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Electromagnetic Models In Active And Passive Microwave Remote Sensing of Terrestrial Snow Leung Tsang 1, Xiaolan Xu 2 and Simon Yueh 2 1 Department of Electrical Engineering, University of Washington, Seattle, WA 2 Jet Propulsion Laboratory, Pasadena, CA

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Radiative Transfer Equation 2

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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

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DMRT Models 4 QCAFoldy LaxBicontinuous ModelSpheres, pair distribution functions Computer generation of spheres Computer generation of snow microstructures 2 Size parameters Particle diameter (2a); Stickiness ( τ ) Particle diameter (2a); Stickiness ( τ ) ; b Solution methodAnalytical QCANumerical solution of Maxwell equation using Foldy-Lax equations Numerical solutions of Maxwell equations using DDA / FFT

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Quasi-Crystalline Approximation (QCA) Lorentz-Lorenz law; Generalized Ewald-Oseen theorem Phase matrix, pair distribution function and structure factor Structure factor is the Fourier transform of 5

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Diameter = 1.4 mm; Stickiness parameter τ =0.1; stickiness, adhere to form aggregates QCA sticky has weaker frequency dependence than Mie scattering Scattering Rate: QCA Compared With Classical Mie Scattering 6

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Scattering Properties 1-2 polarization frame Phase matrix Scattering coefficient Mean cosine of scattering: angular distribution 7

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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 8

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Scattering Properties Comparison Scattering properties Independent scattering QCAFoldy LaxBicontinuous Frequency dependence 4.0As low as 2.8 Consistent with QCA As low as 2.5 mean cosine0 Dipole pattern Up to 0.3 Consistent with QCA Up to 0.6 Cross-pol in phase matrix 00Up to 15 dB below like-pol Up to 7 dB below like-pol 9

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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

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Random Shuffling Use Bonding States ◦ (a) Unbonded ◦ (b) Single-bond ◦ (c) Double-bond ◦ (d) Triple bond Kranendonk-Frenkel algorithm to calculate the probability, dependent on stickiness Aggregates formed from sequence of bonding Computer Generation Of Dense Sticky Particles Simulated sticky particles fv = 40% 11

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Solutions of Maxwell Equations using Foldy-Lax equations 12 field on particle i incident field Green’s function Mie scattering coefficients field on particle j

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Comparison Between Classical RT, DMRT / QCA and NMM3D NMM3D and QCA in agreement Weaker frequency dependence than independent scattering 13

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Model Comparison 14 Scattering properties Independent scattering QCAFoldy LaxBicontinuous Frequency dependence 4.0As low as 2.8 Consistent with QCA As low as 2.5 mean cosine0Up to 0.3Consistent with QCA Up to 0.6 Cross-pol in phase matrix 00Nonzero Dipole interactions Up to 15 dB below like-pol Nonzero Dipole interactions Up to 7 dB below like-pol QCAFoldy LaxBicontinuous ModelSpheres,Computer generation of spheres Computer generation of snow microstructures Size parametersParticle diameter (2a); Stickiness ( τ ) Particle diameter (2a); Stickiness ( τ ) ; b Solution methodAnalytical QCANumerical solution of Maxwell equation using Foldy-Lax equations Numerical solutions of Maxwell equations using DDA / FFT

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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

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Bicontinuous Model: Computer Generation of Terrestrial Snow Generation: superimposing a large number of stochastic waves Cutting level α determined by fraction volume 16

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Bicontinuous Model: Generation 17 Computer generated snow pictures vs. real snow picture A. Wiesmann, C. Mätzler, and T. Weise, "Radiometric and structural measurements of snow samples," Radio Sci., vol. 33, pp. 273-289, 1998. Depth Hoar (30%): 3 cm * 3 cm picture

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Volume integral equation Discrete Dipole Approximation (DDA): in each cube Matrix equations Matrix-vector product by FFT Numerical Solution Of Maxwell Equation 18

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Bicontinuous Parameters Bicontinuous parameters ( α,, b) One to one relation between α and f V Parameter : inverse size ◦ Grain sizes decrease as increases ◦ ζ follows Gamma distribution with mean value Parameter b determines the size distribution ◦ Size distribution uniform for large b ◦ Broad size distributon for small b 19

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SSA and Correlation function of Bicontinuous Medium 20

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Real Snow Parameters Real snow parameters ◦ Fraction Volume (f V ) or density ( ρ ): f V = ρ snow / ρ ice ◦ Auto Correlation Function (ACF) ◦ Specific Surface Area (SSA) ◦ Grain size Two grain size parameters ◦ D 0 : Equivalent grain size relating to SSA ◦ D max : Prevailing grain size, visually determined ◦ Empirical fit: D max =2.73D 0 21

