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, WA
2Jet Propulsion Laboratory, Pasadena, CA
Radiative Transfer Equation
P  sˆ, sˆ' 
dI  r , sˆ 
  e I  r , sˆ    dsˆ' P  sˆ, sˆ '  I  r , sˆ ' 
ds
I  r , sˆ'  : Intensity at r in direction sˆ
 e : extinction coefficient
P  sˆ, sˆ'  : scattering from direction sˆ ' to direction sˆ
2
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
QCA
Foldy Lax
Bicontinuous
Model
Spheres, pair
distribution functions
Computer generation of
spheres
2 Size
parameters
Particle diameter
(2a);
Stickiness (τ)
Analytical QCA
Particle diameter (2a);
Stickiness (τ)
Computer generation
of snow
microstructures
<ζ>; b
Solution method
4
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 g  r  and structure
factor

Structure factor is the Fourier transform of h  r   g  r   1
H () 
f11 ()  
f 22 ()  
1
kK r
Nmax
1
kK r
Nmax
2n  1
 n(n  1) T
n 1
2n  1
 n(n  1) T
n 1
q()  n0 (1  n0 (2 )3 H ())
P11 ( )  f 11 ( ) q( )
2
P22 ( )  f 22 ( ) q( )
2
5
(M )
n
(M )
n
1
(2 )3



dr ( g (r )  1)e ipr
X n( M ) n (cos )  Tn( N ) X n( N ) n (cos ) 
X n( M ) n (cos )  Tn( N ) X n( N ) n (cos ) 
Scattering Rate: QCA Compared With
Classical Mie Scattering

Diameter = 1.4 mm; Stickiness parameter τ=0.1;

stickiness, adhere to form aggregates
QCA sticky has weaker frequency dependence than Mie scattering

Scattering Coefficient
3
10
2
10
1
s
 [1 / m]
10
0
10
-1
10
s By Mie Scattering
-2
10
s By QCA Sticky Particles
-3
10
s By Non-Sticky Particles
-4
10
1
2
10
10
Frequency [GHz]
6
Scattering Properties
1-2 polarization frame
 Phase matrix



I s  V0 P kˆs , kˆi I i
where I is the Stokes vector, P is the phase matrix
 P11
P
P   21
 P31

 P41
P12
P13
P22
P23
P32
P33
P42
P43
P14 
P24 
P34 

P44 
Scattering coefficient
 Mean cosine of scattering: angular distribution

 S   p  sˆ, sˆ'  d '    d  sin   P11     P22    

0

7
'
'
'
 p  sˆ, sˆ  sˆ  sˆ  d 
'
'
 p  sˆ, sˆ  d 



0
d  sin  cos   P11     P22    


0
d  sin   P11     P22    
Phase Matrix: Angular Dependence
QCA More Forward Scattering

Frequency = 17.5 GHz; Diameter = 1.4 mm; Stickiness parameter
τ=0.1

QCA predicts more forward scattering than Mie
Normalized Phase Matrices
0.35
P11(Mie) / s
0.3
P22(Mie) / s
P11(QCA) / s
0.25
P22(QCA) / s
0.2
0.15
0.1
0.05
0
0
20
40
60
80
100
 [deg]
8
120
140
160
180
Scattering Properties Comparison
Scattering
properties
Frequency
dependence
mean cosine
Cross-pol in
phase matrix
9
Independent
scattering
4.0
0
Dipole pattern
0
QCA
Foldy Lax
Bicontinuous
As low
as 2.8
Up to
0.3
0
Consistent with
QCA
Consistent with
QCA
Up to 15 dB
below like-pol
As low as 2.5
Up to 0.6
Up to 7 dB below
like-pol
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
Computer Generation Of Dense Sticky Particles


Random Shuffling
Use Bonding States
◦ (a) Unbonded
◦ (b) Single-bond
◦ (c) Double-bond
◦ (d) Triple bond


