CONTENTS
Chapter 1 Introduction ......................................................................................................... 1
Chapter 2 Physical principle of the radiative transfer model ........................................... 3
2.1 Theory of radiative transfer method .................................................................................. 3
2.1.1 Basic radiative transfer equations for MC method......................................................... 3
2.1.2 Basic algorithm for MC radiative transfer simulation ................................................... 4
2.1.3 Example of a simple MC radiative transfer simulation .................................................. 6
2.2 Radiative properties in the atmosphere.............................................................................. 92.2.1 Coordinate system and atmosphere-surface model ........................................................ 92.2.2 Gaseous absorption ........................................................................................................ 92.2.3 Rayleigh scattering ....................................................................................................... 102.2.4 Mie scattering: Aerosol and cloud particles................................................................. 13
2.2.5 Moment of particle size distribution and volume mixing ratio ..................................... 15
2.2.6 Weighted average optical properties by particle size distribution................................ 16
2.2.7 Hygroscopic growth of aerosol ..................................................................................... 182.3 Radiative properties in plant canopies ............................................................................. 212.3.1 Optical properties of a leaf ........................................................................................... 212.3.2 Radiative transfer modeling of leaf canopy .................................................................. 24
2.3.3 Canopy structure of forest and grassland ..................................................................... 26
Chapter 3 Basic algorithms................................................................................................. 283.1 Generation of random numbers ....................................................................................... 28
3.1.1 Generation of uniform pseudorandom number............................................................. 28
3.1.2 Conversion method (inverse function method) ............................................................. 29
3.1.3 LUT method .................................................................................................................. 31
3.1.4 Rejection method (selection-rejection method)............................................................. 32
3.1.5 Weighting method.......................................................................................................... 33
3.1.6 Metropolis method ........................................................................................................ 34
3.1.7 Composite method......................................................................................................... 34
3.1.8 Summary of random number generator........................................................................ 35
3.2 Emissions from radiation source...................................................................................... 36
3.2.1 Emission energy and distribution of modeled photon................................................... 36
3.2.2 Solar radiation .............................................................................................................. 37
3.2.3 Point radiation source .................................................................................................. 38
3.2.4 Thermal emission .......................................................................................................... 38
3.2.5 Modification of spatial distribution of radiation source based on importance ............ 41
3.2.6 Radiation source of isotropic angular distribution....................................................... 42
3.3 Ray tracing ....................................................................................................................... 50
3.3.1 Intersection of modeled photon with geometric object ................................................. 50
3.3.2 Relationship between modeled photon and geometric object in 3-D scene.................. 51
3.3.3 Relationship between modeled photon and surface with roughness............................. 56i3.3.4 Space subdivision method ............................................................................................. 58
3.3.5 Boundary volume method ............................................................................................. 58
3.4 Collision........................................................................................................................... 60
3.4.1 Collision probability and determination of collision point........................................... 60
3.4.2 Treatment of absorption ................................................................................................ 62
3.4.3 Treatment of mixed media ............................................................................................. 65
3.4.4 Russian roulette method and photon-splitting method ................................................. 67
3.5 Scattering ......................................................................................................................... 69
3.5.1 Scattering process ......................................................................................................... 69
3.5.2 Modeling of scattering by MC method.......................................................................... 73
3.6 Reflection......................................................................................................................... 78
3.6.1 Definitions..................................................................................................................... 78
3.6.2 Reflection from surface with arbitrary BRDF .............................................................. 80
3.6.3 Isotropic reflection: Lambertian reflection................................................................... 82
3.6.4 Diffuse-specular mixture model: DSM model............................................................... 84
3.6.5 Rahman-Pinty-Verstraete (RPV) BRDF model............................................................. 90
3.6.6 Li-Sparse-Ross-Thick (LSRT) BRDF model ................................................................. 98
3.7 Refraction....................................................................................................................... 108
3.7.1 Snell’s law and Fresnel equations .............................................................................. 108
3.7.2 Treatment of refraction in atmosphere........................................................................ 109
3.7.3 Refractive index in earth’s atmosphere ....................................................................... 111
3.8 Calculation of irradiance and heating rate ......................................................................113
3.8.1 Definition of radiative quantities and sampling of quantities by MC method.............113
