# Emissivities of rough surface over layered media in microwave remote sensing of snow

**ABSTRACT** The rough surfaces in Greenland are exhibited as sastrugi. The roughness heights are less than 8 cm for much of the year except in late winter and spring, when they increase to 25 cm or less. Roughness profiles were also related to snow and firn ventilation. WindSat, launched in January 2003, was the first spaceborne polarimetric radiometer to measure all 4 elements of Stokes vector, viz., the vertical polarized brightness temperatures, the horizontal polarized brightness temperatures, and the real and imaginary part of the cross-correlations of the vertical and horizontal polarizations. It was shown by Tsang (1984, 1990) that azimuthal asymmetry will create nonzero third and fourth Stokes parameter in passive microwave remote sensing. Thus the third and fourth Stokes parameters contain information of the azimuthal structure. Usually the third and fourth Stokes parameters are quite small between 0.5 K to 1 K over land and less than plusmn2.5 K over ocean. However, measurements of third and Stokes parameters over Greenland show surprising values of 10 K for the third Stokes parameter and between -10 K and 20 K for the fourth Stokes parameter. In this paper, we use physically based electromagnetic model to study the passive polarimetric remote sensing of snow in Greenland by consider the scattering and emission from a random rough surface over multi-layered media. We consider the random rough surface varied in only one horizontal direction so that azimuthal asymmetry exists in the 3-D problem. Dyadic Green's functions of multilayered medium (Tsang et al., 2000) is used to formulate the surface integral equation so that the polarization dependence of emission and scattering is accounted for systematically. The surface integral equations are solved by using the method of moments in conjunction with fast numerical algorithms such as the multilevel UV method. Numerical results of brightness temperatures are illustrated for all four Stokes parameters to demonstrate the si-

gnatures of sastrugi in passive microwave remote sensing. To account for the large third and fourth Stokes parameters, we also consider the case of anisotropic scatterers in volume scattering. Full multiple volume scattering are studied with numerical solutions of the radiative transfer equations for non-spherical scatterers with preferred orientation.

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**ABSTRACT:**WindSat has systematically collected the first global fully polarimetric passive microwave data over both land and ocean. As the first spaceborne polarimetric microwave radiometer, it was designed to measure ocean surface wind speed and direction by including the third and fourth Stokes parameters, which are mostly related to the asymmetric structures of the ocean surface roughness. Although designed for wind vector retrieval, WindSat data are also collected over land and ice, and this new data has revealed, for the first time, significant land signals in the third and fourth Stokes parameter channels, particularly over Greenland and the Antarctic ice sheets. The third and fourth Stokes parameters show well-defined large azimuth modulations that appear to be correlated with geophysical variations, particularly snow structure, melting, and metamorphism, and have distinct seasonal variation. The polarimetric signatures are relatively weak in the summer and are strongest around spring. This corresponds well with the formation and erosion of the sastrugi in the dry snow zone and snowmelt in the soaked zone. In this paper, we present the full polarimetric signatures obtained from WindSat over Greenland, and use a simple empirical observation model to quantify the azimuthal variations of the signatures in space and time.IEEE Transactions on Geoscience and Remote Sensing 10/2008; · 3.47 Impact Factor

Page 1

Emissivities of Rough Surface over Layered Media in

Microwave Remote Sensing of Snow

1Peng Xu, 2Leung Tsang, 3Li Li and 1Kun Shan Chen

1Center for Space and Remote Sensing Research, National Central University, Chung-Li, Taiwan 32054

2Department of Electrical Engineering, University of Washington, Box 352500, Seattle, WA 98195-2500, USA

3Naval Research Laboratory, 4555 Overlook Avenue, SW Washington, DC 20375

tsang@ee.washington.edu

Abstract—The rough surfaces in Greenland are exhibited as

sastrugi. The roughness heights are less than 8 cm for much of

the year except in late winter and spring, when they increase to

25 cm or less. Roughness profiles were also related to snow and

firn ventilation. WindSat, launched in January 2003, was the

first spaceborne polarimetric radiometer to measure all 4

elements of Stokes vector, viz., the vertical polarized brightness

temperatures, the horizontal polarized brightness temperatures,

and the real and imaginary part of the cross-correlations of the

vertical and horizontal polarizations. It was shown by Tsang

(1984, 1990) that azimuthal asymmetry will create nonzero third

and fourth Stokes parameter in passive microwave remote

sensing. Thus the third and fourth Stokes parameters contain

information of the azimuthal structure. Usually the third and

fourth Stokes parameters are quite small between 0.5K to 1K

over land and less than ±2.5 K over ocean. However,

measurements of third and Stokes parameters over Greenland

show surprising values of 10K for the third Stokes parameter

and between -10K and 20K for the fourth Stokes parameter.

