1. Particle Scattering Single Dipole scattering (‘tiny’ particles)
– Rayleigh Scattering
Multiple dipole scattering – larger particles
(Mie scattering)
Extinction –
Rayleigh particles and the example
of microwave measurement of cloud
liquid water
Microwave precipitation
Scattering phase function – radar/lidar equation
backscattering properties
e.g. Rayleigh backscatter & calibration
of lidar, radar reflectivity

2. Analogy between slab and particle scattering
Insert 13.10/ 14.1
slab
particle
Slab properties are governed by oscillations (of dipoles) that
coherently interfere with one another creating scattered
radiation in only two distinct directions - particles scatter
radiation in the same way but the interference are less
coherent producing scattered stream of uneven magnitude
in all directions

3. Radiation from a single dipole*
Scattered wave is spherical in wave form (but amplitude not
even in all directions)
Scattered wave is proportional to the local dipole moment (p=E)
Basic concept of polarization
Key points to note:
parallel & perpendicular
polarizations
scattering angle
Any polarization state can be
represented by two linearly
polarized fields superimposed
in an orthogonal manner on one
another
* Referred to as Rayleigh scattering

4. Scattering Regimes
From Petty (2004)

5. Scattering Geometry

6. Rayleigh Scattering Basics
Single-particle behavior only governed by size parameter and index of refraction m!

7. Rayleigh “Phase Function”
Vertical Incoming Polarization
Horizontal Incoming Polarization
Incident Light Unpolarized

8. Polarization by Scattering
Fractional polarization for Rayleigh Scattering
The degree of polarization is
affected by multiple scattering.
Position of neutral points
contain information about the
nature of the multiple scattering
and in principle the aerosol
content of the atmosphere
(since the Rayleigh component
can be predicted with models).

9. Rayleigh scattering as observed POLDER:
Radiance
Strong spatial variability
Scattering
angle
0.04
Pol. Rad
650 nm
Smooth pattern
Signal governed by scattering angle
(Deuz₫ et al., 1993, Herman et al., 1997)
Proportional to Q

10. Radiation from a multiple dipole
particle r
ignore dipole-dipole
interactions 
rcos
At P, the scattered field is
composed on an EM field
from both particles
size
parameter P
For those conditions for which =0, fields reinforce each other such that I4E2

11. Scattering in the forward
corresponds to =0 –
always constructively add
Larger the particle (more
dipoles and the larger is
2r/ ), the larger is the
forward scattering
The more larger is 2r/,
the more convoluted (greater
# of max-min) is the scattering
pattern

13. Phase Function of water spheres (Mie theory)
High Asymmetry Parameter
Properties of the phase
function
asymmetry parameter
g=1 pure forward scatter
g=0 isotropic or symmetric (e.g
Rayleigh)
g=-1 pure backscatter
forward scattering & increase with x
rainbow and glory
Smoothing of scattering function by polydispersion
Low Asymmetry Parameter

14. P() 180 Properties of the phase function fig. 14.19
180
P()
forward scattering & increase with x
rainbow and glory
smoothing of scattering function by polydispersion
single
particle
Properties of the phase
function
fig
asymmetry parameter
g=1 pure forward scatter
g=0 isotropic or symmetric (e.g
Rayleigh)
g=-1 pure backscatter
poly-dispersion

16. Particle Extinction Particle scattering is defined
in terms of cross-sectional
areas & efficiency factors
σext = effective area projected
by the particle that
determines extinction
Similarly σsca, σabs
Geometric
cross-section
r2
The efficiency factor then follows

17. Particle Extinction (single particle)
=1
Note how the spectrum
exhibits both coarse
and fine oscillations
Implications of these
for color of scattered light
How Qext2 as 2r/
extinction paradox
‘Rayleigh’ limit x 0 (x<<1)

18. Extinction Paradox shadow area r2 combines the effects
of absorption and any
reflections (scattering)
off the sphere.

19. insert 14.10
Poisson spot – occupies a unique
place in science – by
mathematically demonstrating
the non-sensical existence of
such a spot, Poisson hoped to
disprove the wave theory of
light.

20. Mie Theory Equations Exact Qs, Qa for spheres of some x, m.
a, b coefficients are called “Mie Scattering coefficients”, functions of x & m. Easy to program up.
“bhmie” is a standard code to calculate Q-values in Mie theory.
Need to keep approximately x + 4x1/3 + 2 terms for convergence

21. Mie Theory Results for ABSORBING SPHERES

22. Volumes containing clouds of many particles
Extinctions, absorptions and scatterings by all
particles simply add- volume coefficents
half of 14.9
L-4 L
L-1 L2
n( r)= the particle size distribution
# particles per unit volume
per unit size r
Exponential distribution (rain)
Modified Gamma distribution (clouds)
Lognormal distribution
(aerosols, sometimes clouds)

