stokes vector-mueller matrix radiative transfer in an atmosphere-ocean system by george w. kattawar...

Stokes Vector-Mueller Matrix Radiative Transfer in an Atmosphere-Ocean System

by

George W. KattawarDept. of Physics

Texas A&M University

Why do Polarimetry?

• It is the only correct way to do radiative transfer • In 1947 Karl von Frisch showed that honey bees (Apis mellifera)

used polarized light for navigation• More recently, Rüdiger Wehner showed that certain ants also use

the sun as a compass and studied the neurophysiology of their vision

• Talbot Waterman showed that a variety of crustaceans, squids, octopuses, and fishes are able to detect the orientation of the electric vector of linearly polarized light

• Humans can see a faint image called Haidinger’s brush when viewing the clear zenith sky at sunrise or sunset

• Circular or elliptical polarization is rare in nature; however, it occurs just inside the critical angle for internally reflected light. Also a family of beetles called Scarabaeidae convert unpolarized light into left circularly polarized light

Haidinger's Brush

Polarization DirectionLeft-handed circular

polarizationRight-handed circular

polarization

Plankton as viewed by a squid

Planktonic animal as seen through "regular" visionAs seen when placed between two crossed linear polarizing filters

As seen by putting the two polarizers at 45° to each other

Contrast enhancement using polarization

Photo taken with a flash lamp and no polarization optics

Photo taken with circular polarized light for illumination and a circular analyzer for viewing

What are the optically significant constituents of ocean water?Water (of course)Dissolved Organic Compounds (CDOM, gelbstoff, gilvin)BacteriaPhytoplanktonLarger Organic Particles (zooplankton, “marine snow” – amorphous aggregates of smaller particles)Inorganic particles (quartz sand, clay minerals, metal oxides)

Stokes vector and polarization parameters

I is the radiance (this is what the human eye sees)

Q is the amount of radiation that is polarized in the 0/90 orientation

U is the amount of radiation polarized in the +/-450 orientation

V is the amount of radiation that is right or left circularly polarized

DOP= Degree of polarization=

Q2 +U2 +V2 / I

DOLP = Degree of linear polarization =

Q2 +U2 / I

DOCP = Degree of circular polarization = |V|/I

Orientation of plane of polarization = tan-1(U/Q)

Ellipticity= Ratio of semiminor to semimajor axis of polarization ellipse=b/a

=tan[(sin-1(V/I))/2]

Nissan car viewed in mid-wave infrared

This data was collected using an Amber MWIR InSb imaging array 256x256. The polarization optics consisted of a rotating quarter wave plate and a linear polarizer. Images were taken at eight different positions of the quarter wave plate (22.5 degree increments) over 180 degrees. The data was reduced to the full Stokes vector using a Fourier transform data reduction technique.

I Q U V

Scattering plane

Scattering Geometry

Incident beam E⊥i

EP

i

Scattering angle

EP

s

E⊥s

Scattering object

Scattering amplitude matrix

EPs

E⊥s

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

eikr

−ikr

S2 S3S4 S1

⎛

⎝⎜

⎞

⎠⎟

EPi

E⊥i

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Stokes vector-Mueller matrix formulationThe electric field can be resolved into components as follows:

E = El l + Er r

Where El and Er are complex oscillatory functions. The four component Stokes vector can now be defined as follows:

I =ElEl∗+ ErEr

∗=I l + Ir

Q=ElEl∗−ErEr

∗=I l −Ir

U =ElEr∗+ ErEl

∗

V =i (ElEr∗−ErEl

∗)