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Bicontinuous Model: Parameters Dependences on 22

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Bicontinuous Model: Parameters Dependence on parameter b: b increases 23

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Bicontinuous Model: Correlation Function Close To Exponential Spatial auto correlation function 24

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Bicontinuous Model: Log Scale Correlation Function 25

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Bicontinuous Model: Specific Surface Area In Microwave Regime Analytical expression Numerical procedure: Use digitized picture, discretize according to microwave resolutions Count surface area 26 Δx [mm] 0.40.50.60.8 SSA [cm 2 /g]83.270.365.950.1 Example: =6000 [m -1 ], b=1.5, f V =30% Bicontinuous SSA=71.8 [cm 2 /g] Example: =6000 [m -1 ], b=1.5, f V =30% Bicontinuous SSA=71.8 [cm 2 /g]

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Bicontinuous Model: Phase Matrix Mean cosine: 27

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Brightness temperature increases with for the same κ S ◦ Physical temperature is 250 K ◦ Optical thickness = κ S d; All curves have same κ S Passive remote sensing: Effects Of ‘Mean Cosine’ 28

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Mean Cosine Comparisons Mean cosine > 0, means forward scattering is stronger than backward scattering 29 Models Mean cosine μ 1- μ Meaning Bicontinuous0.1 ~ 0.60.4 ~ 0.9 Forward scattering Rayleigh Phase Matrix 01.0Dipole scattering HUT0.960.04Strong forward scattering

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Data Validation With SnowSCAT Data collected ◦ At IOA snow pit ◦ Radar backscattering and ground data: Dec. 28, 2010~Mar. 1, 2011 Data ◦ Time series backscattering ◦ Time series SWE ◦ SSA ◦ Density ◦ Depths of multilayer structure ◦ Grain sizes 30

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Comparisons With SnowSCAT Time series data for 9 different days in the same IOA snow pit Ground truth of data point #8 ◦ Bottom layer is the thickest layer ◦ Bottom layer has the largest grain size Typical values of measured SSA ◦ SSA measured in a different year from snow depth, density and grain size ◦ Bottom layers: 59 ~ 124 [cm 2 /g] ◦ Top and intermediate layers: 100 ~ 790 [cm 2 /g] 31 # layer12345 Depth [cm]11020722 Density [g/cm 3 ]0.1120.1480.2120.1920.204 Grain size [mm]0.50.8 1.53.0 Date12/28/1001/04/1101/12/1101/18/1101/26/1102/01/1102/08/1102/23/1103/01/11 SWE [mm] 546173.576839799113114 Snow Depth [cm] 313945524953696059

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Data Validation With SnowSCAT Bicontinuous input parameters 32 # layer12345 [m -1 ] 3000020000 100006000 b1.0 1.5 1.2 fVfV 12.2%16.1%23.0%20.9%22.1%

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Data Validation With SnowSCAT Bicontinuous extracted parameters 33 Layer [m -1 ] bOptical thickness Mean cosine μ Correlation length [mm] Analytical SSA [cm 2 /g] Numerical SSA [cm 2 /g] 1300001.01.6×10 -4 0.190.051309222 2200001.08.7×10 -3 0.140.080228200 3200001.50.0150.050.085238188 4100001.50.0120.110.1711795.2 560001.20.170.310.287257.4

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Data Validation With SnowSCAT Co-polarization at 16.7 GHz 34

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DMRT Models Comparison 35 Scattering properties Independent scattering QCAFoldy LaxBicontinuous Frequency dependence 4.0As low as 2.8 Consistent with QCA As low as 2.5 mean cosine0, dipole patternUp to 0.3Consistent with QCA Up to 0.6 Cross-pol in phase matrix 00Nonzero Dipole interactions Up to 15 dB below like-pol Nonzero Dipole interactions Up to 7 dB below like-pol QCAFoldy LaxBicontinuous ModelSpheres, pair distribution functions Computer generation of spheres Computer generation of snow microstructures Size parametersParticle diameter (2a); Stickiness ( τ ) Particle diameter (2a); Stickiness ( τ ) ; b Solution methodAnalytical QCANumerical solution of Maxwell equation using Foldy-Lax equations Numerical solutions of Maxwell equations using DDA / FFT

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Summary Bicontinuous model ◦ Computer Generation of snow microstructures ◦ Three parameters α,, 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 Compare with SnowSCAT scatterometer data Using ground truth snow measurements 36

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