Chapter 1 Introduction
Models that treat three-dimensional (3-D) radiative transfer processes are needed for thedetailed computation of the spatial and angular distributions of the radiative energy in an
atmosphere-ocean-land system with clouds and in forest and city canopies. Such a radiation
model is especially important for realistic simulations of the 3-D distribution of radiative
heating in cloud fields simulated by cloud-resolving dynamical models. In addition, for the
purpose of wide-area observation of the atmosphere and land using satellite remote sensing data,
it is important to understand the relationships between satellite-observed signals (radiance) andthe properties of clouds, vegetation, and land surfaces. Although simplified treatments havebeen used conventionally, more detailed, realistic treatments are becoming possible with the aidof improved computational power.
There are several complicated problems related to modern radiative transfer calculation
methods in geophysical research, including 3-D radiative transfer in a cloudy atmosphere andplant/city canopy, light scattering by leaves and non-spherical particles, polarization, radiativetransfer in a spherical atmosphere, and scattering of a laser beam. Most of these can be modeledby Monte Carlo (MC) methods. In particular, the MC methods for 3-D radiative transfer
problems are popular not only in geophysics but also in astrophysics, medical sensing,mechanical engineering, and many other fields (Taniguchi et al., 1994).
The MC method is widely used as a method for numerical integration. Its advantage is thatit can be applied very easily to solving complicated problems that are difficult to solve withother deterministic (analytical/numerical) methods. In general, deterministic methods work
more efficiently for rather simple problems than the MC method, but the merit of the MC
method increases with increasing problem complexity (e.g., for problems with higher
dimensions of integration). The deterministic methods require larger computational resources(computation time and memory) for a complicated problem, while the computational resources
needed for the MC method do not strongly depend on the complexity of the problem.
Deterministic solvers for one-dimensional (1-D) radiative transfer were established in the
1980s. Accurate 3-D radiative transfer calculations were very difficult without the MC method,which has a long history of application to radiative transfer. The MC method was used witheven smaller computational resources in the 1970s for 3-D radiative transfer calculations in a
simple 3-D geometry (McKee and Cox, 1974; Aida, 1977; Oikawa and Saeki, 1977; Davies,
1984). The rapid growth of computational power enabled researchers to deal with the problem
using a more complicated atmosphere (Barker and Davies, 1992; Cahalan et al., 1994a; Barker
et al., 1998; O’Hirok and Gautier, 1998; Macke et al., 1999; Fu et al., 2000; Barker et al., 2003;
Iwabuchi and Tsuboki, 2004). In the 1970s, plant ecologists tried to develop MC models for theradiative transfer in the plant canopy (Oikawa and Saeki, 1977). Subsequently, advances inmodeling studies were made by researchers in geophysics and planetary physics (Ross andMarshak, 1988; Antyufeev and Marshak, 1990; North, 1996; Govaerts and Verstraete, 1998). As
a result, similar theories and methods have been used in recent MC-based models for the1

atmosphere and plant canopy.
The development of deterministic (analytical/numerical) models for 3-D radiative transfer
began in the 1990s, and several models are mature enough for actual applications withreasonable computation time (Gastellu-Etchegorry et al., 1996; Evans, 1998; Marshak andDavis, 2005). However, MC models are often more efficient, especially for complicatedproblems. One of the advantages of the MC model is that the model treats the non-approximate
physical processes behind the radiative transfer and can simulate very accurately the physicalprocesses.
The authors have developed MC-based 3-D radiative transfer models that can be used forvarious applications such as for the radiative energy budget in the atmosphere and plant canopy
and in simulations for remote sensing using optical measurements. This technical report
describes the basic theory simply and uses algorithms to provide details. In writing this report,
we gave special attention to covering various possible algorithms in great detail, to enable
readers to write the computation code by themselves. Due to this effort, explanations of thefundamental theories of radiative transfer are not the main focus of the report, since they can befound in many textbooks and in the literature.
In the following, Chapter 2 describes briefly the basic theory and algorithms of the MCmodel. Chapter 2 also describes the radiative properties of atmospheric components and plant
canopies. Chapter 3, the main part of this report, describes the respective algorithms used in the
MC models. The fundamental processes of radiative transfer are fully described, including
source emissions, scattering and absorption in the atmosphere and plant canopy, reflection fromthe land surface and ocean, and computation methods for estimating the radiative quantities ofinterest. Chapter 4 presents methods and techniques for improving computational efficiency. In
some cases, the acceleration methods presented here enable significantly faster computation.
Chapter 5 introduces three actually-working models for the atmosphere and plant canopy andpresents several examples of their applications. Chapter 6 presents conclusions.

2Chapter 2 Physical principle of the radiativetransfer model

2.1 Theory of radiative transfer method
Here, we summarize the theory of the photon transport simulation based on the MC method. Wethen introduce the fundamental concept of the MC algorithm and simple code for MC radiativetransfer simulation.

2.1.1 Basic radiative transfer equations for MC method
The differential equation for 3-D radiative transfer under the steady state is expressed as theradiation I (W m–2 sr–1) budget of an infinitesimally small arbitrary area:Ω⋅ ∇I (r,Ω) = − β e (r)I (r,Ω) +βs (r)I (r,Ω')P(r,Ω',Ω)dΩ' + β a (r)BT (r) (2.1.1)
4 π ∫4 π

The first term on the right side of (2.1.1) is an attenuation of I(r, Ω) in the photon path (unitvector Ω), and the second term is the incoming radiation toward the direction Ω via a scatteringfrom the direction Ω′ with I(r, Ω ). The third term is the contribution of the thermal radiation. βe
and βs (m–1) are extinction and scattering coefficients for the unit volume, respectively, and P isa scattering phase function, which is defined as the scattering ratio of the radiation with thedirection Ω to Ω′.
The radiative transfer equation is also expressed by the integral equation (Marchuk et al.,1980):

f (x) = ∫ X k(x', x) f (x')dx'+ψ (x)(2.1.2)x = (r, Ω) and k(x′, x) is a kernel for photon transport, which is a product of the collision(collision kernel) at point x′ and the state transition (transition kernel) from point x′ to point x,
and is expressed ask(x', x) =β s (r')P(µ)exp[−τ (r',r)]β e (r) ⎛r − r' ⎞
δ
Ω
−
⎜
⎟
2
r − r' ⎠2πβ e (r') r − r'⎝

(2.1.3)µ is a cosine between (r – r )/| r – r | and the scattering direction Ω. By substituting f(x) to(2.1.2), the integral equation of the radiative transfer can be formulated as a von Neumann
series:3

∞f (x) = ∑ ∫ X k n (x', x) f (x')dx'(2.1.4)n= 0

As shown in (2.1.4), the collision density in the focused area is a summation of the contributionof radiation from zero to infinite orders of scattering.
The MC radiative transfer calculation is an integration technique based on the MC method.
The radiation is sampled by simulating the physical processes described in (2.1.1) and (2.1.4),
such as the emission, collision, and scattering of photons. MC radiative transfer simulation does
not directly solve the above sets of equations, but rather traces the trajectories of the “modeled
photon” in the simulated space, where the modeled photon is defined as a sampling unit in the
MC radiative transfer simulation and is not identical to an actual light photon defined as aquantum unit. The modeled photon corresponds to a group containing a number of photons.
The MC radiative transfer calculation has two different approaches: forward-type andbackward-type MC. While the forward type simulates the photon trajectory along the actual
light path, the backward type simulates the photon trajectory in reverse, following the actual
light path from the local point to the light source. The backward type is superior to the forward
type when we would like to accurately determine the radiative quantity in the focused point withcostless time. On the other hand, the forward type is useful in simulating many physicalquantities over the simulated space in a one-time simulation. Hereafter, we mainly focus on thedescription for the forward-type MC method. Some of the algorithms and simulation techniques
are common to both approaches.

2.1.2 Basic algorithm for MC radiative transfer simulation
In the MC simulation, the spatial distribution of radiative quantities such as irradiance, radiance,and heating rate are sampled by tracing the modeled photon, the position, direction, and weightof which are modified by the physical laws governing the radiative transfer theory (Fig. 2.1.1.).
The fundamental flow of the algorithm is depicted in Fig. 2.1.2. The procedure is as follows:i) Start the photon tracing at the given light source. Solar radiation or thermal radiationfrom the earth could be a natural source of light energy. An artificial source, such as alaser, could also be a potential source of light.
ii) Obtain a random number, simulate the path of photon travel using the random number,and move the photon ahead by the determined path length.
iii) After the photon travel, if the photon has gone outside the simulation space, stop thephoton tracing and return to procedure i). If the photon still exists in the simulation
space, return to ii) and continue the photon tracing.
Here, the path for photon travel is determined using the random number ρ:τ = −ln ρ(2.1.5)The traveling path of the modeled photon is described in 3.4 in detail. When the modeled4

photon collides with the media, it is scattered or absorbed. The partition of the scattering andabsorption depends on the optical properties of the media. Although the MC radiative transfer
model could perform these processes via various numerical techniques, the scattering direction
must be determined following the statistical distribution of the phase function P. The
determination of the scattering direction is described in 3.5 in detail.
Radiative quantities such as irradiance, radiance, and heating rate are sampled by tracing themodeled photon. In general, these quantities are calculated by

F = ∑ ΨiΞ(ri )i⎧1 r ∈ A
Ξ(r) = ⎨⎩0 r ∉ A(2.1.6)

where A is the sampling area and Ψ(r) is a contribution function of these quantities. In theactual simulation, these quantities are evaluated using various methods based on (2.1.6).
For example, when we want to know the irradiance at a certain surface in the focused area,
we can calculate it by counting the weight of the modeled photons that pass through that surface.
Also, when we want to know the radiance, we can calculate it by sampling the contribution of
the modeled photon at every scattering event using the local estimation method (LEM). The
heating rate can be sampled from the change in the weight of photons after every scattering. A
detailed description of the sampling method is provided in the following chapter.

Fig. 2.1.1 Basic concept of photon tracing in the MC method.

5

Fig. 2.1.2 Flowchart of the MC radiative transfer algorithm.

2.1.3 Example of a simple MC radiative transfer simulation
We introduce a simple MC simulation as shown in Fig. 2.1.2. Let us consider a 1-D rod-likeatmosphere (Fig. 2.1.3), in which a photon can move only along the z-axis. A photon entersfrom the top of the rod, and photon tracing starts. The entering photon travels the path length
determined by (2.1.5) and collides with a medium. Figure 2.1.4 shows an example of thesimulation results.
The results (Fig. 2.1.4) are obtained from the atmospheric medium under isotropic scattering(50% forward and 50% backward scattering) and no absorption. As the atmospheric optical
thickness gradually increases, the reflectance at the top of the rod-like atmosphere gradually
increases and the transmittance at the bottom of the atmosphere decreases. In addition toatmospheric optical thickness, these results (reflectance and transmittance) depend on an
atmospheric scattering phase function and absorption.
The simulation code is shown on the next page. This source code of just 40 lines enables us
to simulate the conceptual behavior of the atmospheric reflected and transmitted flux undervarious atmospheric conditions (under various quantities of single scattering albedo, surface
reflectance, etc.).

6

Fig. 2.1.3 Radiative transfer in 1-D rod-like atmosphere.

Reflectance/Transmittance1
ReflectanceTransmittance

0.8
0.6
0.4
0.2
0

0

1

2

3

4

5

Optical thickness

Fig. 2.1.4 Relationship between reflectance/transmittance and optical thickness
derived from the MC simulation.

7

Fortran 77 simulation code
c Simple one-dimensional monte carlo simulation
implicit none
integer i,np
real rand,zu,zb,z,tau,sr,path,w,dir,omg,sumr,sumt
np=10000
tau=1.0
sr=0.5omg=1.0
zu=tauzb=0.0sumr=0.0
sumt=0.0! total photon! optical thickness! surface reflectance! single scattering albedo! upper boundary! lower boundary! summation of reflectance! summation of transmittance

do i=1,np! photon loop
w=1.0
! initial photon weightz=zu! photon set in the upper boundary
dir=-1.0! photon direction (+1 upward -1 downward)do! single photon interaction loop
path=-log(rand(0)) ! photon free path
z=zu+dir*path! determination of next position z
if(z.le.zb)then! if next position of z is lower than lower boundary
z=zb! z is set to the lower boundary
dir=1.0! upward photon direction
sumt=sumt+w! transmittance sampling
w=w*sr! new photon weight
elseif(z.gt.zu)then ! if z is larger than upper boundary
sumr=sumr+w! reflectance samplingexit! exit do loopelse
! else case scattering occurs
if(rand(0).gt.0.5)then ! random number [0,1] is larger than 0.5
dir=1.0*dir! photon direction unchanged (forward scattering)else! random number [0,1] is less than 0.5
dir=-1.0*dir ! photon direction change (backscattering)end ifw=w*omg! new photon weightend ifend doend dowrite(*,*) "Reflectance top & Transmittance bottom"
write(*,*) sumr/np,sumt/np
stop
end

8

2.2 Radiative properties in the atmosphereWe describe the optical properties in an atmosphere-surface system. In particular, we emphasize
the optical properties of Rayleigh scattering and Mie scattering.

2.2.1 Coordinate system and atmosphere-surface model
In the MC method, various kinds of coordinate systems are potentially available. The typicalcoordinate systems are1) Cartesian coordinate system, which is conventionally expressed by (x, y, z)2) Spherical coordinate system, which is usually employed when considering the sphericalshape of the planetary surface
3) Coordinate system fitted in a topographical shape, which uses a rectangular grid in the (x,y) direction. The gridding of the vertical direction (z) depends on the (x, y) grid system
and is determined along the shape of the surface topography.
A constant optical property is assumed as a single voxel, but it consists of a mixture of several
atmospheric media. The typical compounds are1) Absorbing gases: H2O, CO2, O3, CH4, O2, other uniformly mixed gases
2) Scattering gases (Rayleigh scattering): O2, N2, CO2
3) Aerosols: water-soluble species, dust, soot, organic carbon, etc.
4) Cloud particles: cloud water, cloud ice
In each grid, extinction, single scattering albedo, and phase function are given based on thesecompounds. In addition, since aerosol and cloud particles have a particle size distribution, it ispossible to divide these particles into “bins” for each size distribution. In this case, each bin isdealt with as one compound of the media. However, this method requires massive amounts ofcomputational memory and, in most cases, the single parameter of the so-called “effective
particle” works well as a representative of the particle size distribution. Therefore, detailed
particle size information is not usually necessary.
The surface properties are provided as a surface elevation and normal vector at each (x, y)
point of the grid. The surface reflectance is modeled as a Lambertian or bidirectional reflectance
distribution function (BRDF). Physical descriptions of these models are given in 3.6.

2.2.2 Gaseous absorption
The spectra of gas absorptions have strong and spike-like peaks, which are determined by theenergy level transitions of the gases. The frequent oscillation of these spike-like peaks spreads
over a certain waveband, forming the spectral distribution of the absorption coefficient as anaggregation of such spike-like peaks. The absorption peaks are collected in the HITRAN

9database and are available to the public. In the actual atmosphere, since the width of anabsorption peak varies with the atmospheric temperature and pressure, the absorption
coefficients are computed as a function of the temperature and pressure.
In the radiative transfer simulation, a line-by-line calculation within a specific waveband isideally necessary; however, wavelength band models such as the correlated k-distribution(CKD) method are usually used due to the computational limitation (Shibata, 1999). So far,several waveband models have been developed for climate/meteorological and remote sensing
purposes. MODTRAN (a later version of LOWTRAN, Kneizys et al., 1988) was developed forthe latter purpose. The modeling for climate/meteorology is dedicated to achieving the required
accuracy with fewer bands (Fu and Liou, 1992; Chou and Lee, 1996; Kato et al., 1999;
Nakajima et al., 2000).
The CKD method calculates the radiative quantity in a single waveband by the weighted
average of the radiative calculation in multiple terms. For the given kth absorption coefficientβa(r, k) and weight w(k), w(k) is normalized asK∑ w(k) = 1(2.2.1)k=1

When F(k) is defined as a radiative quantity of the kth term, the waveband-integrated radiative
quantity is calculated asK

F = ∑ w(k)F(k)(2.2.2)

k=1

In the MC modeling, there are two approaches to solve the CKD method: the approach to
distribute the number of photons proportionally to the weight of each CKD term and theapproach to distribute the number of photons with arbitrary rules (e.g., equal distribution) andEq. (2.2.2).

2.2.3 Rayleigh scatteringa. Scattering cross section
The scattering cross section for Rayleigh scattering by atmospheric molecules is expressed as(Thomas and Stamnes, 1999)σ=

24 π 3 (ns2 −1)

2

λ4 N s2 (ns2 + 2)

2

F(air)(2.2.3)where λ is a wavelength (cm). σ is nearly proportional to λ–4. Ns, which is a number representing
molecular density (molecules cm–3) in the standard atmosphere (288.15 K, 1013.25 hPa), is
derived from Avogadro’s number A (= 6.0221367 × 1023 mol–1) and molar volume.

10

6.0221367 ×10 23 273.15
Ns =×
22.4141×1000 288.15= 2.546899 ×1019(2.2.4)

ns is a refractive index (see 3.7). F is called a depolarization term or King factor:

F(air) =6 + 3ρ
6 − 7ρ

(2.2.5)The parameter ρ is a depolarization factor, which is an anisotropy indicator of the molecular
structure. The King factor is the most uncertain parameter and the following values have been
proposed:
Penndorf (1957): F(air) = 1.0608
Young (1981): F(air) = 1.048(2.2.6a)(2.2.6b)Of the two, the value given by Young (1981) has been the most widely used. However, in theory,
F is related to the wavelength λ. According to Bodhaine et al. (1999), the weighted average ofthe depolarization of the molecules proposed by Bates (1984) has the best accuracy. Bates(1984) proposed the following equations:

1
λ4
1
1
F(O2 ) = 1.096 +1.385 ×10 −3 2 +1.448 ×10 −4 4λλF(Ar) = 1.0
F(CO2 ) = 1.15
F(N 2 ) = 1.034 + 3.17 ×10 −4(2.2.7a)(2.2.7b)(2.2.7c)(2.2.7d)Bodhaine et al. (1999) recommended using the following equation based on the equations given
by Bates (1984):

F(air, CO 2 ) =78.084F(N 2 ) + 20.946F(O 2 ) + 0.934 ×1.0 + CCO ×1.15
278.084 + 20.946 + 0.934 + CCO(2.2.8)2

CCO2 is the CO2 concentration (parts per volume, e.g., 360 × 10–6 for 360 ppm).
Finally, σ is determined by substituting it for the King factor derived from (2.2.8) and Ns
derived from (2.2.4). The refractive index of the atmosphere is calculated using the equation
mentioned in 3.7. These sets of equations are the most accurate procedure for σ determination atpresent. In these equations, it is possible to calculate σ under a change in the CO2 concentration.
Bodhaine et al. (1999) also proposed a simpler empirical equation for σ (10–32 m2) under aconstant CO2 concentration (360 ppm, λ (µm)):

σ=1.0455996 − 341.29061λ−2 − 0.90230850λ2
1+ 0.0027059889λ−2 − 85.968563λ2(2.2.9)The error for this equation is less than 0.01% in 0.25–0.85 µm and 0.05% in 0.85–1.0 µm,

11which is still accurate compared with what has so far been proposed.
-28

Scattering Cross Section (m^2)

10

-29

10

-30

10

-31

10

-32

10

0.2

0.4

0.6

0.8

1.0

Wavelength (µm)

Fig. 2.2.1 Scattering cross section of Rayleigh scattering.

b. Scattering coefficient and optical thickness
Under hydrostatic equilibrium, it is easy to calculate the vertically integrated optical thickness
when the scattering coefficient is given. The optical thickness is a function of atmosphericpressure:τ (λ ) = σ

PA
M dry g(2.2.10)P, Mdry, and g are pressure, average molecular weight, and gravitational acceleration,
respectively. According to Bodhaine et al. (1999), when we assume CCO2 to be 360 ppm,
Mdry (0 ppm CO2) = 28.95949 gm mol–1
Mdry (360 ppm CO2) = 28.96491 gm mol–1

M dry ≅ 28.9595 +15.0556(CO2 )g is a function of the latitude φ and elevation z (m). In Bodhaine et al. (1999), g (cm s–1) was
calculated by following the method given by List (1968):

g = g0 − (3.085462 ×10 −4 + 2.27 ×10 −7 cos 2φ )z+(7.254 ×10 −11 +1.0 ×10 −13 cos 2φ )z 2−(1.517 ×10−17+ 6.0 ×10

−20

cos 2φ )z(2.2.11)3

g0 is the gravitational acceleration at z = 0 m.

g0 = 980.616(1− 0.0026373cos 2φ ) + 0.0000059 cos 2 2φ

12(2.2.12)

According to Bodhaine et al. (1999), zc (m), which provides the height for effective g in (2.2.10),
is a useful parameter and computed byzc = 0.73737z + 5517.56(2.2.13)

Once the optical thickness is derived from (2.2.10), the scattering coefficient can be calculatedasβ s (λ, z) =dτ (λ )
A dP
= −σ
dz
M dry g dz

(2.2.14)

When the atmosphere is far from the hydrostatic equilibrium condition, the scattering
coefficient can be computed under the given vertical profiles of the temperature and pressure.
The scattering coefficient is then a function of the molecular density N:

β s (λ, z) = σN

(2.2.15)

N is calculated as

N=

A P
⋅
R* T

(2.2.16)

where R* (= 8.3143 J K–1 mol–1) is the universal gas constant. When Nb and Nt are the density
values at heights zb and zt, respectively, N(z) is formulated under the assumption of an
exponential vertical profile:⎡ z − zb⎤
N(z) = N b exp⎢−(ln N b − ln N t )⎥⎣ zt − zb⎦(2.2.17)Hence, the optical thickness between zb and zt is calculated asτ (zb , zt ) = ∫ z σN(z)dzztb⎡ z − zb⎤zt
= σN b ∫ z exp⎢−(ln N b − ln N t )⎥dzb
⎣ zt − zb⎦N
Nb − N t=σ b(zt − zb )N t ln N b − ln N t(2.2.18)

2.2.4 Mie scattering: Aerosol and cloud particles
The aerosol and cloud particles are usually approximated as having a spherical shape with a sizeof 0.005–100 µm. In this case, a scattering event can be mathematically formulated by Mietheory (Mie scattering). The optical properties of Mie scattering are characterized by the
wavelength λ, particle radius r, and complex refractive index n. The complex refractive indices13

of water, ice, ammonium sulfate, soot, soil particles, etc., were investigated based on laboratory
measurements and observations (Hale and Querry, 1973; Palmer and Williams, 1974; Downing
and Williams, 1975; Shettle and Fenn, 1979; d’Almeida et al., 1991). The refractive index ofparticles such as water weakly depend on the temperature. A method to calculate the single
sphere extinction coefficient, scattering cross section, and scattering phase function wasprovided by Bohren and Huffman (1983) or elsewhere. Nowadays, we can use state-of-the-art
simulation code that can calculate not only spherical particles but also non-spherical particle
cases.
Various sizes of aerosol and cloud particles are generally mixed together in the atmosphere.
Let the number density be N (#/m3) and particle size distribution be n(r) (m–3 µm–1).

dN
dr∞
N = ∫ n(r)drn(r) =(2.2.19a)(2.2.19b)

0

The particle size distribution can be approximated by an analytical function (Appendix A1), ora simulated distribution by the bin method and the observed distribution are also used.
75%H2SO4
Dust-like
Vocanic Ash
Soot
Yellow Sand

75%H2SO4 (OPAC)
Insoluble (OPAC)
Mineral (OPAC)
Soot (OPAC)

NH4NO3 (Lacis)(NH4)2SO4 (Lacis)
Water Soluble (rstar)
Water Soluble (OPAC)
3.0

2.6
Water (DW1975)
Water (HQ1973)
Water (Lacis)
Water (PW1974)
Ice (wave16)

Real Part

2.0

2.5

2.5

1.8

Real Part

2.2

Real Part

2.4

2.0

1.6
1.4

2.0

1.5

1.5

1.2

75%H2SO4
Dust-like

1.0

2

1

10
100
Wavelength (µm)

46 8 Volcanic
2 Ash
4

1.0

75%H2SO4 (OPAC)
Insoluble (OPAC)
6 8 Mineral
2 (OPAC)
4
6 8

1
10
Soot
Soot (OPAC)
Yellow Wavelength
Sand
(µm)

1000

100

NH4NO3 (Lacis)
Sea Salt (Lacis)(NH4)2SO4 (Lacis)
Sea Salt (rstar)
Water
Sea Salt (OPAC)
1 Soluble (rstar)
10
100
1000
Water Soluble(OPAC) (µm)
Wavelength

2

0

10

1

10

0

10

-1

0

10

10

-1

10

-1

10

-2

-2

-4

10

-5

10

Water (DW1975)
Water (HQ1973)
Water (Lacis)
Water (PW1974)
Ice

-6

10

-7

10

-8

10

-9

10

10
Imaginary Part

-3

10

Imaginary Part

10

-2

10

Imaginary Part

Sea Salt (Lacis)
Sea Salt (rstar)
Sea Salt (OPAC)

-3

10

-4

10

-5

10

-6

10

10
100
Wavelength (µm)

1000

-5

10

-6

10

-7
-8

10

10

-9

-8

1

-4

10

10

-7

10

-10

10

-3

10

2

4

6 8

1

2

4

6 8

10

2

4

6 8

100

2

10

1

Wavelength (µm)

Fig. 2.2.2 Reflective indices of various materials.

