Development of a Precise and
Fast Multistream Scattering-Based
DMRT model with Jacobian
Miao Tian
A.J. Gasiewski
University of Colorado
Department of Electrical Engineering
Center for Environmental Technology
Boulder, CO, USA
IGARSS 2011
July, 2011
Vancouver, CA
Motivation for a UMRT
Attribute
UMRT
DOTLRT *
Fast, Stable Analytic Matrix Inversion
Yes
Yes
Fast Jacobian
Yes
Yes
Phase Matrix
Reduced Mie or DMRT
(4x4)
Reduced HG (2x2)
Polarization
Tri-polarization
+ 4th Stokes
Single-polarization
Interface Refraction / Internal Reflection
Yes
No
Radiation Stream Interpolation
Yes, cubic spline
No
Thermal Emission Approximation.
Linear dependence
Constant
Level/Layer Centric
Level Centric
Layer Centric
DMRT
Yes
No
* Voronovich, A.G., A.J. Gasiewski, and B.L. Weber, "A Fast Multistream Scattering
Based Jacobian for Microwave Radiance Assimilation,“ IEEE Trans. Geosci.
Remote Sensing, vol. 42, no. 8, pp. 1749-1761, August 2004.
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UMRT: Medium Model
• In UMRT, a planar-stratified model is used.
• Categorized into
1) Upper-half, Sparse medium:
a) General atmosphere
b) Weakly homogeneous and slightly
dissipative
Upper-half, Sparse
Medium
c) Independent scattering: scattering
intensity is the sum of scattering
intensities from each particle.
d) Particle size distribution function
e) Sparse medium radiative transfer
Lower-half, Dense
Medium
2) Lower-half, Dense medium:
a) Ice, snow, soil and etc.,
b) Inhomogeneous and strongly
dissipative
c) Multiple volumetric scattering:
scattering intensity is related to the
presence of other particles.
d) Pair distribution function (Percus-Yevick Approximation).
e) Dense medium radiative transfer (DMRT) by Tsang and Ishimaru
• Specular interface and Snell’s law are applied at each layer boundary.
(from Golden et al., 1998)
• Not included yet: rough surface interfaces (ice ridges, rough soil, etc…)
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Mie Phase Matrix: Flowchart
Sphere
Parameters
(m, x)
Scattering
Angle (Θ)
Interpretation
of (i1,2)
Isotropic
Sphere
Rotation
Matrix Lr(i1,2)
Mie
Coefficients
(an, bn)
Angular
Functions
Reduced Mie
Phase Matrix
P’(θs,θi)
Modified
Stokes matrix
L(Θ)
2
  d

  n( D)dD
Modified
Stokes Matrix
0
L(θs,θi;Δϕ)
(πn, τn)
0
Mie Phase
Matrix
P(θs,θi; Δϕ)
Under the assumptions: 1) Isotropy and 2) Sphericity
f vh  f hv  0


 L ( )  



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July, 2011
f vv
0
0
0
2
0
f hh
0
0


0
0


 
Re{ f vv f hh }  Im{ f vv f hh }

Im{ f vv f hh } Re{ f vv f hh } 
0
2
0
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Mie Stokes Matrix
For Mie Scattering (van de Hulst, 1981; Bohren and Huffman, 1983),
i  2n  1
f vv ()   
[an n (cos )  bn n (cos )]
k n 1 n(n  1)
i  2n  1
f hh ()   
[an n (cos )  bn n (cos )]
k n 1 n(n  1)
where
an
and bn are the Mie coefficients;
n
and  n are the angle-dependent functions.
Interpretation of i1 and i2
cos  s sin  i  cos  i sin  s cos 
cos i1 
sin 
cos  i sin  s  cos  s sin  i cos 
cos i2 
sin 
where
2  acos[cos( i1, 2 )]     2
i1, 2  
0    
acos[cos( i1, 2 )]
1)
and
cos   cos  s cos  i  sin  s sin  i cos 
sin  

