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, 1.4http://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 theorydata spaces color mixturesabsorption

Reading

The Electromagnetic Spectrum (review)

Units:Micrometer = 10-6 mNanometer = 10-9 m

Light emitted by the sun

The Sun

Light from Sun – Light Reflected and Emitted by Earth

Wavelength, μm

W m

-2 μ

m -1

W m

-2 μ

m-1 sr

-1

The sun is not an ideal blackbody – the 5800 K figure and graph are simplifications

Atmospheric Constituents

10

50

90

Troposphere

Stratosphere

Mesosphere

Thermosphere

Almost allH2O

Hei

ght (

km)

Temperature (K)

280200

Ozone

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 scatteringFor absorption, O2, O3, and N2 are important in the UVCO2 and H2O are important in the IR (NIR, MIR, TIR)

Solar spectra before and after passage through the atmosphere

Atmospheric transmission

Modeling the atmosphere

dz

e

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.

L

L ze

L

LsL ze

)(

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

Radiosonde dataA

ltitu

de (k

m)

Alti

tude

(km

)

Relative Humidity (%) Temperature (oC)

20

15

10

5

0

20

15

10

5

00 20 40 60 80 100 -80 -40 0 40

Mt Everest

Mt Rainier

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 – - 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.

Ig

Ls↑

Itoa

0.5º

Is↓

L

Terms and units used in radiative transfer calculations

DN = a·Ig·r + b

Radiative transfer equation

Ig is the irradiance on the groundr is the surface reflectancea & b are parameters that relate to instrument and atmospheric characteristics

This is what we want

Parameters that relate to instrument and atmospheric characteristics

DN = g·(e·r · i·Itoa·cos(i)/ + e· r·Is↓/ + Ls↑) + o

g amplifier gainatmospheric transmissivitye emergent anglei incident angler reflectanceItoa solar irradiance at top of atmosphereIg solar irradiance at ground Is↓ down-welling sky irradianceLs↑ up-welling sky (path) radianceo amplifier bias or offset

Radiative transfer equation

DN = a·Ig·r + b

The factor of Consider a perfectly reflective (r=100%) diffuse “Lambertian” surface that

reflects equally in all directions.

Lambert

The factor of 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/ 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 ?

∫ ∫ L sin cos ddL2

0 0

•Incoming directional radiance L at elevation angle is isotropic

•Reflected directional radiance L cos is isotropic

•Area of a unit hemisphere:

∫ ∫ sin dd 2

0 0

The factor of Consider a perfectly reflective (r=100%) diffuse “Lambertian” surface that

reflects equally in all directions.

i Itoa cos(i)

Itoa

gi Itoa cos(i)

i

r reflectance

r (i Itoa cos(i)) / reflected light

“Lambertian” surface

e

e

Ls↑ (Lp)Highlighted terms relate to the surface

i

i Itoa cos(i)

Itoa

gi Itoa cos(i)

i

r reflectance

r (i Itoa cos(i)) / reflected light

“Lambertian” surface

e

e

Measured Ltoa

DN(Itoa) = a Itoa + b

Ltoae r (i Itoa cos(i)) / +

e r Is↓ / + Ls↑

Ls↑ (Lp)Highlighted terms relate to the surface

Is↓

Ls↑=r Is↓ / i

Lambert

Next lecture: Atmospheric scattering and other effects

Mauna Loa, Hawaii