14

10
100
Wavelength (µm)

1000

2.2.5 Moment of particle size distribution and volume mixing ratio
The geometric average radius, and cross-section-mean and volume-mean radii are defined by1∫ rn(r)dr
N
⎛1
⎞1/2
2
rsec = ⎜ ∫ r n(r)dr ⎟
⎝N
⎠
⎛1
⎞1/ 3
3
rvol = ⎜ ∫ r n(r)dr ⎟⎝N⎠rgeo =(2.2.20a)(2.2.20b)(2.2.20c)

The most important parameters for particle size distribution are the effective radius reff and theeffective dispersion veff (Hansen and Travis, 1974):∫ r ⋅ πr n(r)dr = rr =∫ πr n(r)dr r∫ (r − r ) ⋅ πr n(r)dr = 1 ∫ r n(r)dr −1
v =
r r
N
r ∫ πr n(r)dr
2

eff

3
vol
2
sec

2

2

2

4

eff

eff

2
eff

2

(2.2.21)3
eff vol(2.2.22)

These parameters are the weighted average and dispersion of the particle size distribution. The
relationships between the effective radius, cross-section-mean radius, and volume-mean radius
can be expressed by⎛ r ⎞2 ⎛ r ⎞3
κ = ⎜⎜ sec ⎟⎟ = ⎜⎜ vol ⎟⎟⎝ reff ⎠ ⎝ reff ⎠(2.2.23)

Here, κ is a coefficient related to the width of the particle size distribution. According to thefield observation of the particle size distribution by Martin et al. (1994), κ in maritime boundary
layer clouds is 0.8 and in continental ones is 0.67.
The atmospheric particle volume in a unit volume V (volume mixing ratio in m3/m3) and themass density M (kg m–3) are expressed by

4 3
4π 3
πr n(r)dr =rvol N
3
3
M = ρV

V=∫(2.2.24a)(2.2.24b)

ρ is a density (kg m–3). If V or M and N are given as a priori knowledge, the effective radius canbe derived from (2.2.23a) and (2.2.24a,b):⎡ 3⎤1/ 3 ⎡ 3 M ⎤reff = ⎢V=⎢⎥⎣ 4 πκN ⎥⎦⎣ 4 πκN ρ ⎦1/ 3(2.2.25)

15

Thus, the parameter κ is related to V or M and N. From (2.2.20) and (2.2.24), the geometriccross section (m2 m–3) of particles contained in the unit atmospheric volume is calculated by

3 V

3 M

eff

eff

∫ πr n(r)dr = Nπr = 4 r = 4 r ρ
2

2
sec

(2.2.26)

The particle size distribution is usually approximated by an analytical function. Clouds andaerosols can be widely approximated by the power-law, gamma, and lognormal distributions(Hess et al., 1998). The characteristics of the typical distribution function and its relationship to
M, V, and N are summarized in Appendix A1.

2.2.6 Weighted average optical properties by particle size distribution
The extinction coefficient, scattering coefficient, and scattering phase function, which areaveraged with the weighting of the particle size distribution, are

βe = ∫ Q˜ e (r)πr 2 n(r)drβ s = ∫ Q˜ s (r)πr 2 n(r)dr(2.2.27a)(2.2.27b)∫ P˜ (Θ, r)Q˜ (r)πr n(r)dr∫ Q˜ (r)πr n(r)dr2

P(Θ) =

s(2.2.27c)

2s

The extinction/scattering efficiency factor, which is defined by the extinction cross section in aunit cross section, and phase function can be calculated by Mie theory. When the average
extinction/scattering efficiency factor weighted by the particle size distribution is formulated by∫ Q˜ (r)πr n(r)dr , Q = ∫ Q˜ (r)πr n(r)drQ =∫ πr n(r)dr∫ πr n(r)dr
2

2

e

e

s

2

s

2

(2.2.28)

the average extinction cross section, scattering cross section, and single scattering albedo of thesingle particle are calculated from (2.2.23b):r32
σ e = Qe πrsec= Qe π vol
reff
r3
2
σ s = Qs πrsec= Qs π volreffσQ
ω= s = sσ e Qe(2.2.29a)(2.2.29b)(2.2.29c)

In summary, when the particle size distribution is given in a scattering particle medium, therequired parameters for the optical parameter to be determined are16

(Qe ,Qs , P(Θ)) and ( N, rsec )Note that this can be rewritten in a different form expressed by M and V when using(2.2.23–24).
As shown in the above description, the average optical properties weighted by the particle
size distribution are usually required to integrate using the weight of the particle size
distribution and geometric cross section (see (2.2.27–28)). That is,∞rmax
Y = ∫ 0 Y˜ (r)πr 2 n(r)dr ≅ ∫ r Y˜ (r)πr 2 n(r)dr(2.2.30)minHere, Y(r) does not strongly depend on r and irregularly varies (e.g., scattering efficiency factor).
n(r) is generally distorted toward the positive direction. Therefore, the transformation given
below is useful:

dN
dN
=r
= rn(r)d ln rdr∞
ln rmax
Y = ∫ Y˜ (r)πr 3n(r)d ln r ≅ ∫Y˜ (r)πr 3n(r)d ln r−∞

ln rmin(2.2.31)(2.2.32)

Here, it should be emphasized that it is not recommended to refer to n(r) when determining the
integral domain. The maximum rmax and minimum rmin of the integral domain should bedetermined by minimizing r3n(r).
Generally, it is necessary to consider a mixture of several aerosols and cloud water/cloud ice
for an atmospheric medium. The optical properties of the mixing media are described inAppendix A2. In the radiative transfer simulation, waveband average parameters are usuallyemployed. Appendix A3 summarizes the method to derive the waveband average parameters.
Furthermore, an example of the determination of the aerosol vertical profile is described inAppendix A4.

Fig. 2.2.3 Cloud water phase function (left) under the assumption of a lognormal
distribution for the particle volume. Phase function (right) for various widths of thelognormal distribution for the particle volume. 10 µm is assumed for the effective
radius.

17

2.2.7 Hygroscopic growth of aerosol
There are two types of aerosols: hygroscopic aerosols (e.g., (NH4)2SO4, NH4NO3, NaCl) andnon-hygroscopic aerosols (e.g., soil, desert dust, black carbon). The size of an hygroscopic
aerosol increases when absorbing atmospheric water vapor. Then, the density and refractive
index approach the parameters for water. Hygroscopic growth with an increase in relative
humidity does not begin until the relative humidity reaches a critical point (RHD). When therelative humidity rises above the RHD, hygroscopic growth begins and follows a curve
determined by the relative humidity. On the other hand, when the aerosols release theircontained water with a decrease in the relative humidity, all of the water is released at a criticalpoint (RHC). Generally, the RHC is less than the RHD. Due to such hysteresis of thehygroscopic aerosol, the increase ratio and mixing ratio cannot be determined simply by the
relative humidity: they depend on the pathway of the aerosol particles. In the actual atmosphere,
most particles lay above the hysteresis curve of the relative humidity (Hess et al., 1998).
The growth factor (radius increase ratio) B and mixing ratio are defined by

r
r0
m
x= 0
m

B=

(2.2.33a)(2.2.33b)

where m is the mass of particles and the subscript 0 indicates the dissolved substance under a
dry condition. Here, the relationship between B and ρ isB3 =m ρ0 ρ0=m0 ρ ρx(2.2.34)

The maximum B is approximately 4 at a relative humidity = 0.99. Strictly speaking, the relative
humidity η (between 0 and 1) of the equilibrium state increases slightly due to the Kelvin effect(surface tension). However, since this phenomenon is negligibly small, the relative humidity
approximately becomes equal to water activity:η ≅ aw(2.2.35)

Tang and Munkelwitz (1994, 1996) derived empirical formulae for the various materials:

aw = 1.0 + ∑ Ci x i(2.2.36a)iρ = 0.9971+ ∑ Ai x i(2.2.36b)i

These equations are applicable to the density ρ calculation by using (2.2.36b) and the mixing
ratio calculated by (2.2.36a) when the relative humidity η = aw and η > RHC.
B is also calculated using (2.2.34). When the material is the same, B is determined only by ηand the increase ratio of the particle size depends only on B (r = r0B). The aerosol size
distribution under hygroscopic growth is expressed by18

dN dN1r== n0 (r0 ) , r0 =dr Bdr0 B
B
dN
dN
∴
=
d ln r d ln r0n(r) =(2.2.37a)(2.2.37b)

For example, when the number density of the particle size distribution under the dry condition isa lognormal distribution (Appendix A1), the dissolved case of size distribution becomes
lognormal and the mode radius is proportional to B. The standard deviation σ is unchanged.
In the case of the refractive index, the molar ratio is expressed using the mixing ratio x:

fs =

xM w
(1− x)M s + xM w

(2.2.38)

where Mw and Ms are the molar weights of the water and dissolved substrate, respectively.
Letting the molar refractions of water and the dissolved substrate be Rw and Rs, respectively,
the real part of the refractive index nr is obtained fromnr2 −1(1− fs )Rw + fs Rs=ρ2
nr + 2(1− fs )M w + fs M s(2.2.39)

The molar refractions are calculated from the following relationships:

Rw =M w nw2 −1
M s ns2 −1R=,sρ w nw2 + 2
ρ s ns2 + 2(2.2.40)

Tang and Munkelwitz (1994, 1996) calculated an accurate value for nr at λ = 0.633 µm. By
using it, the ratio to interpolate the refractive index is calculated at λ = 0.633 µm:α=nr − nw
ns − nw(2.2.41)

At this wavelength, nw = 1.333. When this value is applied to the real and imaginary parts of the
all wavelengths, the refractive index is interpolated asn = (1− α )n w + αn s(2.2.42)

where ns and nw are complex refractive indices for a dissolved substrate and water, respectively.
There are other ways to interpolate the refractive index, using the weighted average of the mass
(α = x) and weighted average of the volume (α = 1/B3); however, the method just described(2.2.39) is more reasonable. The parameters for typical materials are summarized in Table 2.2.2.
Because Mie scattering of the hygroscopic aerosol depends not only on the radius but also
on the refractive index, it is necessary to prepare a 2-D LUT (radius and α). When the particle
size distribution and relative humidity are given, x, ρ, and α can be calculated from the relative
humidity. Then, the scattering properties of a hygroscopic aerosol are obtained from the19weighted average of the particle size distribution for the interpolated scattering properties fromthe LUT. The LUT can be prepared for a dry particle radius and relative humidity. However, it isdifficult to change the relationship between the relative humidity and radius of the dissolved
particles.

Table 2.2.1 Parameters of various hygroscopic materials.
RHD
RHC
C1
C2
C3
C4
A1
A2
A3
A4
Dry Density, ρ0
Refractive Index, ns
Ms
aEb = a × 10b(NH4)2SO4
0.8
0.4–0.37–2.715E–1
3.113E–1–2.336E0
1.412E0
5.92E–1–5.036E–2
1.024E–2
0
1.76
1.526
132.154

NH4NO30.62
0.25–0.32–3.65E–1–9.155E–2–2.826E–1
0
4.05E–1
9.0E–2
0
0
1.725
1.554
80.052

20

NaCl
0.753
0.48–0.46–6.366E–1
8.624E–1–1.158E+1
1.518E+1
7.41E–1–3.741E–1
2.252E0–2.06E0
2.165
1.544
58.44

H2SO4
NA
NA
–5.196E–1
9.746E–1–9.693E0
9.405E0
7.367E–1–4.934E–1
1.754E0–1.104E0
1.834
1.429
98.086

2.3 Radiative properties in plant canopies
We now discuss the photon interaction within a plant canopy. The definitions for the vegetation
parameters and the method of forest scene generation are also described.

2.3.1 Optical properties of a leaf
Photons interact with the leaves, stems, and branches within a plant canopy. Because moststudies have been dedicated to understanding the influence of environmental conditions on the
spectral features of a leaf, we primarily focus on the optical properties of a leaf. Since modelingstudies for the optical properties of stems and branches have already been conducted, theobserved data from such studies are generally used for radiative transfer simulation. The
primary difference between a leaf and stem/branch is the existence of chlorophyll-a andchlorophyll-b.

a. Photon reflection, transmission, and absorption in a leaf
The reflectance and transmittance of a single leaf are determined by the leaf’s structure andbiochemical composition. Figure 2.3.1 shows an example of simulated single leaf reflectanceand transmittance between 400 and 2500 nm. Every leaf has similar spectralreflectance/transmittance patterns.
ReflectanceTransmittance

0.0

0.8

0.2

0.6

0.4

0.4

0.6

0.2

0.8

0.0

500

1000

1500
2000
Wavelength

Transmittance

Reflectance

1.0

1.0
2500

Fig. 2.3.1 Typical single leaf reflectance and transmittance. These are calculatedusing the LIBERTY model (Dawson et al., 1998) for the biochemical data of a larch
needle.

21

>5+147?5012-,.47,90<,5@@7<7594'"#)*+,-,.*/++012
3145678979:)5++;+,<5=-,4579

'"!
!"&
!"%
!"$
!"#
!"!

(!!

'!!!
'(!!
#!!!
31?5+5984*0A9BC#(!!

Fig. 2.3.2 Relative absorption coefficient for the pigment in a leaf.
In the visible spectral region, due to the strong absorption of chlorophyll-a andchlorophyll-b and carotenoids, the reflectance and transmittance are low. Chlorophyll-a andchlorophyll-b have strong peaks around 400 nm and 670 nm (Fig. 2.3.2), and the carotenoids
have a strong peak at 440–450 nm. Consequently, the reflectance has a convex shape, with areflectance/transmittance peak around 550 nm. This is the reason why leaves have a dark green
color. A fresh leaf has a rather light green color because of lower amounts of chlorophyll-a andchlorophyll-b. In the autumn, the leaf’s color becomes red or yellow due to the lack of thepigment required for photosynthesis.
In the near-infrared region (700–1400 nm), there is almost no absorption by the pigment.
Therefore the incoming photons are scattered many times within the cell wall before exiting.
The intensities of the reflected and transmitted radiation are determined by the leaf structuresuch as the thickness of the cell and the cell size.
In the shortwave infrared region (1400–2500 nm), there are two strong water absorption
bands at 1450 nm and 1900 nm, causing a reduction in the reflectance and transmittance of theleaf. Also, other pigments such as protein, cellulose, and water have a continuous weakabsorption, causing a gradual decrease in reflectance and transmittance with longer wavelengths(Fig. 2.3.2).

b. Chlorophyll fluorescenceIn the visible spectral region, there is a photon emission process called chlorophyll fluorescence.
Chlorophyll fluorescence is a phenomenon that occurs in photosynthesis II. In thephotosynthesis cycle, the photon energy absorbed by the plant is converted to organic energy.

22However, not all of the photon energy absorbed by the chlorophyll in the leaf is converted. The
extra energy that is not converted is dissipated as thermal heat or reemitted. This reemittanceprocess is chlorophyll fluorescence. In chlorophyll fluorescence, chlorophyll molecules in an
excited condition emit photons. The peak wavelength of chlorophyll fluorescence is around 680
nm (red). The energy of chlorophyll fluorescence is 0.5–3% of the absorbed energy (Hikosaka,
2003), making it negligibly small in radiative transfer modeling.

c. Xanthophyll cycle and spectral change
The xanthophyll cycle is an energy dissipation process for thermal energy that occurs when theabsorbed photosynthetically active radiation is larger than the leaf’s photosynthesis ability. In
the biochemical reaction processes of the xanthophylls, the absorption peak of the carotenoid
moves from 443 nm to 451 nm (Yoshimura, 2001). The reflectance changes caused by thisprocess are generally small at the canopy scale. It may be possible to detect this change withlaboratory measurements. However, at the canopy scale, this has a very small effect on canopy
reflectance and transmittance because the canopy contains leaves in various conditions (sunlit
leaves, shaded leaves with various leaf angles) and the amount of absorbed energy is different
for each leaf.

d. Modeling of leaf surface reflectance and transmittanceAs described above, a leaf’s reflectance and transmittance are determined by the pigment
concentration and leaf structure. Several models have been proposed to simulate leaf-levelreflectance and transmittance. Jacquemoud and Baret (1990) proposed PROSPECT, a leaf
reflectance and transmittance model for broadleaf. Dawson et al. (1998) proposed the LIBERTY
model, a reflectance and transmittance model for needle leaves. These models have different
assumptions for leaf structures. PROSPECT assumes that the leaf has a layered plate structure,and the LIBERTY model views the leaf as a series of floating spheres regularly positioned in theleaf. Both models simulate the hemispherical reflectance and transmittance of a leaf. Here, themodel structure used in PROSPECT is described as an example.
PROSPECTThe PROSPECT model assumes that the leaf is a series of layered plates. This model is anextension of the plate model by Allen et al. (1969). When the maximum incident angle is θm, thehemispherical reflectance (RN,θm) and transmittance (TN, θm,) of the Nth plate are calculated by

RN ,θ m = xRN ,90 + y
TN ,θ m = xTN ,90(2.3.1)(2.3.2)where x and y are given in the following equations:x=t(θ m ,n)t(90,n)(2.3.3)23

y = x(t(90,n) −1) + 1− t(θ m ,n)(2.3.4)t(θm,n) is the average transmittance over the incident angle from 0 to θm in the boundary layer
with the refractive index (n). This is analytically calculated by Fresnel’s law and Stern’s theory(Stern, 1964). RN,90 and TN,90 are calculated using the following equations:

RN ,90T
1= N N ,90 −N =−NN−Nb − b90a90 − a90 a90b90 − a−1
90b90
2
2
(1+ ρ 90
− τ 90+ δ90 )a90 =2 ρ 9022(1− ρ 90 + τ 90
+ δ90 )
b90 =
2τ 90
N
90

2
2
2
δ90 = (τ 90− ρ 90−1) 2 − 4 ρ 90t(90,n) 2 k 2 (n 2 − t(90,n))ρ 90 = 1− t(90,n) + 4
n − k 2 (n 2 − t(90,n)) 2t(90,n) 2 kn 2
τ 90 = 4n − k 2 (n 2 − t(90,n)) 2(2.3.5)(2.3.6)(2.3.7)(2.3.8)(2.3.9)(2.3.10)k is a parameter related to the pigment absorption and can be converted to K by the following
equation:k − (1− K)e−K − K 2 ∫ K x −1e−x dx = 0∞(2.3.11)The leaf absorption coefficient is derived as a weighted average of the content (C) andabsorption coefficient (K) of each pigment:

K = ∑ K iC i

(2.3.12)

The refractive index of the leaf is diverse. However, the values between 1.3 and 1.5 are widelyused (Gausman et al., 1974). In the PROSPECT model, the leaf reflectance and transmittanceare calculated from the maximum incident angle of the photon (θm), refractive index (n),absorption coefficient (K), and leaf thickness (N).

2.3.2 Radiative transfer modeling of leaf canopy
a. Leaf area index and leaf area density
The leaf area density u(r) is defined as half of the total leaf area in the unit volume (m2 /m3 =m–1) at the position r. The leaf area index (l) is half of the total leaf area in the unit ground area
and can be obtained by integrating the leaf area density u(r) from the ground to the top of the
canopy height (H):24l = ∫ 0 u(r)dz
H(2.3.13)b. Leaf angle distribution
Generally, plants allocate leaves to gain the maximum photosynthesis. The leaf angle
distribution is diverse, varying with the plant species and leaf position in the canopy. It is
helpful to use the statistical distribution function. The distribution function gL(r, Ω L) is defined
as a function of the position r and the normal vector Ω L. gL(r, Ω L) is normalized by

1 2 π π2∫ ∫ gL (r,Ω L )sinθ L dθ L dϕ L = 1
2π 0 0(2.3.14)

In most cases, the leaf angle does not depend on the azimuth angle and (2.3.14) is expressed by∫ g (r,θ )dµ = 1
1

0

L

L

(2.3.15)

L

where µL = cosθL. At present, several types of leaf angle distribution functions have been
proposed. The simplest function is a uniform leaf distribution:gL (r,θ L ) = 1(2.3.16)

The ellipsoidal distribution is defined by:

Fig. 2.3.3 Ellipsoidal distribution function (θm = 45°, ε. = 0.9).

