Presentation for chapters 5 and 6

LIST OF CONTENTS

1. Surfaces - Emission and Absorption2. Surfaces - Reflection3. Radiative Transfer in the Atmosphere-Ocean System4. Examples of Phase Functions5. Rayleigh Phase Function6. Mie-Debye Phase Function7. Henyey-Greenstein Phase Function8. Scaling Transformations9. Remarks on Scaling Approximations

SURFACES - EMISSION AND ABSORPTION

• Energy emitted by a surface into whole hemisphere - spectral flux emittance:

• Energy absorped when radiation incident over whole hemisphere – spectral flux absorptance:

• Kirchoff’s Law for Opaque Surface:

Energy emitted relative to that of a blackbody

ss TvTv ,2,,2,

2. SURFACES - REFLECTION

• Ratio between reflected intensity and incident energy – Bidirectional Reflectance Distribution Function (BRDF):

• Lambert surface – reflected intensity is completely uniform .• Specular surface – reflected intensity in one direction

• In general: BRDF has one specular and one diffuse component:

SURFACE REFLECTION

Analytic reflectance expressions

o

no

kkono

2,

Seeliger-Lommelyreciprocit of principleobey model This

,

FormulaMinnaert 11

Transmission through a slab• Transmitance

• Transimitted intensity leaving the medium in downward direction

TRANSMISSION THROUGH A SLAB

For collimated beam:• Transmitted intensity is

• Flux transmitted

,,coscoscos oo

sv

voo

svvt vFeFI s

dT

dveFF o

vo

svvt

s cos,,cos dT

TRANSMISSION THROUGH A SLAB

• Flux transmitance is

dve

FFv

ov

osv

vto

s cos,,

cos2,,

d

d

T

T

RADIATIVE TRANSFER EQUATION

direction scattered directionincident '

','4

14

'

vvs

v IpdvaBaIddI

RADIATIVE TRANSFER EQUATION

• For Zero scattering

2

2

1

21 ,,12

solution generalWith

PPtP

P

PP

vvs

v

etdtBePIPI

TBIddI

RADIATIVE TRANSFER IN THE ATMOSPHERE-OCEAN SYSTEM

• The refractive index is in the atmosphere and in the ocean.

• In aquatic media, radiative transfer similar to gaseous media• In pure aquatic media Density fluctuations lead to Rayleigh-like

scattering.

• In principle: Snell’s law and Fresnel’s equations describe radiative coupling between the two media if ocean surface is calm.

• Complications are due to multiple scattering and total internal reflection as below

1rm 34.1rm

RADIATIVE TRANSFER IN THE ATMOSPHERE-OCEAN SYSTEM

• Demarcation between the refractive and the total reflective region in the ocean is given by the critical angle, whose cosine is:

• where

• Beams in region I cannot reach the atmosphere directly• Must be scattered into region II first

EXAMPLES OF PHASE FUNCTIONS

• We can ignore polarization effects in many applications eg:• Heating/cooling of medium,Photodissociation of molecules’Biological dose

rates

• Because: Error is very small compared to uncertainties determining optical properties of medium.

• Since we are interested in energy transfer-> concentrate on the phase function

RAYLEIGH PHASE FUNCTION

• Incident wave induces a motion (of bound electrons) which is in phase with the wave ,nucleus provides a ’restoring force’ for electronic motion

• All parts of molecule subjected to same value of E-field and the oscillating charge radiates secondary waves

• Molecule extracts energy from wave and re-radiates in all directions

• For isotropic molecule, unpolarized incidenradiation:

RAYLEIGH PHASE FUNCTION

• Expanding in terms of incident and scattered angles:

• Azimuthal-averaged phase function is:

RAYLEIGH PHASE FUNCTION

• By expressing in terms of Legendre Polynomials:

• Asymmetry factor for Rayleigh phase function is zero (because of orthogonality of Legendre Polynomials):

• Only non-zero moment is

MIE-DEBYE PHASE FUNCTION

• Scattering by spherical particles

• Scattering by larger particles:-> Strong forward scattering – diffraction peak in forward direction!

• Why?• For a scattering object small compared to

wavelength:-> Emission add together coherently because all oscillating dipoles are subject to the same field

MIE-DEBYE PHASE FUNCTION

• For a scattering object large compared to wavelength:

• All parts of dipole no longer in phase

• We find that:

• Scattered wavelets in forward direction: always in phase

• Scattered wavelets in other directions: mutual cancellations, partial interference

HENYEY-GREENSTEIN PHASE FUNCTION

• A one-parameter phase function first proposed in 1941:

• No physical basis, but very popular because of the remarkable feature:

• Legendre polynomial coeffients are simply:

• Only first moment of phase function must be specified, thus HG expansion is simply: