Miao Tian AJ Gasiewski University of Colorado Department of Electrical Engineering Center for Environmental Technology Boulder CO USA Part I Motivation Part II Unified Microwave Radiative Transfer ID: 429158
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Slide1
Development of a Precise and Fast Multistream Scattering-Based DMRT model with Jacobian
Miao TianA.J. Gasiewski
University of Colorado
Department of Electrical Engineering
Center for Environmental Technology
Boulder, CO, USASlide2
Part I: MotivationPart II: Unified Microwave Radiative Transfer
UMRT ModelingFull Mie Phase Matrix 2.1) Mie Stokes Matrix 2.2) Symmetry 2.3) Mie Phase MatrixFull Dense Medium Radiative Transfer Phase matrix 3.1) Summary
3.2) DMRT-QCA Phase MatrixUMRT Solution 4.1) Matrix Symmetrization 4.2) Solution to Single Layer 4.3) Recursive Solution to Multiple Layers 4.4) Critical Angle and Radiation Stream Interpolation 4.5) Jacobian ProcedurePart III: Summary and Future Work
OutlineSlide3
Motivation
4
Radiative Transfer (RT)
Polarized Stokes vector (rather than scalar) radiation formulation Uniform treatment of all media Analytic solution for an Rayleigh scattering atmosphere (Chandrasekhar, 1960)
Matrix-based
discrete ordinate-eigenanalysis (DOE) for multiple layer structure
Discrete ordinate tangent linear radiative transfer (DOTLRT) by Voronovich
et al
., 2004: numerically fast and stable solution to all matrix operations required by DOE. Current DOTLRT:
Matrix operations such as computed by Taylor expansion (Siewert
et al., 1981)Matrix inversion is not stable for highly opaque layers (Stamnes et al., 1988).
Sparse medium (e.g., general atmosphere, rain, fog, cloud and etc.,)
Single polarizationLayers with constant temperatureNon-refracting layers (i.e., no abrupt surfaces)Layer centric (rather than level centric)
Goal I:
Finally, the new UMRT should have more general applicability in above means.
Develop a unified microwave radiative transfer (UMRT) model from the DOTLRT
Incorporate with the dense medium radiative transfer theory (DMRT) Slide4
Motivation (2/2)
U
narguable significance to global climatic system
The Center for Environmental Technology has a large amount new Arctic sea ice measurement data from:AMISA 2008 - Summer experiment
ARCTIC 2006 - Late winter
experiment
Specifically, the geophysical case is chosen to be Arctic sea ice.
The UMRT will be validated by comparing with field measurements.
Table.1
Comparison
UMRTDOTLRT
Fast, Stable Analytic Matrix InversionYes
YesFast JacobianYesYesPhase MatrixFull Mie, DMRT (4x4)Reduced HG (2x2)Polarization
Tri-polarization
Single-polarizationCritical Angle Effect
YesNo
Radiation
Stream Interpolation
Yes, cubic spline
No
Thermal Emission
Approx.
Linear
dependence
Constant
Level/Layer
Centric
Level
Centric
Layer
Centric
DMRT
Yes
NoSlide5
Motivation for a UMRT
Attribute
UMRT
DOTLRT *Fast, Stable Analytic Matrix InversionYesYesFast Jacobian
Yes
Yes
Phase Matrix
Reduced Mie or
DMRT (4x4)Reduced HG (2x2)
PolarizationTri-polarization
+ 4th StokesSingle-polarizationInterface Refraction / Internal ReflectionYes
NoRadiation Stream Interpolation
Yes, cubic splineNoThermal Emission Approximation.Linear dependenceConstantLevel/Layer CentricLevel Centric
Layer Centric
DMRTYes
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.Slide6
UMRT: Medium Model
(from Golden
et al.
, 1998)Upper-half, Sparse Medium
Lower-half, Dense Medium
In UMRT, a planar-stratified model is used.
Categorized into
Specular interface and Snell’s law are applied at each layer boundary.
1) Upper-half, Sparse medium:
Not
included yet: rough surface interfaces (ice
ridges,
rough soil, etc…)
a) General atmosphere
b) Weakly homogeneous and slightly
dissipative
c)
Independent scattering
: scattering
intensity is the sum of scattering
intensities from each particle.
d) Particle size distribution function
e) Sparse medium radiative transfer
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 IshimaruSlide7
Parameters
(
m
, x)
Scattering
Angle
(
Θ
)
Mie Coefficients
(
a
n, bn
)
Angular Functions
(
π
n
,
τ
n
)
Modified Stokes matrix
L
(
Θ
)
Modified Stokes Matrix
L
(
θ
s
,
θ
i
;
Δϕ
)
Rotation
Matrix
L
r
(
i
1,2
)
Interpretation
of
(
i
1,2
)
Isotropic Sphere
Sphere
Reduced Mie Phase Matrix
P
’
(
θ
s
,
θ
i
)
Mie Phase Matrix
P
(
θ
s
,
θ
i
;
Δϕ
)
Mie Phase Matrix: Flowchart
Under the assumptions: 1) Isotropy and 2) SphericitySlide8
Mie Stokes Matrix
For Mie Scattering (van de Hulst, 1981; Bohren and Huffman, 1983),
where and are the Mie coefficients;
and are the angle-dependent functions.