11

Simulated sticky particles fv = 40%
Kranendonk-Frenkel algorithm to calculate the probability ,
dependent on stickiness
Aggregates formed from sequence of bonding
Solutions of Maxwell Equations using
Foldy-Lax equations
E E
ex
i
Eiex
E inc
G0
12
N
inc
 G0  T j E exj
field on particle i
incident field
Green’s function
j 1
j i
Tj
E exj
Mie scattering coefficients
field 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
QCA
Foldy Lax
Bicontinuous
Model
Spheres,
Computer generation of
spheres
Size parameters
Particle diameter
(2a);
Stickiness (τ)
Analytical QCA
Particle diameter (2a);
Stickiness (τ)
Computer generation
of snow
microstructures
<ζ>; b
Solution method
14
Numerical solution of
Maxwell equation using
Foldy-Lax equations
QCA
Foldy Lax
Scattering
properties
Frequency
dependence
mean cosine
Independent
scattering
4.0
0
As low as
2.8
Up to 0.3
Cross-pol in
phase matrix
0
0
Consistent with
QCA
Consistent with
QCA
Nonzero
Dipole interactions
Up to 15 dB below
like-pol
Numerical solutions of
Maxwell equations
using DDA / FFT
Bicontinuous
As low as 2.5
Up to 0.6
Nonzero
Dipole interactions
Up to 7 dB below
like-pol
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
Bicontinuous Model: Computer
Generation of Terrestrial Snow

Generation: superimposing a large number of
stochastic waves
1
S (r ) 
cos(  r   )

N
N
n 1
n
n
 n  [0, 2 ] uniformly distributed
 n vector :

, ,  
 ,  are uniformly distributed in [0,  ] &[0, 2 ]
 follows -distribution, whose mean value is  .

Cutting level α determined by fraction volume


1 (ice), if S r  

  S r   


0 (air), if S r  



16
Bicontinuous Model: Generation

Computer generated snow pictures vs.
real snow picture
fV  30%
  4500  m 1 
Horizontal
Plane
b  1.5
Y
l Plane
X
5mm
10mm
15mm
20mm
X
17
Depth Hoar (30%): 3 cm * 3 cm picture
A. Wiesmann, C. Mätzler, and T. Weise, "Radiometric and
structural measurements of snow samples," Radio Sci., vol.
33, pp. 273-289, 1998.
Numerical Solution Of Maxwell
Equation

Volume integral equation

E r E

inc

r 
k2
  
'
'
'
Matrix equations
pi  i E

18

Discrete Dipole Approximation (DDA): in
each cube
p j   V  Pj


d r G r , r  P r   r r  1 L  E r



V

V


inc
i
N
  i  Aij  p j
j 1
j i
Matrix-vector product by FFT
Bicontinuous Parameters

Bicontinuous parameters (α, <ζ>, b)
One to one relation between α and fV
fV 

1
1  erf   
2
Parameter <ζ> : inverse size
◦ Grain sizes decrease as <ζ> increases
◦ ζ follows Gamma distribution with mean value <ζ>
p   

 b  1
1
  b  1

b 1
b
  

 
exp

b

1












Parameter b determines the size distribution
◦ Size distribution uniform for large b
◦ Broad size distributon for small b
19
SSA and Correlation function of Bicontinuous
Medium
2 
b2
exp  2 2 
b 1
fV ice
SSA   3

ACF  d    Cm   Cs  d  
m 1
where Cs  d    cos  
  d

b

1


  arctan 
20
b 1
m
 sin  b 
sin  b
Real Snow Parameters

Real 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
Measured SSA 
6
ice D0
◦ Dmax: Prevailing grain size, visually determined
◦ Empirical fit: Dmax=2.73D0
21
Bicontinuous Model: Parameters

fV  30%
fV  30%
fV  30%
  3500  m 1 
  4500  m 1 
  5500  m 1 
Horizontal
Vertical
b Plane
1.5 Plane
22
X
5mm5mm
10mm
10mm
15mm
15mm
20mm
20mm
Y
Z
X
Vertical
Horizontal
b Plane
1.5 Plane
Y
5mm5mm
10mm
10mm
15mm
15mm
20mm
20mm
Y
Z
e
Dependences on <ζ>
X
X
Horizontal
b  1.5 Plane
5mm
10mm
15mm
20mm
X
Bicontinuous Model: Parameters
Dependence on parameter b: b increases
fV  30%
fV  30%
fV  30%
  4500  m 1 
  4500  m 1 
  4500  m 1 
X
23
Horizontal
Plane Horizontal
Vertical
Plane
Plane
Vertical
P
b  1.0 Plane
bHorizontal
 2.0
Y
X
X
5mm5mm
10mm
10mm
15mm
15mm
20mm
20mm
Y
5mm5mm5mm
10mm
10mm
10mm
15mm
15mm
15mm
20mm
20mm
20mm
X
Z
5mm
10mm
15mm
20mm
Z
Z
Vertical
Plane
b  0.5
Y