In this paper, we use physically based electromagnetic model to

study the passive polarimetric remote sensing of snow in

Greenland by consider the scattering and emission from a

random rough surface over multi-layered media. We consider

the random rough surface varied in only one horizontal direction

so that azimuthal asymmetry exists in the 3-D problem. Dyadic

Green’s functions of multilayered medium (Tsang et al., 2000) is

used to formulate the surface integral equation so that the

polarization dependence of emission and scattering is accounted

for systematically. The surface integral equations are solved by

using the method of moments in conjunction with fast numerical

algorithms such as the multilevel UV method. Numerical results

of brightness temperatures are illustrated for all four Stokes

parameters to demonstrate the signatures of sastrugi in passive

microwave remote sensing. To account for the large third and

fourth Stokes parameters, we also consider the case of

anisotropic scatterers in volume scattering. Full multiple volume

scattering are studied with numerical solutions of the radiative

transfer equations for non-spherical scatterers with preferred

orientation.

Keywords-Stokes parameters; scattering from rough surface;

emissivity; layered media

I.

INTRODUCTION

Active and passive microwave remote sensing have been

applied to Greenland ice sheet. The remote sensing is

important as the microwave signatures yield information of

the physical parameters of the snow. By observing the

changes in the physical parameters over time, interannual

changes across the Greenland ice sheet can be monitored.

Understanding the changes of the snow surface can be related

to global climate change. Active microwave remote sensing

data from the Ku band NASA Scatterometer (NSCAT) and

from the C band European Remote sensing satellite Advanced

Microwave instrument scatterometer [1] to study the

dependence of radar cross section on incidence angle and

azimuthal angle, spatial gradient and temporal rate of change.

An empirical model is used to parameterize the scattering

signature over ice sheets. The model parameters are empirical

parameters rather than physical parameters. Interpretation of

passive microwave measurements of SSM/I over Greenland

have also been made using a layered model of snow and ice

[2].

The changes in the radar cross section can be related to

snow melt and to changes in accumulation. Ashcraft and

Long [1] were also able to demonstrate strong azimuth

dependence of the backscattering signatures by examining the

radar cross section as a function of azimuthal angle for a range

of azimuthal angles. The azimuth angle dependence can be

used to relate to the surface profiles particularly to the wind-

formed erosion features known as sastrugi. Sastrugi crests are

parallel to the wind direction. WindSat, launched in January

2003, was the first spaceborne polarimetric radiometer to

measure all 4 elements of Stokes vector, viz., the vertical

polarized brightness temperatures, the horizontal polarized

brightness temperatures, and the real and imaginary part of the

cross-correlations of the vertical and horizontal polarizations

1-4244-1212-9/07/$25.00 ©2007 IEEE.1436

Page 2

[3]. It was shown by Tsang [4, 5] that azimuthal asymmetry

will create nonzero third and fourth Stokes parameter in

passive microwave remote sensing. Thus the third and fourth

Stokes parameters contain information of the azimuthal

structure. This has the distinct advantage that such azimuthal

structure is determined at a single azimuthal angle in

polarimetric passive remote sensing. Usually the third and

fourth Stokes parameters are quite small between 0.5K to 1K

over land. However, measurements of third and Stokes

parameters over Greenland show surprising values of 10K for

the third Stokes parameter and between -10K and 20K for the

fourth Stokes parameter. In this paper, we use numerical

solution of Maxwell equations to study the problem of

polarimetric passive microwave remote sensing of a random

rough surface over multi-layered structure.

The sastrugi has rms height of 8 cm to 25 cm and exceed

the wavelength significantly from 10 GHz to 37 GHz. It is

well known that Greenland firn and Antarctica firn has multi-

layering structures that cause strong multiple reflections. The

multi-layering structures have thicknesses of the order of

several centimeters for each layer with each layer having a

different density and grain from the adjacent layers. Thus for

the case of dry snow, microwaves can penetrate up to several

meters at 10GHz, 19 Ghz and 37 Ghz depending on the grain

size. Hence there can be 20 to 30 layering boundaries within a

depth of several meters. West et al. [6] used a multilayered

model to study such reflections and found significantly

decrease of emissivities due to these centimeters layering.

