Electromagnetic Models In Active And Passive

Microwave Remote Sensing of Terrestrial Snow

Leung Tsang1, Xiaolan Xu2 and Simon Yueh2

1Department of Electrical Engineering, University of Washington, Seattle, WA2Jet Propulsion Laboratory, Pasadena, CA

Radiative Transfer Equation

2

' ' '

'

' '

ˆ,ˆ ˆ ˆ ˆ ˆ, , ,

ˆ ˆ, : Intensity at in direction

: extinction coefficient

ˆ ˆ ˆ ˆ, : scattering from direction to direction

e

e

dI r sI r s ds P s s I r s

dsI r s r s

P s s s s

'ˆ ˆ,P s s

Dense Media Radiative Transfer Equation (DMRT)Model 1) QCA

◦ Analytical Approximate Solution of Maxwell Equations

Model 2) Foldy Lax equations◦ Numerical Maxwell Equation Model (NMM3D)

Since 2009, Model 3) Bicontinuous medium: ◦ Numerical Maxwell Equation Model (NMM3D)◦ Bicontinuous media; Realistic microstructure of

snow◦ Comparisons With SnowSCAT

3

DMRT Models

4

QCA Foldy Lax BicontinuousModel Spheres, pair

distribution functions

Computer generation of spheres

Computer generation of

snow microstructures

2 Size parameters

Particle diameter (2a);

Stickiness (τ)

Particle diameter (2a);

Stickiness (τ)

<ζ>; b

Solution method

Analytical QCA Numerical solution of Maxwell equation using Foldy-Lax

equations

Numerical solutions of

Maxwell equations using DDA / FFT

Quasi-Crystalline Approximation (QCA)

Lorentz-Lorenz law; Generalized Ewald-Oseen theorem Phase matrix, pair distribution function and

structure factor Structure factor is the Fourier transform of

5

3

1( ) ( ( ) 1)(2 )

ip rH dr g r e

g r

1h r g r

)()()( 21111 qfP

)()()( 22222 qfP

max( ) ( ) ( ) ( )

111

1 2 1( ) (cos ) (cos )( 1)

NM M N N

n n n n n nnr

nf T X T XkK n n

max

( ) ( ) ( ) ( )22

1

1 2 1( ) (cos ) (cos )( 1)

NM M N N

n n n n n nnr

nf T X T XkK n n

))()2(1()( 300 Hnnq

Diameter = 1.4 mm; Stickiness parameter τ=0.1; stickiness, adhere to form aggregates QCA sticky has weaker frequency dependence than

Mie scattering

101

102

10-4

10-3

10-2

10-1

100

101

102

103

Frequency [GHz]

s [1 /

m]

Scattering Coefficient

s By Mie Scattering

s By QCA Sticky Particles

s By Non-Sticky Particles

Scattering Rate: QCA Compared With Classical Mie Scattering

6

7

Scattering Properties1-2 polarization frame Phase matrix

Scattering coefficientMean cosine of scattering: angular

distribution

0

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

ˆ ˆ,

where is the Stokes vector, is the phase matrix

s is iI V P k k I

I PP P P PP P P P

PP P P PP P P P

' '11 220

' ' '11 220

' '11 220

ˆ ˆ, sin

sin cosˆ ˆ ˆ ˆ,

ˆ ˆ, sin

S p s s d d P P

d P Pp s s s s d

p s s d d P P

0 20 40 60 80 100 120 140 160 1800

0.05

0.1

0.15

0.2

0.25

0.3

0.35

[deg]

Normalized Phase Matrices

P11(Mie) / s

P22(Mie) / s

P11(QCA) / s

P22(QCA) / s

Phase Matrix: Angular DependenceQCA More Forward Scattering

Frequency = 17.5 GHz; Diameter = 1.4 mm; Stickiness parameter τ=0.1

QCA predicts more forward scattering than Mie

8

Scattering Properties ComparisonScattering properties

Independent scattering

QCA Foldy Lax Bicontinuous

Frequency dependence

4.0 As low as 2.8

Consistent with QCA

As low as 2.5

mean cosine 0Dipole pattern

Up to 0.3

Consistent with QCA

Up to 0.6

Cross-pol in phase matrix

0 0 Up to 15 dB below like-pol

Up to 7 dB below like-pol

9

Dense Media Radiative Transfer Equation (DMRT)Model 1) QCA

◦ Analytical Approximate Solution of Maxwell Equations

Model 2) Foldy Lax equations◦ Numerical Maxwell Equation Model (NMM3D)