25

gL (r,θ L ) =B(2.3.17)1− ε cos 2 (θ L − θ m )2

where B is a normalization factor. Figure 2.3.3 is an example of the ellipsoidal distribution. This
distribution function can be characterized by two parameters (the center of the leaf angle
distribution θm and eccentricity ε).

c. Extinction coefficient for a leaf canopy
The extinction coefficient for a leaf canopy is described by the projection area in the photon
direction Ω. The G-function is expressed byG(r,Ω) =1 2 π π2∫ ∫ gL (r,Ω L )Ω L ⋅ Ω sinθ L dθ L dϕ L
2π 0 0(2.3.18)

Here, G becomes 1/2 when the leaf angle distribution is uniform. The extinction coefficient in aunit volume βe is expressed byβe (r,Ω) = G(r,Ω)u(r)(2.3.19)

The canopy optical thickness can be calculated by integrating (2.3.19) from the ground to the
top of the canopy:Hτ = ∫ βe (r,θ = 0)dz

(2.3.20)02.3.3 Canopy structure of forest and grasslanda. Modeling of tree structure
There are several ways of modeling a canopy structure for forest light environmental simulation(Fig. 2.3.4). The simplest canopy is a plane-parallel approximation. In this canopy, the leaves,stems, and branches are distributed homogeneously in a layer (Fig. 2.3.4 left). Just as in the caseof a plane-parallel atmosphere, it is possible to simulate a multilayer canopy, in which eachlayer has different optical properties.
A 3-D canopy scene is composed of individual trees, in which all the trees are modeled asgeometric objects (Fig. 2.3.4 center). In this case, the leaves and branches are distributed within
the canopy objects. Canopies modeled using cones, cylinders, and ellipsoids could be good
approximations, and it is possible to make more complicated shapes with combinations of theseprimitive geometric objects.
In a plane-parallel canopy and 3-D object canopy, the canopy optical and structural
conditions can be defined using the leaf area density u and leaf angle distribution gL.
Furthermore, when we need to prepare a more realistic canopy scene, it can be developed by26

detailed modeling of the individual leaf, stem, and branch positions and property allocations(Fig. 2.3.4 right). This enables us to conduct very realistic radiative transfer simulations, but the
preparation of a canopy scene is not an easy task and the radiative transfer calculation itself
requires vast computational resources.

Fig. 2.3.4. Various canopy scenes. Plane parallel (left), 3-D geometric scene(center), very realistic scene (right).

b. Preparation of canopy scene
To determine the 3-D canopy scene, it is necessary to determine the individual tree sizes andpositions. When observation data are available, these data could be used to model the canopy
scene. In other cases, it is necessary to determine them ecologically. When the output from a
dynamic vegetation model is available, it could be another source of information for the canopy
structure. Figure 2.3.5 shows an example of a canopy scene.

Fig. 2.3.5

Example of a canopy scene.

27

Chapter 3 Basic algorithms

3.1 Generation of random numbers
We discuss the generation of a uniform pseudorandom number and random numbers that follow
the arbitrary distributed probability density function (PDF). Generally, all random numbers that
follow the arbitrary distributed PDF can be generated from the uniform pseudorandom number.

3.1.1 Generation of uniform pseudorandom number
There are several methods to generate the uniform pseudorandom number in [0,1]. The mostcommon method is a linear congruential generator. In this method, the nth integer is calculatedby the following recurrence equation:

I n = aI n−1 + b (mod m )(3.1.1)

where m is a modulus and a is a multiplier. The uniform pseudorandom number in [0,1] is
obtained by

ρ=I m(3.1.2)

When m is sufficiently large, a long-range random number can be produced. There is a special
case (b = 0). In this case, the recurrence equation is expressed by

I n = aI n−1 (mod m )(3.1.3)This is called a multiplicative congruential generator. It has been theoretically and empirically
proven that a multiplicative congruential generator has the same accuracy as a linear
congruential generator. Note that m and a should be selected carefully. Park and Miller proposeda minimal standard generator (Press et al., 1992: Numerical Recipes (NR)):a = 7 5 = 16807, m = 2 31 −1 = 2147483647(3.1.4)

The recurrent cycle of this algorithm is 231 – 2 = 2.1 × 109.
The use of Eq. (3.1.3) sometimes causes an overflow in a computer simulation, in which theresults depend on the computer system. A convenient method to overcome this issue isintroduced in NR (“ran0”). The most disadvantageous point of the linear congruential generator
is a serious correlation. This problem is well discussed in Fushimi (1989) and NR. (“ran1”) in
NR is a method to overcome this problem by shuffling two random numbers.
At present, various criteria and tests have been proposed to measure the robustness of a28

random number. The proposed method for generating a random number has been wellinvestigated, and it is known that the maximum-length linearly recurring sequence is one of thebest methods: it can generate a long cycle and uniform random numbers. This method is called ashift register with feedback loops. The recurrence equation for the method is expressed by

gn = a1gn−1 + a2 gn−2 ++ a p gn− p (mod 2)(3.1.5)The initial value of gn (n = 1, 2,…, p) can be selected freely, unless all coefficients are 0. This
recurrence equation has 1-bit information (0 or 1) and is defined above the Gallois body
(Fushimi, 1989). In a computer, random numbers are generated by an exclusive OR operation.
The cycle of this method is 2p – 1. An N-bit integer should be N 1-bit numbers. Fushimi (1989)
described the following recurrence equation:

gn = gn−32 + gn−521 (mod 2)(3.1.6)

A 32-bit integer is composed of a series of thirty-two 1-bit integers. The cycle of this equation is2521 – 32 – 1 ~ 3 × 10147, that is, significantly larger than one required for actual applications using
a current simulation resource. This method determines a random number by a single exclusive
OR operation between two integer numbers. Therefore, this method is very fast compared toothers, because it uses the bit operation rather than arithmetic calculation. This method is fasterthan other methods described in the NR. Other bit operation-based methods like the Mersenne
Twister are also good random number generators.

3.1.2 Conversion method (inverse function method)
Let us consider random numbers that follow the general PDF. When the PDF (g(x)) is defined in
the interval a–b, g(x) is normalized by∫ g(x)dx = 1
b(3.1.7)a

The random variable X that follows this PDF is derived from the uniform pseudorandom
number ρ in [0, 1]:

G(X) = ∫ a g(x)dx = ρX(3.1.8a)−1

X = G (ρ )(3.1.8b)

If (3.1.8) is analytically solved, this method is valid.

a. Exponential distribution
In the case of exponential distribution with an average of 1,g(x) = e − x , x ≥ 0(3.1.9a)29

G(x) = 1− e − x
G −1 (x) = −log(1− x)(3.1.9b)(3.1.9c)

X = −log(1− ρ ) = −log ρ ′, ρ ′ > 0(3.1.10)If we put

the random number X that follows the exponential function can be generated. If the average is
not 1 but µ, then the random number X is derived by

X = −µ log ρ ′, ρ ′ > 0(3.1.11)b. Gaussian distribution
If we assume a Gaussian distribution with an average of 0 and variance 1, the PDF is expressedbyg(x) =⎛ x2 ⎞
1exp⎜− ⎟2π
⎝ 2⎠(3.1.12)

We can obtain two random numbers that follow the Gaussian (Box-Muller method, NR):

X1 = −2 log ρ1 cos(2 πρ 2 )(3.1.13)

X2 = −2 log ρ1 sin(2 πρ 2 )However, (3.1.13) is not very fast. The rejection method that is described later is an effective
way to improve the speed by avoiding the sine and cosine calculations (Appendix A6).
1) Generation of ρ1, ρ2.
2) Determination of the position of the point in the quadrant, whose center and area are
the origin and 4, respectively.
W1 = 2ρ1 – 1, W2 = 2ρ2 – 1(3.1.14a)3) Calculation of the distance from the origin

R 2 = W12 +W22

(3.1.14b)4) If R2 < 10–12 or > 1, then return 1).
5) Using the random number, the sine and cosine of the azimuth angle are determined byρ1 = R 2 , cos φ = W1 R , sin φ = W1 R

(3.1.14c)6) Finally, the Gaussian random number is determined by

X1 =

W1
W
−2 log R 2 , X2 = 2 −2 log R 2
RR(3.1.14d)

The arbitrary Gaussian distribution with an average µ and standard deviation σ is derived from30

(3.1.13):

X ′ = σX + µ(3.1.15)c. Gamma distribution: Integer α case
The PDF of the gamma distribution with the structure parameter α and scale parameter 1 isexpressed byg(x) =

1 α −1 − x
x e
Γ(α )

(3.1.16)

When α = 1, (3.1.16) becomes an exponential distribution. The sum of two g(x)’s that follow α1and α2 is equal to g(x) with the structure parameter α1 + α2, that is, the so-called reproductive
property.
By using the reproductive property and exponential distribution (3.1.10), the parameter isfollowed by the gamma distribution if α is an integer:

X = −log ρ1 − log ρ 2 −− log ρα(3.1.17)= −log(ρ1ρ 2  ρα )When α is small, this is the best way. However, when α is large, another method would bedesirable. A detailed discussion is provided by Fushimi (1989).

d. Discrete distribution
When the probability parameter X has discrete values x1, x2,…, xn and their probabilities are p1,p2,…, pn, the random number X that follows this discrete distribution is determined by
k−1

k

i=1

i=1

X = xk , if ∑ pi ≤ ρ < ∑ pi

(3.1.18)

3.1.3 LUT method
If Eqs. (3.1.8a) and (3.1.8b) are not analytically solved, a numerical method is required. In thiscase, a look-up table (LUT) for X = G −1 ( ρ ) should first be analytically computed for various
values of ρ. It is possible to prepare the LUT for ρ = G( X) . The former method is more
effective. X is derived by interpolating from the LUT. Thus, the accuracy depends on the
resolution of the LUT. Anyway, it is impossible to obtain an exact solution.
When the number of parameters is less than 2, a solution is easily obtained by linear
interpolation or spline interpolation. When the LUT has a larger dimension, the interpolation
itself requires vast computation. The fastest method is a nearest-neighbor interpolation.
However, that is not reliable and, due to memory limitations, it is impossible to make a31

fine-resolution LUT. Therefore, a multidimensional LUT is not effective for improving the
speed and accuracy.

3.1.4 Rejection method (selection-rejection method)
Let us consider f(x) that meets the following inequality:

f (x) ≥ g(x)(3.1.19)f(x) is called a comparison function. Here, f(x) should meet the following conditions: it can be
integrated analytically and its inverse function should be easily derived. When F(x) is definedby(3.1.20)
the parameter x is randomly determined by the following equations:∫ f ( x′)dx′ = F(x) = ρ∫ f ( x′)dx′ F(b)xab

(3.1.21a)1a∴ x = F −1 (ρ1F(b))(3.1.21b)

This method is called the rejection method or selection-rejection method (Fushimi, 1989; Press
et al., 1992). In von Neumann’s method, the random number that follows f(x) is first determined.
Then, the rejection or selection is randomly determined. The probability for which x in Eq.
(3.1.21b) is followed by the PDF = g(x) is given by g(x)/f(x). In this case, using the newly
generated random number, the following criteria are applied:(3.1.22)

If rejection is chosen, x is determined by (3.1.21b). The advantage of this method is that it
obtains an exact solution, even if the function g(x) is complex. It should be noted that thismethod requires iterative calculation. When the efficiency is close to 1, we can obtain a goodefficiency:∫ f (x)dx = F(b)b(3.1.23)aTherefore, it is important to find a good comparison function. The typical comparison function
is a rectangular, triangular, or trapezoid shape. For example, if g(x) is a smooth function in thegiven interval a–b and the difference between the maximum and minimum is small, the simplest
function is a rectangular distribution function:f (x) = gmax

(3.1.24)32

where gmax is a maximum of g. In this case, F(x) is expressed byF(x) = gmax (x − a)(3.1.25)

and x is calculated by

x = a + ρ (b − a)(3.1.26)

The problem occurs when the function has a strong peak. Under such a condition, the difference
between the maximum and minimum is large and the efficiency deteriorates when we use therectangular function. It is sometimes very difficult to find a good comparison function, andmoreover, it is sometimes difficult to find the maximum of the PDF.
As described in 3.1.2, the determination method for the random position in the circle is one
example of the rejection method. A randomly-distributed point within a 3-D arbitrarily shaped
object can also be determined by the rejection method.

3.1.5 Weighting method
This method adds the weight w to each random number x. If the change in w is allowed, this is avery fast method. Let us consider the PDF h(x) normalized in the same interval with the PDF
g(x):∫ h(x)dx = 1
b(3.1.27)aFirst, the random number that followed h(x) is determined using one of the previously explained
methods (inverse function method, LUT method, rejection method). For this x, the following
weight provides the random number that distributes the PDF g(x):w(x) =g(x)h(x)(3.1.28)

G(x) = ∫ a g( x′)dx′
x

= ∫a

x

g( x′)
h( x′)dx′h( x′)(3.1.29)= ∫ w( x′)h( x′)dx′xa

As an example, when h is a uniform distribution function,h(x) =

1b−a(3.1.30)

The random number x is determined by:33x = a + ρ (b − a)(3.1.31)

Then, the weight is provided byw(x) = (b − a)g(x)(3.1.32)

This is just a function proportional to the PDF g(x).
The disadvantage of this method is that the weight may take a large range of values from 0
to infinity. This may cause a large noise in the MC integration. The possible application of thismethod is as follows: First a value for x that closely follows g(x) is generated using a rejection
method and comparison function h(x). Then, the balance of the probability is adjusted by using
the weight formulated in (3.1.28).

3.1.6 Metropolis method
The Markov chain Monte Carlo (MCMC) method is an MC method in the narrowest sense. The
Metropolis method is a type of MCMC method. It was originally designed for statistical physics
but this method can be applied as a general random number generator that follows the general
PDF.
A Markov chain is a recurrence formula in which the current state depends only on the value
just before the current state. In the Metropolis method, the random number X is determined by
the following procedures:
1) Next position: the new X′ is randomly selected from the current value Xt.
2) Calculation of the transition probability: selection of g(X′)/g(Xt).
3) Renewal of the state: by using the random number, the following value is determined:⎧ X ′ if ρ ≤ g( X ′) g(X t )X t+1 = ⎨⎩ X t otherwise(3.1.33)

The random series X1, X2,… generated by this method has a strong correlation. Thus thesame value for X is frequently generated. This is a major difference compared with the usual
random number sampling methods. Usually, this method is distinguished from other MC
methods. It is necessary to investigate the impact of these strong correlations on the accuracy.

3.1.7 Composite method
When a PDF can be expressed by the composition of the PDFs, the PDF is decomposed byg(x) = p1g1 (x) + p2 g2 (x) ++ pn gn (x)p1 + p2 ++ pn = 1

34(3.1.34)(3.1.35)∫ g (x)dx = 1; i = 1, 2,, nba(3.1.36)iFirst, the PDF pi of the discrete distribution is determined by the random integer i. Then, X isdetermined by gi(x). The various methods described before can be used for the determination.

3.1.8 Summary of random number generator
In summary, we strongly recommend to using the rejection method because of its applicability
to various purposes and efficient speed. This method does not require a large memory and isaccurate. It is not necessary to calculate the inverse function. The only requirement is a criterion
to reject or select the random number by repeating the rejection criteria. It is also applicable tosample random variables based on a 2-D PDF.
The characteristics of the various methods are summarized in Table 3.1.1.
Table 3.1.1
generators.

Advantage and disadvantage for the various random number

Advantage
Inverse function -Accurate
method-Fast for a simple function
LUT method-Approximated value is obtained
under an arbitrary PDF
Rejection
methodWeightingmethodMetropolis
methodDisadvantage-Fails to work if inverse function is
not available-Large memory is necessary-Bad accuracy for a bad LUT
resolution-Accurate value is obtained under -Sometimes difficult to find a goodan arbitrary PDF
comparison function-Very simple-May cause a large noise in MC
calculation-Accurate value is obtained under -Series of random numbers is aan arbitrary PDF.
Markov chain, so it causes a strong
correlation with the nearest random
number.

35

3.2 Emissions from radiation source
Let us consider the photon emissions from a radiation source. We will discuss several radiation
sources, including solar radiation, thermal radiation from the atmosphere-surface, artificial light,
point radiation source like a laser, and radiation from the stars.

3.2.1 Emission energy and distribution of modeled photon
The radiation energy that a single photon conveys in the MC model is generally formulated by
the emission energy from the simulation space divided by the total number of photons. The
transferred energy by a modeled photon with weight 1 is given by

Etot
N tot
Etot = ∑ ∑ ∑ E (i, j, k )
E1 =

i

j

(3.2.1a)(3.2.1b)k

where E is the emission energy from an element (voxel or surface element), (i, j, k) is anelement index corresponding to (x, y, z). In the case of a horizontally uniformly emitted flux inan element, when we assume A (i, j) as a horizontal cross section of the modeled air column (i,j) and F(i, j, k) as the emitted flux from the element (i, j, k), the following relationship exists:

E (i, j, k ) = ∫∫ F(x, y, k)dxdy(3.2.2)= F(i, j, k)A(i, j)

This equation is valid for a radiation source with a finite size, such as solar radiation andthermal emission. We cannot define the emitted flux for a point source because the emitted flux
becomes infinite in the emission point. In the point source case, the emitted power is defined asa finite value. If we consider the various radiation sources, the total emission energy is writtenas

E (i, j, k ) = Esolar (i, j, k) + Ethermal (i, j, k) + Epoint (i, j, k) +

(3.2.3)The energy can be simply divided by distributing the number of photons in a way that is
proportional to the energy of each component. There are several methods to determine theemission position from the random number. For example, first, we could allocate the same
number of photons in a divided subspace and then change the transport energy of a single
photon for every subspace. However, the most reasonable method is a method where the photon
distribution is determined to follow the 3-D distribution of the energy E. Here, the element of
the emission is first determined based on the 3-D distribution of E. Then, the position of theradiation source is determined by the emission energy distribution. Consider that Ecol is E
accumulated in a column; Ecol can be expressed by

36

N +1

Ecol (i, j ) = ∑ E (i, j, k )(3.2.4)k=0

where k = 0 and k = N + 1 represent the surface and the top of the atmosphere, respectively. The
average number of modeled photons in the modeled air column (i, j) is

N col (i, j ) = N tot

Ecol (i, j )Etot(3.2.5)It should be noted that this value is not an integer. Using an integer for Ncol, if the following
relationship is given:

N col ≤ N col (i, j ) < N col +1(3.2.6)the number of modeled photons entering the air column (i, j) is determined using the random
number ρ:⎧ N col⎨⎩ N col +1

if ρ ≥ N col − N col(3.2.7)

if ρ < N col − N col

Using these equations, Eq. (3.2.5) is satisfied.
Next, k, which is an index of vertical number for the column, is determined. From Eq. (3.2.4),
the probability of emission from the kth element is expressed by

E (i, j, k )Ecol (i, j )(3.2.8)k is determined based on this discrete probability distribution and random number. In the next
section, a method to determine the emitted flux for various radiation sources, emission positions,and emission directions is described.

3.2.2 Solar radiation
As well as (3.2.2), the energy of the solar radiation incident at the top of the atmosphericcolumn is expressed by⎧ AF
for k = N +1
Esolar ( k ) = ⎨ solar
for k ≠ N +1
⎩0

(3.2.9)where A is a column cross section and Fsolar is the incoming solar flux at the horizontal plane.
Strictly speaking, solar radiation is a radiation source that has a strongly-peaked cone-like shape.
Fsolar is a function of the sun-earth distance and solar zenith angle. It annually varies within 3%.
It is easy to determine the emission position when the radiation source is only solar radiation.

37

The emission flux is independent of the horizontal plane. When the simulation space is definedas [0, Xmax], [0, Ymax], [zBOA, zTOA], the emission position is determined from two random
numbers:⎛ x ⎞ ⎛ρ x X max ⎞⎟⎜ ⎟ ⎜⎜ y ⎟ = ⎜ρ yYmax ⎟⎜ ⎟ ⎜⎟⎝ z ⎠ ⎝ zTOA ⎠(3.2.10)

From Eqs. (3.2.1) and (3.2.9), the energy of the modeled photon is given by:

E1 =Fsolar X maxYmax
N tot(3.2.11)The direction of the emission is described in 3.2.6.

3.2.3 Point radiation source
Point radiation sources such as artificial lights and lasers are also regarded as cone-shaped
radiation sources. The emission point is localized at a point. Therefore, the spatial distributionfunction of the flux is a delta function. Generally, the emission power should be a known
parameter. The emission direction is determined by random selection of a position within thecone angle. The emission point is trivial.

3.2.4 Thermal emission
a. Thermal emission from atmosphere
The thermal emission from the kth atmospheric layer is determined by the temperature
distribution and emissivity within the layer. According to Kirchhoff’s law, the emission
coefficient is equal to the absorption coefficient βa. The emitted flux is expressed by

Ftherm (k) = 4 π ∫ z β a ( z) B(T ( z))dz
zk(3.2.12)k−1where B is a Planck function. It represents the radiance of thermal radiation from a black body:

Bλ (T ) =c1
λ [exp(c2 λT ) −1](3.2.13)5where λ and T are the wavelength (µm) and absolute temperature (K), respectively. c1 =1.1910437 × 108 and c2 = 1.438786 × 104. The emission coefficient is assumed to be constant.
The vertical distribution of the Planck function is expressed by the nth order polynomial:B( z) = B( zk−1 ) + b1 ( z − zk−1 ) + b2 ( z − zk−1 ) ++ bn ( z − zk−1 )
2

38

n

(3.2.14)n = 0 or 1 is usually an enough approximation. When the model photon is emitted from thisvoxel, the vertical coordinate z of an emission point is determined by the random number:

∫
∫

z
z k−1
zk
z k−1

B( z)dz
B( z)dz=ρ(3.2.15)

For example, when the emission coefficient is constant in the voxel and n = 1 in (3.2.14), b1
is expressed byB( z) = B( zk−1 ) + b1 ( z − zk−1 )B( zk ) − B( zk−1 )b1 =zk − zk−1(3.2.16a)(3.2.16b)

By substituting it to (3.2.12), Ftherm is obtained

Ftherm (k) = 4 πβ a ( k )B( zk−1 ) + B( zk )(zk − zk−1 )2(3.2.17)

Also, the height z of the emission point is obtained by substituting (3.2.15) to (3.2.16.):2∫=[ B(zk−1 ) + b1 (z − zk−1 )]dz
[ B(zk−1 ) + B(zk )](zk − zk−1 )z′
2 ∫ [ B( zk−1 ) + ( B( zk ) − B( zk−1 )) z′]dz′
0
z

z k−1

B( zk−1 ) + B( zk )(3.2.18)2 ∫ [1+ γz′]dz′
z′

=
=

0

2+γ

2 z′ + γz′2=ρ2+γHere, z′ and γ are assumed:z − zk−1
zk − zk−1
B( z k )γ=−1
B( zk−1 )z′ =(3.2.19a)(3.2.19b)

Finally, by solving the following equation:

γz′2 + 2 z′ − (2 + γ )ρ = 0(3.2.20)39

we obtain⎧ 1+ γ (γ + 2)ρ −1⎪
z′ = ⎨γ⎪⎩ρz = zk−1 + ( zk − zk−1 ) z′

for γ ≠ 0

(3.2.21)for γ = 0
(3.2.22)

The angular distribution of emission is isotropic. The determination of the direction is thesame as in the isotropic scattering case. The emission direction is determined using the zenith
angle θ and the azimuth angle φ:cosθ = 1− 2 ρ1
φ = 2πρ 2(3.2.23a)(3.2.23b)

ρ1 and ρ2 are random numbers. The derivation of these variables is described in 3.5.2.

b. Thermal emission from horizontal plane
The emissivity of the thermal radiation from the isotropic reflective surface is given by 1
reflectance, which is dependent of the direction of emission. Using the zenith angle µ0 andalbedo α, the emissivity ε and albedo α have the following relationship:

ε (µ0 ) = 1− α (µ0 )(3.2.24)

The emitted flux from a surface with a temperature Tsfc is expressed by

Ftherm (0) = ∫ 0∫ B(T )ε (µ )µ dµ dφ= 2 πB(T ) ∫ ε (µ )µ dµ= πB(T )[1− 2 ∫ α (µ )µ dµ ]
2π

1

sfc

0

0

0

0

0

1

sfc

0

0

0

0

(3.2.25)

1

sfc

0

0

0

0

where B is a Planck function. When average reflectance and emissivity are defined by

α = 2 ∫ 0 α (µ0 )µ0 dµ0
ε = 1− α1(3.2.26a)(3.2.26b)

the emitted flux in (3.2.25) is

Ftherm (0) = πB(Tsfc )ε

(3.2.27)

Next, let us consider the emission direction. When the emission is isotropic, the direction of
the emission can be determined in the same manner as the Lambertian reflection. The cosines ofthe zenith angle and azimuth angle are determined by40µ0 = ρ1
φ0 = 2πρ 2(3.2.28a)(3.2.28b)

where ρ1 and ρ2 are random numbers. The description of the derivation is described in 3.6.3.
The anisotropic emission from the surface is determined by the rejection method. First, as in(3.2.28a), the zenith angle is randomly determined. Next, the emissivity (3.2.24) is calculated
and the rejection method is applied to the probability ε (µ 0 ) ε max . εmax is the maximum
emissivity. Since the emissivity is less than 1, the efficiency of the rejection method is good.
The azimuth angle can be determined by the same method as (3.2.28b), using a uniform
pseudorandom number.

