Tuesday, 20 January Lecture 5: Radiative transfer theory where light comes from and how it gets to where it’s going http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html (scattering) http://id.mind.net/~zona/mstm/physics/light/rayOptics/refraction/refraction1.html (refraction) http://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, photogrammetry

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

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

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, O2, O3, and N2 are important in the UV CO2 and H2O are important in the IR (NIR, MIR, TIR)

Atmospheric transmission

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

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

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

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

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

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

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

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