Radiative Transfer in Coupled Environmental Systems

Knut Stamnes is professor of physics in the Department of Physics and Engineering Physics, and Director of the Light and Life Laboratory at Stevens Institute of Technology in Hoboken New Jersey. Stamnes began his career in upper atmospheric physics, and has since specialized in atmospheric radiation, remote sensing, and climate-related studies. He is a fellow of the OSA, a member of the AGU, EGU, and SPIE, and was elected member of the Norwegian Academy of Technological Sciences in 2009.Jakob J. Stamnes is professor of physics in the Department of Physics and Technology at the University of Bergen, Norway. He is fellow of the OSA, founding member and fellow of the EOS (European Optical Society), a member of the SPIE, EGU, and the Norwegian Physical Society, and was elected member of the Norwegian Academy of Technological Sciences in 2009.

• Preface XIAcknowledgments XIII1 Introduction 11.1 Brief History 11.2 What is Meant by a Coupled System? 21.3 Scope 31.4 Limitations of Scope 42 Inherent Optical Properties (IOPs) 72.1 General Definitions 72.1.1 Absorption Coefficient and Volume Scattering Function 72.1.2 Scattering Phase Function 82.2 Examples of Scattering Phase Functions 112.2.1 Rayleigh Scattering Phase Function 112.2.2 Henyey–Greenstein Scattering Phase Function 112.2.3 Fournier–Forand Scattering Phase Function 132.2.4 The Petzold Scattering Phase Function 142.3 Scattering Phase Matrix 142.3.1 Stokes Vector Representation IS = [I,Q,U,V]T 162.3.2 Stokes Vector Representation I = [I, I,U,V]T 202.3.3 Generalized Spherical Functions 222.4 IOPs of a Polydispersion of Particles – Integration over the Size Distribution 242.4.1 IOPs for a Mixture of Different Particle Types 252.4.2 Treatment of Strongly Forward-Peaked Scattering 262.4.3 Particle Size Distributions (PSDs) 282.5 Scattering of an Electromagnetic Wave by Particles 292.5.1 Summary of Electromagnetic Scattering 302.5.2 Amplitude Scattering Matrix 312.5.3 Scattering Matrix 322.5.4 Extinction, Scattering, and Absorption 342.6 Absorption and Scattering by Spherical Particles – Mie–Lorenz Theory 352.7 Atmosphere IOPs 412.7.1 Vertical Structure 412.7.2 Gases in the Earth's Atmosphere 422.7.3 Molecular IOPs 432.7.4 IOPs of Suspended Particles in the Atmosphere 452.7.5 Aerosol IOPs 452.7.6 Cloud IOPs 472.8 Snow and Ice IOPs 482.8.1 General Approach 482.8.2 Extension of Particle IOP Parameterization to Longer Wavelengths 502.8.3 Impurities, Air Bubbles, Brine Pockets, and Snow 512.9 Water IOPs 532.9.1 Absorption and Scattering by PureWater 532.9.2 Absorption and Scattering byWater Impurities 542.9.3 Bio-Optical Model Based on the Particle Size Distribution (PSD) 562.10 Fresnel Reflectance and Transmittance at a Plane Interface Between Two Coupled Media 632.10.1 Stokes Vector of Reflected Radiation 652.10.2 Total Reflection 652.10.3 Stokes Vector of Transmitted Radiation 672.11 Surface Roughness Treatment 682.11.1 Basic Definitions 682.11.2 Reciprocity Relation and Kirchhoff's Law 702.11.3 Specular Versus Lambertian and Non-Lambertian Reflection at the Lower Boundary 712.11.4 Scattering, Emission, and Transmission by a Random Rough Surface – Kirchhoff Approximation 722.11.4.1 Rough Dielectric Interface 722.11.5 Slope Statistics for a Wind-Roughened Water Surface 762.12 Land Surfaces 772.12.1 Unpolarized Light 782.12.2 Polarized Light 823 Basic Radiative Transfer Theory 853.1 Derivation of the Radiative Transfer Equation (RTE) 853.1.1 RTE for Unpolarized Radiation 853.1.2 RTE for Polarized Radiation 873.2 Radiative Transfer of Unpolarized Radiation in Coupled Systems 883.2.1 Isolation of Azimuth Dependence 893.3 Radiative Transfer of Polarized Radiation in Coupled Systems 903.3.1 Isolation of Azimuth Dependence 913.4 Methods of Solution of the RTE 933.4.1 Formal Solutions 943.4.2 Single-Scattering Approximation 963.4.3 Successive Order of Scattering (SOS) Method 1003.4.4 Discrete-Ordinate Method 1023.4.5 Doubling-Adding and Matrix OperatorMethods 1053.4.6 Monte Carlo Method 1093.5 Calculation ofWeighting Functions – Jacobians 1103.5.1 Linearized Radiative Transfer 1103.5.2 Neural Network Forward Models 1124 Forward Radiative Transfer Modeling 1174.1 Quadrature Rule –The Double-Gauss Method 1174.2 Discrete Ordinate Equations – Compact Matrix Formulation 1204.2.1 "Cosine" Solutions 1204.2.2 "Sine" Solutions 1224.3 Discrete-Ordinate Solutions 1234.3.1 Homogeneous Solution 1234.3.2 Vertically Inhomogeneous Media 1284.3.3 Particular Solution – Upper Slab 1294.3.4 