3.2.5 Modification of spatial distribution of radiation source based on importance
As mentioned above, the method of distributing the energy proportionally to the emission
energy is the most reasonable method. However, sometimes, we may want to modify theradiation distribution. One example is a case where we need to sample the radiation from anelement with low emittance energy. Since the emitted energy is very small in this case, it isnecessary to use many photons to reduce noise. Another example is a case where we need aradiation contribution from thermal radiation in the middle infrared, although the solar radiation
is greater than the thermal in this spectral domain. In such cases, the photon in the importantpart is relatively increased, and the weight of the photon is modified. Here we describe amethod to overcome it using the importance function defined in the 3-D space.
Consider that the emitted energy E(i, j, k) and importance function G(i, j, k) are given in theelement of 3-D space. Here, G should be positive. When we define the total emittance energy as
Etot, the emission probability and weight in the element (i, j, k) are usually expressed asp(i, j, k ) = E (i, j, k ) Etot
w(i, j, k ) = 1(3.2.29a)(3.2.29b)

p is normalized by∑∑∑ p(i, j, k ) = 1ij(3.2.30)kHere, we modify so that the emission probability distribution is proportional to E and G:

p′(i, j, k ) = G (i, j, k ) E (i, j, k ) Etot

(3.2.31)

Then, the new emission probability distribution isp′′(i, j, k ) =p′(i, j, k )G (i, j, k ) E (i, j, k )=∑∑∑ p′(i, j, k ) ∑∑∑ G(i, j, k )E(i, j, k )
i

j

k

i

j

41

k

(3.2.32)This meets the normalization as∑∑∑ p′′(i, j, k ) = 1ij(3.2.33)k

In each element, the emission probability and weight should be conserved. Hence, theseparameters are constrained by the following relationship:p′′(i, j, k )w′′(i, j, k ) = p(i, j, k )w(i, j, k )(3.2.34)

Here, the new weight is obtained by substituting (3.2.29) and (3.2.32) to (3.2.34):

w′′(i, j, k ) =p(i, j, k )w(i, j, k )p′′(i, j, k )∑∑∑ G (i, j, k )E (i, j, k )= ij(3.2.35)k

G (i, j, k ) Etot

When we use p and w , instead of p and w, we can obtain the photon distribution modified by
the importance function G. This modification does not give any biases, and this is an effective
method to adjust the photon distribution and weight as a variance reduction technique.

3.2.6 Radiation source of isotropic angular distribution
Let us consider the isotropic emission within a finite cone-shaped angular range, like the solar
radiation that is incident at the top of the atmosphere. Here, we describe the determination of thedirection of radiance and emission based on the MC method. The photon direction should bedetermined to equalize the horizontal component of irradiance for all photons. This can beanalytically calculated in the special cases where the center of the cone is directed to the nadir
or zenith direction and the angular range is 0°, 180°, or 360°. If it is not calculated analytically,
it is necessary to use the LUT method or rejection method. The following discussion can beapplied to solar radiation and artificial light.

a. Definition
The coordinates are defined by the zenith angle θ and azimuth angle φ. The center direction of
the emission cone angle range is defined by (Θ, Φ). Then, the zenith angle and azimuth angle
relative to this direction (Θ, Φ) are defined by θˆ,φˆ (Fig. 3.2.1). Since the direction of emission(θ, φ) is derived by rotating (Θ, Φ) with respect to θˆ,φˆ , (θ, φ) is derived from θˆ,φˆ .
When Δ is defined by a half angle of the cone range, it generally has a range like42

0≤Δ≤πHere,
Δ = 0:
Δ = π/2:Δ = π:

collimated radiation
Lambertian
Spherically isotropic

Fig. 3.2.1 Schematics of the cone-shaped angular range and radiation source
isotropically emitted.

b. Irradiance and radiance
Let F0 and R0 be the irradiance for the horizontal plane to the center of the radiation angularrange and the radiance within the angular range. The relationship between F0 and R0 is definedbyF0 = ∫ 0∫cos Δ

υ ( Δ) = ∫ 0

∫

2π

1

R0 cosθˆ d cos θˆdφˆ

(3.2.36)

When we put
2π

1
cos Δ

cos θˆ d cosθˆdφˆ

(3.2.37)= π (1− cos Δ cos Δ )the relationship between irradiance and radiance is written by

F0 = R0υ(Δ)
F
R0 = 0υ ( Δ)(3.2.38a)(3.2.38b)



The radiance is uniquely determined when F0 and Δ are provided. The spheradiance F0 is43

obtained by
2π
1F0 = ∫ 0 ∫ cos Δ R0 d cosθˆdφˆ
= 2 πR0 (1− cos Δ)= F02(1− cos Δ)
1− cos Δ cos Δ⎧
2⎪⎪F0 1+ cos Δ=⎨⎪F 2(1− cos Δ)⎪⎩ 0 1+ cos 2 Δ(3.2.39)

if cos Δ ≥ 0
if cos Δ < 0This is also determined uniquely when F0 and Δ are provided.
The irradiance to the horizontal plane is given by

F0 (Θ, Δ, R0 ) = ∫ 0

2π

∫

1
cos Δ

R0 cos θ d cos θˆdφˆ

(3.2.40)

It should be noted that cosθ is a function of θˆ, φˆ :cosθ = sin Θsin θˆ cos φˆ − cosΘcosθˆ(3.2.41)

The irradiance projection factor to the horizontal plane is defined byυ (Θ, Δ) = ∫ 0

2π

∫

1
cosΔ

cosθ d cosθˆdφˆ

(3.2.42)

The difference between (3.2.42) and (3.2.37) is the difference in the projected plane. Using(3.2.38b) and (3.2.42), the irradiance in (3.2.40) is written as

F0 (Θ, Δ, R0 ) = R0υ (Θ, Δ)= F0

υ (Θ, Δ)
υ ( Δ)(3.2.43)

Generally, it is difficult to obtain the solution of the double integral of (3.2.42). As discussed
later, some part of this function can be integrated analytically.

c. Angular distribution function
Using the solid angle ξ, the normalized angular distribution function Ψ is defined as∫ Ψ (θ , φ ) cosθ dξ = 1
4π(3.2.44)044This function Ψ0 is required to calculate the radiance emitted from the radiation source
described in 3.9. In the isotropic emission case, the angular distribution function is given by⎧ 1⎪
Ψ0 (θ , φ ) = ⎨ υ (Θ, Δ)
⎪0
⎩
⎛θ ⎞
⎛Θ ⎞
Ω = ⎜ ⎟, Ω0 = ⎜ ⎟
⎝φ ⎠
⎝Φ ⎠

if Ω ⋅ Ω0 > cos Δ(3.2.45)

if Ω ⋅ Ω0 ≤ cos Δ

When we apply a judgment of whether the direction is contained in Δ or not, the use of sinΔ isbetter than cosΔ.

d. Irradiance projection factor for horizontal plane
We discuss the calculation method for the factor in (3.2.42). Three cases exist, as shown in Fig.
3.2.2.

Fig. 3.2.2 Three cases.

1) Emission is always in an up or down direction
In this case, Δ ≤ π /2 − Θ ( cos Δ ≥ sinΘ ). Therefore, cosθ is always positive or negative in all
integral domains.

υ (Θ,Δ ) = 2 ∫1π∫ sinΘsin θˆ cos φˆ − cos Θcos θˆ dφˆd cos θˆ

cos Δ 01= 2π cos Θ ∫ cos θˆd cos θˆ(3.2.46)

cos Δ= π cos Θ sin 2 Δ
The irradiance to the horizontal plane is derived from (3.2.37) and (3.2.43):

π cosΘ sin 2 Δ
F0 (Θ, Δ, R0 ) = F0
π (1− cos Δ cos Δ )= F0 cosΘ45(3.2.47)

2) Emission has an up and down direction part 1
In this case, π /2 − Θ < Δ < π − π /2 − Θ ( −sinΘ < cos Δ < sinΘ ). First, the integration is
executed over the condition of cos θˆ ≥ sinΘ :υa (Θ, Δ) ≡ ∫ 0

∫

2π

1
sin Θ

cosθ d cosθˆdφˆ

= π cosΘ (1− sin 2 Θ)= υπ cosΘ(3.2.48)

3

In the other part, the sign of cosθ depends on cos φˆ . It can be written by the single integral

υ b (Θ,Δ ) ≡ ∫

2π

∫

sinΘ

cos Δ

0

cos θ d cos θˆdφˆ

()⎡ φ sinΘsin θˆ cos φˆ − cos Θcos θˆ dφˆ ⎤⎢∫0⎥ˆ
= 2∫ ⎢ π⎥ d cos θ
cos Δ
ˆ
ˆ
ˆ
ˆ
⎢⎣− ∫φ sinΘsin θ cos φ − cos Θcos θ dφ ⎥⎦sinΘ
=22sinΘsin θˆ sin φ + cos Θcos θˆ(π − 2φ ) d cos θˆ
s

sinΘ

s

∫

cos Δ

(

[

s

)

s

(3.2.49)]

wherecosΘcosθˆsin Θsin θˆsin 2 θˆ − cos 2 Θ
sin φ s =sin θˆ sin Θcos φ s =(3.2.50a)(3.2.50b)

When we put

t = cosθˆ
we can obtain an analytical solution for (3.2.49):46

[υb (Θ, Δ) = 4 ∫ cos Δ sin 2 Θ − t 2 dt + 2 cosΘ π ∫ cos Δ tdt − 2 ∫ cosΔ tφ s (t )dt
sin Θ

sin Θ

sin Θ

]

sin Θ

⎡
⎛ t ⎞⎤= 2⎢t sin 2 Θ − t 2 + sin 2 Θsin −1 ⎜⎟⎥⎝ sin Θ ⎠⎦cosΔ
⎣

sin Θ
⎡π
⎤
+2 cosΘ⎢ (sin 2 Θ − cos2 Δ) − 2 ∫ cos Δ tφ s (t )dt ⎥⎣2⎦⎡⎛ cos Δ ⎞⎤22
= sin 2 Θ⎢π − 2 sin −1 ⎜⎟⎥ − 2 cos Δ sin Θ − cos Δ⎝ sin Θ ⎠⎦⎣[+cosΘ π (sin 2 Θ − cos2 Δ) − 4 ∫

sin Θ
cos Δ

tφ s (t )dt= sin 2 Θ[π − 2 sin −1 γ ] − 2 cos ΔsinΘ 1− γ 2

]

⎡
⎤
1
g 2 cos 2 Θ⎥+cosΘsin 2 Θ⎢π (1− γ 2 ) − 4 ∫γ g cos−1dg
1− g 2 sin 2 Θ ⎥⎦⎢⎣(3.2.51)
where⎛ cosΘ t ⎞φ s = cos−1 ⎜⎟⎝ sin Θ 1− t 2 ⎠
cos Δ
γ=
sin Θ

(3.2.52a)(3.2.52b)

Equation (3.2.51) can be calculated numerically by the Gaussian quadrature. The irradianceprojection factor is a sum of (3.2.48) and (3.2.51):

υ (Θ, Δ) = υa (Θ, Δ) + υb (Θ, Δ)(3.2.53)3) Emission has both an up and down direction part 2
In this case, Δ ≥ π − π /2 − Θ ( cos Δ ≤ −sinΘ ). The irradiance projection factor is derived bysubtracting (3.2.46) from the spherical integration:υ (Θ,Δ ) = ∫

2π

0

1

2π

cos Δ

−1

0

−1

∫ cos θ d cos θdφ − ∫ ∫π10− cos Δ= 2π − 2 ∫ ∫1= 2π − 2 ∫cos θ d cos θˆdφˆ

cos θ d cos θˆdφˆπ∫ sinΘsin θˆ cos φˆ − cos Θcos θˆ dφˆd cos θˆ− cos Δ 01
= 2π ⎡⎢1− cos Θ ∫ cos θˆd cos θˆ⎤⎥
−
cos Δ
⎣
⎦
2
= 2π − π cos Θ sin Δ

Example47(3.2.54)

Figure 3.2.3 shows the irradiance projection factor for the horizontal plane as a function of Δ
with a solar zenith angle of 0°, 30°, 60°, and 90°. The four quadratic points give insufficientaccuracy. More than 16 quadratic points give rise to sufficient accuracy. In this figure,υ (Θ,Δ ) υ ( Δ ) in (3.2.43) is plotted.

Fig. 3.2.3

Irradiance projection factor for the horizontal plane as a function of Δ

with a solar zenith angle of 0°, 30°, 60°, and 90° (left) and υ (Θ,Δ ) υ ( Δ ) (right).

f. Determination of emission direction: Rejection method
The emission direction is easily determined using the rejection method. First, a direction in theemitted domain is uniformly selected. Since irradiance to the horizontal plane depends on thecosine of the zenith angle, the rejection method is applied to form a zenith angle dependency (ahigher zenith angle is likely to be rejected). θˆ,φˆ are first determined isotropically:∫ dφˆ ∫ d cosθˆ = ρ∫ dφˆ ∫ d cosθˆ
2π

1

0

2π

cos θˆ
10

cosΔ(3.2.55)θˆ

∴cosθˆ = 1− ρθˆ (1− cos Δ)∴sin θˆ = ρ (1− cos Δ)[2 − ρ (1− cos Δ)]θˆθˆ

(3.2.56a)(3.2.56b)

Note that 1 – cosΔ) should be very accurate when Δ is very small. Also,φˆ = 2 πρ φˆ

(3.2.57)

The better and more powerful way is to use the polar coordinate method (Appendix A6),
which can determine ρθˆ ,cos φˆ,sin φˆ simultaneously. The emission direction Ω = (θ, φ) is adirection rotating Ω 0 = (Θ, Φ) relative to θˆ, φˆ . This is the same formulation as the scattering48

problem. If we put

Ω = (u x ,u y ,u z ) ,Ω0 = (u0 x ,u0 y ,u0 z )TT(3.2.58)

then

⎡
⎛u u
⎛−u
u02 x + u02y ⎞⎢⎜ 0 x 0z⎟
⎜ 0y⎢
Ω = cosθˆ Ω0 + sin θˆ⎢cos φˆ⎜u0 yu0 z u02 x + u02 y ⎟ + sin φˆ⎜ u0 x
⎜
⎟
⎜
⎢
⎜
⎟
⎜
2
2
⎢⎣
⎝ − u0 x + u0 y ⎠⎝⎤u02 x + u02y ⎞⎥⎟⎥u02 x + u02 y ⎟⎥
⎟
⎟⎥
0
⎠⎥⎦
(3.2.59)

Here, if u0 x + u0 y = 1− u0 z ≈ 0 , it is expressed by the more simple form:222⎛sin θˆ cos φˆ ⎞⎜⎟
Ω = sign(u0z )⎜ sin θˆ sin φˆ ⎟⎜⎜
ˆ ⎟⎟⎝ cos θ ⎠(3.2.60)

sign() is a function that has both positive and negative signs.
In an actual case, a z component of (3.2.59) is first derived:cosθ = u z = cos θˆ u0 z − sin θˆ cos φˆ u02 x + u02 y(3.2.61)

Next, the rejection method is applied by considering the cosθ dependency of the angular
distribution:(3.2.62)

Here, Cmax is defined by⎧⎪1
if cosΘ > cos Δ
C max = ⎨⎪⎩max[ cos(Θ − Δ), cos(Θ + Δ) ] otherwise(3.2.63)

If the random number is rejected in (3.2.62), the new direction is determined again using(3.2.56–57, 3.2.61). The rejected probability in a single cycle is less than 1/2. Therefore, on
average, the direction is determined in two cycles at the worst case. Consequently, the rejection
method provides a fast and simple method to determine the direction.

49

3.3 Ray tracing
The ray tracing method is a visualization method for 3-D graphics that traces the photon
trajectory. In the 3-D MC radiative transfer simulation, as shown in Fig 3.3.1, it is necessary tofind the nearest intersection point of the photon trajectory with geometric objects in the space.
Here, we describe a simulation method for a photon traveling in a simulation space with typicalgeometric objects (plane parallel, rectangle, cone, cylinder, ellipsoid, and surface withlarge-scale roughness).

Fig. 3.3.1 Relationship between modeled photon and geometric objects.

3.3.1 Intersection of modeled photon with geometric object
As shown in Fig. 3.3.2, there are three conditions that may exist between the modeled photon
and geometric objects such as a rectangle, cone, cylinder, or ellipsoid. In case A (Fig. 3.3.2),
there is no intersection. In case B, there are two intersection points between the modeled object
and geometric object. If the photon trajectory is a tangent of the object, the intersections aredegenerated. In case C, in which the photon is located within the object, there is one intersection.
The algorithm to find the intersection points has the following procedures:
Case 1: Starting point of the modeled photon is outside the objecti) Judgment whether the photon trajectory has intersection points or notii) Extraction of the intersection points of all objects and determination of the appropriate points
for each object (b1 of case B in Fig 3.3.2)iii) Determination of the nearest point from all candidates selected in procedure ii)
Case 2: Starting point of the modeled photon is inside the objecti) Extraction of the intersection points of the photonii) Extraction of the intersection point along the photon direction (C2 in case C in Fig. 3.3.2)

50

Fig. 3.3.2 Intersection of the modeled photon with the geometric object. A: nointersection point, B: two intersection points with object, and C: one intersection
point when the modeled photon is located within the object.

3.3.2 Relationship between modeled photon and geometric object in 3-D scene
a. Equation for modeled photon trajectory
The modeled photon trajectory, which passes through the point r0(x0, y0, z0) with the direction
vector Ω(vx, vy, vz) is expressed by the parameter t:r(t) = r0 + t Ω (vector expression)(3.3.1)x, y, and z of this equation are expressed byx = x0 + tvx
y = y0 + tvy
z = z0 + tvz(3.3.2a)(3.3.2b)(3.3.2c)b. Relationship between rectangle and photon
A rectangle is expressed by the six equations for plain surfaces in 3-D space:x = x 1; x = x 1+ r x
y = y 1; y = y 1 + r yz = z 1; z = z 1 + r z(3.3.3a)(3.3.3b)(3.3.3c)

In determining the intersection point of the modeled photon with the plain surface (3.3.3a), by
substituting (3.3.3a) to (3.3.2a), t is obtained:51

t = (x1 – x0 )/vx; t = (x1 + rx – x0)/vx(3.3.4)

Then, the y and z candidates for the intersection point are derived by substituting t in (3.3.4) to(3.3.2b) and (3.3.2c):y = y0 + vy (x1 – x0)/vx; y = y0 + vy (x1 + rx – x0)/vx
z = z0 + vz (x1 – x0)/vx; z = z0 + vz (x1 + rx – x0)/vx(3.3.5)(3.3.6)Furthermore, y and z require the following condition:y 1 ≤ y ≤ y 1 + r y, z 1≤ z ≤ z 1 + r z(3.3.7)

If all the above conditions are satisfied, the derived point is an intersection point. If the photon
is located within the rectangular object, the following condition should be satisfied:t≥0(3.3.8)The same calculation can be applied to the other planes (y and z).

c. Relationship between plane-parallel layer and photon
In the radiative transfer simulation, it is often necessary to simulate the horizontally infinite
plane-parallel layer condition in the finite size of a simulation scene. As shown in Fig. 3.3.3, anoutgoing photon from the scene reenters from the opposite side of the face. This calculation canbe achieved by the intersection searching problem between a rectangle and photon. When aphoton incident from the top of the layer has an intersection at the bottom of the layer z = z1, theintersection point (x′, y′) is determined by the same method as the rectangular case. Thus, by
substituting z = z1 to (3.3.2c), t is obtained by:t = (z1 – z0 )/vz

(3.3.9)By substituting t to (3.3.2a) and (3.3.2b), (x′, y′) is obtained byx′ = x0 + tvx
y′ = y0 + tvy(3.3.10a)(3.3.10b)

Here, (x′, y′) may be located outside the simulation scene. When the scene boundary is definedby [0, xd], [0, yd], the new photon position in the scene under the periodic boundary conditions isexpressed by the following equation:
If vx is positive,x = x′ – integer(x′/xd) × xd

(3.3.11a)

If vx is negative,x = x′ – [integer(x′/xd) – 1] × xd

(3.3.11b)52

Here, an integer indicates the numerical operation to cut after the decimal point. The newposition of y is also calculated in the same manner.

Fig. 3.3.3 Relationship between a plane-parallel layer and photon.

