Lecture 5: Radiative transfer theory where light comes from and how it gets to where it’s going Wednesday, 19 January 2010 Ch 1.2 review, 1.3, (scattering) (refraction) (Snell’s Law) Review On Solid Angles, (class website -- Ancillary folder: Steradian.ppt) Reading Here is a website where you can experiment with additive color mixing

What was covered in the previous lecture 2 LECTURES Jan 051. Intro Jan 072. Images Jan 123. Photointerpretation Jan 144. Color theoryprevious Jan 195. Radiative transfertoday Jan 216. Atmospheric scattering Jan 267. Lambert’s Law Jan Volume interactions Feb 029. Spectroscopy Feb Satellites & Review Feb Midterm Feb Image processing Feb Spectral mixture analysis Feb Classification Feb Radar & Lidar Feb Thermal infrared Mar Mars spectroscopy (Matt Smith) Mar Forest remote sensing (Van Kane) Mar Thermal modeling (Iryna Danilina) Mar Review Mar Final Exam Today The atmosphere and energy budgeting Modeling the atmosphere Radiative transfer equation “Radiosity” Friday’s lecture: Color and the spectrum Color perception Additive & subtractive color mixing Ternary diagrams and color transformations Selective absorption of light

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

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 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 ( %) 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) 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 ) i 17 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↑

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

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·I g ·r + b 21 What will be covered in next Tueday’s lecture? Mauna Loa, Hawaii