Model 3) Bicontinuous medium: ◦ Numerical Maxwell Equation Model (NMM3D)◦ Bicontinuous media; Realistic microstructure of

snow◦ Comparisons With SnowSCAT

10

Random Shuffling Use Bonding States

◦ (a) Unbonded◦ (b) Single-bond◦ (c) Double-bond◦ (d) Triple bond

Kranendonk-Frenkel algorithm to calculate the probability , dependent on stickiness

Aggregates formed from sequence of bonding

Computer Generation Of Dense Sticky Particles

Simulated sticky particles fv = 40%

11

Solutions of Maxwell Equations using Foldy-Lax equations

12

01

Nex inc exi j j

jj i

E E G T E

field on particle iexiEincE

0G

jTexjEincident field

Green’s function

Mie scattering coefficientsfield on particle j

Comparison Between Classical RT, DMRT / QCA and NMM3D NMM3D and QCA in agreement Weaker frequency dependence than

independent scattering

13

Model Comparison

14

Scattering

properties

Independent scattering

QCA Foldy Lax Bicontinuous

Frequency dependen

ce

4.0 As low as 2.8

Consistent with QCA

As low as 2.5

mean cosine

0 Up to 0.3 Consistent with QCA

Up to 0.6

Cross-pol in phase matrix

0 0 NonzeroDipole

interactionsUp to 15 dB

below like-pol

NonzeroDipole

interactionsUp to 7 dB below

like-pol

QCA Foldy Lax BicontinuousModel Spheres, Computer generation

of spheresComputer

generation of snow

microstructuresSize

parametersParticle diameter

(2a);Stickiness (τ)

Particle diameter (2a);

Stickiness (τ)

<ζ>; b

Solution method

Analytical QCA Numerical solution of Maxwell equation using Foldy-Lax

equations

Numerical solutions of

Maxwell equations using DDA / FFT

Dense Media Radiative Transfer Equation (DMRT)Model 1) QCA

◦ Analytical Approximate Solution of Maxwell Equations

Model 2) Foldy Lax equations◦ Numerical Maxwell Equation Model (NMM3D)

Model 3) Bicontinuous medium: ◦ Numerical Maxwell Equation Model (NMM3D)◦ Bicontinuous media; Realistic microstructure of

snow◦ Comparisons With SnowSCAT

15

16

Bicontinuous Model: Computer Generation of Terrestrial SnowGeneration: superimposing a large

number of stochastic waves

Cutting level α determined by fraction volume

1

1( ) cos( )

[0, 2 ] uniformly distributed

vector : , ,

, are uniformly distributed in [0, ] &[0, 2 ] follows -distribution, whose mean value is .

N

n nn

n

n

S r rN

1 (ice), if

0 (air), if

S rS r

S r

17

Bicontinuous Model: GenerationComputer generated snow

pictures vs. real snow picture1

30%

4500

1.5

Vf

m

b

A. Wiesmann, C. Mätzler, and T. Weise, "Radiometric and structural measurements of snow samples," Radio Sci., vol. 33, pp. 273-289, 1998.

