Toward modeling for efficient PDMs
Wenbo Sun and Constantine Lukashin
• Introduction
• Status of the adding-doubling radiative transfer model (ADRTM)
• Employing symmetry of Stokes parameters in building PDMs
• Mapping polarization parameters to uniform viewing angle grid
• Effect of ocean wave shadow on the polarization of reflected light
• Effect of ice cloud particle size distributions on the polarization of reflected light
• Summary
Science Team Meeting, April 16-18, 2013, Hampton, Virginia
1. ADRTM: This can calculate full Stokes parameters (I, Q, U, V). 2. Atmospheric profiles: Any atmosphere profile. 3. Spectral gas absorption: Line-by-Line and k-distribution plus ozone cross-section table. 4. Molecular scattering: Rayleigh with depolarization factor. 5. Particulate absorption and scattering: Mie for water clouds (Gamma size distribution); PML FDTD for fine-mode aerosols; CPML PSTD code is developed for coarse-mode aerosols; FDTD, PSTD, GOM for ice clouds. 6. Surface reflection model: Lambert surface for land ; More practical model for land is being considered… Cox & Munk without Gram-Charlier expansion plus foam for ocean; Cox & Munk with Gram-Charlier expansion plus foam for ocean; Wave shadowing effect is integrated in the ocean surface model; Lambert model for water-leaving radiance from ocean water volume. More practical model for water-leaving radiance is being considered… 7. Output: polarization parameters are mapped to uniform angular grids.
Status of the adding-doubling radiative transfer model (ADRTM)
Employing symmetry of Stokes parameters in simplifying the PDMs
),(coscoscos4
),,()1()1(),,(
40 yx
vs
vsWLWCvs ZZPfff
MRRR
)exp(1
),(2
22
2
yx
yx
ZZZZP
sv
svx
x
ZZ
coscos
sincossin
sv
vy
y
ZZ
coscos
sinsin
])1)(1(4
)36(24
)36(24
)3(6
)1(2
1)[2
exp(2
1),(
222224042440
303221
22
ccc
ccZZP
uc
uc
c
cZ
u
uZ
If wind direction is NOT accounted for
If wind direction is accounted for
(Cox and Munk,1956)
(Cox and Munk,1954)
Reflectance and DOP on the principal plane at a wavelength of 670 nm from
Cox-and-Munk models with and without wind-direction dependence.
AOLPs from the ocean wave slope probability distribution model with
wind direction (a) 0o, (b) 90o, and (c) 180o, and for (d) the ocean wave
slope probability distribution model without wind direction dependence.
),()360,( 0 RAZVZAIRAZVZAI
),()360,( 0 RAZVZAQRAZVZAQ
),()360,( 0 RAZVZAURAZVZAU
),()360,( 0 RAZVZAVRAZVZAV
),()360,( 0 RAZVZADOPRAZVZADOP
),(180)360,( RAZVZAAOLPRAZVZAAOLP oo
• Wind direction and wave slope distribution models have little effect on the polarization state of reflected light at TOA.
• We will use the Cox-and-Munk model without wind direction dependence [Cox and Munk 1956] in the future modeling for operational PDMs.
• Using the Cox-and-Munk model without wind direction dependence, we only need to calculate DOP and AOLP over the RAZ of 0o to 180o. These quantities will be obtained by symmetry for the RAZ of 180o to 360o with noting that
. (23b)
Mapping ADRTM output polarization parameters to uniform VZA
ADRTM’s outputs are at Gaussian Quadrature points of VZA.
• For each RAZ, linear extrapolation is done to obtain I, Q, U, and V at
VZA = 0 deg and 90 deg.
• For each RAZ, linear interpolation is done to obtain I, Q, U, and V at
uniform VZA grid point from 0 to 90 deg.
• At each grid point of the uniform VZA and RZA, DOP and AOLP are
derived from the I, Q, U, and V at these uniform points
This simple treatment greatly reduces the stream number of the ADRTM
required for the PDM modeling, makes the modeling times faster.
