1. Lecture 5: Radiative transfer theory where light comes from and how it gets to where it’s going Tuesday, 19 January 2010 Ch 1.2 review, 1.3, 1.4 http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html (scattering) http://id.mind.net/~zona/mstm/physics/light/rayOptics/refraction/refraction1.htmlhttp://id.mind.net/~zona/mstm/physics/light/rayOptics/refraction/refraction1.html (refraction) http://id.mind.net/~zona/mstm/physics/light/rayOptics/refraction/snellsLaw/snellsLaw1.htmlhttp://id.mind.net/~zona/mstm/physics/light/rayOptics/refraction/snellsLaw/snellsLaw1.html (Snell’s Law) Review On Solid Angles, (class website -- Ancillary folder: Steradian.ppt) Last lecture: color theory data spaces color mixtures absorption Reading

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

3. Light from Sun – Light Reflected and Emitted by Earth Wavelength, μm W m -2 μm -1 W m -2 μm -1 sr -1 The sun is not an ideal blackbody – the 5800 K figure and graph are simplifications

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 (0 - 0.04%) Ozone (0 – 12x10 -4 %) Sulfur Dioxide Nitrogen Dioxide Ammonia Nitric Oxide All contribute to scattering For absorption, O 2, O 3, and N 2 are important in the UV CO 2 and H 2 O are important in the IR (NIR, MIR, TIR)

6. Solar spectra before and after passage through the atmosphere

7. Atmospheric transmission

8. Modeling the atmosphere To calculate  we need to know how k in the Beer-Lambert- Bouguer Law (called  here) varies with altitude. Modtran models the atmosphere as thin homogeneous layers. Modtran calculates k or  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.  o is the incoming flux

9. Radiosonde data Altitude (km) Relative Humidity (%) Temperature ( o C) 20 15 10 5 0 20 15 10 5 0 0 20 40 60 80 100 -80 -40 0 40 Mt Everest Mt Rainier

10. Radiant energy – Q (J) - electromagnetic energy Solar Irradiance – I toa (W m -2 ) - Incoming radiation (quasi directional) from the sun at the top of the atmosphere. Irradiance – I g (W m-2) - Incoming hemispheric radiation at ground. Comes from: 1) direct sunlight and 2) diffuse skylight (scattered by atmosphere). Downwelling sky irradiance – I s↓ (W m -2 ) – hemispheric radiation at ground Path Radiance - L s↑ (W m -2 sr -1 ) (L p in text) - directional radiation scattered into the camera from the atmosphere without touching the ground Transmissivity –  - 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·  -1 ) Note: reflectance is sometimes considered to be the reflected radiance. In this class, its use is restricted to the % energy reflected. IgIg L s↑ I toa 0.5º I s↓ L Terms and units used in radiative transfer calculations

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

12. DN = g·(  e ·r ·  i ·I toa ·cos(i)/  +  e · r·I s↓ /  + L s↑ ) + o gamplifier gain  atmospheric transmissivity eemergent angle iincident angle rreflectance I toa solar irradiance at top of atmosphere I g solar irradiance at ground I s↓ down-welling sky irradiance L s↑ up-welling sky (path) radiance oamplifier bias or offset Radiative transfer equation DN = a·I g ·r + b

13. The factor of  Consider a perfectly reflective (r=100%) diffuse “Lambertian” surface that reflects equally in all directions. Lambert

14. The factor of  Consider a perfectly reflective (r=100%) diffuse “Lambertian” surface that reflects equally in all directions. If irradiance on the surface is I g, then the irradiance from the surface is r·I g = I g W m -2. The radiance intercepted by a camera would be r·I g /  W m -2 sr -1. The factor  is the ratio between the hemispheric radiance (irradiance) and the directional radiance. The area of the sky hemisphere is 2  sr (for a unit radius). So – why don’t we divide by 2  instead of  ?

15. ∫ ∫ L sin  cos  d  d  L 22  00 Incoming directional radiance L  at elevation angle  is isotropic Reflected directional radiance L  cos  is isotropic Area of a unit hemisphere: ∫ ∫ sin  d  d  22  00 The factor of  Consider a perfectly reflective (r=100%) diffuse “Lambertian” surface that reflects equally in all directions.

16. i I toa cos(i) I toa  g  i I toa cos(i) ii r reflectance r (  i I toa cos(i) ) /  reflected light “Lambertian” surface ee e L s↑ (L p ) Highlighted terms relate to the surface i

17. i I toa cos(i) I toa  g  i I toa cos(i) ii r reflectance r (  i I toa cos(i) ) /  reflected light “Lambertian” surface ee e Measured L toa DN(I toa ) = a I toa + b L toa  e r (  i I toa cos(i) ) /  +  e r I s↓ /  + L s↑ L s↑ (L p ) Highlighted terms relate to the surface I s↓ L s↑ =r I s↓ /  i Lambert

19. Next lecture: Atmospheric scattering and other effects Mauna Loa, Hawaii