiance L cos  is isotropic
Area of a unit hemisphere:
∫ ∫ sin  d dw=2p
Consider a perfectly reflective (r=100%) diffuse “Lambertian” surface that reflects equally in all directions.
If irradiance on the surface is Ig, then the irradiance from the surface is
r·Ig = Ig W m-2.
The radiance intercepted by a camera would be r·Ig/p W m-2 sr-1.
The factor p is the ratio between the hemispheric radiance (irradiance) and the directional radiance. The area of the sky hemisphere is 2p sr (for a unit radius).
So – why don’t we divide by 2p instead of p?
Lambert

14. Itoa Itoa cos(i) i Ls↑ (Lp) ti te Ls↑=r Is↓ /p i e Ig=ti Itoa cos(i)
Measured Ltoa
DN(Itoa) = a Itoa + b
Itoa
Ltoa=te r (ti Itoa cos(i)) /p + te r Is↓ /p + Ls↑
Itoa cos(i) i
Is↓
Ls↑=r Is↓ /p
Highlighted terms
relate to the surface
Ls↑ (Lp) ti te
Lambert i e
Ig=ti Itoa cos(i)
r (ti Itoa cos(i)) /p
reflected light
r reflectance
“Lambertian” surface

16. Next lecture: Atmospheric scattering and other effects
Rayleigh Scattering
l>>d
Mauna Loa, Hawaii