In this paper, we use the surface integral equation approach

to treat 3D scattering, emission and absorption of a random

rough surface over a multilayered medium. The random

rough surface is assumed to vary in only one horizontal

direction so that azimuthal asymmetry exists in the 3-D

problem. Dyadic Green’s functions of multilayered medium

[7] is used to formulate the surface integral equation so that

the polarization dependence of emission and scattering is

accounted for systematically. Numerically results are

[

×⋅⋅+⋅

∫∫

illustrated for polarimetric passive microwave remote sensing.

The surface roughness profiles are based on that of sastrugi as

measured in Greenland ice sheets.

II.

FORMULATION OF ROUGH SURFACE OVER LAYERED

MEDIA

Fig. 1 Geometry of a rough surface over multilayered media

Consider an oblique tapered plane wave

upon a random rough surface profile

multilayered media as shown in Fig. 1. Note that

)(min

1

xfd >

so that the first flat boundary will not cut across

the rough surface. The region zero upper rough surface is air,

and the other layers are with permittivities of

respectively. Only boundary between region 0 and 1 is a

rough surface.

inc

. Below is

ψ

)(x

impinging

fz =

N

ε

,

ε

,

ε

,

21

L

,

Applying Green’s theorem to both media above and below

rough surface, we have

]

<

>

⋅

=×⋅×∇⋅+

)(for

)( for

0

)(

ˆ

y

) 'r(' ˆn) ' r,(

ˆ

y ) 'r( ' ˆn) 'r,(

ˆ

y')(

ˆ

y

inc

xfz

xfz

rE

ErGHrGi dSrE

ωµ

(1a)

[][]

<

>

⋅

=×⋅×∇⋅+×⋅⋅−×⋅×∇⋅+×⋅⋅−

∫∫∫∫

)(for

)(for

)(

ˆ

y

0

) '

r

( ' ˆ

n

) '

r

,(

ˆ

y

) '

r

(' ˆ

n

) '

r

,(

ˆ

y

') '

r

(' ˆ

n

) '

r

,(

ˆ

y

) '

r

(' ˆ

n

) '

r

,(

ˆ

y

'

1

11111111

xfz

xfz

rE

ErGHrGidSErGHrGidS

RR

ωµωµ

(1b)

[]

<

>

⋅

=×⋅⋅−×⋅×∇⋅+⋅

∫∫

)(for

)( for

0

)(

ˆ

y

) 'r( ' ˆn ) 'r,(

ˆ

y ) 'r( ' ˆn) 'r,(

ˆ

y')(

ˆ

y

inc

xfz

xfz

rH

ErGiHrGdSrH

ωε

(1c)

[][]

<

>

⋅

=×⋅⋅−×⋅×∇⋅−×⋅⋅−×⋅×∇⋅−

∫∫ ∫∫

)(for

)(for

)(

ˆ

y

0

) 'r(' ˆn) 'r,(

ˆ

y) 'r(' ˆn) 'r,(

ˆ

y') 'r(' ˆn) 'r,(

ˆ

y) 'r(' ˆn) 'r,(

ˆ

y'

1

1111111111

xfz

xfz

rH

ErGiHrGdS ErGiHrGdS

RR

ωεωε

(1d)

where [8]

()

∫∫

−+−+−

∇∇+=

∇∇+=

') '

y

() '

x

(exp

1

k

8

1

k

) '

r

,(

1

k

∫∫

) '

r

,(

222

zzikyikxikdkdk

i

IrgIrG

zyx

z

yx

π

(2a)

[]

()

'for ) 'z() 'y() 'x(exp)(

ˆ

h)(

ˆ

h)(

ˆ )

z

(

ˆ

e

1

8

) '

r

,(

111111111

1

2

1

zzz ikyikx ikkkkek

k

dkdk

i

rG

zyxzzz

z

1

yx

<−−−+−−−+−−=

π

(2b)

[]

