Transcript of Vancouver, CA July, 2011IGARSS 2011 Development of a Precise and Fast Multistream Scattering-Based...
- Slide 1
- Vancouver, CA July, 2011IGARSS 2011 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
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- Vancouver, CA July, 2011IGARSS 2011 Part I: Motivation Part II:
Unified Microwave Radiative Transfer 1)UMRT Modeling 2)Full Mie
Phase Matrix 2.1) Mie Stokes Matrix 2.2) Symmetry 2.3) Mie Phase
Matrix 3)Full Dense Medium Radiative Transfer Phase matrix 3.1)
Summary 3.2) DMRT-QCA Phase Matrix 4.UMRT 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 Procedure Part III: Summary and
Future Work Outline
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- Vancouver, CA July, 2011IGARSS 2011 Motivation 4 Radiative
Transfer (RT) Polarized Stokes vector (rather than scalar)
radiation formulation Uniform treatment of all media 1) Analytic
solution for an Rayleigh scattering atmosphere (Chandrasekhar,
1960) 2) Matrix-based discrete ordinate-eigenanalysis (DOE) for
multiple layer structure 3) 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: a)Matrix operations such as computed by Taylor
expansion (Siewert et al., 1981) b)Matrix inversion is not stable
for highly opaque layers (Stamnes et al., 1988). a)Sparse medium
(e.g., general atmosphere, rain, fog, cloud and etc.,) b)Single
polarization c)Layers with constant temperature d)Non-refracting
layers (i.e., no abrupt surfaces) e)Layer centric (rather than
level centric) Goal I: Finally, the new UMRT should have more
general applicability in above means. a)Develop a unified microwave
radiative transfer (UMRT) model from the DOTLRT b)Incorporate with
the dense medium radiative transfer theory (DMRT)
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- Vancouver, CA July, 2011IGARSS 2011 Motivation (2/2)
1)Unarguable significance to global climatic system 2)The Center
for Environmental Technology has a large amount new Arctic sea ice
measurement data from: a)AMISA 2008 - Summer experiment b)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 ComparisonUMRTDOTLRT
Fast, Stable Analytic Matrix InversionYes Fast JacobianYes Phase
MatrixFull Mie, DMRT (4x4)Reduced HG (2x2)
PolarizationTri-polarizationSingle-polarization Critical Angle
EffectYesNo Radiation Stream InterpolationYes, cubic splineNo
Thermal Emission Approx.Linear dependenceConstant Level/Layer
CentricLevel CentricLayer Centric DMRTYesNo
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- Vancouver, CA July, 2011IGARSS 2011 Motivation for a UMRT
AttributeUMRT DOTLRT * Fast, Stable Analytic Matrix InversionYes
Fast JacobianYes Phase Matrix Reduced Mie or DMRT (4x4) Reduced HG
(2x2) Polarization Tri-polarization + 4 th Stokes
Single-polarization Interface Refraction / Internal ReflectionYesNo
Radiation Stream InterpolationYes, cubic splineNo Thermal Emission
Approximation.Linear dependenceConstant Level/Layer CentricLevel
CentricLayer Centric DMRTYesNo * 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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- Vancouver, CA July, 2011IGARSS 2011 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 Snells 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 Ishimaru
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- Vancouver, CA July, 2011IGARSS 2011 Parameters ( m, x )
Scattering Angle () Mie Coefficients (a n, b n ) Angular Functions
( n, n ) Modified Stokes matrix L() Modified Stokes Matrix L( s, i
; ) Rotation Matrix L r (i 1,2 ) Interpretation of (i 1,2 )
Isotropic Sphere Sphere Reduced Mie Phase Matrix P ( s, i ) Mie
Phase Matrix P( s, i ; ) Mie Phase Matrix: Flowchart Under the
assumptions: 1) Isotropy and 2) Sphericity
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- Vancouver, CA July, 2011IGARSS 2011 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 and
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- Vancouver, CA July, 2011IGARSS 2011 The normal incident and
scattered angles based Stoke matrix is Stokes Matrix: Symmetry Is =
? Symmetry of Stokes matrix is divided into two cases. = 3)Finally,
the Rayleigh Stokes matrix is symmetric for all four Stokes
parameters. 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: 1)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. ?
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- Vancouver, CA July, 2011IGARSS 2011 ii ss didi dsds Reduced
Phase Matrix
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- Vancouver, CA July, 2011IGARSS 2011 The reduced Mie phase
matrix can be numerically calculated. 1)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, =2 mm, freq = 3 GHz (32x32 quadrature angles) Reduced
Rayleigh phase matrix (179x179 angles) Same and frequency Each
plot: Up-Left corner:, Forward Scattering Up-Right corner:,
Backward Scattering Reduced Mie PM: Validation
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- Vancouver, CA July, 2011IGARSS 2011 Reduced Mie Phase Matrix
freq = 30 GHz freq = 300 GHzfreq = 1000 GHz freq = 100 GHz
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- Vancouver, CA July, 2011IGARSS 2011 Reduced Henyey-Greenstein
PM
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- Vancouver, CA July, 2011IGARSS 2011 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 b)Ewald-Oseen theorem
with L-L law: the average multiple amplitudes: c)The absorption
coefficient is calculated as a function of d)Percus-Yevick
approximation: structure factor e)Applying all above parameters,
the DMRT-QCA phase matrix is calculated by f)Applying same rotation
and azimuthal integration procedure, the reduced DMRT-QCA phase
matrix can be calculated.