23. Effective Radius & Variance
Mean particle radius – doesn’t have much physical relevance for radiative effects
For large range of particle sizes, light scattering goes like πr2. Defines an “effective radius”
“Effective variance”
Modified Gamma distribution
a = effective radius
b = effective variance

24. Scattering/extinction properties
Cloud optical depths
(visible/nir ’s)
Microwave (Rayleigh) scattering x0 W
wz

25. 1st indirect aerosol effect!
Polydisperse Cloud: Optical Depth,
Effective Radius, and Water Path
(visible/nir ’s)
Cloud Optical Depth
Volume Extinction Coefficient [km-1]
Cloud Optical Depth
Local Cloud Density [kg/m3]
Cloud Effective Radius [μm]
1st indirect aerosol effect!
(Twomey Effect)
ρcloudz

27. Variations of SSA with wavelength
Somewhat Absorbing
Non-Absorbing!

28. Satellite retrieve of cloud optical depth & effective radius
Non-absorbing Wavelength (~1):
Reflectivity is mainly a function of optical depth.
Absorbing Wavelength (<1):
Reflectivity is mainly a function of cloud droplet size (for thicker clouds).

29. The reflection function of a nonabsorbing band (e. g. , 0
The reflection function of a nonabsorbing band (e.g., 0.66 µm) is primarily a function of cloud optical thickness
The reflection function of a near-infrared absorbing band (e.g., 2.13 µm) is primarily a function of effective radius
clouds with small drops (or ice crystals) reflect more than those with large particles
For optically thick clouds, there is a near orthogonality in the retrieval of tc and re using a visible and near-infrared band
re usually assumed constant in the vertical. Therefore:

30. Cloud Optical Thickness and Effective Radius (M. D. King, S
Cloud Optical Thickness and Effective Radius (M. D. King, S. Platnick – NASA GSFC)
Cloud Optical Thickness
Cloud Effective Radius (µm)
Recently came up with a way to visualize both the retrievals and the cloud phase simultaneously with a single image. 1 10
>75 1 10
>75 6 17 28 39 50 2 9 16 23 30
Ice Clouds
Water Clouds
Ice Clouds
Water Clouds
King et al. (2003)

31. Monthly Mean Cloud Effective Radius Terra, July 2006
Liquid water clouds
Larger droplets in SH than NH
Larger droplets over ocean than land (less condensation nuclei)
Ice clouds
Larger in tropics than high latitudes
Small ice crystals at top of deep convection
routine production of global re is NEW thanks to MODIS

32. Aerosol retrieval from space- the MODIS aerosol algorithm
Uses bi-modal, log-normal aerosol size distributions.
5 small - accumulation mode ( m)
6 large - coarse mode (> .5 m)
Look up table (LUT) approach
15 view angles ( degrees by 6)
15 azimuth angles (0-180 degrees by 12)
7 solar zenith angles
5 aerosol optical depths (0, 0.2, 0.5, 1, 2)
7 modis spectral bands (in SW)
Ocean retrievals
compute IS and IL from LUT
find ratio of small to large modes () and
the aerosol model by minimizing
and Im is the
measured radiance.
then compute optical depth from
aerosol model and mode ratio.

33. Land retrievals
Select dark pixels in near IR,
assume it applies to red and blue
bands.
Using the continental aerosol model,
derive optical depth & aerosol models (fine & course modes) that best fit obs (LUT approach including multiple scattering).
The key to both ocean and land retrievals is that the surface reflection is small.

34. “Deep Blue” MODIS Algorithm works over Bright Surfaces
Uses fact that bright surfaces are often darker in blue wavelengths
Uses 412 nm, 470nm, and 675nm to retrieve AOD over bright surfaces.
Still a product in its infancy

35. “Deep Blue” MODIS Algorithm works over Bright Surfaces
Uses fact that bright surfaces are often darker in blue wavelengths
Uses 412 nm, 470nm, and 675nm to retrieve AOD over bright surfaces.
Complements “Dark Target” retrieval well.
Still being improved!

36. MAIAC

37. Scattering phase function

38. spheres
spherical
Non spherical with plane
of symmetry
non spheres

39. Particle Backscatter Differential cross-section
Cd()I0 is the power scattered
into  per unit solid angle
Differential cross-section
Bi-static cross-section
Backscattering cross-section
CbI0 is the total power assuming a particle scatters isotropically by the amount is scatters at =180

40. Polarimetric Backscatter: LIDAR depolarization
Transmit linear
Receive parallel/perpendicular
Ice
Water/Ice/Mix
=0 for
spheres

41. Polarimetric Backscatter: RADAR ZDR
Transmit both horizontal & vertical
Receive horizontal & vertical
for spheres, ZDR~0

42. Lidar Calibration using Rayleigh scattering
Laser backscattering
Crossection as measured
During the LITE experiment
For Rayleigh scattering

43. Lidar Calibration using Rayleigh scattering
Rayleigh scattering is well-understood and easily calculable anywhere in the atmosphere!
ns = 1 + a * (1 + b λ-2)
Stephens et al. (2001)