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
Analogy between slab and particle scattering
Insert 13.10/ 14.1
Slab properties are governed by oscillations (of dipoles) thatcoherently interfere with one another creating scattered radiation in only two distinct directions - particles scatterradiation in the same way but the interference are lesscoherent producing scattered stream of uneven magnitude in all directions
slab
particle
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)
* Referred to as Rayleigh scattering
Basic concept of polarization
Key points to note:• parallel & perpendicular polarizations
• scattering angle
Any polarization state can berepresented by two linearly polarized fields superimposed in an orthogonal manner on one another
Scattering Regimes
From Petty (2004)
Scattering Geometry
Single-particle behavior only governed by size parameter and index of refraction m!
Rayleigh Scattering Basics
Vertical Incoming Polarization
Horizontal Incoming Polarization
Incident Light Unpolarized
Rayleigh “Phase Function”
The degree of polarization is affected by multiple scattering.
Position of neutral points contain information about thenature of the multiple scatteringand in principle the aerosolcontent of the atmosphere(since the Rayleigh component can be predicted with models).
Polarization by Scattering
Fractional polarization for Rayleigh Scattering
Rayleigh scattering as observed POLDER:
0.04
0
Strong spatial variability
Smooth pattern
Signal governed by scattering angle
(Deuz₫ et al., 1993, Herman et al., 1997)
Radiance
Pol. Rad
650 nm
Proportional to Q
Scatteringangle
Radiation from a multiple dipoleparticle
r
rcos
P
At P, the scattered field is composed on an EM field from both particles
cosEEEEI
eEeEE
r)cos(
ii
2122
21
21
2
21
For those conditions for which =0, fields reinforce each other such that I4E2
sizeparameter
ignore dipole-dipoleinteractions
Scattering in the forwardcorresponds to =0 –always constructively add
Larger the particle (moredipoles and the larger is 2r/ ), the larger is theforward scattering
The more larger is 2r/, the more convoluted (greater # of max-min) is the scatteringpattern
Phase Function of water spheres (Mie theory)
Low Asymmetry Parameter
High Asymmetry ParameterProperties of the phase
function
1
12
1cosdcos)(cosPg
asymmetry parameter
g=1 pure forward scatterg=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
Particle Extinction
Geometriccross-sectionr2
Particle scattering is definedin terms of cross-sectional areas & efficiency factors
σext = effective area projected by the particle that determines extinction
Similarly σsca, σabs
The efficiency factor then follows
Particle Extinction (single particle)
Note how the spectrumexhibits both coarseand fine oscillations
Implications of these for color of scattered light
How Qext2 as 2r/ extinction paradox
‘Rayleigh’ limit x 0 (x<<1)
=1
Extinction Paradox
shadow arear2
?? 1r
area shadowQ
2ext
2
2
22
2ext
rrr
r
ndiffractio by filled area area shadowQ
combines the effectsof absorption and any reflections (scattering)off the sphere.
insert 14.10
Poisson spot – occupies a uniqueplace in science – by mathematically demonstratingthe non-sensical existence ofsuch a spot, Poisson hoped todisprove the wave theory oflight.
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
Mie Theory Results for ABSORBING SPHERES
Volumes containing cloudsof many particles
Extinctions, absorptions and scatterings by all particles simply add- volume coefficents
dr),r(Qr)r(n sca,abs,extsca,abs,ext
0
2
n( r)= the particle size distribution # particles per unit volume per unit size
half of 14.9L-4
L2
L
L-1
7-
3-
3-
o
10V
10100V
c.c per droplets N
cm10m r
rNV
3
0
3
3
4
100
10
3
4
r
Exponential distribution (rain)
Modified Gamma distribution (clouds)
Lognormal distribution(aerosols, sometimes clouds)
Modified Gamma distribution
Effective Radius & Variance
a = effective radiusb = effective 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”
(visible/nir ’s)
ρcloudz
Polydisperse Cloud: Optical Depth, Effective Radius, and Water Path
Cloud Optical Depth
Volume Extinction Coefficient [km-1]
Cloud Optical Depth
Local Cloud Density [kg/m3]
1st indirect aerosol effect!