Depth Hoar (30%): 3 cm * 3 cm picture

X

Z

Vertical Plane

5mm10mm15mm20mm

X

Y

Horizontal Plane

5mm10mm15mm20mm

18

Volume integral equation

Discrete Dipole Approximation (DDA): in each cube

Matrix equations

Matrix-vector product by FFT

Numerical Solution Of Maxwell Equation

2 ' ' '

, 1inc

rV V

kE r E r dr G r r P r r L E r

jjp V P

1

Ninci iji ii j

jj i

p E A p

19

Bicontinuous ParametersBicontinuous parameters (α, <ζ>, b)

One to one relation between α and fV

Parameter <ζ> : inverse size◦ Grain sizes decrease as <ζ> increases◦ ζ follows Gamma distribution with mean value <ζ>

Parameter b determines the size distribution◦ Size distribution uniform for large b◦ Broad size distributon for small b

1 1 erf2Vf

111 exp 11

bbbp b

b

20

SSA and Correlation function of Bicontinuous Medium

2

1

1

2 2 exp 213

sinwhere cos

sin

arctan1

V ice

m

m sm

bs

bbSSA

f

ACF d C C d

bC d

b

db

21

Real Snow ParametersReal snow parameters

◦Fraction Volume (fV) or density (ρ): fV = ρsnow / ρice

◦Auto Correlation Function (ACF)◦Specific Surface Area (SSA)◦Grain size

Two grain size parameters◦D0: Equivalent grain size relating to SSA

◦Dmax: Prevailing grain size, visually determined◦Empirical fit: Dmax=2.73D0

0

6Measuredice

SSAD

22

Bicontinuous Model: ParametersDependences on <ζ>

1

30%

4500

1.5

Vf

m

b

1

30%

3500

1.5

Vf

m

b

1

30%

5500

1.5

Vf

m

b

X

Z

Vertical Plane

5mm10mm15mm20mm

X

Y

Horizontal Plane

5mm10mm15mm20mm

X

Z

Vertical Plane

5mm10mm15mm20mm

X

YHorizontal Plane

5mm10mm15mm20mm

X

Z

Vertical Plane

5mm10mm15mm20mm

X

Y

Horizontal Plane

5mm10mm15mm20mm

23

Bicontinuous Model: ParametersDependence on parameter b: b

increases1

30%

4500

1.0

Vf

m

b

1

30%

4500

0.5

Vf

m

b

1

30%

4500

2.0

Vf

m

b

X

ZVertical Plane

5mm10mm15mm20mm

X

Y

Horizontal Plane

5mm10mm15mm20mm

X

ZVertical Plane

5mm10mm15mm20mm

X

Y

Horizontal Plane

5mm10mm15mm20mm

X

Z

Vertical Plane

5mm10mm15mm20mm

X

Y

Horizontal Plane

5mm10mm15mm20mm

24

Bicontinuous Model: Correlation FunctionClose To Exponential

Spatial auto correlation function130%; 6000 ; 1.0

correlation length 0.3[ ]Vf m b

mm

1

1

sin, where cos

sinm b

m s sm

bACF d C C d C d

b

130%; 11500 ; 1.0;

correlation length 0.15 [ ]Vf m b

mm

25

Bicontinuous Model: Log Scale Correlation Function

130%; 6000 ; 1.0

correlation length 0.3[ ]Vf m b

mm

130%; 11500 ; 1.0;

correlation length 0.15 [ ]Vf m b

mm

26

Bicontinuous Model: Specific Surface Area In Microwave Regime Analytical expression

Numerical procedure: Use digitized picture, discretize according to microwave resolutions

Count surface area

22 2 exp 213

V ice

bbSSA

f

Δx [mm] 0.4 0.5 0.6 0.8SSA

[cm2/g]83.2 70.3 65.9 50.1

Example: <ζ>=6000 [m-1], b=1.5, fV=30%Bicontinuous SSA=71.8 [cm2/g]

X

Z

Vertical Plane

10mm

X

Y

Horizontal Plane

5mm10mm15mm20mm

27

Bicontinuous Model: Phase MatrixMean cosine:

' ' '

' '

ˆ ˆ ˆ ˆ,

ˆ ˆ,

p s s s s d

p s s d

1

1.0

0.33, 2.04s

b

m

1

2.0

0.029, 0.762s

b

m

1

0.5

0.40, 3.98s

b

m

28

Brightness temperature increases with for the same κS

◦ Physical temperature is 250 K◦ Optical thickness = κSd; All curves have same κS

Passive remote sensing: Effects Of ‘Mean Cosine’