0 15 30 45 60 75 90
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
VZA (o)
DO
P
90 75 60 45 30 15 0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
VZA (o)
To
tal R
efle
cta
nce
Wave Length = 490 nm
Pristine Atmosphere
Wind Speed = 7.5 m/s
SZA = 23.57o
0 15 30 45 60 75 90
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
VZA (o)
To
tal R
efle
cta
nce
90 75 60 45 30 15 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
DO
P
VZA (o)
Reflectance and DOP on the principal plane calculated with the ADRTM at the
Gaussian quadrature points (black dots) and after the mapping to uniform VZA
grids (solid curve)
10 20 30 40 50 60
5
10
15
20
25
30
VZ
A (
x3
o)
RAZ (x3o)
AO
LP
(o)
0
15
30
45
60
75
90
105
120
135
150
165
18010 20 30 40 50 60
2
4
6
8
10
12
14
16
18
RAZ (x3o)
AO
LP
VZ
A (
18
qu
ad
ratu
re p
oin
ts fro
m 3
.77 to 8
7.5
3o)
0
15
30
45
60
75
90
105
120
135
150
165
180
Before mapping to uniform VZA After mapping to uniform VZA
AOLPs before (left panel) and after(right panel) the mapping to
uniform VZA grids
Effect of ocean wave shadow on the polarization of reflected light
Ocean wave shadowing effect is calculated by multiplying the reflection
matrix by a bidirectional shadowing function
)()(1
1),(
vs
vsS
where erfc(x) is the complementary error
function [Tsang et al., 1985].
)cos1(2
cos
)cos1(2
cosexp
)cos1(2
cos2
1)(
222
221
2
erfc
90 75 60 45 30 15 0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
MLS pristine atmosphere
WindSpeed = 7.5 m/s
SZA = 33.30o
Re
fle
cta
nce
VZA (o)
RAZ = 0o
0 15 30 45 60 75 90
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
RAZ =180o
VZA (o)
Re
fle
cta
nce
90 75 60 45 30 15 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
RAZ = 0o
VZA (o)
DO
P
0 15 30 45 60 75 90
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
RAZ =180o
VZA (o)
DO
P
No Shadowing
Shadowing
Reflectance and DOP on the principal plane from ocean surface model with
shadowing and without shadowing factor
AOLPs from ocean surface model with shadowing (left panel) and without
shadowing (right panel) factor
10 20 30 40 50 60
5
10
15
20
25
30
AO
LP
(o)
RAZ (x3o)
VZ
A (
x3
o)
Shadowing
0
15
30
45
60
75
90
105
120
135
150
165
18010 20 30 40 50 60
5
10
15
20
25
30
AO
LP
(o)
RAZ (x3o)
VZ
A (
x3
o)
0
15
30
45
60
75
90
105
120
135
150
165
180
No Shadowing
0 500 1000 1500 2000 2500
10-2
10-1
100
101
102
103
104
105
FIRE II: 11/22/1991
dN
/dL
(m
-3m
-1)
L (m)
HP(1984): T = -20o
to -25o
Effect of ice cloud particle size distributions on the polarization
of reflected light
Two in-situ-measured ice particle size distributions used in the
modeling for ice clouds
0 15 30 45 60 75 90 105 120 135 150 165 180
0.1
1
10
100
1000
P1
1
Scattering Angle (o)
0 15 30 45 60 75 90 105 120 135 150 165 180
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
P1
2/P
11
A
Phase matrix elements of ice clouds with hexagonal column particle
shape and HP (1984) and FIRE II (1991) size distributions
90 75 60 45 30 15 0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
RAZ = 0o
MLS pristine atmosphere
WindSpeed = 7.5 m/s
SZA = 33.30o
WL = 865 nm
Ice cloud OD = 0.3
VZA (o)
Re
fle
cta
nce
0 15 30 45 60 75 90
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
RAZ =180o
VZA (o)
Re
fle
cta
nce
90 75 60 45 30 15 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
RAZ = 0o
DO
P
VZA (o)
0 15 30 45 60 75 90
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
RAZ =180o
DO
P
A
Reflectance and DOP on the principal plane from thin cirrus with HP (1984)
and FIRE II (1991) size distributions
Summary
1. Wind direction and wave slope distribution models have little effect on
the polarization state of reflected light at TOA.
2. Based on Cox-and-Munk model without wind direction dependence,
DOP is symmetric and AOLP is supplementary to the principal plane.
3. Mapping the ADRTM results to uniform VZA grids makes the modeling
and the PDM more efficient.
4. Ocean wave shadowing slightly reduces the total reflectance and DOP
at large VZA, but affects the AOLP insignificantly.
5. Ice cloud particle size distribution affects the polarization insignificantly.
6. Future work is to compare model results with Parasol/RSP data and to
derive land surface reflection matrix from these data with the model.