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- Vancouver, CA July, 2011IGARSS 2011 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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- Vancouver, CA July, 2011IGARSS 2011 PM Comparison: Mie vs. DMRT
(2) 1)Sticky 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 of
snow using DMRT theory with multiple scattering effects, IEEE,
TGARS, Vol.45, 2007, by L. Tsang et al.,
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- Vancouver, CA July, 2011IGARSS 2011 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 30 GHz 100
GHz
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- Vancouver, CA July, 2011IGARSS 2011 UMRT: Discretizition
a)Separate the up- (+) and down- (-) welling components of
radiation. b)Let c)Use the Gauss-Legendre quadrature with the
Christoffel weights. and 1)Numerical DRTE for first two Stokes
parameters The boundary conditions are: where 2) Symmetrizing
variables
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- Vancouver, CA July, 2011IGARSS 2011 UMRT: Symmetrization For
vertical polarization, let Similarly, for horizontal polarization,
let In UMRT, Note: 1) The matrices and are symmetric. 2) Making use
of Gershgorins circle theorem (see Voronovich et al., 2004), it was
shown that 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.
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- Vancouver, CA July, 2011IGARSS 2011 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: where
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- Vancouver, CA July, 2011IGARSS 2011 UMRT: Solution for Single
Layer Applying B.C, at z = h ru inc u inc tu inc zz Similarly,
b)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,
a)Symmetry: 1)When x, tanh(x) and coth(x) are bounded to 1. 2)When
sinh(x), sinh -1 (x) is small (invertible).
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- Vancouver, CA July, 2011IGARSS 2011 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. (b) Under the assumption of mirror symmetry, the up- and
down- welling self-radiation are also equal. u inh tu inh (c) u inh
ru inh Layer Centric Assume: 2)In UMRT, the up- and down- welling
radiations are both assumed to be linear with height. By applying
conventional block (2x2) matrix inversion,
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- Vancouver, CA July, 2011IGARSS 2011 UMRT: Solution for Single
Layer (3) and Validation of above solutions can be done by reducing
the case to DOTLRT: if t 3 = 0, then t 1 = t 2 = 0. From where, we
find Similarly, Finally, the inhomogeneous solution in UMRT is
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- Vancouver, CA July, 2011IGARSS 2011 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
-tu inh (b) u inh -v inh -u inh -ru inh (c) v inh -tv inh Level
Centric Example TypeThickness Bot. Temp. Top Temp. Rain Rate
Particle dia. Freq. Case I Rain1 km300 K273 K10 mm/hr1.4 mm13.4 GHz
Case II Rain1 km273 K300 K10 mm/hr1.4 mm13.4 GHz
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- Vancouver, CA July, 2011IGARSS 2011 UMRT: Recursive Solution
for Multilayer (2)
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- Vancouver, CA July, 2011IGARSS 2011 UMRT: Recursive Solution
for Multilayer (3) Note: 1)Matrices and for all individual layers
should be first obtained. 2)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
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
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- Vancouver, CA July, 2011IGARSS 2011 UMRT: Jacobian Procedure
UMRT Jacobian Procedure Key:
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- Vancouver, CA July, 2011IGARSS 2011 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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- Vancouver, CA July, 2011IGARSS 2011 Future Work Future Work:
1)Jacobian capability 2)Full model validation Table.1
ComparisonUMRTDOTLRT Fast, Stable Analytic Matrix Inversion Yes
JacobianYes Phase MatrixFull Mie (4x4)Reduced HG (2x2)
PolarizationTri-polarizationSingle-polarization Critical Angle
EffectYesNo Radiation Stream Interpolation Yes, cubic splineNo
Thermal Emission Approx. Linear dependence Constant Level/Layer
CentricLevel CentricLayer Centric
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- Vancouver, CA July, 2011IGARSS 2011 QUESTIONS?
- Slide 31
- Vancouver, CA July, 2011IGARSS 2011 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)
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- Vancouver, CA July, 2011IGARSS 2011 P 33 is asymmetric to 90 o
and P 33 (at 0 o ) is greater than P 33 (180 o ). P 34 is symmetric
to 90 o. 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)
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- Vancouver, CA July, 2011IGARSS 2011 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*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.
- Slide 34
- Vancouver, CA July, 2011IGARSS 2011 UMRT: Critical Angle and
Interpolation Critical angle : the angle of incidence above which
total internal reflection occurs. It is given by Atmosphere ( n atm
) Ice ( n ice ) Snow ( n snow ) 1)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 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.
- Slide 35
- Vancouver, CA July, 2011IGARSS 2011 b) Natural raindrop
distribution: Marshall and Palmer (MP) SDF c) Ice-sphere
distribution: Sekhon and Srivastava SDF d) The absorption
coefficient is 3)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. 1)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
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- Vancouver, CA July, 2011IGARSS 2011 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.
- Slide 37
- Vancouver, CA July, 2011IGARSS 2011 Mie Stokes Matrix Figure
from Scattering of Electromagnetic Waves, vol. I, p.7 by L. Tsang,
et al., 2000 Under 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 as
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- Vancouver, CA July, 2011IGARSS 2011 Reduced Mie PM: 10 GHz
1)Below ~60 GHz: Forward scattering ~ Backward scattering 2) Above
~60 GHz: Forward scattering > Backward scattering 3) P 12 is 90
o rotation to P 21. 4) P 34 = -P 43 5) P 44 is greater than P
33
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- Vancouver, CA July, 2011IGARSS 2011 zz zz