They are all real numbers and satisfy the relation

I2 = Q2 + U2 + V2

Is

Qs

Us

Vs

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

M11 M12 M13 M14M21 M22 M23 M24M31 M32 M33 M34M41 M42 M43 M44

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Ii

Qi

U i

V i

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

M is called the Mueller matrix

Symmetry Operations

Hi: I am a particle Reciprocal position Mirror image

S2 S3

S4 S1

⎛

⎝⎜

⎞

⎠⎟

S2 −S4−S3 S1

⎛

⎝⎜

⎞

⎠⎟

S2 −S3−S4 S1

⎛

⎝⎜

⎞

⎠⎟

IS

QS

US

VS

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

a1 b1

b1 a2

a3 b2 −b2 a4

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

I iQi

Ui

Vi

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

For randomly oriented particles with their mirror images in equal numbers

ΦΨ

I i

Θ

Y

l

r

X

I f

Z

If =R(−Ψ)L(Q,Φ)R(−Φ)I i

R(Φ) =

1 cos2Φ sin2Φ −sin2Φ cos2Φ 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

; L(Q,Φ) =

S11 S12

S12 S22

S33 S34 −S34 S44

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Rotation matrix leaves I, Q2+U2 and V invariant

Rotation Matrices

Polarizer and analyzer settings for Mueller matrix

measurements S

11

S

12

S

13

S

14

S

21

S

22

S

23

S

24

S

31

S

32

S

33

S

34

S

41

(OO) (HO-VO) (PO-MO) (LO-RO)

(OH-OV)(HH-VV) -

(HV+VH)

(PH-MV) -

(PV+MH)

(LH-RV) -

(LV+RH)

(OP-OM) (HP-VM) -

(HM+VP)

(PP-MM) -

(PM+MP)

(LP-RM) -

(LM+RP)

(OL-OR)

S

42

(HL-VR) -

(HR+VL)

S

43

(PL-MR) -

(PR+ML)

S

44

(LL-RR) -

(RL+LR)

Diagram showing the input polarization (first symbol) and output analyzer orientation (second symbol) to determine each element of the MSMM denoted by S ij. For example, HV denotes horizontal input polarized light and a vertical polarization analyzer. The corresponding symbols denoting polarization are V, vertical; H, horizontal; P, +45, M, -45; R, right handed circular polarization; L, left handed circular polarization; and O, open or no polarization optics. This set of measurements was first deduced by Bickel and Bailey..

Consider a beam of light passing though a small volume of water with no absorption or scattering

The Radiative Transfer Process

Let’s add some absorption to the water

Now add scattering:

Light can scatter out of the beam into a different direction

And into the beam from a different direction

Monte Carlo Method: the Good News and the Bad News

First the Good News

• Can handle virtually any geometry

• Can calculate both vector and scalar results simultaneously

• Can handle several different phase functions, single scattering albedos, and ground albedos simultaneously in a single run

• Detectors can be placed at any position and in any direction in the atmosphere-ocean system. There is no bin-averaging with this method

• Easy to calculate order of scattering contributions to final result. This allows a more reasonable meaning of “level of line formation” in spectroscopy of planetary atmospheres

Now the Bad News

• Since calculation is statistical in nature, results are always subject to statistical uncertainty

• Slow compared to other methods; however, Monte Carlo calculations are trivial to parallelize

Biased SamplingConsider the expectation value of some function f(x)

f (x) = f (x)p(x)dx∫

p(x) is the probability density function and f(x) is called the estimator

Now suppose we want to sample from %p(x)

f (x) = f (x)p(x)dx= f (x)

p(x)%p(x)

%p(x)dx∫∫

In+1 =S11 In +S12 Qncos2Φ+Unsin2Φ( )

Bivariate density function

With backward Monte Carlo we cannot sample above density function since we do not know In, Qn, and Un so we are forced to use biased sampling. We therefore sample Φuniformly between 0 and 2and Qaccording to the density function S11. To remove the bias we must therefore divide Eq. (1) by S11 This leads to the use of the reduced Mueller matrix where all elements are divided by S11.