29

Mean Cosine ComparisonsMean cosine > 0, means forward

scattering is stronger than backward scattering

Models Mean cosine μ

1-μ Meaning

Bicontinuous

0.1 ~ 0.6 0.4 ~ 0.9

Forward scattering

Rayleigh Phase Matrix

0 1.0 Dipole scattering

HUT 0.96 0.04 Strong forward scattering

30

Data Validation With SnowSCATData collected

◦At IOA snow pit◦Radar backscattering and ground data:

Dec. 28, 2010~Mar. 1, 2011Data

◦Time series backscattering◦Time series SWE◦SSA◦Density◦Depths of multilayer structure◦Grain sizes

31

Comparisons With SnowSCAT Time series data for 9 different days in the same IOA snow pit

Ground truth of data point #8◦ Bottom layer is the thickest layer◦ Bottom layer has the largest grain size

Typical values of measured SSA◦ SSA measured in a different year from snow depth, density and grain size◦ Bottom layers : 59 ~ 124 [cm2/g]◦ Top and intermediate layers : 100 ~ 790 [cm2/g]

# layer 1 2 3 4 5Depth [cm] 1 10 20 7 22

Density [g/cm3] 0.112 0.148 0.212 0.192 0.204Grain size [mm] 0.5 0.8 0.8 1.5 3.0

Date 12/28/10

01/04/11

01/12/11

01/18/11

01/26/11

02/01/11

02/08/11

02/23/11

03/01/11

SWE [mm]

54 61 73.5 76 83 97 99 113 114

Snow Depth [cm]

31 39 45 52 49 53 69 60 59

32

Data Validation With SnowSCATBicontinuous input parameters

# layer 1 2 3 4 5<ζ> [m-1] 30000 20000 20000 10000 6000

b 1.0 1.0 1.5 1.5 1.2fV 12.2% 16.1% 23.0% 20.9% 22.1%

33

Data Validation With SnowSCATBicontinuous extracted

parametersLaye

r<ζ> [m-1] b Optical

thicknessMean

cosine μCorrelation length [mm]

Analytical SSA [cm2/g]

Numerical SSA

[cm2/g]1 30000 1.0 1.6×10-4 0.19 0.051 309 222

2 20000 1.0 8.7×10-3 0.14 0.080 228 200

3 20000 1.5 0.015 0.05 0.085 238 188

4 10000 1.5 0.012 0.11 0.17 117 95.2

5 6000 1.2 0.17 0.31 0.28 72 57.4

34

Data Validation With SnowSCATCo-polarization at 16.7 GHz

60 80 100 120-11

-10

-9

-8

-7

SW E [mm]

vv [d

B]

Co-polarization @ 16.7 GHz

MeasurementDMRT

DMRT Models Comparison

35

Scattering

properties

Independent scattering

QCA Foldy Lax Bicontinuous

Frequency dependen

ce

4.0 As low as 2.8

Consistent with QCA

As low as 2.5

mean cosine

0, dipole pattern

Up to 0.3 Consistent with QCA

Up to 0.6

Cross-pol in phase matrix

0 0 NonzeroDipole

interactionsUp to 15 dB

below like-pol

NonzeroDipole

interactionsUp to 7 dB below

like-pol

QCA Foldy Lax BicontinuousModel Spheres, pair

distribution functions

Computer generation of spheres

Computer generation of

snow microstructures

Size parameters

Particle diameter (2a);

Stickiness (τ)

Particle diameter (2a);

Stickiness (τ)

<ζ>; b

Solution method

Analytical QCA Numerical solution of Maxwell equation using Foldy-Lax

equations

Numerical solutions of

Maxwell equations using DDA / FFT

36

Summary Bicontinuous model

◦ Computer Generation of snow microstructures◦ Three parameters α, <ζ>, b◦ Correlation function close to exponential◦ correlation function and SSA◦ Grain size indirectly, empirically related to correlation

function and SSA◦ Computer Generate structures and solve Maxwell

equations numerically using DDA Compare with SnowSCAT scatterometer data Using ground

truth snow measurements