Particular Solution – Lower Slab 1334.3.5 General Solution 1344.3.6 Boundary Conditions 1355 The Inverse Problem 1375.1 Probability and Rules for Consistent Reasoning 1375.2 Parameter Estimation 1405.2.1 Optimal Estimation, Error Bars and Confidence Intervals 1405.2.2 Problems with More Than One Unknown Parameter 1475.2.3 Approximations: Maximum Likelihood and Least Squares 1575.2.4 Error Propagation: Changing Variables 1605.3 Model Selection or Hypothesis Testing 1635.4 Assigning Probabilities 1685.4.1 Ignorance: Indifference, and Transformation Groups 1685.4.2 Testable Information:The Principle of Maximum Entropy 1735.5 Generic Formulation of the Inverse Problem 1815.6 Linear Inverse Problems 1825.6.1 Linear Problems without Measurement Errors 1835.6.2 Linear Problems with Measurement Errors 1855.7 Bayesian Approach to the Inverse Problem 1865.7.1 Optimal Solution for Linear Problems 1895.8 Ill Posedness or Ill Conditioning 1915.8.1 SVD Solutions and Resolution Kernels 1925.8.2 Twomey–Tikhonov Regularization – TT-Reg 1975.8.3 Implementation of the Twomey–Tikhonov Regularization 1985.9 Nonlinear Inverse Problems 2005.9.1 Gauss–Newton Solution of the Nonlinear Inverse Problem 2015.9.2 Levenberg–Marquardt Method 2036 Applications 2056.1 Principal Component (PC) Analysis 2056.1.1 Application to the O2 A Band 2066.2 Simultaneous Retrieval of Total Ozone Column (TOC) Amount and Cloud Effects 2076.2.1 NILU-UV Versus OMI 2096.2.2 Atmospheric Radiative Transfer Model 2106.2.3 LUT Methodology 2106.2.4 Radial Basis Function Neural Network Methodology 2106.2.5 Training of the RBF-NN 2116.2.6 COD and TOC Values Inferred by the LUT and RBF-NN Methods 2116.2.7 TOC Inferred from NILU-UV (RBF-NN and LUT) and OMI 2136.2.8 Summary 2146.3 Coupled Atmosphere–Snow–Ice Systems 2156.3.1 Retrieval of Snow/Ice Parameters from Satellite Data 2166.3.2 Cloud Mask and Surface Classification 2186.3.2.1 Snow Sea Ice Cover and Surface Temperature 2186.3.3 Snow Impurity Concentration and Grain Size 2196.4 Coupled Atmosphere–Water Systems 2256.4.1 Comparisons of C-DISORT and C-MC Results 2266.4.2 Impact of Surface Roughness on Remotely Sensed Radiances 2266.4.3 The Directly Transmitted Radiance (DTR) Approach 2286.4.4 The Multiply Scattered Radiance (MSR) Approach 2296.4.5 Comparison of DTR and MSR 2306.5 Simultaneous Retrieval of Aerosol and Aquatic Parameters 2326.5.1 Atmospheric IOPs 2336.5.2 Aquatic IOPs 2346.5.3 Inverse Modeling 2356.6 Polarized RT in a Coupled Atmosphere–Ocean System 2376.6.1 C-VDISORT and C-PMC Versus Benchmark – Aerosol Layer – Reflection 2396.6.2 C-VDISORT and C-PMC Versus Benchmark – Aerosol Layer – Transmission 2396.6.3 C-VDISORT and C-PMC Versus Benchmark – Cloud Layer – Reflection 2426.6.4 C-VDISORT and C-PMC Versus Benchmark – Cloud Layer – Transmission 2426.6.5 C-VDISORT Versus C-PMC – Aerosol Particles – Coupled Case 2456.6.6 C-VDISORT Versus C-PMC – Aerosol/Cloud Particles – Coupled Case 2456.6.7 Summary 2496.7 What if MODIS Could Measure Polarization? 2496.7.1 Motivation 2496.7.2 Goals of the Study 2506.7.3 Study Design 2506.7.4 Forward Model 2526.7.5 Optimal estimation/Inverse model 2526.7.6 Results 2546.7.7 Concluding Remarks 260A Scattering of ElectromagneticWaves 263A.1 Absorption and Scattering by a Particle of Arbitrary Shape 264A.1.1 General Formulation 264A.1.2 Amplitude Scattering Matrix 265A.1.3 Scattering Matrix 266A.1.4 Extinction, Scattering, and Absorption 268A.2 Absorption and Scattering by a Sphere – Mie Theory 271A.2.1 Solutions of VectorWave Equations in Spherical Polar Coordinates 272A.2.2 Expansion of Incident PlaneWave in Vector Spherical Harmonics 275A.2.3 Internal and Scattered Fields 277B Spectral Sampling Strategies 287B.1 The MODTRAN Band Model 289B.2 The k-Distribution Method 290B.3 Spectral Mapping Methods 293B.4 Principal Component (PC) Analysis 294B.5 Optimal Spectral Sampling 294C Rough Surface Scattering and Transmission 297C.1 Scattering and Emission by Random Rough Surfaces 297C.1.1 Tangent Plane Approximation 298C.1.2 Geometrical Optics Solution 300C.1.2.1 Stationary-Phase Method 301D Boundary Conditions 313D.1 The Combined Boundary Condition System 313D.2 Top of Upper Slab 315D.3 Layer Interface Conditions in the Upper Slab 317D.3.1 Interface Between the Two Slabs (Atmosphere–Water System) 319D.4 Layer Interface Conditions in the Lower Slab 325D.5 Bottom Boundary of Lower Slab 325D.5.1 BottomThermal Emission Term 327D.5.2 Direct Beam Term 327D.5.3 Bottom Diffuse Radiation 328D.5.4 Bottom Boundary Condition 329References 331Index 347