3.8.2 Method I: Random sampling of transmittance.............................................................114
3.8.3 Method II: Analytical sampling of transmittance ........................................................115
3.8.4 Method III: Hybrid method..........................................................................................117
3.8.5 Method IV: Use of equivalence theorem for absorption ..............................................118
3.9 LEM ............................................................................................................................... 120
3.9.1 Formulas for local estimate ........................................................................................ 120
3.9.2 Definition of PDF for angular distribution ................................................................ 123
3.10 Treatment of atmospheric sphericity............................................................................ 126
3.10.1 Characteristics of sphericity effect ........................................................................... 126
3.10.2 Correction of sphericity effect for line of sight ......................................................... 127
3.11 Spectral integration ...................................................................................................... 133
3.11.1 Extraterrestrial solar irradiance and photon flux density ........................................ 133
3.11.2 Calculation of spectral integration ........................................................................... 134
3.12 Canopy photosynthesis ................................................................................................ 137
3.12.1 Single leaf photosynthesis rate.................................................................................. 137
3.12.2 APAR calculation ...................................................................................................... 138
Chapter 4 Improvement of numerical efficiency ............................................................ 140
4.1 MCS method .................................................................................................................. 140ii

4.1.1 Base of MCS method................................................................................................... 140
4.1.2 Discussion on efficiency of MCS method.................................................................... 141
4.2 Variance reduction methods........................................................................................... 143
4.2.1 Exponential transformation method ........................................................................... 143
4.2.2 Modification of contribution function......................................................................... 144
4.2.3 Truncation approximations for anisotropic scattering ............................................... 147
4.2.4 Collision-forcing method ............................................................................................ 153
4.2.5 Numerical diffusion..................................................................................................... 155
4.2.6 Semi-analytical calculation of direct-beam and single-scattering components ......... 155
4.3 Parallelization ................................................................................................................ 157
4.3.1 Parallel programming................................................................................................. 157
4.3.2 Parallel computing by MC method I: Photon split method ........................................ 157
4.3.3 Parallel computing by MC method II: Space split method......................................... 158
Chapter 5 Introduction of radiative transfer models and examples of applications... 160
5.1 Simple code for MC radiation........................................................................................ 160
5.2 Introduction of atmospheric radiative transfer model.................................................... 163
5.2.1 Radiative transfer model MCARaTS........................................................................... 163
5.2.2 Calculation examples using MCARaTS ...................................................................... 166
5.3 Introduction of plant canopy radiative transfer model................................................... 170
5.3.1 Plant canopy radiative transfer model ....................................................................... 170
5.3.2 Simulated results ......................................................................................................... 173
Chapter 6 Conclusion ........................................................................................................ 176Acknowledgements ............................................................................................................... 177
Appendix................................................................................................................................ 178
A1 Analytical functions for size distribution....................................................................... 178
A1.1 Power-law distribution ................................................................................................ 178
A1.2 Gamma distribution..................................................................................................... 178
A1.3 Modified/generalized gamma distribution .................................................................. 180
A1.4 Lognormal distribution (number density) ................................................................... 181
A1.5 Lognormal distribution (volume density) .................................................................... 182
A2 Optical properties of mixed media................................................................................. 183
A3 Spectrally averaged optical properties ........................................................................... 185
A4 Vertical profile of aerosols ............................................................................................. 186
A5 Representation of photon transport by direction vector................................................. 188
A5.1 Movement of photon .................................................................................................... 188
A5.2 Rotation of direction due to scattering ........................................................................ 188
A5.3 Angle between two vectors .......................................................................................... 189
A6 Determination of random point in a circle: Polar coordinate method............................ 190
A7 Mathematical functions.................................................................................................. 191
A7.1 cos–1x............................................................................................................................ 191
A7.2 exp(–x) ......................................................................................................................... 192
iii

A7.3 1 – exp(–x) ................................................................................................................... 192
A7.4 sin(x) and cos(x) .......................................................................................................... 192
References............................................................................................................................... 194iv