Fig. 3.3.4

Definition of cone parameters.

d. ConeThe equation for a cone is expressed by(x – x1)2 + (y – y1)2 = c2 (z – z1)2 (cone side)z = z1; (x – x1)2 + (y – y1)2 < r2 (cone bottom)(3.3.12)(3.3.13)

where (x1, y1, z1) are the coordinates of the cone apex. Figure 3.3.4 shows the definition of thecone parameters. By substituting (3.3.2) to (3.3.12), we obtain a quadratic equation for t:(vx2 + vy2 – c2vz2)t2 + 2{vx(x0 – x1) + vy(y0 – y1) – c2vz(z0 – z1)}t+ {(x0 – x1)2 + (y0 – y1)2 – c2(z0 – z1)2} = 0(3.3.14)

When the first, second, and third terms of (3.3.14) are expressed by A, B, and C, respectively, D
53

is defined by

D = B 2 − 4AC(3.3.15)

D should be ≥0 to have intersection points with the cone side. Then, t is determined byt=−B ± D
2A(3.3.16)

A candidate for the intersection point is determined by substituting t to (3.3.2). However, theintersection is limited to a point with the following condition:
z1 – h ≤ z ≤ z1

(3.3.17)

The intersection with the bottom face of a cone is calculated by finding the intersection between
the plane z = z1 – h and photon trajectory under the condition in (3.3.13).
When a photon is inside the cone, the solution should meet (3.3.8).

e. Cylinder
As shown in Fig. 3.3.5, the equations for the side and top/bottom of a cylinder are expressed by(x – x1)2 + (y – y1)2 = r2 ; z2 ≤ z ≤ z1 (cylinder side)z = z1; (x – x1)2 + (y – y1)2 < r2; z = z2; (x – x1)2 + (y – y1)2 < r2(cylinder top and bottom)(3.3.18)(3.3.19)(x1, y1, z1), and (x1, y1, z2) are counterpoints of the top and bottom circles. By substituting (3.3.2)
to (3.3.18), the quadratic equation for t is expressed by(vx2 + vy2)t2 + 2{vx(x0 – x1) + vy(y0 – y1)}t + {(x0 – x1)2 + (y0 – y1)2 – r2} = 0(3.3.20)
When the first, second, and third terms of (3.3.20) are expressed by A, B, and C, respectively, D
is defined by (3.3.15). D should be ≥0 to have intersection points with the cylinder side. Since t
is calculated from (3.3.16), a candidate for the intersection point is determined by substituting t
to (3.3.2). However, the intersection is limited to a point with the following condition:z2 ≤ z ≤ z1

(3.3.21)

An intersection with the top and bottom circles is an intersection of the photon trajectory withthe plane z = z1; z = z2 under the (3.3.19) condition. When a photon is inside the cylinder, thesolution should meet (3.3.8).

54

Fig. 3.3.5 Definition of cylinder parameters.

f. Ellipsoid
The equation for an ellipsoid is expressed by( x − x1 ) + ( y − y1 ) + (z − z1 ) = 1
2

a2

2

a2

2

b2

(3.3.22)

where (x1, y1, z1) is the center point of the ellipsoid. The quadratic equation of t is expressed bysubstituting (3.3.2) to (3.3.22):(b2vx2 + b2vy2 + a2vz2)t2 + {2(x0vxb2 – x1vxb2 + y0vyb2 – y1vyb2 + z0vza2 – z1vza2)t+ b2(x0 – x1)2 + b2(y0 – y1)2 + a2(z0 – z1) 2 – a2b2 = 0(3.3.23)
When the first, second, and third terms of (3.3.23) are expressed by A, B, and C, respectively, D
is defined by (3.3.15). D should be ≥0 to have intersection points with the ellipsoid. Since t canbe derived from (3.3.16), a candidate for the intersection point is determined by substituting t to
(3.3.2). Moreover, when a photon is located in the ellipsoid, the photon that meets (3.3.8) is anintersection.

Fig. 3.3.6

Definition of ellipsoid parameters.

55

3.3.3 Relationship between modeled photon and surface with roughness
When the photon is incident at the ground surface, calculation of the intersection between thephoton and ground surface is required. If the surface is perpendicular to the z-axis, theintersection point can be calculated by (3.3.2–6).
In general, the ground surface has large-scale roughness. When we consider it, it isnecessary to find the intersection between a sloped surface and the photon. When the surface
topography is given by the digital elevation model (DEM), the topographic information is given
by a grid point of the quadrate.
First, we describe a method to interpolate the surface elevation z at the arbitrary point (x, y)using the bilinear method, with the nearest four grid points.
In the bilinear method, the surface elevation z is calculated from the nearest four grid points:z=(y2 − y)za − (y1 − y)zb
(y2 − y1 )(3.3.24)

Here, za, and zb are calculated by(x2 − x)z1 − (x1 − x)z2
(x2 − x1 )(x − x)z4 − (x1 − x)z3
zb = 2(x 2 − x1 )za =(3.3.25)(3.3.26)(3.3.24)–(3.3.26) are equations for a curved ground surface. A quadratic equation for the
ground surface is given by substituting (3.3.24)–(3.3.26) to (3.3.2):v x v (z1 − z 2 + z 3 − z 4 )t 2
⎧ y 0 v x (z1 − z 2 + z 3 − z 4 ) + x 0 v v (z1 − z 2 + z 3 − z 4 )⎫⎪⎪+ ⎨− y 2 v x (z1 − z 2 ) − y1 v x (z 3 − z 4 )⎬t⎪− x v (z − z ) − x v (−z + z ) − v x y
⎪
1 y
2
3
z d d
⎩ 2 y 1 4⎭

(3.3.27)⎧ y 0 x 0 (z1 − z 2 + z 3 − z 4 ) − y 0 x 2 (z1 − z 4 ) − y 0 x1 (−z 2 + z 3 )⎫⎪⎪+ ⎨− x 0 y 2 (z1 − z 2 ) − x 0 y1 (−z 2 + z 3 )⎬=0⎪y x z − y x z + y x z − y x z − z x y
⎪
2 1 2
1 1 3
1 2 4
0 d d
⎩ 2 21⎭
where xd = (x2 – x1), yd = (y2 – y1). t is derived by solving these equations. The intersection pointis then obtained from (3.3.2). However, the obtained x, y should meet the following conditions.
x 1 ≤ x ≤ x 2; y 1 ≤ y ≤ y 2(3.3.28)

To determine the intersection point with the ground surface, it is necessary to examine(3.3.27) and (3.3.28) over all quadrate grids. As shown in photon trajectory A in Fig. 3.3.7(center), the photon intersection has two solutions: intersection from the top side and bottom

56

side. In this case, the true intersection point can be determined by selecting the nearest
intersection.
When there are N × M number of grid points in the (x, y) plane, they form (N – 1) × (M – 1)
number of quadrates. Therefore, a large computation time is needed if we employ a finer
resolution or large-scale DEM data. In this case, the following method reduces and simplifies
the computation:i) The maximum and minimum elevations are determined from the DEM before starting thesimulation.
ii) The photon intersections with two planes, which are perpendicular to the z-axis with heights
equal to the maximum and minimum of the DEM (Fig. 3.3.7 center), are derived.
iii) The calculation of the intersection is only conducted in a large rectangular area (Fig. 3.3.7
bottom), which is determined by including the two points calculated from ii).
The efficiency of this method depends on the ratio of the distance between grids to thedifference between the maximum and minimum elevations. When the difference in elevation issmall and the distance between grids is large, this method drastically improves the computation
efficiency. This method is also efficient when the incident angle of the photon is small.
Further improvement can be achieved by the two additional methods described in the next
section.

Fig. 3.3.7 Intersection between ground surface and photon. Top: Relationshipbetween arbitrary point (x, y) and four nearest points. Center: Relationship between
surface elevation and photon position. Bottom: Calculation area constrained by the
maximum and minimum elevations.

57

3.3.4 Space subdivision method
The space subdivision method divides the simulation space into subdivisions and memorizes
where the object in each subdivision is (Araya, 2003). This method reduces the number ofintersection calculations. When there are a large number of objects, the computation time
improves. Furthermore, hierarchical nesting of the subdivisions is effective.
In the case of a ground surface with large-scale roughness, for example, a simulation scene
is divided vertically to include the DEM grid columns. Then, the photon is moved on a
subdivision-by-subdivision basis. The photon trajectory is judged by whether the photon reaches
the boundary of a subdivision or collides with the ground surface. This method is fairly simple(3.3.2) since there is only one surface object per subdivision. Therefore, it achieves fastercomputation.

3.3.5 Boundary volume method
The boundary volume method uses simple virtual objects that wrap around complicated objects(Araya, 2003). This method first examines the photon intersection with virtual objects. If thephoton has no intersection with a virtual object, the calculation of its intersection with thecomplicated objects in the virtual object can be skipped. If the photon has an interaction with avirtual point, the calculation of its possible intersection with the complicated object executes.
Thus, it reduces the computation time by first determining cases where the photon has nointersection. The hierarchical nesting of subdivisions is also effective in this method.
As an example, it is applicable for fast determination of the photon intersection with theground surface. When we use a rectangle as a boundary volume that includes the surface
elements, the z-axis boundary of this rectangle is determined by the maximum and minimum
among the four positions of the surface grids. If the photon has no intersection with thisrectangle, it also has no intersection with the ground surface. This method is effective when it isused in combination with the space subdivision method.
As another example, when we examine the intersection with a cone or ellipsoid, it makes it
faster to examine a cylinder with an infinite length that contains the object before examining theactual object (cone, ellipsoid). In this case, it is not necessary to consider the z-axis. The
problem degenerates to a 2-D (x, y) problem. Examination of the intersection is the same as thatof the calculation method for the side of a cylinder. This can be simplified as follows. If we let
the photon position r = (x, y)T , photon direction Ω = (ux, uy)T, and c = (p, q)T, the radius of thecircle R is shown in Fig. 3.3.8. Then, the condition for an intersection between the photon andcylinder is2

cos α =

c − r − R2c−r

⎛ c−r ⎞≥ Ω ⋅⎜⎟⎝ c−r ⎠

When it is expressed by the (x, y) component( p − x)2 + (q − y)2 − R 2 ≥ ( p − x)u x + (q − y)u y
58(3.3.29)∴( p − x)2 + (q − y)2 ≥ R 2 + [( p − x)u x + (q − y)u y ]2(3.3.30)

Evaluation of (3.3.30) can be done more quickly than the cone and ellipsoid described in 3.3.2.
This is effective when a single tree consists of a combination of several objects.

Fig. 3.3.8 Relationship between cylinder and photon trajectory.

59

3.4 Collision
A photon is scattered or absorbed after collision with a particle in a medium. This collision
process can be represented as a simple probabilistic process. However, not only one modelingmethod but also several modifications can be used. Especially, we should pay attention to how
the absorption is treated, how the photon weight is modified at the collision, and how several
kinds of media components are treated. This section discusses this topic.

3.4.1 Collision probability and determination of collision point
a. Extinction coefficient and photon path length
Light beam intensity decreases by scattering and absorption in a medium. The radiance of thebeam at the distance s from the source at the incident point in a homogeneous volume follows
the Lambert-Beer law:

I = I 0 exp(−βe s)(3.4.1)

where βe is the volume extinction coefficient (extinction cross section per unit volume) (m–1),
which is closely associated with the photon mean free path length (MFP):

MFP = 1 βe

(3.4.2)For inhomogeneous media, when the light transports from r0 to r1,

I = I 0 e −τ (r0 ,r1 )τ (r0 , r1 ) =(3.4.3)∫ β (t)dt; t = r′ − re(3.4.4)0

r0 →r1

where τ is the optical thickness along the light path. The integral on the right hand side of(3.4.4) can be numerically computed.

b. Collision probability and sampling of collision point
From (3.4.3), the ratio I/I0

T (τ ) = e − τ(3.4.5)is the transmittance, which can be understood as the probability that no collision occurs until thephoton reaches the point with an optical thickness of τ. The contrary probability of this is thecollision probability, i.e., the probability that the photon experiences a single collision before its
arrival at the point with τ. The collision probability C is written as

60

C(τ ) = 1− T (τ ) = 1− e −τ

(3.4.6)This function C is the cumulative distribution function of collision, and its derivative is the
PDF:

dC(τ ) − τ= e = T (τ )dτ

(3.4.7)

It is clear that this PDF is the transmittance itself.
Let us assume that the photon collides once at a random location with a value of optical
thickness between 0 and infinity. The sampled random optical thickness is determined by auniform random number ρ:C(τ ) = ρ ; T (τ ) = 1− ρ = ρ˜
∴ τ = −ln(1− ρ ) = −ln ρ˜(3.4.8a)(3.4.8b)where (1 – ρ) is still a uniform random number; then, it is replaced with ρ in the 2nd equation
above. By substituting the randomly determined τ into (3.4.4), the collision point r1 isdetermined.

Fig. 3.4.1 Relationships between optical thickness, transmittance, and collision
probabilityc. Sampling collision point in prescribed limited path
The probability of a collision at a point with an optical thickness between τ1 and τ2 is given by
the difference in the collision probability C or the transmittance T:

ΔC(τ 1 , τ 2 ) = C(τ 2 ) − C(τ 1 ) = e − τ 1 − e − τ 2 = T (τ 1 ) − T (τ 2 ) = ΔT (τ 1 , τ 2 )(3.4.9)This probability can also be written as61

(

ΔC(τ 1 , τ 2 ) = e −τ 1 1− e − (τ 2 −τ 1 )).

(3.4.10)= T (τ 1 )C(τ 2 − τ 1 )That is, it is the collision probability for the optical thickness of τ2 – τ1 multiplied by thetransmittance for τ1.
Let us consider about the probability of collision at a point with the optical thickness
between τ1 and τ. For this “limited” collision, the cumulative distribution function is as follows:C ′(τ ; τ 1 → τ 2 ) =T (τ 1 )C(τ − τ 1 ) C(τ − τ 1 ) 1− e −(τ −τ 1 )==
T (τ 1 )C(τ 2 − τ 1 ) C(τ 2 − τ 1 ) 1− e −( τ 2 −τ 1 )(3.4.11)

If a collision occurs at a random point with the optical thickness between τ1 and τ2, then thecorresponding optical thickness τ is determined by a random number:C ′(τ ; τ 1 → τ 2 ) = ρT (τ − τ 1 ) = 1− ρC(τ 2 − τ 1 )∴e −( τ −τ 1 ) = 1− ρ (1− e −(τ 2 −τ 1 ) )(3.4.12a)(3.4.12b)τ 1 = 0, τ 2 = ∞(3.4.13)(3.4.12c)Ifthen (3.4.12c) is the same as (3.4.8). We have obtained the probability of a collision within anarbitrary segment of the path as (3.4.10), and a random collision point within the segment canbe determined by (3.4.12). By applying this, we can, for example, sample the heating rate byforced collision in an arbitrary path segment. The probability of absorption at a random point
determined with this method is the collision probability in (3.4.19) multiplied by the single
scattering co-albedo:

ΔC(τ 1 , τ 2 )(1− ω ) = T (τ 1 )C(τ 2 − τ 1 )(1− ω ) .

(3.4.14)3.4.2 Treatment of absorption
The scattering should occur at a randomly determined point, and this is the only method to treat
the scattering in the MC model. Absorption can be treated in two different ways. Here, we
specify the optical thickness of the media with βe, βa, βs, ω, and P.

a. Absorption at collision point
This method treats the collision as a mixture of absorption and scattering. With thetransmittance T(τ) for the path between the points r0 and r1, the optical thickness for thecollision will be (see also Fig. 3.4.2)

62∫ [β (t) + β (t)]dt; t = r′ − rτ = τe =sa0(3.4.15)

r0 →r1

If the photon collides, the collision is either of scattering or absorption. The probability that thecollision is scattering is given asω = β s βe

(3.4.16)

Then, the photon weight after the scattering should be

w1 = w0ω(3.4.17)

The absorption (heating) weight should be

Δw = w0 − w1 = w0 (1− ω )(3.4.18)

This heating should be sampled at the randomly determined collision point.

Fig. 3.4.2 Schematic of collision treatment: In the case where absorption is
included in the collision.

b. Continuous absorption in path segments
All energy should be scattered if the absorption is treated separately from the collision.
Representing the transmission function by T(τ),τ = τs =∫ β (t)dt; t = r′ − r .
s(3.4.19)0

r0 →r1

If the photon transports by the path length l′ in a medium with a homogeneous absorption
coefficient, the photon weight should become63w′ = w0 e −β a l′

(3.4.20)

The weight that should be absorbed in the course of the path is given by

Δw = w′ − w0 = w0 (1− e −β a l′ )(3.4.21)

Where should the heating of this weight be deposited? If a single point for absorption is being
sampled, representing the path length to the point by la, using (3.4.12c), we get

e −β a la = 1− ρ (1− e − β a l′ )∴la = −

ln[1− ρ + ρ e − β a l′ ] .

(3.4.22)βa

In this way, the heating rate is sampled for every voxel that the model photon penetrates. This isa kind of collision-forcing method.
If one tries to calculate heating rates in a medium with a very small extinction coefficient,the method described above is better than the method of sampling the absorption at the collision
points. The method in this subsection can sample the heating every time the photon penetrates avoxel, without depending on the degree of absorption. In contrast, with the method of sampling
the heating at the collision point, the frequency of sampling is too low if the extinction
coefficient is small. However, Iwabuchi (2006) pointed out that the two methods have almost
the same efficiency if one uses the collision-forcing method based on the scaling transformationin combination with the method for sampling the heating at collision points. In addition, themethod for sampling the heating at collision points is easy to implement in the code and can beused in combination with the maximum cross section (MCS) method (4.1). The method is thus
recommended by the authors.

Fig. 3.4.3 Schematic for the treatment of collision: In a case where the absorption
is calculated from the path length.

64

3.4.3 Treatment of mixed media
The actual atmosphere is composed of many component gases and particles. Several
scattering/absorption components are mixed together, including Rayleigh scattering by gases,
gaseous absorption, Mie scattering by aerosol particles and cloud water/ice particles, andscattering by larger particles such as raindrops, snowflakes, and graupel particles. From themicroscale point of view, aerosols and hydrometeors are polydispersions of particles withvarious sizes, and their size distributions vary spatially and temporarily. As for the forest canopy,
several kinds of leaves or leaves of various sizes are mixed together in each tree in the actual
forest.
In the case where a medium is composed of N components, the optical properties βe, βa, βs,ω, and P of the mixed media are given as follows:
N

N

N

i=1

i=1
N

i=1

βe = ∑ β˜ e (i), β a = ∑ β˜ a (i), β s = ∑ β˜ s (i)N∑ β˜ s (i) ∑ω˜ (i)β˜e (i)β s i=1
ω= = N= i=1 Nβe
∑ β˜e (i) ∑ β˜e (i)i=1(3.4.24)

i=1

N

N

∑ β˜ (i)P˜ (i) ∑ω˜ (i)β˜ (i)P˜ (i)sP=(3.4.23)

i=1

e

=

βs

i=1

(3.4.25)N∑ω˜ (i)β˜e (i)i=1where the following equations are derived for the extinction coefficients:

βe = β a + β sβ˜ (i) = β˜ (i) + β˜ (i)ea(3.4.26)s

The determination of the random collision point should be based on the total extinction
coefficient βe, if one tries to model the radiative transfer in the mixed media by the MC method.
It is difficult to determine the scattering direction using the mixed phase function P at thevoxel of the collision. This is because a very large amount of computer memory is required if
one generates the phase function LUT for each voxel, because each phase function should betabulated at 10,000 or more points. Note that the number of voxels is usually very large, as large
as 200 × 200 × 50 or more in a 3-D atmospheric radiation model! A resolution is to determine
randomly a component i for scattering at the collision event using a random number and then to
determine the scattering direction using the phase function P˜ (i) of the component. A phasefunction LUT is required with respect to the component (not to the voxel). The size distributionof particles varies voxel by voxel. Thus, in practice, an LUT is prepared for various components
and various size distributions, and indexes for each voxel and each component are saved in65computer memory to refer to the type of phase function. In this case, P˜ ( m(i) ) should be usedinstead of P˜ (i) , where m is the index of the phase function LUT. By selecting the kind ofcomponent in the collision (absorption or scattering), for example, it is possible to sample theheating rate for each component.
If scattering and absorption are included in the “collision,” there are two possible ways torandomly determine the component: one is to select the kind of collision (scattering plus
absorption), and the other is to select the kind of scattering. Both are physically correct andproduce no bias.

a. To select the kind of collision
In this case, a random number determines a component i active for the collision. The probabilityof i is given asβ˜ e (i) N
pcol (i) =, ∑ pcol (i) = 1βe
i=1(3.4.27)

The single scattering albedo can differ by component. The photon weight just after the
scattering should becomew′ = wω˜ (i)(3.4.28)

The heating rate at this collision isΔw = w(1− ω˜ (i))(3.4.29)

That is, the heating rate differs by the randomly selected component i. This method results inlarge noise in heating rates and radiance (sampled by the LEM), because of varying the single
scattering albedo by component. For example, let us consider a two-component mixture ofgaseous absorption for i = 1 with ω˜ (1) = 0 and scattering due to cloud particles for i = 2 withω˜ (2) = 0.999999 . The sampled heating should be⎧w
for i = 1
Δw = ⎨−6⎩w ×10 for i = 2(3.4.30)

If the scattering due to cloud particles is dominant, the gaseous absorption rarely occurs [β˜e (1) << β˜e (2) ]; then, a very small heating due to cloud particles is usually sampled, whereas asignificantly larger heating by a factor of 106 due to gaseous absorption is sampled rarely. Thissituation results in a very large noise in the integration result, and the convergence is too slow.
That is why the authors do not recommend this method.

66

b. To select the kind of scattering
The probability of scattering is given simply by ω in (3.4.24). The weight just after the
scattering should be

w′ = wω

(3.4.31)

The heating rate should be

Δw = w(1− ω )(3.4.32)

After scaling the weight in this way and sampling the heating, the active scattering component
is determined randomly. The scattering probability for the ith component is(3.4.33)

The difference from the method using (3.4.27) is the use of a scattering coefficient instead of an
extinction coefficient. It is easily understood from (3.4.31–32) that the weight and the heating
sampled are always independent of the active component for the scattering. Since ω is uniquely
given voxel by voxel, the constant rates for absorption and scattering should always be sampled
at every voxel. This method is thus efficient.

3.4.4 Russian roulette method and photon-splitting method
The photon weight decreases at the collision event or in the course of transport in the absorption
media. However, tracing photons with a very small weight is not effective for efficientcomputation. Therefore, the Russian roulette method is useful; the weight is randomly cut off if
the weight is smaller than a prescribed threshold (Booth, 1985; Kawrakow and Rogers, 2001).
A random number is generated if the weight w is smaller than an arbitrary value W (can be>1), and then, the weight is modified as⎧0 if ρ w ≥ w W
w′ = ⎨⎩W if ρ w < w W(3.4.34)

If the new weight is 0, the model photon is “dead,” so tracing of the model photon terminates. In
other words, the survival of the model photon is randomly determined. The probability ofsurvival is w/W, and the total energy is conserved:

0 × (1− w W ) +W × w W = w(3.4.35)

where the original energy is w. This method is the so-called Russian roulette method. It is
usually reasonable to apply this method under the condition that the photon weight is smallenough, e.g., w < W/2. This method is energy conservative and unbiased. It can be applied at67

any step of the MC simulation and at any location in the model domain.
A similar energy conservative weight control method is the photon-splitting method, which
splits a model photon into N model photons with the weightw′ = w N(3.4.36)More generally, the divided weight is not necessarily the same. An application example is tosplit the model photon (photon packet) at the scattering event and redirect the subpackets into
different directions. This photon-splitting method is indeed energy conservative and can beapplied at any step of an MC simulation and at any location in the domain, without any bias.
Both methods are methods to control photon population and weights. Table 3.4.1
summarizes the characteristics of these two methods. When do these methods work well? The
answer would depend on the importance of the radiative quantities of interest. If one needs toincrease the frequency of sampling, then photon splitting works well. In contrast, if one needs todecrease the sampling frequency, the Russian roulette method is useful. If, for example,
higher-order scattering is not important, then the Russian roulette method can reduce the photon
population by setting W as large enough after the photon packet experiences several scattering
events. The photon-splitting method is not required in the usual radiative transfer model for theatmosphere and canopy, and the method could add complexity to the simulation code.
Variance of the photon weight generally results in noise in the integrated quantities. The
appropriate value for W used in the Russian roulette method would be between 0.1 and 5.
Table 3.4.1 Changes in the photon weight and population, due to the Russianroulette method and the photon-splitting method.
WeightPopulation

Russian rouletteincrease (if it survives)
decrease

68

Photon splitting
decreaseincrease

3.5 Scattering
When a photon enters the atmosphere or a plant, it interacts with atmospheric molecules, cloudparticles, aerosols, leaves, stems, etc. The scattering is an electromagnetic interaction between
them. We summarize the basic concept of light scattering for atmospheric molecules, aerosols,
clouds, and leaves. Then, the MC modeling of these scattering events is described.

