stokes vector-mueller matrix radiative transfer in an atmosphere-ocean system by george w. kattawar...
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Stokes Vector-Mueller Matrix Radiative Transfer in an Atmosphere-Ocean System
by
George W. KattawarDept. of Physics
Texas A&M University
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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
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Haidinger's Brush
Polarization DirectionLeft-handed circular
polarizationRight-handed circular
polarization
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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
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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
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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)
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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]
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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
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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
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
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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
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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
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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
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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..
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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
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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
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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)
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Comparison with Monte Carlo
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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
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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
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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
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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
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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
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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
Monte Carlo (scalar)
Monte Carlo (vector)
MulticomponentApprox.MulticomponentApprox. (scalar)
downwelling upwelling
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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)
Monte Carlo (vector)
Multicomponent Approx.
Multicomponent Approx. (scalar)
cosine of the critical angle
downwelling upwelling
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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
Monte Carlo (scalar)
Monte Carlo (vector)
Multicomponent Approx.
Multicomponent Approx.(scalar)cosine of the criticalangle
downwelling upwelling
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-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 = 0.9 (Monte Carlo)
g=0 (Rayleigh) (Monte Carlo)
Top of atmosphere
Bottom of ocean
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-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 (Monte Carlo)
g = 0.9 (Monte Carlo)
g = 0.8 (M.C.A.)
0.9 (MCA)
g=0 (Rayleigh) (Monte Carlo)
Downward radianceUpward radiance
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-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 = 0.9 (Monte Carlo)
g = 0.8 (MCA)
g = 0.9 (MCA)
g=0 (Rayleigh) (MonteCarlo)
Upward radiance Downward radiance
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-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 (Monte Carlo)
g = 0.9 (Monte Carlo)
g = 0.8 MCA
g = 0.9 MCA
g=0 (Rayleigh) (Monte Carlo)
Upward radiance Downward radiance
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Backward Monte Carlo Calculation
Earth
Atmosphere
Incoming irradiance
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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
SS: AOI=85.3 degrees
PP: AOI=85.3 degrees
SS: AOI=89.09 degrees
PP: AOI=89.09 degrees
Principal plane of the sun viewing towards the sun
Conservative Rayleigh scattering atmosphere, tau = 0.15
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-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=85.3 deg, Phi=0 deg
AOI=89.1deg, Phi=o deg
AOI=48.5 deg, Phi=180 deg
AOI=85.3 deg, Phi=180 deg
AOI=89.1 deg, Phi=180 deg
Percent error=(SS-PP)x100/SS
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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