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!