Transcript Presentation for chapter 6

Presentation for chapters 5
and 6
LIST OF CONTENTS
1.
2.
3.
4.
5.
6.
7.
8.
9.
Surfaces - Emission and Absorption
Surfaces - Reflection
Radiative Transfer in the Atmosphere-Ocean System
Examples of Phase Functions
Rayleigh Phase Function
Mie-Debye Phase Function
Henyey-Greenstein Phase Function
Scaling Transformations
Remarks on Scaling Approximations
SURFACES - EMISSION AND ABSORPTION
•
Energy emitted by a surface into
whole hemisphere - spectral flux
emittance:
Energy emitted
relative to that
of a blackbody
•
Energy absorped when radiation
incident over whole hemisphere –
spectral flux absorptance:
•
Kirchoff’s Law for Opaque Surface:
 v,2 , Ts    v,2 , Ts 
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
Minnaert Formula
  , o    n 
k 1
o

k 1
This model obey principle of reciprocit y
Lommel - Seeliger
2n
  , o  
  o
Transmission through a slab
• Transmitance
• Transimitted intensity leaving the medium in
downward direction
TRANSMISSION THROUGH A SLAB
For collimated beam:
• Transmitted intensity is





 s v 
s
s
I     Fv  cos   cos  o    o e
 Fv cos  oTd  v,  o ,  
 



vt
• Flux transmitted






 s v 

s
Fvt  Fv cos  o e
  Td  v, o ,   cos d 





TRANSMISSION THROUGH A SLAB
• Flux transmitance is

vt
F


Td  v,  o ,2   s
Fv cos  o





  s v 



 e
  Td  v,  o ,   cos d 





RADIATIVE TRANSFER EQUATION



dI v
av 




'
  I v  1  a B 
d p ' ,   I v  ' 

d s
4 4

  

' incident direction
 scattered direction
RADIATIVE TRANSFER EQUATION
• For Zero scattering
dI v
  I v  Bv T 
d s
With general solution
I  P2   I  P1 e
  P1 , P2 

  P2 
dtBt e

 
P1
t  P , P2 
RADIATIVE TRANSFER IN THE
ATMOSPHERE-OCEAN SYSTEM
mr  1 in the atmosphere and
mr  1.34 in
•
The refractive index is
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
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: