Wednesday 19 January 2010 Lecture 5 Radiative transfer

Wednesday, 19 January 2010 Lecture 5: Radiative transfer theory where light comes from and how it gets to where it’s going Reading Ch 1. 2 review, 1. 3, 1. 4 http: //hyperphysics. phy-astr. gsu. edu/hbase/atmos/blusky. html (scattering) http: //id. mind. net/~zona/mstm/physics/light/ray. Optics/refraction 1. html (refraction) http: //id. mind. net/~zona/mstm/physics/light/ray. Optics/refraction/snells. Law 1. html (Snell’s Law) Review On Solid Angles, (class website -- Ancillary folder: Steradian. ppt) Here is a website where you can experiment with additive color mixing http: //javaboutique. internet. com/Color. Finder/

What was covered in the previous lecture • • LECTURES Jan 05 1. Intro Jan 07 2. Images Jan 12 3. Photointerpretation Jan 14 4. Color theory previous Jan 19 5. Radiative transfer today Jan 21 6. Atmospheric scattering Jan 26 7. Lambert’s Law Jan 28 8. Volume interactions Feb 02 9. Spectroscopy Feb 04 10. Satellites & Review Feb 09 11. Midterm Feb 11 12. Image processing Feb 16 13. Spectral mixture analysis Feb 18 14. Classification Feb 23 15. Radar & Lidar Feb 25 16. Thermal infrared Mar 02 17. Mars spectroscopy (Matt Smith) Mar 04 18. Forest remote sensing (Van Kane) Mar 09 19. Thermal modeling (Iryna Danilina) Mar 11 20. Review Mar 16 21. Final Exam Friday’s lecture: • Color and the spectrum • Color perception • Additive & subtractive color mixing • Ternary diagrams and color transformations • Selective absorption of light Today • The atmosphere and energy budgeting • Modeling the atmosphere • Radiative transfer equation • “Radiosity” 2

The Electromagnetic Spectrum (review) Units: Micrometer (mm) = 10 -6 m Nanometer (nm) = 10 -9 m This chart shows the spectrum The Sun Light emitted by the sun This graph is a spectrum 3

W m-2 μm-1 sr-1 W m-2 μm -1 Light from Sun – Light Reflected and Emitted by Earth Wavelength, μm The sun is not an ideal blackbody – the 5800 K figure and graph are simplifications 4

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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 – 12 x 10 -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 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). Fo is the incoming flux This profile can be measured made using a balloon, or a standard atmosphere can be assumed. 9

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

Radiant energy – Q (J) - electromagnetic energy Terms and units used in radiative transfer calculations 0. 5º 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). Itoa L Ls↑ 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) Is↓ Ig Note: reflectance is sometimes considered to be the reflected radiance. In this class, its use is restricted to the % energy reflected. 11

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 t e i r Itoa Ig Is↓ Ls↑ o amplifier gain atmospheric transmissivity emergent angle incident angle reflectance solar irradiance at top of atmosphere solar irradiance at ground down-welling sky irradiance up-welling sky (path) radiance amplifier bias or offset 13

The factor of p Consider a perfectly reflective (r=100%) diffuse “Lambertian” surface that reflects equally in all directions. Lambert 14

The factor of p 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 2 p sr (for a unit radius). So – why don’t we divide by 2 p instead of p? 15

The factor of p Consider a perfectly reflective (r=100%) diffuse “Lambertian” surface that reflects equally in all directions. 2 p p/2 ∫ ∫ L sin cos d dw=p. L 0 0 • Incoming directional radiance L at elevation angle is isotropic • Reflected directional radiance L cos is isotropic • Area of a unit hemisphere: 2 p p/2 ∫ ∫ sin d dw=2 p 0 0 16

Measured Ltoa DN(Itoa) = a Itoa + b Itoa i Ltoa=te r (ti Itoa cos(i)) /p + te r Is↓ /p + Ls↑ Itoa cos(i) ti Ls↑ (Lp) te i e Ig=ti Itoa cos(i) r (ti Itoa cos(i)) /p reflected light r reflectance “Lambertian” surface 17

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What was covered in today’s lecture? • The atmosphere and energy budgeting • Modeling the atmosphere • Radiative transfer equation • “Radiosity” 20

What will be covered in next Tueday’s lecture? • Atmospheric scattering and other effects - where light comes from and how it gets there - we will trace radiation from its source to camera - the atmosphere and its effect on light - the basic radiative transfer equation: DN = a·Ig·r + b Mauna Loa, Hawaii 21