2)
1  cos 2 
The Stokes rotation matrix
L r is
 cos 2  sin 2  0.5 sin 2
2
2

L r     sin  cos   0.5 sin 2
cos 2
 sin 2 sin 2
0
0
 0
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0
0

0
1
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Stokes Matrix: Symmetry
The normal incident and scattered angles based Stoke matrix is
L( s , i ;  )  L r (  i2 ) L() L r (i1 )
Symmetry of Stokes matrix is divided into two cases.
Is
L( s , i ;  ) = L( i , s ;  ) ?
L( s ,    i ;  ) = L( i ,    s ;  ) ?
1)
The subtraction results of both cases have similar expressions for general Stokes matrices.
0 SR13 SR14 
 0
 0
0 SR23 SR24 
Subtraction Results (General) ＝ 
SR31 SR32 0 SR34 


SR
SR
SR
0
42
43
 41

Thus, Stokes matrix is generally symmetric for the first two Stokes parameters.
2)
For the Mie Stokes matrix, the above subtraction results has following formation:
0
0
Subtraction Results (Mie) ＝ 0

0
0 0
0 
0 0
0 
0 0 SR34 
0 SR43 0 
Thus, the Mie Stokes matrix is symmetric for the first three Stokes parameters.
3)
Finally, the Rayleigh Stokes matrix is symmetric for all four Stokes parameters.
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Reduced Phase Matrix
θi
θs
dθi
dθs
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Reduced Mie PM: Validation
The reduced Mie phase matrix can be numerically calculated.
 P11 P12 0 0 
'
 d
P P
0 0
0
P( s , i ;  ) Mie 
P ( s , i )   21 22
0 0 P33 P34 


0
0
P
P
43
44 

2
1)
 P11 P12

P ( s ,  i ) Ray leigh   P21 P22
0 0
0 0

'
0 0
0 0
0 0
0 P44 
The 3rd and 4th Stokes parameters are independent of the first two Stokes parameters and
can be calculated separately.
Reduced Mie phase matrix: MP(rate)=10 mm/hr,
<D>=2 mm, freq = 3 GHz (32x32 quadrature angles)
Each plot: Up-Left corner:
Up-Right corner:
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Reduced Rayleigh phase matrix (179x179 angles)
Same <D> and frequency
 i  0o ,  s  0o  cos   1    0o, Forward Scattering
 i  0o ,  s    cos   1     , Backward Scattering
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Reduced Mie Phase Matrix
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freq = 30 GHz
freq = 100 GHz
freq = 300 GHz
freq = 1000 GHz
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Reduced Henyey-Greenstein PM
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DMRT-QCA Phase Matrix
1) Incorporates latest version of DMRT-QCA by Tsang, et al., 2008.
2) A prominent advantage: simplification of the phase matrix calculation
Summary of the DMRT-QCA procedure:
a) Lorentz-Lorentz (L-L) law: effective propagation constant K
b) Ewald-Oseen theorem with L-L law: the average multiple amplitudes: X n( N ) , X n( M )
c) The absorption coefficient is calculated as a function of K , X n( N ) , X n( M )
d) Percus-Yevick approximation: structure factor q ()
e) Applying all above parameters, the DMRT-QCA phase matrix is calculated by
i
1
f vv ()  
1  R  kKr
i
1
f hh ()  
1  R  kKr


P ( )  



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f vv
0
0
0
2
0
f hh
0
0




f) Applying same rotation

and azimuthal integration
0
0
  q 
procedure, the reduced
Re{ f vv f hh }  Im{ f vv f hh }

DMRT-QCA phase matrix
Im{ f vv f hh } Re{ f vv f hh } 
0
2

2n  1 ( N ) ( N )
(M )
(M )
T
X

(cos

)