2) The Stokes rotation matrix is
1) Interpretation of
i
1
and
i
2
where
andSlide9
The normal incident and scattered angles based Stoke matrix is
Stokes Matrix: Symmetry
Is = ?
Symmetry
of Stokes matrix is divided into two cases.
=
Finally, the Rayleigh Stokes matrix is symmetric for all four Stokes parameters.
Thus,
Stokes matrix is generally symmetric for the first two Stokes parameters
.
For the Mie Stokes matrix, the above subtraction results has following formation:The subtraction results of both cases have similar expressions for general Stokes matrices. Subtraction Results (General)
Subtraction Results (Mie)
Thus,
the Mie Stokes matrix is symmetric for the first three Stokes parameters
.
? Slide10
θ
iθ
s
dθi
d
θ
s
Reduced Phase MatrixSlide11
The reduced Mie phase matrix can be numerically calculated.
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) Reduced Rayleigh phase matrix (179x179 angles) Same <D> and frequency
Each plot:
Up-Left
corner: , Forward Scattering
Up-Right
corner: , Backward Scattering
Reduced Mie PM: ValidationSlide12
Reduced Mie Phase Matrix
freq
= 30 GHz
freq = 300 GHzfreq = 1000 GHz
freq
= 100 GHzSlide13
Reduced Henyey-Greenstein PMSlide14
DMRT-QCA Phase Matrix
Incorporates latest version of DMRT-QCA by Tsang, et al., 2008. A prominent advantage: simplification of the phase matrix calculation
Summary of the DMRT-QCA procedure:
Lorentz-Lorentz (L-L) law
:
effective
propagation
constant Ewald-Oseen theorem with L-L law: the average multiple amplitudes:The absorption coefficient is calculated as a function of
Percus-Yevick approximation: structure factor
Applying all above parameters, the DMRT-QCA phase matrix is calculated by
Applying same rotation and azimuthal integration procedure, the reduced DMRT-QCA phase matrix can be calculated.Slide15
PM Comparison: Mie vs. DMRT (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 theoryValidation to “Modeling active microwave remote sensing
o
f snow using DMRT theory with multiple scattering
effects
”,
IEEE, TGARS, Vol.45, 2007, by L. Tsang et al.,Slide16
PM Comparison: Mie vs. DMRT (2)
S
ticky particle case:
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
o
f snow using DMRT theory with multiple scattering
effects”, IEEE, TGARS, Vol.45, 2007, by L. Tsang et al.,Slide17
Reduced DMRT Phase Matrix
Sticky
particle case:
mean diameter: 0.14 cm fractional volume: 25%
DMRT-QCA
phase matrix over 16 quadrature angles
10 GHz
3
0 GHz
100 GHzSlide18
UMRT: Discretizition
Separate the up- (+) and down- (-) welling components of radiation.Let
Use
the Gauss-Legendre quadrature with the Christoffel weights .
and
Numerical DRTE for first two Stokes parameters
The boundary conditions are:
where
Symmetrizing variables Slide19
UMRT: Symmetrization
For vertical polarization, let
Similarly, for horizontal polarization, let
In UMRT,
Note:
1) The matrices and are symmetric.
2) Making use of Gershgorin’s circle theorem (see
Voronovich
et al
., 2004), it was shownthat the matrices and are positive definite, since following condition always hold in RT:
The argument of symmetric, positive definite (SPD) matrices and holds
throughout the
entire UMRT algorithm.Slide20
Discrete Ordinate-Eigenanalysis
Step 1. Make following linear transformation.
Step 2
. The DRTE becomes
, where
Step 3
. Decouple the two equations
Step 4
.1 There are two primary methods to solve equations of step 3. For example, Tsang
(L. Tsang,
et al., 2000) uses:
Step 4.2 In UMRT, we use the solution given by A. Voronovich
et al., 2004.by making use of the matrix identity for symmetric matrices:
whereSlide21
UMRT: Solution for Single Layer
Applying B.C, at z = h
ru
inc
u
inc
tu
inc
Δ
z
Similarly,
Positive definite: eigenvalues are non-negative, thus guarantees that exists.
Details can be found in DOTLRT, Voronovich
et al
., 2004
To calculate the reflective and transmissive matrices,
Symmetry:
When
x , tanh
(
x
)
and
coth
(
x
)
are bounded to 1.
When
sinh
(
x
)
,
sinh
-1
(
x
)
is small (invertible). Slide22
UMRT: Solution for Single Layer (2)
Inhomogeneous solution of the DRTEIn DOTLRT, the up- and down- welling radiations are both assumed independent with height.
(b)
Under the assumption of mirror symmetry, the up- and down- welling self-radiation are also equal
.
u
inh
t
u
inh
(c)
u
inh
u
inh
ru
inh
Layer Centric
Assume:
In
UMRT
, the up- and down- welling radiations are both assumed to be linear with height
.