(1)

Comparison with Monte Carlo

Percent error between vector and scalar transmitted radiance for a Rayleigh scattering atmosphere of varying optical thickness

-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90

Angle of view in degrees

Percent Error

tau = 0.5tau = 1.0tau = 3.0tau = 5.0

Error = [(vector-scalar)/vector]x100 Solar angle of incidence = 85 degrees

Optical thickness of atmosphere = 0.15

Optical thickness of ocean = 1.0

Direct sunlight

Region of total

Internal reflectionStrong elliptical polarization

skylight

n = 1.338

Multicomponent approach to light propagation with full Stokes vector treatment

• Method developed by Eleonora Zege’s group in Minsk, Belarus

• Applicable to scattering media containing clouds, mists, and ocean water

• Computes all 16 elements of Green matrix for any set of polar and azimuthal angles

• In each altitude dependent sub-layer, aerosol scattering and absorption as well as molecular scattering and absorption are accounted for

• Cloud layers can be handled

• Underlying surface can be included with a bi-directional reflection matrix with the small-angle component in the vicinity of the specular reflection

• Can handle both smooth and wind-ruffled interface characterized by specifying both wind velocity and azimuth

Flow chart of RayXP program

Original VRTE

Small angleVRTE Diffusion

VRTE

3-separatesmall-angleequations

Equation forsmall-angle

non-diagonal terms

Conventional VRTEwith smooth kernel

General solution

Radiance at the Bottom of the Ocean / Top of the Atmosphere

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

Radiance

Monte Carlo (scalar)

Monte Carlo (vector)

Multicomponent Approx. (vector)

Multicomponent Approx. (scalar)

downwelling radiance at bottom of ocean

upwelling radiance at top of atmosphere

cosine of zenith angle

Radiance Just Above the Interface

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1cosine of zenith angle

Radiance



MulticomponentApprox.MulticomponentApprox. (scalar)

downwelling upwelling

Radiance Just Below the Interface

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1cosine of zenith angle

RadianceMonte Carlo (scalar)


Multicomponent Approx.

Multicomponent Approx. (scalar)

cosine of the critical angle


Radiance in the Middle of the Ocean

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

1.80E-01

2.00E-01

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

cosine of zenith angle

Radiance



Multicomponent Approx.

Multicomponent Approx.(scalar)cosine of the criticalangle


-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

0 20 40 60 80 100 120 140 160 180

Angle of view

Percent error

g = 0.8 (Monte Carlo)


g=0 (Rayleigh) (Monte Carlo)

Top of atmosphere

Bottom of ocean

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

25.0

0 20 40 60 80 100 120 140 160 180

Angle of view

Percent error



g = 0.8 (M.C.A.)

0.9 (MCA)


Downward radianceUpward radiance

-30.0

-25.0

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0

Angle of view

Percent errorg = 0.8 (Monte Carlo)


g = 0.8 (MCA)

g = 0.9 (MCA)

g=0 (Rayleigh) (MonteCarlo)

Upward radiance Downward radiance

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

0 20 40 60 80 100 120 140 160 180

Angle of view

Percent error



g = 0.8 MCA

g = 0.9 MCA


Upward radiance Downward radiance

Backward Monte Carlo Calculation

Earth

Atmosphere

Incoming irradiance

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 20 40 60 80 100

Angle of view from vertical

Transmitted radiance

SS: AOI=48.5 degrees

PP: AOI=48.5 degrees





Principal plane of the sun viewing towards the sun

Conservative Rayleigh scattering atmosphere, tau = 0.15

-20.00

-10.00

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

0 20 40 60 80 100

Angle of view from vertical

Percent error in transmited radiance

AOI=48.5 deg, Phi=0 deg


AOI=89.1deg, Phi=o deg




Percent error=(SS-PP)x100/SS

Conclusions

Mueller matrix imaging allows us to detect objects at further distances and see surface features in turbid media when compared to normal radiance measurements

It can be a very powerful tool in identifying and characterizing both anthropogenic and natural aerosols including bioaerosols

It may also prove to be a very useful tool for the screening of precancerous skin lesions

We now have codes that can calculate the full MSMM (Green matrix) for any geometry and source detector configuration

stokes vector-mueller matrix radiative transfer in an atmosphere-ocean system by george w. kattawar...

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