T
X

n
n
n
n
n  n (cos )
n
(
n

1
)
n 1
N max
2n  1 ( N ) ( N )
Tn X n  n (cos )  Tn( M ) X n( M ) n (cos )

n 1 n( n  1)
N max
0
can be calculated.
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PM Comparison: Mie vs. DMRT (1)
1)
Non-sticky particle case
mean diameter: 0.14cm
fractional volume: 25%
frequencies: 13.4GHz, 17.5GHZ, 37GHz
2) DMRT-QCA predicts more forward
scattering than that of the Mie theory
Validation to “Modeling active microwave remote sensing
of snow using DMRT theory with multiple scattering
effects”, IEEE, TGARS, Vol.45, 2007, by L. Tsang et al.,
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PM Comparison: Mie vs. DMRT (2)
1)
Sticky particle case:   0.1
mean diameter: 0.14cm
fractional volume: 25%
frequencies: 13.4GHz, 17.5GHZ, 37GHz
2) DMRT-QCA sticky case predicts much
greater forward scattering than that of the Mie
theory
Validation to “Modeling active microwave remote sensing
of snow using DMRT theory with multiple scattering
effects”, IEEE, TGARS, Vol.45, 2007, by L. Tsang et al.,
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Reduced DMRT Phase Matrix
30 GHz
10 GHz
DMRT-QCA phase matrix over 16
quadrature angles
Sticky particle case:   0.1
mean diameter: 0.14 cm
fractional volume: 25%
100 GHz
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UMRT: Discretizition
1) Numerical DRTE for first two Stokes parameters

M
M
M
M

dTBvi








i
 keviTBvi  k aT z     j Pvv _ ijTBvj    j Pvv _ ijTBvj    j Pvh _ ijTBhj    j Pvh _ ijTBhj

dz
j 1
j 1
j 1
 j 1


M
M
M
M

dTBvi








 i
 keviTBvi  k aT z     j Pvv _ ijTBvj    j Pvv _ ijTBvj    j Pvh _ ijTBhj    j Pvh _ ijTBhj

dz
j

1
j

1
j

1
j

1



M
M
M
M

dTBhi








i
 kehiTBhi  k aT z     j Phv _ ijTBvj    j Phv _ ijTBvj    j Phh _ ijTBhj    j Phh _ ijTBhj

dz
j 1
j 1
j 1
 j 1


M
M
M
M

dTBhi








 i
 kehiTBhi  k aT z     j Phv _ ijTBvj    j Phv _ ijTBvj    j Phh _ ijTBhj    j Phh _ ijTBhj

dz
j 1
j 1
j 1
 j 1





where P _ ij  P (i ,  j )  P _ ij  P (i , j ), P _ ij  P (i , j )  P _ ij  P (i ,  j )
The boundary conditions are: z  0;
T

i Bi

Bi
z  H; T
M
  s T
j 1

j ij Bj
M
 ( i    j s ji )Ts
j 1
 Tcb  2.73K
a)
Separate the up- (+) and down- (-) welling components of radiation.
b)
Let i
c)
Use the Gauss-Legendre quadrature with the Christoffel weights
 cos θi and   j    j
i.
2) Symmetrizing variables




uvi  i i TBvi
, uhi  i i TBhi
, vvi  i i TBvi
, vhi  i i TBhi
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UMRT: Symmetrization
For vertical polarization, let
A0ij 
kevi
 ij 
i
 i j  
 i j  
 i j  
 i j  
Pvv _ ij , B0ij  
Pvv _ ij , C0ij  
Pvh _ ij , D0ij  
Pvh _ ij ,
i  j
i  j
i  j
i  j
Similarly, for horizontal polarization, let
Eoij  
 i j  
 i j  
 i j  
 i j  
k
Phv _ ij , F0ij  
Phv _ ij , Goij  e  ij 
Phh _ ij , H oij  
Phh _ ij ,
i  j
i  j
i
i  j
i  j
In UMRT,
u v    A0  C 0  B 0  D 0  u v   f 
u v   f 