3.5.1 Scattering process
The description of atmospheric scattering follows that given by Hansen and Travis (1974) andShibata (1999) and for leaf scattering follows that given by Shultis and Myneni (1988).

a. Atmospheric scattering processes
The scattering from spherical particles with homogeneous electromagnetic properties, such as acomplex refractive index or complex electrical properties can be analytically introduced by Mietheory. The derivation of Mie theory is required to solve Maxwell’s equation (differential
equation) in 3-D space. The solution is described in the spherical harmonics function. The
scattering intensities depend on a complex refractive index and size parameter x. Because detailsof the derivations of the electrical and magnetic vectors from Mie theory are discussed
elsewhere, we skip the details of these derivations.
The size parameter x is defined by the ratio of the wavelength to the sphere’s circularperimeter:x = 2πa/λ(3.5.1)

Based on this parameter, the scattering can be divided into three scattering domains: Rayleighscattering (x << 1), aerosol and cloud particle scattering (x ~ 1), and geometrical optics
scattering, such as from soil particles on the planetary surface (x >> 1). As shown in Fig. 3.5.1,
the z-axis is defined by the photon direction, and the electrical field vectors of the incident
photon and scattering photon are expressed by the vector sum of the parallel direction with aplane included in the scattering angle (p components) and the perpendicular direction with aplane included in the scattering angle (s components). The incident electromagnetic wave isassumed to be a plane wave, and we only discuss the scattered electromagnetic wave that is
sufficiently apart from the scattering media. In this case, the scattering electrical vector isexpressed using the scattering matrix S.

⎛ E p ⎞ exp(−ikr) ⎛ S2
⎜ ⎟=⎜ikr ⎝ S4
⎝ Es ⎠

S3 ⎞⎛ E 0 p ⎞⎟⎜⎟S1 ⎠⎝ E 0s ⎠(3.5.2)where k and r are the wave number of the incident wave and distance from the scattering point,69

respectively. The ratio of the scattering to incident intensity, or the so-called single scattering
albedo ω, is expressed by

ω=

βs
βe

(3.5.3)where βs and βe are the scattering and extinction coefficients in a unit volume (m–1).
Atmospheric molecular scattering (Rayleigh scattering)
The ion radius of an atmospheric molecule is approximately ~0.1 nm, which is sufficiently
small compared with the photon wavelength (~1 µm). In this case, since the electrical vector
near a single molecule can be assumed to be spatially homogeneous, the electrical field aroundthe molecule induced by the incident wave becomes a dipole field. Scattered photons by theRayleigh scattering are a radiation corresponding to the vibration from the time-dependent
dipole field. When we use the polarization factor (αp), the scattering matrix for Rayleighscattering is written byS=

ik 3α p ⎛ cosΘ 0⎞⎜⎟1⎠
4 πε0 ⎝ 0

(3.5.4)The differential scattering cross section, which is defined by the ratio of the scattering intensity
to the incident intensity, is expressed by2

2⎛ α p k 2 ⎞ 2 1+ cos2 Θ
E p + Es
dσ s
2
=r
=⎜
⎟
2
2
dΩ
2
⎝ 4 πε0 ⎠E 0 p + E 0s

(3.5.5)

We assume that E0p is equal to E0s. The phase function, which is defined by the spatial
distribution of the scattering intensity, is expressed using the differential scattering cross section(3.5.5) normalized by the cross section:p(Θ) =

4 π dσ s 3= (1+ cos 2 Θ)σ s dΩ 4(3.5.6)

The phase function requires the following normalization condition:1∫ p(Θ)dΩ = 1
4π 4 π

(3.5.7)In the case of Rayleigh scattering, the phase functions for all molecules are equal. Therefore, thephase function of the ensemble of the molecules is also equal to (3.5.6). Also, since Rayleighscattering is a perfect elastic scattering, the extinction coefficient is equal to the scattering
coefficient, and the single scattering albedo is 1.

70

Fig. 3.5.1 Relationship between electrical vectors of incident wave and scattering
wave.

Aerosol and cloud scattering
For aerosol and cloud particles, the components S1 and S2 in the scattering matrix S are
expressed by

2n + 1 ⎡Pn1 (cosΘ)dPn1 (cosΘ) ⎤S1 (Θ) = ∑+ bn (x)⎢an (x)⎥sinΘdΘ⎦
n=1 n(n + 1) ⎣∞1
1
2n + 1 ⎡Pn (cosΘ)dP (cosΘ) ⎤S2 (Θ) = ∑+ an (x) n⎢bn (x)⎥sinΘdΘ⎦
n=1 n(n + 1) ⎣∞(3.5.8)(3.5.9)where Pnl is an associated Legendre function and the coefficients an and bn are functions of thesize parameter x. When the incident wave is solar radiation and E0p is equal to E0s, thedifferential scattering cross section can be calculated by
2

2

2
2
E p + Es
S1 (Θ) + S2 (Θ)
dσ s
2
=r
=
2
2
dΩ
2k 2
E 0 p + E 0s

Therefore, the single particle phase function is expressed by71(3.5.10)p(Θ) =

4 π dσ s2π
22=S1 (Θ) + S2 (Θ)
2σ s dΩ σ sk()

(3.5.11)The scattering phase function P from the ensemble composed of various sizes of particles iscalculated by the weighting average of the phase function corresponding to the particle number
density n(a) and scattering cross section as a function of the particle radius (a):a

1 max
P(Θ) =∫ p(Θ)σ s (a)n(a)daβ s a min(3.5.12)

where βs is calculated bya max

β s = ∫ σ s (a)n(a)da(3.5.13)a min

b. Scattering from leaves
There are several ways of modeling the scattering from leaf canopies. We describe the photon
scattering phenomenon from a disk-like leaf using the geometrical optics approach. The
polarization and diffraction are ignored. The reflectance and transmittance from a leaf aremodeled using the Lambertian law (bi-Lambertian). Then, the phase function from a single leaf
is expressed using the normal vector of the leaf (ΩL), vector for the incident direction (Ω′), andvector for the scattering direction (Ω)):

RL ΩL ⋅ Ω
if (ΩL ⋅ Ω)(ΩL ⋅ Ω') < 0ω 0π
T Ω ⋅Ω
p(Ω'→ Ω;ΩL ) = L L
if (ΩL ⋅ Ω)(ΩL ⋅ Ω') > 0ω 0πp(Ω'→ Ω;ΩL ) =(3.5.14a)(3.5.14b)

where the single scattering albedo (ω0) is expressed by the sum of the spherical reflectance andtransmittance of the leaf:

ω0 = RL + TL

(3.5.15)The scattering of an ensemble of leaves with different leaf angles is calculated by the weighting
average of the single leaf phase function corresponding to the leaf angle distribution function
gL(Ω L) and the scattering factor (|Ω • Ω L|):P(Ω'→ Ω;ΩL ) =2∫ gL (ΩL ) Ω'⋅ΩL p(Ω'→ Ω;ΩL )dΩL
G(Ω') 4 πHere, gL(Ω L) can be normalized by72(3.5.16)

1∫ gL (ΩL )dΩL = 1
2π 2 π

(3.5.17)

G is a projected area of the leaves toward the Ω direction for a unit volume and unit leaf area:

1∫ gL (ΩL ) Ω'⋅ΩL dΩL
2π 2 πG(Ω') =

(3.5.18)

Meanwhile, the phase function is analytically calculated when the leaf angle distribution isconstant (gL(Ω L) = 1):

P(Θ) =

88T(sinΘ − ΘsinΘ) + L cosΘ
3π
3ω 0(3.5.19)

The phase function of a spherical atmospheric particle is a function of the scattering angle. In
leaf canopies, however, the phase function is generally a function of the incident and scattering
directions. The uniform leaf angle distribution case (3.5.19) is the only exception, with thephase function depending only on the scattering angle.

3.5.2 Modeling of scattering by MC method
a. Weight of photon after scattering (analog/non-analog absorption method)
If the absorption of the scattering media is not zero (1 – ω), the photon energy is partiallyabsorbed. In an MC simulation, two methods to express this event exist. In the analog
absorption method, the photon tracing will be stopped when the random number ρ < (1 – ω). In
the non-analog absorption method, the weight of the photon is reduced by the rate of absorption.
In the latter case, the photon weight is calculated by the single scattering albedo (ω):
w i +1 = ω w i(3.5.20)b. Determination of scattering direction
The scattering direction of the photon should be determined by following the phase function. If
the phase function is only a function of the scattering angle (Θ), the azimuth angle Φ relative tothe scattering coordinate is expressed using the random number ρΦ (see Appendix A6):
Φ = 2πρΦ(3.5.21)

The scattering angle is calculated by (for more details, see Appendix A5)

1
1Θ
P(Θ)dΩ
=∫∫ P(Θ)sin ΘdΘ = ρΘ
4π 4 π2 073(3.5.22)(3.5.22) is sometimes calculated (or Θ is sometimes obtained) analytically (e.g., isotropic
scattering, Rayleigh scattering).
Generally speaking, it is rare to obtain the form of Θ = f(ρΘ) analytically. Even if it isfeasible to obtain the form of Θ =f(ρΘ), it may require vast computation. In such cases, the LUT
method and rejection method are practical.
In the LUT method, the relationship between ρΘ and Θ is prepared prior to starting the MC
simulation. Then, the scattering angle Θ is determined by interpolating the LUT values
corresponding to ρΘ. This method is suitable for the Mie scattering of aerosol and cloudparticles and leaf scattering. Figure 3.5.2 shows an example of the relationship between the
random number and scattering alngles (Rayleigh scattering (3.5.6), aerosol scattering (3.5.12),
and leaf scattering (3.5.19)).
In the rejection method, first, the scattering angle Θ is determined by the pseudorandom
number, according to a comparison function R(Θ). Then, the ratio P(Θ)/R(Θ) is compared withthe pseudorandom number in [0, 1]. When the random number is larger than P(Θ)/R(Θ), thescattering angle is reflected. This procedure is continued until the scattering direction is
accepted (see 3.1). In the rejection method, the computation efficiency depends on theefficiency of the comparison function. The rejection method is an accurate method. When thesize of the LUT is constrained by the computational resource, this method would be a goodsolution.
In leaf scattering, the scattering phase function is a function of the photon incident direction(θi, ϕi) and scattering direction, except for some special cases. In this case, the LUT becomes
3-D (θi, Θ, ϕr) when we use the relative azimuth angle (ϕr = ϕi – Φ).

Fig. 3.5.2 Relationship between pseudorandom number and scattering angle.
Aerosol (yellow sand, re = 0.375 µm, λ = 0.575 µm, n = 1.54999995 +0.00433298014i), leaf (reflectance = 0.2; transmittance = 0.1), and Rayleighscattering.

74

Isotropic scattering
In an isotropic scattering case,P(Θ) = 1(3.5.23)

Therefore, (3.5.22) can be analytically solved. Then, the scattering angle Θ is obtained bycosΘ = 1 – 2ρΘ(3.5.24)

Rayleigh scattering
In a Rayleigh scattering case, the phase function is given by (3.5.6). When cosΘ = µ, therelationship between µ and ρΘ is calculated byµ3 + 3µ – 4 + 8ρΘ = 0(3.5.25)

As a solution of the third-order equation, we obtain

µ=X−

1
X

(3.5.26)

X = 3 −(4 ρ Θ − 2) + (4 ρ Θ − 2) +1)
2

The solution is analytically calculated by (3.5.26). However, it requires time-consuming
calculation. On the other hand, the rejection method is efficient. The maximum of the Rayleighscattering phase function is 3/2. Therefore, we can use the following comparison function:R(Θ) = 3 2First, the scattering angle is determined by (3.5.24). The probability to follow the phase function
is given byP(Θ) 3 4 (1+ cos Θ)
=
R(Θ)
32
2

= 1− 2 ρ Θ (1− ρ Θ )The rejection or selection is determined using another random number ρr. Therefore, we use two
random numbers every time. One is used to determine the scattering angle, and the other is used
to judge the following criteria:ρ r ≤ 1− 2 ρ Θ (1− ρ Θ )∴ ρ r′ ≥ 2 ρ Θ (1− ρ Θ )On average, the scattering angle is determined by 1.5 iterations.

75

Leaf scattering: A step-by-step determination method
In this method, the scattering direction is determined by the following three steps:
1) Determination of the normal vector Ω L of the leaf surface
2) Modification of the probability of the leaf normal vector Ω L to be proportional to Ω⋅ ΩL3) Determination of the scattering angle by Ω L and a random numberThis method is also applicable to non-spherical particles, such as ice crystals. This method issimilar to the method of Antyufeev and Marshak (1990). In their method, a leaf normal vector isdetermined by a new probability function, which is easily calculated. Then, the weight of thephoton is modified to follow the actual distribution. Here, we describe a method to calculate thescattering angle without any changes in photon weight.
1. Determination of normal vector Ω L of leaf surface using pseudorandom number
In most cases, since the leaf angle distribution is random in the azimuth direction and it only
depends on the zenith angle, the azimuth angle ϕL is easily determined by (3.5.21). The zenith
angle is determined by the following equation:
θ

1 L∫ gL (θL )sinθL dθL = ρθ L
2π 0

(3.5.27)

In this method, an LUT is first prepared using (3.5.27). Then θL is determined by the LUT.
Under some conditions, an analytical solution is obtained (e.g., gL = cos(θL)). Also, it is possible
to use the rejection method.
2. Modification of probability of leaf normal vector Ω L to be proportional to Ω ⋅ ΩL
The scattering angle determined by step 1 is modified to follow the ratio of the projectedarea to the leaf normal direction Ω L with respect to Ω . In this procedure, the rejection method isthe best method. In this case, when θp is defined as the angle between the incident direction andleaf normal direction, θp is expressed byΩ'⋅ΩL = cosθ p(3.5.28)

Then, it is compared with the pseudorandom number ρ:
When cos θ p ≥ ρ , the leaf angle determined by procedure 1 is accepted.
When cos θ p < ρ , the leaf angle determined by procedure 1 is rejected and we return toprocedure 1.

76

3. Scattering angle is determined by Ω L and random number
Once the leaf normal vector Ω L is determined by procedures 1 and 2, the leaf reflectanceand transmittance are determined. The criteria of the reflectance and transmittance are as
follows:
Reflection

RL
RL≥ ρ ; Transmission<ρ
RL + TL
RL + TL(3.5.29)

Then, if the leaf is a bi-Lambertian surface, the scattering direction is determined byµ = ρ Θ , φ = 2πρφ(µ = cos(θ ))(3.5.30)

In a reflection case, the scattering angle is accepted if Ω'⋅ΩL < 0 . If Ω'⋅ΩL ≥ 0 , –Ω isemployed as a scattering direction. In a transmission case, the scattering angle is accepted ifΩ'⋅ΩL ≥ 0 . If Ω'⋅ΩL < 0 , –Ω is employed.
This method does not require a large LUT. It is effective when the leaf angle distribution
depends on the individual tree and position. In the non-Lambertian leaf surface case, thecomputation is rather complex.

77

3.6 Reflection
In this section, we describe the general treatment of reflection in bidirectional reflectance
distribution function (BRDF) models.

3.6.1 Definitions
a. BRDFThe BRDF is a distribution function for reflectance that depends on the incident and reflected
directions. The definition of the BRDF (R) is

R(θ 0 , φ0 ,θ1 , φ1 ) =dI1 (θ1 , φ1 )I 0 (θ 0 , φ0 )cos θ 0 dξ

(3.6.1a)where (θ0, φ0) and (θ1, φ1) are the incident direction and reflectance direction, respectively. Also,dξ and I0, dI1 are the solid angle, and incident and reflected radiation intensities (Thomas andStamnes, 1999).
The reflected radiation intensity dI1 is expressed in the differential form because itcorresponds to the solid angle of the incident radiation dξ. When the incident radiation is
integrated over all directions, it becomes the radiation intensity for the (θ1, φ1) direction. Thus,dξ = dφ0 d cosθ 0
then

I1 (θ1 , φ1 ) = ∫ dI1 (θ1 , φ1 )= ∫ 0 dφ 0 ∫ 0 d cos θ 0 R(θ 0 , φ0 ,θ1 , φ1 )I 0 (θ 0 , φ0 )cosθ 02π

1(3.6.1b)

As shown in (3.6.1a), the BRDF is defined by the radiance divided by irradiance. Therefore, it is
not normalized. Indeed, the BRDF becomes infinite for a flat specular surface. The unit for the
BRDF is sr–1.
The BRDF is usually expressed by three parameters, using the relative azimuth angle φ:R(θ 0 , φ0 ,θ1 , φ1 ) = R(θ 0 ,θ1 , φ )⎧⎪ φ0 − φ1
if φ 0 − φ1 ≤ π
φ =⎨⎪⎩2 π − φ0 − φ1 if φ0 − φ1 > π(3.6.2)It should be noted that the BRDF is sometimes defined in a different form. In particular, thebidirectional reflectance factor (BRF) is defined as the product of (3.6.2) and π. This issometimes confusing.

78

b. Relationship between BRDF and albedo
The albedo is defined as the fraction of the reflected irradiance to the incident irradiance. This isa non-dimensional parameter and is always less than zero. The albedo α(θ0, φ0) depends on theincident direction and is closely related to the BRDF:

α (µ0 , φ0 ) = ∫ 0 ∫ 0 R(µ0 , φ0 , µ1 , φ1 )µ1dφ1dµ1
1

2π(3.6.3)where µ* = cosθ* .
c. Bidirectional reflectance probability function (BR-PDF)
The following function shows the 2-D PDF of the reflected irradiance in a horizontal BRDF
plane:P(µ0 , φ0 ; µ1 , φ1 ) ≡R(µ0 , φ0 , µ1 , φ1 )µ1
α (µ 0 , φ 0 )(3.6.4)

We define this function as the “bidirectional reflectance probability density function (BR-PDF).”This function is normalized as

∫ ∫
1

2π

0

0

P(µ0 , φ0 ; µ1 , φ1 )dφ1dµ1 = 1(3.6.5)d. Isotropic reflection
When the BRDF has no directionality, the reflection is called Lambertian reflection. In this case,the albedo is also constant.

R(µ 0 , φ0 , µ1 , φ1 ) = R0

(3.6.6)α (µ0 , φ0 ) = R0 ∫ 0 ∫ 0 µ1dφ1dµ1 = πR0
α∴ R0 =π1

2πP(µ0 , φ0 , µ1 , φ1 ) =µ1
π(3.6.7a)(3.6.7b)

The BR-PDF becomes(3.6.8)It is easily confirmed that Eq. (3.6.8) meets the normalization (3.6.5).
e. Anisotropy of BRDF
There are several ways to measure the anisotropy of the BRDF. For example, the momentum µ1,
which is used as an anisotropy factor for the scattering phase function, is one possibility.

79However, this factor implicitly assumes the non-anisotropy for the azimuth direction. Here, we
introduce χ to measure the anisotropy of the BRDF by calculating the average of the absolutedeviation. The average deviation between P (BR-PDF of the BRDF model) and isotropic case
(3.6.8) is expressed byχ=

1 1 2πµP(µ0 , φ0 , µ1 , φ1 ) − 1 dφ1dµ1
∫
∫
0
0
2
π

(3.6.9)This is a dimensionless parameter with a range of [0–1]. In an isotropic case, χ = 0. In the
specular reflection case (BRDF is a Dirac delta function), χ = 1.
f. Azimuthally averaged BRDF
An azimuthally averaged reflection function is defined by

R (µ0 , φ 0 , µ1 ) ≡1 2π∫ R(µ0 , φ0 , µ1, φ1 )dφ1
2π 0(3.6.10)

3.6.2 Reflection from surface with arbitrary BRDF
For simplicity, only a horizontal plane is considered. There are several methods for thetreatment of reflection. For example, the reflection direction is first determined and the weightof the photon is then changed (weighting method). However, this method is not cost effective.
The change in the photon weight means that the irradiances of the modeled photons differ fromeach other. After multiple reflections, the weight of the photon disperses from 0 to infinity. Thatcauses large noise in the MC integration.
In the MC simulation discussed in this report, the number of photons in the finite domain isconserved. Each model photon has a constant irradiance for a horizontal plane. Withoutabsorption, the irradiance does not change. In the reflection case, the reflected direction should
be determined not to horizontally change the irradiance of a single modeled photon. Here, we
consider a method where the weight of the photon does not change after the reflection. In thiscase, the weight of the photon from the direction (µ0, φ0) is determined byw′ = wα (µ0 , φ0 )(3.6.11)This equation indicates that the weight of the photon does not change by the reflected direction.
α is always less than 1.
Next, a determination of the reflected direction will be discussed. The reflected direction (µ1,φ1) is determined by following the BR-PDF. Let µ1 bep(µ1 ) = ∫ 0 P(µ1 , φ1 )dφ1
2π(3.6.12)

Then, from Eq. (3.6.5), p(µ1) meets the following normalization:80∫ p(µ )dµ = 1110(3.6.13)1µ1 is solved by∫ p(µ′ )dµ′ = ρ
µ1

1

0

1

(3.6.14)µ1

where ρµ1 is a random number. Generally, the left term of (3.6.14) cannot be analytically solved.
Therefore, the LUT method or rejection method can be used. φ1 can be determined by thefollowing equation:∫ P(µ , φ′ )dφ′ = ρ∫ P(µ , φ′ )dφ′
φ1

0
2π

1

0

1

1

1

1

(3.6.15)

φ1

1

Just as is the case with µ1, the left term of this equation is not analytically solved. Therefore, the
LUT method or rejection method can be used.
It should be noted that the PDF defined in (3.6.15) is a product of Eq. (3.6.4) and µ1. Then,(3.6.15) can be only applied to a horizontal surface. When considering a sloped surface, it isnecessary to determine the direction using a PDF not multiplied by µ1. Then, the rejection
method is applied by using the cosine to the horizontal surface (we do not describe this indetail).
The radiance is commonly calculated by the LEM (see 3.9). The scattering, reflection, andemission processes can be sampled by several methods. In this report, we calculate these by thefollowing procedures:
1. Scaling the weight using (3.6.11)
2. Rescaling the weight by Russian roulette
3. Radiance sampling by LEM
4. Determination of direction after events
If the order of these procedures changes, the formulation changes slightly. However, the above
order is recommended because it is preferable to execute the Russian roulette before the LEM toreduce the variation in weight. A photon with a weight of 0.01 is clearly less important than thatwith a weight of 1.0. It is a waste of CPU time to execute the LEM for a photon with a small
weight.
In the sloped surface case, first, the reflection direction is determined along coordinates thatare normal to the slope; then, the direction is transformed into Cartesian coordinates. For
example, let the slope surface normal be n(Θ, Φ) and the incident direction be Ω(θ, φ). The
incident angle relative to the slope surface normal Ω (θ ′, φ′) is a vector where Ω(θ, φ) isrotated –Φ toward the z-axis and is then rotated –Θ toward the y-axis. Then, the weight scaling is
executed just as on the case for a horizontal surface. Finally, a coordinate transformation in thereverse mode is executed and the direction vector is determined for Cartesian coordinates.