X
X
Bicontinuous Model: Correlation Function
Close To Exponential

Spatial auto correlation function

ACF  d    Cm   Cs  d  , where Cs  d    cos  
m
m 1
24
b 1
 sin  b 
sin  b
fV  30%;   11500  m 1  ; b  1.0;
fV  30%;   6000  m 1  ; b  1.0
correlation length  0.15 [ mm]
correlation length  0.3 [mm]
Bicontinuous Model: Log Scale
Correlation Function
25
fV  30%;   11500  m 1  ; b  1.0;
fV  30%;   6000  m 1  ; b  1.0
correlation length  0.15 [ mm]
correlation length  0.3 [mm]
Bicontinuous Model: Specific Surface Area
In Microwave Regime
Analytical expression
2 
SSA   3
Numerical procedure: Use digitized picture, discretize according to
Vertical Plane
microwave resolutions
Δx [mm]
0.4
0.5
0.6
0.8
SSA [cm2/g]
83.2
70.3
65.9
50.1
Y

Example: <ζ>=6000 [m-1],
b2
exp  2 2  b=1.5, f =30%
V
b 1
Bicontinuous SSA=71.8 [cm2/g]
fV ice
Z

10mm

Count surface area
X
26
5
10
15
20
Bicontinuous Model: Phase Matrix

Mean cosine:
p  sˆ, sˆ  sˆ  sˆ  d 


 p  sˆ, sˆ  d 
'
'
'
27
'
'
b  0.5
b  1.0
b  2.0
  0.40,  s  3.98  m 1 
  0.33,  s  2.04  m 1 
  0.029,  s  0.762  m 1 
Passive remote sensing: Effects Of
‘Mean Cosine’


Brightness temperature increases with for the same κS
◦ Physical temperature is 250 K
◦ Optical thickness = κSd; All curves have same κS
28
Mean Cosine Comparisons

29
Mean 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
Data Validation With SnowSCAT

Data collected
◦ At IOA snow pit
◦ Radar backscattering and ground data: Dec. 28,
2010~Mar. 1, 2011

Data
◦
◦
◦
◦
◦
◦
30
Time series backscattering
Time series SWE
SSA
Density
Depths of multilayer structure
Grain sizes
Comparisons With SnowSCAT

Time series data for 9 different days in the same IOA snow pit
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

Ground truth of data point #8
◦
◦

# layer
1
2
3
4
5
Depth [cm]
1
10
20
7
22
Density [g/cm3]
0.112
0.148
0.212
0.192
0.204
Grain size [mm]
0.5
0.8
0.8
1.5
3.0
Typical values of measured SSA
◦
◦
◦
31
Bottom layer is the thickest layer
Bottom layer has the largest grain size
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]
Data Validation With SnowSCAT

32
Bicontinuous 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
1.2
fV
12.2%
16.1%
23.0%
20.9%
22.1%
Data Validation With SnowSCAT

33
Bicontinuous extracted parameters
Layer
<ζ> [m-1]
b
Optical
thickness
Mean
cosine μ
Correlation
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
Data Validation With SnowSCAT

Co-polarization at 16.7 GHz
Co-polarization @ 16.7 GHz
-7
vv [dB]
-8
Measurement
DMRT
-9
-10
-11
34
60
80
100
SWE [mm]
120
DMRT Models Comparison
QCA
Foldy Lax
Bicontinuous
Model
Spheres, pair
distribution functions
Computer generation of
spheres
Size parameters
Particle diameter
(2a);
Stickiness (τ)
Analytical QCA
Particle diameter (2a);
Stickiness (τ)
Computer generation
of snow
microstructures
<ζ>; b
Solution method
Scattering
properties
Frequency
dependence
mean cosine
Cross-pol in
phase matrix
35
Independent
scattering
4.0
Numerical solution of
Maxwell equation using
Foldy-Lax equations
QCA
Foldy Lax
0, dipole pattern
As low as
2.8
Up to 0.3
0
0
Consistent with
QCA
Consistent with
QCA
Nonzero
Dipole interactions
Up to 15 dB below
like-pol
Numerical solutions of
Maxwell equations
using DDA / FFT
Bicontinuous
As low as 2.5
Up to 0.6
Nonzero
Dipole interactions
Up to 7 dB below
like-pol
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

36
Compare with SnowSCAT scatterometer data Using ground truth snow
measurements