d uh   E 0  G 0  F 0  H 0  uh   f   U  D  uh   f 
v    
 
v      


f
dz  vv   B 0
D0
A0
C 0  v     D U   v   f 
  vh   f    vh   f 
vh   F

H0
E0
G0   
 


 0 
 4 M 1  DOTLRT
4 M 1
4 M 1
4 M 4 M
Note: 1)
2)
The matrices U and D are symmetric.
Making use of Gershgorin’s circle theorem (see Voronovich et al., 2004), it was shown
that the matrices
U  D and U  D are positive definite, since following condition always hold in RT:
ke  k s
The argument of symmetric, positive definite (SPD) matrices
entire UMRT algorithm.
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U  D and U  D holds throughout the
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Discrete Ordinate-Eigenanalysis
Step 1. Make following linear transformation.
'
'
u  u  v, v  u  v
Step 2. The DRTE becomes
'
'
'
d u  
0
 (U  D) u   0  A u 
 '   
 '  
  '  , where A  U  D
dz  v   (U  D)
0
  v   B 0   v 
B U  D
Step 3. Decouple the two equations
'
'
'
'
'
du
d 2u
dv
  Av ,
 A
 A Bu
dz
dz 2
dz
'
'
'
'
'
dv
d2v
du
  Bu ,


B

B
A
v
dz
dz 2
dz
Step 4.1 There are two primary methods to solve equations of step 3. For example, Tsang
(L. Tsang, et al., 2000) uses:  1, 2
  AB
Step 4.2 In UMRT, we use the solution given by A. Voronovich et al., 2004.
u ( z ) 1  1 1  c  s A 1  1 u (0)
t 
 v( z )   2  1 1 
1 1   v(0) 




 B s c  
by making use of the matrix identity for symmetric matrices:
where
c  cosh( AB z)
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s  sinh( AB z )( AB)

Bg( AB)  g( B A) B
1
2
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UMRT: Solution for Single Layer
Applying B.C, at z = h
t
t
t u inc  1  c  B s  s A  c  c  B s  s A  c   u inc 
t
t 

 0  2


 c  B s  s A  c c  B s  s A  c  r u inc 
t
tuinc
 ( c  B s  s A  c ) r  (c  B s  s A  c )
t
Δz
U , D, f
t
t
1
 r  ( c  B s  s A  c ) (c  B s  s A  c )
uinc
t
t  2(c  B s  s A  c ) 1
Similarly,
ruinc
t
a)
A  M 1 1 M 1
Symmetry:
1
b) Positive definite: eigenvalues are non-negative, thus guarantees that
Details can be found in DOTLRT, Voronovich et al., 2004
1
1
t
1
2
exists.
t
g( AB)  ( M 1 1 M 2 )g( 2 )( M 2 1 M 1 )
2
2
To calculate the reflective and transmissive matrices,
1
1
1
1
1
1
a  1 M 2  1 M 2  2 coth( 0.5 2 z ),    2 sinh(  2 z )
2
t
1
1
2
2
1
t
2
1
t
2
1
2
1
1
t
1
bt  M 2 1   2 tanh( 0.5 2 z ) M 2 1 , b r  M 2 1   2 tanh( 0.5 2 z ) M 2 1
2
1
2
1
1
1
2
t
r  t  M 1 bt b r M 1
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2
2
2
2
t
t  2 P  2M 1 bt  a M 1
1
2
1)
2)
When x   , tanh(x) and coth(x) are bounded to 1.
When sinh(x)   , sinh-1(x) is small (invertible).
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UMRT: Solution for Single Layer (2)
Inhomogeneous solution of the DRTE
1)
In DOTLRT, the up- and down- welling radiations are both assumed independent with height.
1





d u  0  U  D u 
u inh   A f 
f
 0 
  v       v   1  u inh  v inh




v


dz  
 D U     f   inh   A f 
Under the assumption of mirror symmetry, the up- and down- welling self-radiation are also equal.
uinh
u
U , D, f
(b)
v  u 
uinh tuinh
U , D, f
ruinh
uinh
(c)
2)
u inh  u   t u inh  r u inh
 u   u inh  t u inh  r u inh
1
 (I  r  t) A f


emission

Layer Centric
In UMRT, the up- and down- welling radiations are both assumed to be linear with height.
Assume: u  t1 z  C1 ,
v  t2 z  C2 , f  t3 z  C3 
d u   t 1   U  D  u   f 
 
 v    


v
t
dz    2   D U     f 
By applying conventional block (2x2) matrix inversion,
1
1

 t 1  ( I  U D)(U  DU D) 1 t 3 
 t1  t 2
1
1
t 2   

1
   ( I  U D)(U  DU D) t 3 
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UMRT: Solution for Single Layer (3)
1
1
1
1
1