() ) 'z( ) 'y() 'x(exp)(

ˆ

h)(

ˆ

)(

ˆ )

z

(

ˆ

e

8

) 'r,(

11111

TM

1111

TE

1

2

1

zikyikxikkkhRkekR

k

dkdk

i

rG

zyxzzz

z

yxR

++−+−−+−=

∫∫

π

(2c)

ε

ε4

Region 3

Region 4

Region N-1

Region N

ε

εN

N-1

- d N-1

3

- d 3

Region 2

Region 1

Region 0

k

y

i

i v

θ

i

2 ε

ε1

- d 2

- d 1

0 ε

φi

x

h i

z

1-4244-1212-9/07/$25.00 ©2007 IEEE. 1437

Page 3

Next we eliminate dependence of

)exp(yiky

because of uniform in y direction. Let

w

H represent the magnetic field without the y dependence of

)exp(

yiky

, i.e.

) 'y exp() 'z, 'x(' ˆn) 'r(' ˆnikHH

yw

×=×

[]

()

[]

()

The 2-d Green functions are

(

(

4

++−−⋅+−⋅−−=+−

++−−⋅+−⋅=+−

∫

∫

∞

∞−

∞

∞−

) 'z() 'x( exp)(

ˆ

h)(

ˆ

e

ˆ

y)(

ˆ )

z

(

ˆ

h

ˆ

y

1

4

) '

z

, '

x

(

) '

z

() '

x

(exp)(

ˆ)

h

(

ˆ

h

ˆ

y

)(

ˆ )

z

(

ˆ

e

ˆ

y

1

4

) '

z

, '

x

(

11111

TM

k

iy

1111

TE

k

iy

1

1

1

11111

TM

k

iy

1111

TE

k

iy

1

1

zikxikkkRkekR

k

dk

k

π

zxq

zikx ikkkRkekR

k

dk

i

π

zxp

zxzzz

z

xR

zxzkzkzkk

z

xR

iyiyiyiy

)3 (

)3 (

b

a

)

)

−+−=

−+−=

22

1

) 1 (

0

) 2(

1

22) 1 (

0

) 2(

) '

z

() '

x

(

) '

z

() '

x

(

4

i

zxkHg

zxkH

i

g

t

t

, where

+=−=

+=−=

2

1

2

x

2

iy

2

1

2

t1

2

z

2

x

2

iy

22

t

z

kkkkk

kkkkk

)4 (

)4 (

b

a

Then (1a) - (1d) can be simplified as

),(00) 'x(''' ˆ ) 'x('

ψ

)(

2

1

) 2() 2(

zxEg dxgndsx

ywi

P

tt

=⊕⊕⊕⋅∇−

∫∫

χψ

(5a)

( )

x

( )

x

( )

x

( )

x

( )

x0) 'x(

ˆ

y''

ˆ

x

ˆ

z'

ˆ

y''1

) '

x

(

ˆ

y

') '

x

('

ˆ

y

''

ˆ

x

ˆ

z

''' ˆ ) '

x

('

ψ

)(

2

1

1

2

t

11

2

t

) 2(

1

1

2

t

2

t1

1

2

t

2

) 2(

1

1

2

t

2

t

ε

1

1

2

t

1

) 2(

1

=⋅⊕

+

∂

∂

⋅+⋅+

−⊕

⋅+⊕⋅−

ψ

+

∂

∂

⋅+⋅∇−−

∫∫∫∫

∫∫∫∫∫

qdx

k

i

x

f

p dxipd

k

k

gd

k

k

k

p dx

k

k

gdx

k

k

qd

k

ik

x

f

qdxgndsx

RRR

iy iy

R

R

iy

R

P

tt

ξ

ωµ

ϕωµϕ ωµϕ

ωε

χχ

ε

ψψ

(5b)

),() 'x('' ' ˆ ) 'x(')(

2

1

00

) 2() 2(

zxHgdxgndsx

ywi

P

tt

=⊕⋅∇−⊕⊕

∫∫

ξϕϕ

(5c)

( )

x

( )

x

( )

x

( )

x

( )

x

0) 'x(

ˆ

y') '

x

('

ˆ

y

''

ˆ

x

ˆ

z

''' ˆ ) '

x

(')(

2

1

) '

x

(

ˆ

y

''

ˆ

x

ˆ

z

'

