CONICAL ELECTROMAGNETIC WAVES DIFFRACTION FROM SASTRUGI TYPE SURFACES OF LAYERED SNOW DUNES ON GREENLAND ICE SHEETS IN

PASSIVE MICROWAVE REMOTE SENSING

Wenmo ChangLeung Tsang

Department of Electrical EngineeringUniversity of Washington

Outline

• Motivation– Observations in passive microwave remote sensing– Large 3rd and 4th Stokes parameters

• Scattering physics– Rough surfaces : Large slope and large height– Total internal reflection in layered media

• Electromagnetic methodology– Maxwell equations for rough surface– Radiative transfer theory for layered media

• Results and discussion

Greenland’s snow profile

• Wind induced– Sastrugi surface

• Large RMS height– 20 cm– 7 wavelengths @ 10.7 GHz– 12 wavelengths @ 18.7 GHz

• Large slope

Photo courtesy of Quintin Lake www.quintinlake.com

Four Stokes parameters in passive microwave remote sensing

• Microwave polarimetric signatures

2

2

*

*

1

2 Re

2 Im

vv

hh

v h

v h

ETET

IU E E

V E E

Observations

• WindSat data over the Summit site

• Large 3rd and 4th Stokes parameters

• Up to 15 K for 10.7 GHz, 18.7 GHz and 37 GHz

Li, L.; Gaiser, P.; Twarog, E.; Long, D.; Albert, M.; , "WindSat Polarimetric View of Greenland," Geoscience and Remote Sensing Symposium, 2006. IGARSS 2006. IEEE International Conference on , vol., no., pp.3824-3827, July 31 2006-Aug. 4 2006

Outline

• Motivations• Scattering physics• Electromagnetic methodology• Results and discussion

Physical model• Large height and large slope coupled with

subsurface total internal reflection• 1-D roughness: azimuthal asymmetry

ki ks

θiφi

θs

φs

↑ Sastrugi surface

Underlying snow layer 1

Underlying snow layer 2

More underlying snow layers

Multilayered snow↓

Computer generation of Sastrugi surfaces

• Statistical data needed for Sastrugi profile

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

1.5

-2 -1.98 -1.96 -1.94 -1.92 -1.9 -1.88 -1.86 -1.84 -1.82 -1.80

0.05

0.1

0.15

0.2

0.25Photo courtesy of Quintin Lake www.quintinlake.com

Large angle transmission

20°

15°

60.2°

Incident angle=55°

Total internal reflection

ε1=1.8, dense snow

ε2=1.3, less dense snow

ε0=1, air

2

2

*

*

1

2 Re

2 Im

vv

hh

v h

v h

ETET

IU E E

V E E

Critical angle=58.2°

Large slope

2 1 1 2

2 1 1 2

1 2

1 2

1 exp( )

1 exp( )

z zv v

z z

z zh h

z z

v h

k kR j

k k

k kR j

k k

• Phase shifts of v- and h-pol are different

• Non-zero U and V are generated

Scattering physics: results based on Maxwell equations

• Air to snow– Ɵi=55 deg

– ɸi=30 deg

– εrsnow =1.8

– εrunder =1.3

10 20 30 40 50 60 70 80 90-50

-40

-30

-20

-10

0

10

20

30

X: 37.6Y: 19.8

Bistatic transmission coefficients

Tra

nsm

itted

ene

rgy

/ dB

t / deg

X: 60.44Y: 16.62

vv

hv

Specular

• Two peaks in transmission• Specular transmission angle in snow: 37.6 deg• A secondary peak: around 60.4 deg

• Critical angle between snow and underlying layers: 58.2 deg• The 60.4-deg transmission will have total internal reflection

ki ks

θiφi

θs

φs

Underlying snow layers

kt

θt

Outline

• Motivations• Scattering physics• Electromagnetic methodology• Results and discussion

Challenges in electromagnetic model• Height of profile

– Past: small to moderate height– New: large height up to 7 wavelengths

• Fluctuations of microwave signatures in simulations– Past coherent 3-D MoM (Tsang et al., 2008)

• Fluctuations due to roughness• Fluctuations due to coherent multiple reflections of

layering

– Present model has less fluctuations

Present hybrid model

• Previous 3D MoM coherent model (Tsang et al., 2008) for comparisons

• (1) Maxwell equations for rough surface scattering to numerically derive rough surface’s bistatic coefficients

• (2) Radiative transfer theory for layered media• Combine (1) and (2) : rough surface’s bistatic

coefficients from Maxwell equations used as boundary conditions for radiative transfer

Rough surface’s boundary conditions

• Numerical methods to solve Maxwell equations (integral equations)– Conical diffraction– Field components obtained

( )

( )

( , ( ))

ˆ · ( , )

( , ( ))

ˆ · ( , )

y

t t y z f x

y

t t y z f x

H x z f x x

n H x z x

E x z f x x

n E x z x

1

1

1

1

( , ) ( , )