1
1 






(
I

U
D
)(
U

D
U
D
)
C

(
I

U
D
)(
U

D
U
D
)
t1 
3
C

t
C
U
D
3
1
1
and



1
1
1
1
C 2   D U  C 3  t 1  
  
  ( I  U D)(U  DU D) 1 C 3  ( I  U D)(U  DU D) 1 t 1 
 
Validation of above solutions can be done by reducing the case to DOTLRT: if t3 = 0, then t1 = t2 = 0.
From where, we find
1
1
(U  D)[( I  U D)(U  DU D) 1 ]
1
1
1
1
1
1
1
1
1
1
1
 (U  DU D)(U  DU D)  I  ( I  U D)(U  DU D)  (U  D)  A
1
Similarly,
1
1
(U  D)[( I  U D)(U  DU D) 1 ]
1
1
1
 (U  DU D)(U  DU D)  I  ( I  U D)(U  DU D)  (U  D)  B
1
Finally, the inhomogeneous solution in UMRT is
1
d u   t 1   U  D  u   f 
t1   A t 3 


t 2   1  dz  v   t 2    D U   v    f 
  
   A t 3 
 

1
1 1

u inh   A f  B A t 3 
 u inh  v inh
1 1
 v inh    1
   A f  B A t 3 
IGARSS 2011
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UMRT: Recursive Solution for Multilayer
In UMRT, the up- and down- welling self-radiations of a single layer are not the same, thus:
u
-vinh
zh
r, t
U, D
v
(a)
uinh -tuinh
-rvinh
U, D
-uinh
z 0
 u   t u inh
u inh
at top of the layer
-uinh
(b)
at bottomof the layer
U, D
vinh -tvinh -ruinh
(c)
 r v inh
at bottomof the layer
at top of the layer
 v   r u inh
v inh
-vinh
 t v inh
at bottomof the layer
at top of the layer
u   u inh ( z  h)  t u inh ( z  0)  r v inh ( z  h)

 Level Centric
 v  v inh ( z  0)  r u inh ( z  0)  t u inh ( z  h)
Example
Type
Thickness
Bot.
Temp.
Top
Temp.
Rain
Rate
Particle
dia.
Freq.
Case I
Rain
1 km
300 K
273 K
10 mm/hr
1.4 mm
13.4 GHz
Case II
Rain
1 km
273 K
300 K
10 mm/hr
1.4 mm
13.4 GHz
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UMRT: Recursive Solution for Multilayer (2)
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UMRT: Recursive Solution for Multilayer (3)
Finally, we have following recursive solutions in UMRT:
( n 1)
U*
(n)
1
( n)
(n)
 u *  t ( I  R r ) (U *  R v* )
( n 1)
R
( n)
1
(n)
 r  t(I  R r) R t
Note:
1)
Matrices r , t , u  ,and v for all individual layers should be first obtained.
2)
The initial conditions are
U
(0)
*
 f
(0)
(0)
and R
S
 i j
sij
i  j
Critical angle and Interpolation:
1)
Only the incident streams that are inside the critical angle will pass through the interface.
2) Such refractive streams are bent from the quadrature angles. UMRT employs the cubic spline
interpolation to compensate them back to the quadrature angles
3) The incident streams whose angle are greater than critical angle will be remove for upwelling
radiation streams and added back to the corresponding downwelling radiation streams.
Finally, the UMRT solutions are modified as
U
R
IGARSS 2011
( n 1)

( n 1)
(n)
1
 u cai  t ( I  R r ) (U
( n)
(n)

