vancouver, ca july, 2011igarss 2011 development of a precise and fast multistream scattering-based...

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

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

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)

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

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.

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

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

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

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. ?

Vancouver, CA July, 2011IGARSS 2011 ii ss didi dsds Reduced Phase Matrix

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

Vancouver, CA July, 2011IGARSS 2011 Reduced Mie Phase Matrix freq = 30 GHz freq = 300 GHzfreq = 1000 GHz freq = 100 GHz

Vancouver, CA July, 2011IGARSS 2011 Reduced Henyey-Greenstein PM

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.

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.,

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.,

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

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

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.

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

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).

Vancouver, CA July, 2011IGARSS 2011 UMRT: Solution for Single Layer (2) Inhomogeneous solution of the DRTE 1)In DOTLRT, the up- and downwelling radiations are both assumed independent with height. (b) Under the assumption of mirror symmetry, the up- and downwelling self-radiation are also equal. u inh tu inh (c) u inh ru inh Layer Centric Assume: 2)In UMRT, the up- and downwelling radiations are both assumed to be linear with height. By applying conventional block (2x2) matrix inversion,

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

Vancouver, CA July, 2011IGARSS 2011 UMRT: Recursive Solution for Multilayer In UMRT, the up- and downwelling 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

Vancouver, CA July, 2011IGARSS 2011 UMRT: Recursive Solution for Multilayer (2)

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

Vancouver, CA July, 2011IGARSS 2011 UMRT: Jacobian Procedure UMRT Jacobian Procedure Key:

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).

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

Vancouver, CA July, 2011IGARSS 2011 QUESTIONS?

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)

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)

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 downwelling directions, ( ) thus: where and are the multiple reflections between the extra top layer and the stack.

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.

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

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.

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

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

Vancouver, CA July, 2011IGARSS 2011 zz zz

vancouver, ca july, 2011igarss 2011 development of a precise and fast multistream scattering-based...

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