81

3.6.3 Isotropic reflection: Lambertian reflection
Here, we describe the determination method for the reflection direction in isotropic scattering.
a. Isotropic scattering for horizontal surface
When we consider the isotropic reflection from a horizontal surface with the solid angle ξ, the
PDF of the reflected direction is normalized by1
1 2π 1
cos
θ
d
ξ
=
∫
∫ ∫ cosθd cosθdφ = 1
π 2π
π 0 0

(3.6.16)

The reflected direction for the azimuth direction is uniform. Therefore, φ is determined by

φ = 2 πρ φ(3.6.17)

The following is a method to determine the zenith angle.
Analytical method
The zenith angle of the reflection direction θ can be calculated using the random number ρ:

1 2 π cos θ∫ ∫ cosθ ′d cosθ ′dφ = ρ′π 0 0∴cos2 θ = 1− ρ ′ ≡ ρθ∴cosθ = ρθ

(3.6.18)(3.6.19)

Rejection method
The direction is first determined isotropically:
cos θ
1 2π
dφ ∫ 0 d cosθ ′ = ρθ∫
02π∴cosθ = ρθ

(3.6.20)(3.6.21)

Then, the cosθ dependency of the direction is determined by the rejection method. This can beachieved by using an additional random number:(3.6.22)

If rejection is selected, the direction is again calculated by (3.6.21). The rejection probability is
1/2. Therefore, on average, the direction is determined in 2 cycles of the procedures. Thismethod requires more random numbers than the analytical method. However, it does not require82calculating the square root. If the fast algorithm of the random number generator is used, therejection method may be faster than the analytical method.
Consequently, both the analytical method and rejection method are practical. The rejection
method is easily applicable to the arbitrary shape of the BRDF.

b. Isotropic reflection for arbitrary slope surface
In the Lambertian case, the PDF of the reflected direction is independent of the incident angle.
Therefore, the same method as for the horizontal surface is used (analytical or rejection method)to calculate the zenith and azimuth angles relative to the normal vector of the slope surface.
Then, the direction relative to the Cartesian coordinates is calculated by coordinatetransformation.
Let the zenith and azimuth angles of the reflected direction relative to the slope normaln(Θ,Φ) be θˆ, φˆ , respectively. Then, the PDF of the reflected direction is normalized by1
1 2π 1cos θˆ dξ = ∫ 0 ∫ 0 cosθˆ d cos θˆdφˆ = 1
∫
2
π
π
π

(3.6.23)

Therefore, θˆ, φˆ can be calculated as well as a horizontal surface.
The reflected direction in Cartesian coordinates can be calculated using the same formula as
the scattering (Appendix A5.2):⎡⎛ cos Θcos Φ⎞⎛ − sinΦ⎞ ⎤
⎢
⎥
ˆ
ˆ
ˆ
ˆ
Ω = cos θ n + sinθ ⎢cos φ ⎜ cos ΘsinΦ ⎟ + sin φ ⎜ cos Φ ⎟ ⎥
⎜
⎟
⎜
⎟
⎢⎣
⎝ − sinΘ ⎠⎝ 0 ⎠ ⎥⎦

(3.6.24)c. Isotropic reflection from surface perpendicular to a horizontal surface
A surface that is perpendicular to a horizontal surface is a special case of isotropic reflection for
an arbitrary slope surface. In this case, Θ = π/2, sinΘ = 1, cosΘ = 0 are substituted in (3.6.24).
Then, we obtain⎡⎛cosΦ⎞⎛0⎞⎛−sin Φ⎞⎤
⎢
⎜
⎟
⎜
⎟
⎜
⎟⎥
Ω = cosθˆ ⎜ sinΦ ⎟ + sinθˆ⎢cos φˆ⎜ 0 ⎟ + sin φˆ⎜ cosΦ ⎟⎥
⎜
⎟
⎜ ⎟
⎜
⎟
⎢⎣
⎝ 0 ⎠
⎝−1⎠
⎝ 0 ⎠⎥⎦⎛ cosθˆ cosΦ − sinθˆ sin φˆ sin Φ⎞⎜⎟= ⎜ cosθˆ sin Φ + sinθˆ sin φˆ cosΦ⎟⎜⎜⎟⎟ˆ cos φˆ
−sin
θ
⎝
⎠

83

(3.6.25)

3.6.4 Diffuse-specular mixture model: DSM model
Here, we describe the diffuse-specular mixture model (DSM model). The direction of specularreflection is determined from surface roughness, which is derived from the modeled distributionof microfacets. Since surface roughness varies with the surface wind speed in the case of awater surface, it is modeled by the method described here. The BRDF and albedo are expressedbyR(µ 0 , φ0 , µ1 , φ1 ) = fd Rd + (1− fd )Rs (µ0 , φ0 , µ1 , φ1 )α (µ0 ) = fdα d + (1− fd )α s (µ0 )(3.6.35a)(3.6.35b)where fd, Rd, and Rs are the fraction of diffuse reflection, and the BRDFs of diffuse and specularreflections. αd and αs are the albedos of the diffuse and specular surfaces, respectively.
In the case of diffuse reflection, the following relationship exists:

Rd =

αd
π

(3.6.36)

a. Specular reflection with surface roughness
Here, we introduce the water surface reflection model by Nakajima and Tanaka (1983). The
angle 2β between the reverse direction of the photon incident direction Ω 0 = (µ0, φ0)T andreflection Ω 1 = (µ1, φ1)T is expressed bycos 2 β = Ω0 ⋅ Ω11
cos β =
(1+ Ω0 ⋅ Ω1 )2

(3.6.37a)(3.6.37b)

Also, the Fresnel reflectance is expressed by RF(β). The direction of a microfacet of Ω n = (µn,φn)T has the following relationship:

Ω0 + Ω1
Ω0 + Ω1µ +µµ0 + µ1µn = 0 1 =2 cos β2(1+ Ω0 ⋅ Ω1 )Ωn =tan 2 θ n =(3.6.38a)(3.6.38b)

1− µ n222= (∂z ∂x) + (∂z ∂y)2µn(3.6.38c)

The slope of the microfacet is given by a 2-D Gaussian distribution and isotropic over theazimuth direction:⎡ (∂z ∂x)2 + (∂z ∂y)2 ⎤⎛ ∂z ∂z ⎞1⎥P⎜ , ⎟ =exp⎢−
2
σ2⎝ ∂x ∂y ⎠ πσ
⎢⎣
⎥⎦
84

(3.6.39a)p(µ n ) =⎛ 1− µ n2 ⎞1
exp⎜− 2 2 ⎟πσ 2µ n3
⎝ σ µn ⎠(3.6.39b)

The BRDF is expressed by

Rs (µ0 , φ0 , µ1 , φ1 ) =RF (β )
p(µ n ) fs (µ0 , µ1 )4 µ0µ1µ n(3.6.40)

where fs is a shadowing factor:

fs (µ0 , µ1 ) =1
1+ F(µ0 ) + F(µ1 )(3.6.41a)2⎤
1 ⎡ exp(−υ* )⎥F(µ* ) = ⎢
−
erfc
υ
(
)
*
2 ⎢⎣ πυ*2⎥⎦(3.6.41b)υ* =(3.6.41c)µ*

σ 1− µ*2

erfc is the complementary error function

erfc(υ* ) =2 ∞ −t 2∫ e dtπ υ*(3.6.41d)

The slope σ is given following the airborne measurements (Nakajima & Tanaka, 1983):

σ 2 = 0.00534 u10(3.6.42a)Cox & Munk (1954) proposed

σ 2 = 0.00512 u10 + 0.003(3.6.42b)b. Albedo of DSM model
The albedo of the DSM model can be calculated by Eq. (3.6.35). Here, the unknown parameteris the albedo of the specular reflection. The albedo of a completely flat surface (σ = 0) is

α s (µ0 ) = RF (θ 0 )(3.6.43)

The albedo of a rough surface (σ > 0) can be calculated by integrating the solid angle ξ over the
direction (µ0, φ0 + π):

α s (µ0 ) = ∫ 2 π R(µ0 , φ0 , µ1 , φ1 )µ1dξ
=A(µ0 )µ0(3.6.44)85

A(µ 0 ) = ∫ µ ′ dµ′ ∫ φ′ dφ ′ B(µ 0 , µ′, φ ′)(3.6.45a)

B(µ 0 , µ′, φ ′) =(3.6.45b)

1

π

min

min

⎛ 1− µ n2 ⎞1 RF (β )exp⎜− 2 2 ⎟ fs (µ0 , µ1 )2 πσ 2 µ n4
⎝ σ µn ⎠Here, the direction (µ1, φ1) is given by rotating (µ′, φ′) from (µ0, φ0 + π):µ1 = sinθ 0 sin θ ′ cos φ ′ − µ0µ′(3.6.46a)

Then, β and µn are derived from (3.6.37–38). When µ′ approaches 1, it is convenient to use thenew parameter as follows:µ˜ ′ ≡ 1− µ′(3.6.46b)

Then, it can be expressed bysin θ ′ = 1− µ′2 = µ˜ ′(2 − µ˜ ′)(3.6.46c)

The lower limitation of φ′, in which Eq. (3.6.46c) should have the upper direction, is given byµ 0µ ′sin θ 0 sin θ ′⎧πif γ < −1⎪ −1
φ ′min = ⎨cos γ if −1 ≤ γ ≤ 1⎪0if γ > 1⎩
γ=(3.6.47a)(3.6.47b)

When the domain of the integration is assumed to be c times (typically three times) the
microfacet slope σ, µ is expressed byµ′ > µ′min =1− c 2σ 2
1+ c 2σ 2(3.6.48a)or

1− µ′ < 1− µ′min =2c 2σ 2
1+ c 2σ 2(3.6.48b)

The function A is pre-computed and stored in an LUT before the MC simulation.

c. Calculation of angle distribution function of DSM model
The radiance can be calculated according to the BRDF divided by the albedo (normalized angle
distribution function, see 3.9). Based on the definition of (3.6.35),

86

R(µ 0 , φ0 , µ1 , φ1 ) fd α d π + (1− fd )Rs (µ0 , φ0 , µ1 , φ1 )=α (µ0 )α (µ0 )

(3.6.49a)

However, for a smooth surface case (σ ≅ 0), specular reflection has a strong spike-like peak. Dueto its strong peak, it causes a large noise and never converges. Therefore, the slope of the
micro-facet σ should be larger than a certain threshold. This causes a small artifact. If theincident direction of the modeled photon is isotropic, the BRDF can be replaced by a smootherfunction. Here, we introduce an approach to modify the minimum σ closely connected with theisotropic characteristics of the incident light.
However, when σ is set larger than a certain threshold, Eq. (3.6.49a) is not effective inusing the radiance calculation. Also, if you use σ with a different value from that in the albedo
calculation, this contravenes the conservation law.
Here, we introduce a method that only changes the shape of the angular distribution under
the same albedo condition. When the direction of the incident photon is isotropic, the diffuse
fraction fd becomes larger. At the same time, the albedo of the diffuse reflection is modified
under conserving the average albedo. From (3.6.35b),

α (µ0 ) = fdα d + (1− fd )α s (µ0 )= fd′α ′d + (1− fd′)α s (µ0 )⎛ f ⎞f
α ′d = d α d + ⎜1− d ⎟α s (µ0 )fd′⎝ fd′ ⎠

(3.6.49b)(3.6.49c)

Equation (3.6.49a) can be transformed byR′(µ0 , φ0 , µ1 , φ1 ) fd′ α d′ π + (1− fd′ )Rs (µ0 , φ0 , µ1 , φ1 )=α (µ0 )α (µ0 )

(3.6.49d)

The peak of this specular function has a weaker and smoother angular distribution than theoriginal one. The average albedo is the same as the original one.

d. Determination of direction in DSM modelFirst, the reflection pattern (diffuse or specular) is determined. The probability of each reflection
is given by

fd

αd
,
α

α
α

(1− fd ) s

In the case of specular reflection, the reflected direction from a flat surface isΩ1 = −Ω0 + 2µ0 Ωn(3.6.50)

where Ω n is a zenith angle.

87

If the surface is not flat, the direction is determined with the PDF by (3.6.40) and (3.6.44):

Qs ( µ 0 , µ1 ,φ1 − φ 0 ) =Rs ( µ 0 ,φ 0 , µ1 ,φ1 ) µ1
α s ( µ0 )R ( β ) fs ( µ 0 , µ1 ) ⎛ ∂z ∂z ⎞= FP⎜ , ⎟
4 A( µ 0 ) µ n4
⎝ ∂x ∂y ⎠(3.6.51)First, the direction of the microfacet Ω n is randomly determined following P. The rejection
method is then applied to make a correct distribution, with an acceptance probability ofS(µ0 , µ1 , φ1 − φ0 ) =RF (β ) fs (µ0 , µ1 )µ n4

(3.6.52)

Since the slope of the micro-facet follows the 2-D Gaussian distribution with a deviation of
σ /2, it can be determined by the two pseudorandom numbers by using the Box-Muller method(see 3.1.2 and NR):222
σ 2 ⎡⎛ ∂Z ⎞ ⎛ ∂Z ⎞ ⎤⎢tan θ n =⎜ ⎟ +⎜ ⎟ ⎥2 ⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥⎦2

(3.6.53)∂Z= −2 ln ρ1 cos 2 πρ 2∂x∂Z
= −2 ln ρ1 sin 2 πρ 2∂y(3.6.54a)(3.6.54b)

To calculate it effectively, a point within the unit circle is randomly selected by using two
random numbers (ρ3, ρ4). To avoid a finite slope, the center of the circle should be rejected. Thisis repeated until it meets the following limitation:

W1 = 1− 2 ρ 3
W2 = 1− 2 ρ 4
2rmin≤ r 2 ≡ W12 +W22 ≤ 1

2
rmin≈ 0.001(3.6.55a)(3.6.55b)(3.6.55c)

When we put

ρ1 = r 2
cos φ n = cos 2 πρ 2 = W1 r
sin φ n = sin 2 πρ 2 = W2 r

(3.6.56a)(3.6.56b)(3.6.56c)

tan 2 θ n = −σ 2 ln r 2
1∴ µn =
1− 2σ 2 ln r 2∴sin θ n = µ n −2σ 2 ln r 2(3.6.57a)then(3.6.57b)(3.6.57c)

88

Finally, we can avoid the sine and cosine calculations. Then, we can obtain Ω n. The reflection
direction is determined byΩ1 = −Ω0 + 2 cos β Ωncos β = Ω0 ⋅ Ωn > 0(3.6.58a)(3.6.58b)

If the scalar product of (3.6.58b) is negative, reflection does not occur. Therefore, the selected
direction should be rejected and a new direction is selected. Also, if the reflected direction is a
downward direction, the selected direction should be rejected.
Next, the rejection method is applied based on the function S in (3.6.52). The maximum of S
(Smax(µ0)) should be determined in advance. This is on the principal plane. Because r has a lower
limitation constrained by (3.6.55c), the slope of the microfacet has such a limitation:

µn ≥

1
2
1− σ 2 ln rmin

≡ µ min ≡ cos θ max(3.6.59)

Therefore, the reflected direction is also constrained within a certain range. The “relative zenith
angle” is defined by⎪⎧−θ1 if φ1 − φ0 < π 2θˆ1 ≡ ⎨⎪⎩θ1 if φ1 − φ0 ≥ π 2(3.6.60)

The edges of this range can be written byθˆ1 = θ 0 − 2θ max , if φ1 − φ0 = 0θˆ1 = θ 0 + 2θ max , if φ1 − φ0 = π(3.6.61a)(3.6.61b)

where

cos 2θ max = 1− 2 sin 2 θ max
sin 2θ max = 2 sin θ max cos θ max
The absolute value of Eq. (3.6.61) is always less thanπ/2. Smax can be prepared as an LUT using
the golden rule (NR). The reflected direction can be obtained by interpolating with respect to µ0.
A problem arises when S has a strong peak and the maximum value is located within it. To
prevent this problem, Smax/S0 should be adjusted so as not to become too large. If S0 is a value of
S for a microfacet with a vertical direction (µn = 1), then⎧Smax
if Smax < DS0⎪⎛ Smax⎞E ⎤′ =⎨ ⎡Smax⎪S0 ⎢D −1+ ⎜ S − D +1⎟ ⎥ if Smax ≥ DS0
⎝ 0
⎠ ⎥⎦
⎩ ⎢⎣

89

(3.6.62)where D and E are constant values (the preferable values are D = 4, E = 0.5). Based on this, therejection method is applied:

S
′
Smax

(3.6.63)

Fig. 3.6.1 shows the simulation results of the BRF for the reflected direction (zenith andazimuth angles) under a solar zenith angle = 0° and a slope dispersion of 0.04.

Fig. 3.6.1 BRF from analytical calculation of DSM model (left) and from MC
simulation (right).

3.6.5 Rahman-Pinty-Verstraete (RPV) BRDF model
Rahman, Pinty, and Verstraete (1993) (RPV) proposed a semi-empirical model to express the
BRDF for vegetation and soil. The RPV model has three geometric parameters (incident zenith,reflected zenith, and relative azimuth), which is the three parameter case in (3.6.2). The BRDF
of the RPV model is expressed byρ 0 ⎧⎪ cos k−1 θ 0 cos k−1 θ1
σ ⎫⎪R(θ 0 ,θ1 , φ ) = ⎨F(γ )[1+ H (G)] +⎬π ⎪⎩ (cos θ 0 + cosθ1 )1−kcos θ 0 ⎪⎭(3.6.65)

Here, ρ0 and σ are constant. F is a function similar to the Henyey-Greenstein function:F(γ ) =1− Θ2

[1+ Θ + 2Θcos γ ]2(3.6.66)

3/2

It expresses the anisotropy of the forward and backward scattering. γ is a phase angle (angle
between incident Ω 0 and reflected direction Ω 1):

cos γ = Ω0 ⋅ Ω1 = cos θ 0 cosθ1 + sin θ 0 sin θ1 cos φ

90(3.6.67)

The function H expresses the shape of hotspot effects:

1+ H (G) = 1+

1− ρ 0δ+G(3.6.68)

Generally, δ = 1. In a special case, δ can be used to fit the observation data. G is a geometricfactor and is given by

G = tan 2 θ 0 + tan 2 θ1 − 2 tan θ 0 tan θ1 cos φ(3.6.69)

The hotspot is located at θ0 = θ1, φ = 0 and has a strong peak. Except for the hotspot effect, theRPV model has a smooth angular distribution.
The RPV model has five parameters (ρ0, k, Θ, δ, σ) depending on the vegetation and soil
conditions. For the simplicity of the simulation, we use σ = 0. The BRDF of RPV is thenrewritten by:

R(θ 0 ,θ1 , φ ) =ρ 0 cos k−1 θ 0 cos k−1 θ1F(γ )[1+ H (G)]π (cosθ 0 + cos θ1 )1−k(3.6.70)

Figure 3.6.2 shows an example of this model.

View Zenith Angle
Fig. 3.6.2 BRF (= BRDF*π) calculated by the RPV model for three different
incident zenith angles (0°, 60°, 89°).

a. Albedo of RPV model
The albedo is calculated by its definition:

91

1 2π

α ( µ 0 ) = ∫ ∫ R( µ 0 , µ1 ,φ ) µ1 dφ dµ1
0 0

1 π

= 2 ∫ ∫ R( µ 0 , µ1 ,φ ) µ1 dφ dµ1(3.6.71)

0 0

[

]

k
2 ρ0 k−1 1⎧⎪ µ1 ( µ 0 + µ1 ) π⎡ 1− ρ0 ⎤ ⎫⎪=µ0 ∫ ⎨
F(
γ)⎢⎣1+ δ + G ⎥⎦ dφ ⎬ dµ10π( µ0 + µ1 ) ∫0⎩⎪⎭⎪

For simplicity, we put

2ρ0 1∫ B(µ0 , µ1 )dµ1
π 0kµ1 (µ0 + µ1 )] π[B(µ 0 , µ1 ) =∫ C(µ0 , µ1, φ )dφ(µ0 + µ1 ) 0⎡ 1− ρ 0 ⎤C(µ0 , µ1 , φ ) = F(γ )⎢1+⎣ δ + G ⎥⎦A(µ 0 ) =(3.6.72a)(3.6.72b)(3.6.72c)

Then, the albedo is expressed byα (µ0 ) = µ0 k−1 A(µ0 )(3.6.73)

The function A cannot be solved analytically. The integrand C in (3.6.72b) with respect to φ issufficiently smooth. Therefore, it is possible to integrate using the Gaussian integration method(Press et al., 1992). The integrand with respect to µ1 is not smooth in the hotspot region (µ1 =µ0). Therefore, to obtain accurate results, it is necessary to use the two integration ranges
0 ≤ µ1 < µ0 and µ0 ≤ µ1 ≤ 1.
This requires vast computation if the double integral discussed above is performed at every
reflectance event. It is better to prepare an LUT for A with respect to µ0. Fortunately, thefunction A with respect to µ0 is very smooth, so that we can obtain sufficient accuracy with 4 to
6 quadratic points (the interpolation error is less than 0.3%). When the parameters are dependent
on the location (x, y), the function A becomes a 3-D (x, y, µ0) LUT.

Cosine of Incident Zenith Angle
Fig. 3.6.3 Relationship between the albedo and incident zenith angle. Sevendifferent types of surfaces are shown.