(n)
 R vcai )
(n)
 r  t ( I  R r ) 1 R t
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UMRT: Jacobian Procedure
UMRT Jacobian Procedure
𝑢′ , 𝑢 ′′ , 𝑣 ′ , 𝑣 ′′
Key:
(n 1)
V
( n 1)

v
''
(n )
u
u
(n)
R
𝑟
v
(n)
𝑛
𝑀1 , 𝑀1𝑇 , 𝑏𝑡−1 ,
v
( n 1)
U
,𝑡
u
𝑛
𝑢∗ , 𝑣∗
𝑛
''
r ,t
(n )

𝑛
'
𝜁 −1 , 𝑎−1 , 𝑏𝑟
'
𝐴, 𝐵 w.r.t:
𝑘𝑠 , 𝑘𝑎 , 𝑇𝑜 , 𝑡𝑇
(n 1)
R
𝑀2 , 𝑀2𝑇 , Λ1 ,
Λ2
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Summary
UMRT is developed based on the DOTLRT concept, however, it
has following key improvements:
1) The symmetry property of the polarized reduced Mie phase matrix
is exploited so that the applicability of the fast and stable matrix
operation (based on symmetry and positive definiteness) is
applicable to both sparse and dense media.
2) Mie phase matrix is applied so that radiation coupling is included in
a fully polarimetric solution.
3) DMRT-QCA phase matrix is included for dense medium layers
describing (e.g.) vegetation, soil, ice, seawater, etc…
4) The physical temperature of a layer is linear in height, allowing the
precise solutions for piecewise linear temperature profiles, thus
extending the applicability of DOTLRT to a level-centric grid.
5) The refractivity profile is accounted for by including the critical
angle effect and applying cubic spline interpolation to a refractive
transition matrix (not discussed).
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QUESTIONS?
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Mie Stokes Matrix: Validation
(a)
(b)
Normalized Mie Stokes matrix elements for single particle: particle diameter =1.4 mm, material
permittivity (ice) = 3.15-j0.001: a) freq = 13.4 GHz; b) freq = 37 GHz.
Replica of Fig. 3 in “Modeling active microwave remote sensing of snow using DMRT theory with multiple scattering effects”,
IEEE, TGARS, Vol.45, 2007, by L. Tsang et al.,
IGARSS 2011
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Mie Stokes Matrix: Validation (2)
(a)
(b)
P33 is asymmetric to 90o and P33(at 0o) is greater than P33(180o).
P34 is symmetric to 90o.
Qualitative validation by the description in Thermal Microwave Radiation Applications for Remote
Sensing, 2006, ch.3, A. Battaglia et al., edited by C. Matzler.
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UMRT: Recursive Solution for Multilayer
 n 1
u*
vinc
U
n 1
R
r, t
v*=u*
v
r, t
u
n 
'
U
r
tvinc v "
'
"
u
n 
n 
U
R
n 
R
In DOTLRT, the self-radiation of each layer is same in the up- and down- welling directions, (u 
(n)
(n)
thus:
'
'
 '
U
'
where
'
( n 1)

'

 u *  t (U *  u ) and u'  R ((nv)   v' )  R (u *  v )
v  r (U *  u )
(n)
u and v are the multiple reflections between the extra top layer and the stack.
U
( n 1)
*
"
u and v , which must satisfy:
IGARSS 2011
(n)
1
 u *  t ( I  R r ) (U
If there exists an external downwelling
radiation vinc, it will result in the presence
of two additional up- and downwelling field
"
 v )
( n)
*
(n)
 R u* )
"


u"  R " v
( n 1)
(n)
(n)