ˆ

y

''1

1

2

t

2

1

) 2(

1

2

t

2

t

1

1

2

t

1

) 2(

1

1

2

t

111

2

t

1

) 2 (

1

2

t

2

t

1

=⋅+⊕⋅−

+

∂

∂

⋅+ ⋅∇−−⊕

⋅−⊕

ψ

+

∂

∂

⋅−⋅−

−−

∫∫∫∫∫

∫∫∫∫

p dx

k

k

g dx

k

k

qd

k

ik

x

f

qdxgndsx

qdx

k

i

x

f

pdxipd

k

k

gd

k

k

k

RR

iy

R

P

tt

RRR

iy iy

ξξϕϕϕϕ

χ

ωε

ωεψωεψ

ωµ

(5d)

where

(

ϕ

()

()

[]

()

()

[]

Using MoM with rooftop basis function and Galerkin’s

method, (5) can be expressed as matrix equation

+

00

BBDC

ηη

where

+=

2

total

t

k

C

η

ρ

The

BABA

are similar with that decoupled case

[9] of non-layer except that k is replaced by

∇⋅+=

==

∇⋅+=

==

=

=

)(

2

)(

2

,

ˆ

n1)(

)(,)(

,

ˆ

n1)(

)(,)

xfz

ywttx

yw

xfz

yw ttx

yw

zxEfx

xfzxEx

zxHfx

xfzxHx

χ

ψ

ξ

) 6 (

)6 (

)6 (

)6 (

d

c

b

a

=

ψ

+−−

0

0

1

00

) 1 (

total

) 1 () 1 ()1 (

2

) 1 (

total

2

) 0 () 0 (

) 1 () 1 (

total

) 1 (

total

)1 () 1 (

) 0 () 0 (

ywi

ywi

RR

RR

H

E

A

AB

DCABB

AB

ξ

ϕ

χ

ρ

ρ

(7)

(

τ

)

+−=

)1 () 1 () 1 (

total

)1 (

2

) 1 () 1 (

1

R

iy

R

CkC

k

A

k

k

AA

η

ρ

τ

and

2

t

2

t1

k

k

=

τ

,

ε

ε

ρ

1

=

.

) 1 () 1 () 0 () 0 (

tk , and

()

()

(

)

Let the incident wave be horizontal, vertical, left, right hand

circular, 45, -45 degrees polarizations, and solve equation (7)

for each case. We use fast solver such as UV [10]. Then we

get the Stokes parameters in direction

=

RC LC

TT

V

−

'

+−

'

−

'

+−

'

−

(

+

)x

−

'

x

∆=

⋅∆=

⋅++⋅∆=

⋅++⋅∆=

⋅∆=

22

1

) 1 (

1

22

'

)

2

1

) 1 (

mn

1

2

2

t

)1 (

mn

1

2

t

'1

2) 1 (

mn

1

2

t

'1

2) 1 (

mn

1

2)1 (

mn

)()(

(

)'(

4

ˆ

y

),(

ˆ

y

'

ˆ

x

ˆ

z

ˆ

y

),(

'

ˆ

x

ˆ

z

),(

ˆ

y

zzxxkH

zz

fzzxx ik

C

ZXq

k

ik

D

p

dx

d

k

ik

fpiC

ZXq

dx

d

k

ik

fZXqB

pA

t

xt

R

R

R

iy

xR

R

R

iy

xR

R

R

R

η

(8)

),( ϕθ

:

(

θ

)

−

−

=

−

v

h

v

h

S

TT

T

T

U

T

T

I

4545

,ϕ

(9)

1-4244-1212-9/07/$25.00 ©2007 IEEE.1438

Page 4

III. VECTOR RADIATIVE TRANSFER EQUATIONS FOR NON-

SPHERICAL SCATTERERS

The vector radiative transfer equation are [11, 12]

ϕθ

θ

),,(coszIk

dz

∫∫

⋅+

+⋅−=

π

2

π

ϕθ

(

ϕθ

;

ϕ

,

θ

(

θθ

d

ϕ

d

ϕ

,

θ

(

ϕθ

00

), ', ') ', ''sin''

)()

),,(

zIP

zCTF

zI d

e

(10)

Note that the extinction matrix is a function of both θ and φ.

In [11, 13], the equations are solved within the first order

scattering approximation. Large third and fourth Stokes

parameters can result. In this paper, we solve the equations

exactly using quadrature methods including the effects of full

multiple scattering.

ACKNOWLEDGMENT

The research in this paper was supported by NASA.

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