(Twomey Effect)
Cloud Effective Radius [μm]
Variations of SSA with wavelength
Non-Absorbing!
Somewhat Absorbing
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).
• 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 c and re using a visible and near-infrared band
• re usually assumed constant in the vertical. Therefore: erLWP 3
2
Cloud Optical Thickness and Effective Radius (M. D. King, S. Platnick – NASA GSFC)
King et al. (2003)King et al. (2003)
Ice CloudsIce Clouds Ice CloudsIce Clouds>75>75 >75>7511 11 10101010 5050 303066 22 16162828 39391717 99 2323
Cloud Optical ThicknessCloud Optical Thickness Cloud Effective Radius (µm)Cloud Effective Radius (µm)
Water CloudsWater CloudsWater CloudsWater Clouds
Monthly Mean Cloud Effective RadiusTerra, July 2006
Liquid water cloudsLiquid water clouds
–Larger droplets in SH Larger droplets in SH than NH than NH
–Larger droplets over Larger droplets over ocean than land (less ocean than land (less condensation nuclei)condensation nuclei)
Ice cloudsIce clouds
–Larger in tropics than Larger in tropics than high latitudeshigh latitudes
–Small ice crystals at Small ice crystals at top of deep top of deep convectionconvection
Aerosol retrieval from space- the MODIS aerosol algorithm
Uses bi-modal, log-normal aerosol size distributions.• 5 small - accumulation mode (.04-.5 m)• 6 large - coarse mode (> .5 m)
Look up table (LUT) approach• 15 view angles (1.5-88 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
• then compute optical depth from aerosol model and mode ratio.
and Im is the measured radiance.
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.
“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
“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!
MAIAC
Scattering phase function
spheres
non spheres
0
0
0
0
4434
3433
22
11
2
00
00
000
000
V
U
Q
I
SS
SS
S
S
Rk1
V
U
Q
I
2
sca
sca
sca
sca
Non spherical with planeof symmetry
spherical
0
0
0
0
4434
3433
2212
1211
2
00
00
00
00
V
U
Q
I
SS
SS
SS
SS
Rk1
V
U
Q
I
2
sca
sca
sca
sca
Particle Backscatter
Differential cross-section
Bi-static cross-section
Backscattering cross-section
)(PC
)(C scad
4
)(C)(C dbi 4
)(CC db 1804
Cd()I0 is the power scatteredinto per unit solid angle
CbI0 is the total power assuming a particle scatters isotropically by the amount is scatters at =180
Polarimetric Backscatter: LIDAR depolarization
• Transmit linear• Receive parallel/perpendicular
11 120
12 22
00
0
, 11 22
, 11 22
, 11 22
, 1
1 1 1 11 1,
1 1 1 12 2
( )
( )
1
1
depolarization ratio
measured sca
r
sca
measured
measured r
measured r
measured
I MI
M M
S SI I
S S
II
Q
I S S
I S S
linear
I S S
I S
1 22 12( 2 )S S =0 for spheres
Ice
Water/Ice/Mix
for spheres, ZDR~0
Polarimetric Backscatter: RADAR ZDR
• Transmit both horizontal & vertical
• Receive horizontal & vertical
Lidar Calibration using Rayleigh scattering
Laser backscatteringCrossection as measuredDuring the LITE experiment
For Rayleigh scattering
1 1
1
( ) 8
( ) 3b b
sca sca
C m ster
C m
Stephens et al. (2001)
Lidar Calibration using Rayleigh scattering
R 24 3
Ns24
(ns2 1)2
(ns2 2)2
6 36 7
ns = 1 + a * (1 + b λ-2)
Rayleigh scattering is well-understood and easily calculable
anywhere in the atmosphere!