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.