ˆ ˆ· ·

(x, z) ( , )

ˆ ˆ· ·

=y y

t t y t t y

y y

t t y t t y

E x z E x z

n E n E

H H x z

n H n H

Four types of surface unknowns Continuity boundary conditions

Numerical solutions• Numerical methods to solve Maxwell

equations (integral equations)– Conical diffraction– Field components obtained

( )

( )

, , for r in region 0ˆ, · , ; , , ; ,

0, for r in region 1

, , for r in region 0ˆ, · , ; , , ; ,

0, for r in region 1

yyi t t

z f x

yyi t t

z f x

E x zE x z ds x n g x z x z g x z x z x

H x zH x z ds x n g x z x z g x z x z x

1 1 1 1( )

1

1 1 1 1

1

0, for r in region 0, ; , , ; ,

, for r in region 1

0, for r in region 0ˆ · , ; , , ; ,

, , for r in region 1

tz f x

y

t t

y

ds x g x z x z g x z x z xE x z

ds x n g x z x z g x z x z xH x z

Numerical requirements• Physical Parameters

– RMS height: 20 cm• 7.1 wavelengths @ 10.7 GHz• 12.4 wavelengths @ 18.7 GHz

• Numerical parameters– Surface length: 4 m

• 142 wavelengths @ 10.7 GHz• 249 wavelengths @ 18.7 GHz

• Number of surface unknowns: up to 20,000• Linear solver

– Direct solver based on LU decomposition– In the future: multi-level UV

Bistatic coefficients

• Bistatic scattering and transmission coefficients

2 22

2

4 2, ; , lim

coscos

s s

s s i ir s

i inc ii

r E E

PE A

/2 /2* * *

/2 /2 00

1 1Re Re 2 cos

2 2 4

L L Linci i xi yi yi xi iL L zz

g kP dx E H dx E H E H

01( , ,0)cos ( , , , ) ( , ,0)cost

t t t t i t t i i u i i iI d I

11( , ,0) cos ( , , , ) ( , ,0)cosr

d s s s i s s i i u i i iI d I

Boundary conditions for radiative transfer

• ‘Boundary condition’ for radiative transfer of layered media

• Multiple reflection– Iterative scheme

• Solid angle integral

Matrices formed by the numerical bistatic coefficients

t rFlatSurface Subsurface FlatSurfau ce dI I I

At upper rough boundary

At lower flat subsurface

Periodic profile• Single realization• Hybrid model compared with coherent 3-D MoM

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

Azimuthal angle /

T v / K

Hybrid

Coherent 3-D MoM

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

Azimuthal angle /

T h / K

Hybrid

Coherent 3-D MoM

0 10 20 30 40 50 60 70 80 90-20

-15

-10

-5

0

5

10

15

20

Azimuthal angle /

U /

K

Hybrid

Coherent 3-D MoM

0 10 20 30 40 50 60 70 80 90-20

-15

-10

-5

0

5

10

15

20

Azimuthal angle /

V /

K

Hybrid

Coherent 3-D MoM

Geometry

Outline

• Motivations• Scattering physics• Electromagnetic methodology• Results and discussion

Sastrugi surface at 10.7 GHz

• Averaging over 5 realizations

Geometry

0 10 20 30 40 50 60 70 80 90150

160

170

180

190

200

210

220

230

240

250

Azimuthal angle /

10.7 GHz

T v

0 10 20 30 40 50 60 70 80 90140

150

160

170

180

190

200

210

220

230

Azimuthal angle /

10.7 GHz

T h

0 10 20 30 40 50 60 70 80 90-20

-15

-10

-5

0

5

10

15

Azimuthal angle /

10.7 GHz

U

0 10 20 30 40 50 60 70 80 90-10

-8

-6

-4

-2

0

2

4

6

8

10

Azimuthal angle /

10.7 GHz

V • 3rd and 4th Stokes parameters up to -15 K / +10 K

Sastrugi surface at 18.7 GHz

• Averaging over 5 realizations

0 10 20 30 40 50 60 70 80 90120

140

160

180

200

220

240

260

Azimuthal angle /

18.7 GHz

T v

0 10 20 30 40 50 60 70 80 90120

140

160

180

200

220

240

Azimuthal angle /

18.7 GHz

T h

0 10 20 30 40 50 60 70 80 90-4

-2

0

2

4

6

8

Azimuthal angle /

18.7 GHz

U

0 10 20 30 40 50 60 70 80 90-2

0

2

4

6

8

10

12

Azimuthal angle /

18.7 GHz

V

• 3rd and 4th Stokes parameters up to -2 K / +10 K

Geometry

Summary• Hybrid model

– 2-D MoM for rough surface– Radiative transfer for layer media– Combine rough surface boundary conditions with

radiative transfer

• Numerical results of the model– Large 3rd and 4th Stokes parameters up to -15 K / 10 K– Less fluctuations– Both 10.7 GHz and 18.7 GHz show up to 15K

Thank You for Your Attention!