Tuesday 20 January Lecture 5 Radiative transfer theory

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/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) 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

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

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). Fo is the incoming flux This profile can be measured made using a balloon, or a standard atmosphere can be assumed.

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)

Terms and units used in radiative transfer calculations 0. 5º 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 Itoa L Ls↑ 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.

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 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

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 then cos irradiance d dw=p. L If irradiance on the surface is I , the from the surface is g 0 0 r·Ig = Ig W m-2. • Incoming directional radiance L at elevation angle is-2 isotropic The radiance intercepted by a camera would be r·Ig/p W m sr-1. • Reflected directional radiance L cos is isotropic The factor p is the ratio between the hemispheric radiance (irradiance) and the directional radiance. area of the sky hemisphere is 2 p sr (for a unit • Area of a unit. The hemisphere: radius). So – why don’t we divide 2 pbyp/22 p instead of p? ∫ ∫ sin Lambert d dw=2 p 0 0

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 Highlighted terms relate to the surface ti Ls↑=r Is↓ /p Ls↑ (Lp) te i e Ig=ti Itoa cos(i) Lambert Is↓ r reflectance r (ti Itoa cos(i)) /p reflected light “Lambertian” surface

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