By applying conventional block (2x2) matrix inversion, Slide23
UMRT: Solution for Single Layer (3)
andValidation of above solutions can be done by reducing the case to DOTLRT: if t3 = 0, then
t
1 = t2 = 0.From where, we find
Similarly,
Finally, the inhomogeneous solution in UMRT isSlide24
UMRT: Recursive Solution for Multilayer
In UMRT, the up- and down- welling self-radiations of a single layer are not the same, thus:
(a)
-v
inh
-rv
inh
-u
inh
-
t
u
inh
(b)
u
inh
-v
inh
-u
inh
-ru
inh
(c)
v
inh
-tv
inh
Level Centric
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 GHzSlide25
UMRT: Recursive Solution for Multilayer (2)Slide26
UMRT: Recursive Solution for Multilayer (3)
Note:
Matrices
and for all individual layers should be first obtained.
The initial conditions are
Finally, we have following recursive solutions in UMRT:
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 angles3) 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 asSlide27
UMRT: Jacobian Procedure
UMRT Jacobian Procedure
Key:
,
,
,
,
w.r.t
:
,
,
,
,
,
,
,
,
,
,
,
,
,
Slide28
Summary
UMRT is developed based on the DOTLRT concept, however, ithas following key improvements:
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.Mie phase matrix is applied so that radiation coupling is included in a fully polarimetric solution.DMRT-QCA phase matrix is included for
dense
medium
layers describing (e.g.) vegetation, soil, ice, seawater, etc…
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.The refractivity profile is accounted for by including
the critical angle effect and applying cubic
spline interpolation to a refractive transition matrix (not discussed).Slide29
Future Work
Future Work: Jacobian capabilityFull model validation
Table.1
ComparisonUMRTDOTLRTFast, Stable Analytic Matrix InversionYes
Yes
Jacobian
Yes
Yes
Phase MatrixFull Mie (4x4)Reduced
HG (2x2)
PolarizationTri-polarizationSingle-polarizationCritical Angle EffectYes
NoRadiation
Stream InterpolationYes, cubic splineNoThermal Emission Approx.Linear dependenceConstantLevel/Layer CentricLevel Centric
Layer CentricSlide30
QUESTIONS?Slide31
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.,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.
Mie Stokes Matrix: Validation
(a)
(b)Slide32
P33 is asymmetric to 90
o and P33(at 0o) is greater than P33(180
o
). P34 is symmetric to 90o.
Q
ualitative validation by the description in Thermal Microwave Radiation Applications for Remote
Sensing
,
2006, ch.3, A. Battaglia et al., edited by C. Matzler.
Mie Stokes Matrix: Validation (2)
(a)
(b)Slide33
UMRT: Recursive Solution for Multilayer
v
inc
tv
inc
If there exists an external downwelling
radiation
v
inc
, it will result in the presence
of two additional up- and downwelling field
and , which must satisfy:
v
*
=u
*
u
*
In
DOTLRT
, the self-radiation of each layer is same in the up- and down- welling directions, ( )
thus:
where and are the multiple reflections between the extra top layer and the stack.Slide34
UMRT: Critical Angle and Interpolation
Critical angle: the angle of incidence above which total internal reflection occurs. It is given by
Atmosphere
(natm)
Ice
(
n
ice
)Snow(n
snow)
…
…
…
…
…
…
In both UMRT and DOTLRT, once the number
M
is chosen, the quadrature angles are then fixed in each layer (for self-radiation).
Superposition to the self-radiation of this layer
…
…
Only the refractive streams whose incident streams are inside the critical angle will successfully pass through the interface.
Such refractive streams are bended and generally away from the quadrature angles. Therefore need to be correctly compensated back to the fixed quadrature angles.
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.Slide35
b)
Natural raindrop distribution: Marshall and Palmer (MP) SDF c) Ice-sphere distribution: Sekhon and Srivastava SDFd) The absorption coefficient is
Polydispersed spherical particles
a) With size distribution function (SDF):
Mie Scattering and Extinction
Details can be found in M. Jansen, Ch.3 by A. Gasiewski, 1991.
Spherical Mie scattering
2) Monodispersed spherical particle
where and are spherical Bessel and Hankel functions of the 1
st
kind.
is the size parameter and Slide36
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 byMicrowave Radiometry, edited by M. Janssen, 1993.Slide37
Mie Stokes Matrix
Figure from Scattering of Electromagnetic Waves, vol. I, p.7 by L. Tsang, et al., 2000Under the assumptions: 1) Isotropy and 2) Sphericity
For Mie Scattering (van de Hulst, 1981; Bohren and Huffman, 1983),
where and are the Mie coefficients;
and are the angle-dependent functions asSlide38
Reduced Mie PM: 10 GHz
Below ~60 GHz: Forward scattering ~ Backward scattering
2) Above ~
60 GHz: Forward scattering > Backward scattering3) P12 is 90o rotation to P
21
.
4)
P
34 = -P43
5)
P44 is greater than P33
Slide39
Δ
z
Δ
z