1
 r  t(I  R r) R t
 R
v  r u  t v inc 
( n 1)
"
R v inc  r v inc  t u 
"
(n)
July, 2011
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UMRT: Critical Angle and Interpolation
Critical angle: the angle of incidence above which total internal reflection occurs. It is given by
n 
C  arcsin  2 
 n1 
Atmosphere
(natm)
Superposition to the
self-radiation of this 
layer
Snow
(nsnow)
Ice
(nice)
…
…
…
…
…
…
…
…
 nair 

n
 snow 
C _ sa  arcsin 
 nsnow 

 nice 
C _ is  arcsin 
1)
In both UMRT and DOTLRT, once the number M is chosen, the quadrature angles are then fixed
in each layer (for self-radiation).
2)
Only the refractive streams whose incident streams are inside the critical angle will successfully
pass through the interface.
3)
Such refractive streams are bended and generally away from the quadrature angles. Therefore
need to be correctly compensated back to the fixed quadrature angles.
4)
The incident streams whose angle are greater than critical angle will be remove for upwelling
radiation streams and added back to the corresponding downwelling radiation streams.
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Mie Scattering and Extinction
1)
Spherical Mie scattering
jn (mx)[ xjn ( x)]' jn ( x)[ mxjn (mx)]'
jn (mx)[ xhn ( x)]'hn ( x)[ mxjn (mx)]'
jn ( x)[ mxjn (mx)]'m 2 jn (mx)[ xjn ( x)]'
bn  
hn (mx)[ mxjn (mx)]'m 2 jn (mx)[ xhn ( x)]'
jn () and hn () are spherical Bessel and Hankel functions of the 1st kind.
x  2a  is the size parameter and m  
an  
where
2)
Monodispersed spherical particle
2 
2 
2
2
ηe x,m  2  (2n  1) Re{an  bn }, ηs x,m  2  (2n  1) Re{ an  bn }
x n 1
x n 1
3)
Polydispersed spherical particles
a) With size distribution function (SDF): nD   N 0 D  e
P   D Q


2
ke  e ( x, m) D n( D)dD, ks   s ( x, m) D 2 n( D)dD
40
40
b) Natural raindrop distribution: Marshall and Palmer (MP) SDF
c) Ice-sphere distribution: Sekhon and Srivastava SDF
d) The absorption coefficient is
k a  ke  k s
Details can be found in M. Jansen, Ch.3 by A. Gasiewski, 1991.
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Validation of Mie Absorption and Scattering
Replica of Fig.3.6-7 in Chapter 3 (right side), originally produced by A. Gasiewski, Atmospheric Remote Sensing by
Microwave Radiometry, edited by M. Janssen, 1993.
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Mie Stokes Matrix
Under the assumptions: 1) Isotropy and 2) Sphericity
f vh  f hv  0


 L ( )  



f vv
0
0
0
2
0
f hh
0
0


0
0


 
Re{ f vv f hh }  Im{ f vv f hh }

Im{ f vv f hh } Re{ f vv f hh } 
0
2
0
For Mie Scattering (van de Hulst, 1981; Bohren and Huffman, 1983),
i  2n  1
f vv ()   
[an n (cos )  bn n (cos )]
k n 1 n(n  1)
i  2n  1
f hh ()   
[an n (cos )  bn n (cos )]
k n 1 n(n  1)
where
an
and bn are the Mie coefficients;
n
and  n are the angle-dependent functions as
2n  1
n
 n 1 
 n 2 ,  n  n n  (n  1) n 1
n 1
n 1
 0  0,  0  1,   cos 
n 
Figure from Scattering of Electromagnetic Waves, vol. I, p.7 by L. Tsang, et al., 2000
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Reduced Mie PM: 10 GHz
1)
Below ~60 GHz:
Forward scattering ~ Backward scattering
2)
Above ~60 GHz:
Forward scattering > Backward scattering
3) P12 is 90o rotation to P21.
4) P34 = -P43
5) P44 is greater than P33
IGARSS 2011
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t u inc
u  ( z  z )
v  ( z  z )
Δz
U , D, f
U , D, f
u  ( z  0)
Δz
v  ( z  0)
u inc
r u inc
v inh ( z  z )
u  ( z  z )
U , D, f
r v inh ( z  z ) t u ihn ( z  0)
r, t
U , D, f
u  ( z  z )

u ihn ( z  z )
U , D, f
u inh ( z  0)
IGARSS 2011
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