92

The albedo should have the value of [0–1]. The albedo calculated from Eq. (3.6.73)
sometimes takes values out of this range (especially in small µ0 cases). To prevent suchabnormal values, we multiply 1/α for the BRDF when the albedo is larger than 1. Then, thealbedo becomes 1 by compulsion.
Integration of function C
Multiple calculations are required for the integration of C(φ) over φ = 0–π at a given pair of (µ1,µ0) . Therefore, as fast and accurate an integration scheme as possible is necessary. Here, we use
the Gaussian integration method.
To achieve effective integration, it would be better to calculate the coefficient with respect
to (µ1, µ0). Again, we rewrite the form of function C:⎡ 1− ρ 0 ⎤C(µ0 , µ1 , φ ) = F(γ )⎢1+⎣ δ + G ⎥⎦(3.6.74)

F and G in (3.7.74) are

F(γ ) =1− Θ2

[1+ Θ2 + 2Θcos γ ](3.6.75a)

3/2µ02 + µ12 − 2µ0µ1 cos γ
G=µ0µ1(3.6.75b)cos γ = µ0µ1 + (1− µ02 )(1− µ12 ) cos φ(3.6.75c)

Therefore, when we put⎧2⎪d1 = 1− Θ
⎪d2 = 1+ Θ2⎪⎨d 3 = µ0µ1⎪
22⎪d 4 = (1− µ0 )(1− µ1 )⎪22⎩d5 = µ0 + µ1

(3.6.76)

then F, G, and C are simply expressed by

d1
3/2[d2 + 2Θcos γ ](3.6.77)d5 − 2d 3 cos γ
d3(3.6.78)F(γ ) =G=⎡⎤d1
(1− ρ 0 )d 3
1+
⎢
⎥
3/2
[d2 + 2Θcos γ ] ⎣ δd3 + d5 − 2d3 cos γ ⎦2 cos γ = 2(d 3 + d 4 cos φ )C(µ0 , µ1 , φ ) =93(3.6.79)(3.6.80)

The integration by the Gaussian method is

C (µ0 , µ1 ) = ∫ 0 C(µ0 , µ1 , φ ′)dφ ′πN= π ∑ wiC (µ0 , µ1 , cos φ (i))(3.6.81)i=1

where w is a weight of the Gauss-Legendre integration. cosφ in each integration point should be
prepared by an LUT for effective calculation.

b. BRDF calculation
The BRDF can be directly calculated by the incident and reflected zenith angles and relative
azimuth angle. RPV is rewritten by

R(µ 0 , µ1 , φ ) =ρ 0 F(γ )[1+ H (G)]π [µ0µ1 (µ0 + µ1 )]1−k(3.6.82)

When it is divided by the albedo, the strong µ0 dependency is diminished. Using (3.6.73), we
obtainF(γ )[1+ H (G)]R(µ 0 , µ1 , φ )ρ0=⋅
α (µ0 )
πA(µ 0 ) [µ1 (µ0 + µ1 )]1−k(3.6.83)

The function H can be rewritten by1− ρ 0
δ+G
G = tan 2 θ 0 + tan 2 θ1 − 2 tan θ 0 tan θ1 cos φ
H (G) =

(3.6.84a)(3.6.84b)

Here, if we use the following relationship:

Ω0 ⋅ Ω1 = µ0µ1 + sin θ 0 sin θ1 cos φ(3.6.85)

we obtaintan θ 0 tan θ1 cos φ =sin θ 0 sin θ1 cos φ Ω0 ⋅ Ω1
=
−1
µ0µ1
µ0µ1

When Eqs. (3.6.86) and (3.6.87) are substituted to (3.6.84b),

94(3.6.86)⎧ 2
1− µ02 1= 2 −1⎪tan θ 0 =
⎪
µ02
µ0
⎨
2
⎪tan 2 θ = 1− µ1 = 1 −1
1
⎪⎩
µ12
µ12

(3.6.87)µ02 + µ12 − 2µ0µ1 (Ω0 ⋅ Ω1 )1
1
Ω0 ⋅ Ω1G=+ −2=µ02 µ12µ0µ1µ0µ1(3.6.88)

we obtain

Finally, we can avoid the sine/cosine/tangent calculation.

H (G) =F(γ ) =(1− ρ 0 )µ0µ1

δµ0µ1 + µ02 + µ12 − 2µ0µ1 (Ω0 ⋅ Ω1 )1− Θ2

[1+ Θ + 2Θ(Ω ⋅ Ω )]2

0

(3.6.89)(3.6.90)

3/2

1

By using the above equations, we can calculate Eq. (3.6.83) from the incident and reflected
zenith angles and relative azimuth angle. When µ1 is nearly 0, we should treat it carefully.

c. Determination of reflection direction in RPV model
When a set of RPV model parameters (ρ0, k, Θ, δ) is given and these parameters are constant in
the simulation space, a 3-D LUT is an effective way to simulate. However, if the RPV model
parameters (ρ0, k, Θ, δ) vary in space, a seven-dimensional (7-D) LUT is required. This requiresa large memory, even for a coarse LUT. Also, the interpolation from a 7-D LUT is numericallyvery intensive and does not promote accuracy.
At the time of the reflection event, five parameters (ρ0, k, Θ, δ, µ0,) are already fixed and µ1
and φ should be determined. We propose a method to determine µ1 and φ simultaneously. The
comparison function should be a close match to the BR-PDF. Therefore, the µ1 dependency of
the BR-PDF is approximated by a linear function.
Determination of comparison function
The BR-PDF is approximated by a linear function as follows:

P(µ0 , µ1 , φ ) ≈ a1 (µ0 )µ1 + a0 (µ0 )(3.6.91)

In the isotropic case, P = µ1/π, which is well expressed by (3.6.91), a1 and a0 are determined by
the normalization conditions:95⎛ a1 (µ0 )⎞+ a0 (µ0 )⎟ = 1⎠
2∫ ∫ [a (µ )µ + a (µ )]dµ dφ = 2π ⎜⎝
2π

0

1

0

1

∴a1 (µ0 ) =

0

1

0

0

1

1
− 2a0 (µ0 )π(3.6.92a)(3.6.92b)

These coefficients can be determined by normalization conditions after regressing to thefunction B (average BRDF over the azimuth angle φ) (see 3.6.71–72).
It is necessary to obtain the coefficient of the following parameters:

B(µ 0 , µ1 ) ≈ a1 (µ0 )µ1 + a0 (µ0 )(3.6.93a)

It should be noted that the comparison function always takes positive values for the range of µ1= 0–1. Finally, it becomesa1 (µ0 ) =a1 (µ0 )π (a1 (µ0 ) + 2a0 (µ0 ))(3.6.93b)where the comparison function should be positive; therefore, it is forced by−1
1< a1 (µ0 ) <π
π(3.6.93c)

a0 can be determined by1⎛1⎞
a0 ( µ 0 ) = ⎜ − a1 ( µ 0 )⎟⎠2⎝π(3.6.93d)

Here, we can find a comparison function that is “larger than the original BR-PDF in all
points”:

P(µ0 , µ1 , φ ) ≡R(µ0 , µ1 , φ )µ1
α (µ0 )

(3.6.94)

We can find the above equation divided by (3.6.91). To meet the criteria for a comparison
function that is larger than the original BR-PDF in all points, we multiply the following
equation by (3.6.91):⎧µ1
R(µ0 , µ1 , φ ) ⎫b(µ0 ) = max⎨⋅⎬
α (µ 0 ) ⎭⎩ [a1 (µ0 )µ1 + a0 (µ0 )](3.6.95)

The function max{} is a maximum of all µ1, φ. R/α can be calculated by (3.6.83). In the case of
the RPV model, it is located on the principal plane (φ = 0 or π). When the BR-PDF is a linear
function such as (3.6.91a), the factor b becomes 1. If the BRDF has a strong peak, multiple
iterations are necessary. Instead of b, we use the following equations:96

if b(µ0 ) < bmin⎪⎧b(µ0 )b′(µ0 ) = ⎨⎪⎩bmin −1+ [b(µ0 ) − bmin +1] S if b(µ0 ) ≥ bmin

(3.6.96)

S is a tuning parameter. That is, if the BR-PDF is far from the linear function, it should beapproximated by a similar angular distribution shape to the BR-PDF. bmin = 4, S = 0.5 are thebest values for most cases. In the case of the RPV model, there is no prominent peak and b =usually 1–5. Consequently, the comparison function becomesf (µ0 , µ1 , φ ) = b′(µ0 )[a1 (µ0 )µ1 + a0 (µ0 )](3.6.97)

The coefficients a1 and b′ should be prepared as a 3-D LUT. If the reflection event occurs, it
is interpolated by the LUT with respect to µ0. Since the comparison function is usuallyestimated roughly, it is not necessary to derive these coefficients accurately. We evaluate a0 from
Eq. (3.6.97).
Use of rejection method
In the following description, we ignore the argument µ0. The comparison function in Eq.
(3.6.97) can be written asf (µ1 , φ ) = b′(a1µ1 + a0 )(3.6.98)µ1 that follows this equation can be determined by solving:

2 π ∫ 0 (a1µ1′ + a0 )dµ1′ = ρ µ1
µ1

(3.6.99)

Therefore, this is a problem to solve a quadratic equation:

a1µ12 + 2a0µ1 −

ρ µ1
=0
π

(3.6.100)

The solution of this equation within the range of µ1 = 0–1 is1) If a1 ≠ 0µ1 =−a0 + a02 + a1 ρ µ1 π2) Else if a1 = 0

a1µ1 = ρ µ1(3.6.101a)(3.6.101b)

The azimuth angle can easily be determined by

φ = 2 πρ φ(3.6.102)

97

The probability that the above direction is followed by the BR-PDF in the RPV model isgiven byγ=P(µ1 , φ )µ1
R(µ 0 , µ1 , φ )=⋅f (µ1 , φ ) b′(a1µ1 + a0 )α (µ0 )

(3.6.103)

Therefore, we use the additional random number and apply the following criteria:(3.6.104)

If rejected, the new direction is selected again by (3.6.101–102). The reflected direction
followed by the RPV model can be determined by the above prescribed method.

c. Simulation example
Several test simulations were performed. The prescribed method had enough speed (using aPentium IV 3.6 GHz PC, we could achieve a reflection simulation with over 6,000,000 events).
Figure 3.6.4 shows the validation results for the exact BRF and the BRF calculated by the MC
method. The results have a fairly good accordance.

Fig. 3.6.4 Analytical BRF calculation of RPV (left) and BRF by MC simulation(right).

3.6.6 Li-Sparse-Ross-Thick (LSRT) BRDF model
Lucht et al. (2000) proposed a semi-empirical model to express a land surface BRDF, called the
Li-Sparse-Ross-Thick (LSRT) linear kernel model. This model is employed to estimate theretrieval of land surface parameters from MODIS and MISR data. Also, it is combined with theSHARM model by Lyapustin (2000).

98

Here, we describe a method to include the LSRT model in the MC simulation.

a. Original formulas of LSRT BRDF
Let the incident direction be Ω 0 (this is a different definition than in the previous section). The
zenith angle, its cosine, and azimuth angle are expressed by θ0, cosθ0 = µ0, and φ0, respectively.
The reflected direction is also defined by Ω 1 and its components are θ0, µ0, and φ0. The LSRT
model is a three-parameter BRDF model (3.6.2). The hotspot direction is expressed by –µ0 = µ1,
φ = π.
The BRDF can be expressed by the combination of the reflectance contributions from
Lambertian, geometric-optics, and volume scatterings:

πR(µ0 , µ1 , φ ) = k L + kg fg (µ0 , µ1 , φ ) + kv fv (µ0 , µ1 , φ )(3.6.106)

Here, the kernel function is expressed by

fv ( µ 0 , µ1 ,φ ) =fg ( µ 0 , µ1 ,φ ) =(π 2 − γ ) cos γ + sinγ − π− µ 0 + µ1

(3.6.107)4(1+ cos γ ′ )1( t − sin t cos t ) µ1′−1 − µ0′ −1 − ( µ1′ −1 − µ 0′ −1 ) −π
2 µ 0′ µ1′()(3.6.108)
These functions have negative values in some directions. γ is a phase angle (inverse direction ofincident direction and reflected direction) given by()
cos γ ′ = Ω′ ⋅ Ω′ = − ( µ ′ µ ′ + 1− µ ′ 1− µ ′ cos φ )cos γ = Ω0 ⋅ Ω1 = − µ 0 µ1 + 1− µ 02 1− µ12 cos φ
2

0

1

0

1

2

0

1

(3.6.109a)(3.6.109b)

Here, the angle t is expressed bycos t =

22
2
2
2
h tan θ0′ + tan θ1′ − 2tan θ0′ tan θ1′ cos φ + tan θ0′ tan θ1′ sin φ−1bµ1′ − µ 0′ −1(3.6.110)

Here, it is constrained by cos t ≤ 1 . The angle of the prime parameters in (3.6.108–110) is
defined bybtan θ0rbtan θ1′ = tan θ1rtan θ0′ =(3.6.111a)(3.6.111a)

The ratio of the canopy structural parameters is assumed by (Lucht et al., 2000)

99

b/r = 1 and h/b = 2(3.6.112)

Finally, the BRDF is determined by the three parameters kL, kg, and kv.

Fig. 3.6.5 Definition of direction and angles

b. Modifications
When the incident direction is nearly horizontal, the LSRT model outputs a negative value or
infinity. If the value is negative, it is reset to 0. If the albedo is larger than 1, it is reset to 1 andthe BRDF is rescaled as a small value.
The original form of the LSRT model has many trigonometric functions, which requires
rather extensive computation. Therefore, we change the form of (3.6.108) (3.6.110) as an
expression of the direction cosine. The geometric kernel in (3.6.108) is written asfg ( µ 0 , µ1 ,φ ) =⎤1 ⎡⎛ t − sin t cos t ⎞1⎜1−⎟ ( µ1′ − µ 0′ ) − (1+ cos γ ′ )⎥⎢⎠µ 0′ µ1′ ⎣⎝π
2⎦(3.6.113)

Here, we summarize the following function by substituting to (3.6.110):sin θ0′ sin θ1′ cos φ
cos γ ′=−
−1µ 0′ µ1′µ 0′ µ1′
1− µ 0′ 2
1− µ1′ 22
tan 2 θ0′ =
;tan
θ
′
=
1
µ 0′ 2µ1′ 2tan θ0′ tan θ1′ cos φ =(3.6.114)(3.6.115)

Then, we obtaincos t =2h 1− cos γ ′b µ1′ − µ 0′(3.6.116)

fg and t become easier to compute. In the hotspot direction, the following relationship exists:

Ω0′ = − Ω1′ ; µ 0′ = − µ1′ ; cost = 0 ; fg ( µ 0 , µ1 ,φ ) =100

1− µ1′µ1′ 2

The direction cosine in (3.6.113) and (3.6.116) can be derived as follows. First, the following
relationship is found from (3.6.111):⎛ b ⎞2
B + (1− B)µ ; B = ⎜ ⎟⎝r ⎠µ′0 = µ020

⎛ µx 0 ⎞Ω0 = ⎜ µ y 0 ⎟ ; Ω0′ =
⎜ ⎟
⎜µ ⎟
⎝ z0⎠

(3.6.117)⎛ µx 0 ⋅ b r ⎞⎜ µ ⋅ b r⎟y0⎟
B + (1− B) µ 02 ⎜⎜⎟⎝ µz 0⎠1(3.6.118)

The variables indicated by the subscript 1 are expressed in the same manner. Therefore, we
obtaincos γ ′ = Ω0′ ⋅ Ω1′ =B(Ω0 ⋅ Ω1 ) + (1− B) µ 0 µ1

(3.6.119)[ B + (1− B) µ ][ B + (1− B) µ ]
2
0

2
1

For the calculation, we first calculate the scalar product Ω0 ⋅ Ω1 in (3.6.119). From (3.6.118), thetwo directions expressed by the prime parameters are determined and the scalar products arethen determined. These values are substituted to (3.6.116) and we obtain cost; sint, t, and fg are
obtained by (3.6.113). Also, fv is calculated from (3.6.107). Finally, the BRDF is calculated by(3.6.106).
2
fgeo00fgeo20fgeo45
fgeo70Scaled Kernel Value1

fvol00fvol20fvol45
fvol70

0

-1

-2
-80

-60

-40

-20

0

20

40

60

80

View Zenith Angle

Fig. 3.6.6 Kernel function (same as Fig. 2 in Lucht et al. 2000).

101

c. Albedo of LSRT model
The albedo can be written as

α (µ0 ) = ∫ 0 dµ1 ∫ 0 dφR(µ0 , µ1 , φ )µ12π

1= 2 ∫ dµ1µ1 ∫ dφR(µ0 , µ1 , φ )
1

π

0

0

π
2 1
= ∫ dµ1µ1 ∫ dφ ( k L + kg fg (µ0 , µ1 , φ ) + kv fv (µ0 , µ1 , φ ))00π
= k L + kg Ag (µ0 ) + kv Av (µ0 )(3.6.120)

It should be restricted within [0–1]. Here, Ag and Av are the kernel functions of the geometrical
optics and volume scatterings, respectively:π2 1dµ1µ1 ∫ 0 dφ fg (µ0 , µ1 , φ )
∫
0
π
π
2 1
Av (µ0 ) = ∫ dµ1µ1 ∫ dφ fv (µ0 , µ1 , φ )00πAg (µ0 ) =(3.6.121)

As shown in Fig. 3.6.7, these functions sometimes take negative values. These functions areprepared as an LUT with respect to µ0. Unlike the RPV model, the LSRT model does not require
an LUT for each pixel.

Kernel Black-Sky Integral

10Average Albedo
A_geo
A_vol

-1

-2

0.0

0.2

0.4

0.6

0.8

1.0

cos(q0)Fig. 3.6.7 Kernel function of the albedo (same as Fig. 4 in Lucht et al. 2000), withaverage albedo for (kL, kg, kv) = (0.3, 0.07, 0.03).

102

d. Determination of reflected direction for LSRT model
Here, we determine µ1 and φ simultaneously. The following procedure is the same as in the RPVmodel. As much as possible, the comparison function should be selected to have a shape similar
to the original BR-PDF.
Determination of comparison function
The BR-PDF can be approximated by the linear function likeP(µ0 , µ1 , φ ) ≡ R(µ0 , µ1 , φ )µ1 α (µ0 )P(µ0 , µ1 , φ ) ≈ a1 (µ0 )µ1 + a0 (µ0 )(3.6.122a)(3.6.122b)

In the isotropic case, P = µ1/π, which is well expressed by (3.6.91), a1 and a0 are determined by
the normalization conditions:⎛ a1 (µ0 )⎞+ a0 (µ0 )⎟ = 1⎠
2∫ ∫ [a (µ )µ + a (µ )]dµ dφ = 2π ⎜⎝
2π

0

1

0

1

0

1

0

0

1

⎞
1⎛1
∴a0 (µ0 ) = ⎜ − a1 (µ0 )⎟⎠2 ⎝π(3.6.123a)(3.6.123b)

To obtain the coefficients of this function, we first solve the following equation:

P (µ0 , µ1 ) ≡ µ1 ∫ R(µ0 , µ1 , φ )dφπ0(k + k f (µ , µ , φ ) + k f (µ , µ , φ ))dφ
= µ [ k π + k ∫ f (µ , µ , φ )dφ + k ∫ f (µ , µ , φ )dφ ]= µ1 ∫

π

0

L

g g

0

1

v v

0

π

1

L

g

0

1

π

g

0

1

v

0

v

0

(3.6.124)1≈ a1 (µ0 )µ1 + a0 (µ0 )where these functions are prepared as an LUT:

fg (µ0 , µ1 ) = ∫ fg (µ0 , µ1 , φ )dφπ

0(3.6.125)

fv (µ0 , µ1 ) = ∫ fv (µ0 , µ1 , φ )dφπ0

The regression function is written asµ1 [ k L π + kg fg (µ0 , µ1 ) + kv fv (µ0 , µ1 )] ≈ a1 (µ0 )µ1 + a0 (µ0 )From this equation, we obtain the ratio of

103(3.6.126)a0 (µ0 ) a0 (µ0 )=a1 (µ0 ) a1 (µ0 )(3.6.127)

From normalization conditions (3.6.123) and (3.6.127), we obtain

a1 (µ0 ) =a1 (µ0 )π (a1 (µ0 ) + 2a0 (µ0 ))(3.6.128)

Here, when (3.6.122) is used as a comparison function, µ1 should be [0–1] and positive. In thiscase, the albedo should be within (based on the authors’ experience, it would be better if it was
narrower than this range)−

11< a1 (µ0 ) <π
π(3.6.129)

On the other hand, a0 is calculated by (3.6.123b). To meet the criteria that the comparison
function is larger than the original BR-PDF in all points, we multiply the following equation by(3.6.122):⎧µ1
R(µ0 , µ1 , φ ) ⎫b(µ0 ) = max⎨⋅⎬
α (µ 0 ) ⎭⎩ [a1 (µ0 )µ1 + a0 (µ0 )](3.6.130)

The function max{} is applied to all µ1, φ in (3.6.130). b in the BRDF, which has a strong peak,
becomes large and many iterations are necessary in the rejection method. Instead of b, we useb′:⎧⎪b(µ0 )if b(µ0 ) < bminb′(µ0 ) = ⎨⎪⎩bmin −1+ [b(µ0 ) − bmin +1] S if b(µ0 ) ≥ bmin

(3.6.131)

where S is a tuning parameter. When the BR-PDF is far from a linear regression, we use the
BR-PDF, which has a similar shape to the original BR-PDF. bmin = 4, S = 0.5 are better values inmost cases. In the case of the LSRT model, there is usually no prominent peak and b = 1–5.
Consequently, the comparison function becomesh(µ0 , µ1 , φ ) = b′(µ0 )[a1 (µ0 )µ1 + a0 (µ0 )](3.6.132)

When k* varies by pixel, the coefficient should be determined pixel by pixel. The parameters
a1 and b′ are prepared by a 3-D LUT (x, y, µ0). When the reflectance event occurs, a1 and b′ aredetermined by interpolating from the LUT with respect to µ0. Since the comparison function isusually estimated roughly, it is not necessary to derive these coefficients accurately.

104

0.4

5
a1
b'

0.3

4
0.2

a1

3

b'

0.1

0.0
2
-0.1

-0.2

1
0.0

0.2

0.4

0.6

0.8

1.0

cos!!

Fig. 3.6.8 Incident angle dependency of a1, b′. This is an example where (kL, kg, kv)= (0.3, 0.07, 0.03).

Use of rejection method
In the following description, we ignore the argument µ0. The comparison function in Eq.
(3.6.132) can be written byh(µ1 , φ ) = b′(a1µ1 + a0 )(3.6.133)

Since a1µ1 + a0 is normalized as shown in (3.6.123), µ1 that follows this equation can be
determined by solving:

2 π ∫ 0 (a1µ1 + a0 )dµ1 = ρ µ1
µ1

(3.6.134)

Therefore, this is a problem to solve a quadratic equation:

ρ µ1
=0
π

a1µ12 + 2a0µ1 −(3.6.135)

The solution of this equation within the range of µ1 = 0–1 is1) If a1 ≠ 0µ1 =−a0 + a02 + a1 ρ µ1 π2) Else if a1 = 0

a1µ1 = ρ µ1(3.6.136a)(3.6.136b)

On the other hand, the azimuth angle can be easily determined by

φ = 2 πρ φ(3.6.137)

105

The probability that the above direction is followed by the BR-PDF in the LSRT model isgiven by

η=P(µ1 , φ )µ1
R(µ 0 , µ1 , φ )=⋅h(µ1 , φ ) b′(a1µ1 + a0 )α (µ0 )

(3.6.138)

Therefore, we use the additional random number and apply the following criteria:(3.6.139)

If rejected, the new direction is selected again by (3.6.136–137). The reflected direction
followed by the LSRT model can be determined by the above prescribed method.

e. Simulation example
The following is the simulation procedure:
1) Preparation: Development of LUT-Development of LUT with respect to the albedo kernel functions Ag, Av-Development of the coefficients a1, b′ for comparison function-Calculation of average surface emission if thermal emission exists
2) MC simulation: Reflectance event-Determination of Ag, Av and a1, b′ by interpolation of LUT-Calculation of albedo-Scaling of photon weight (albedo should be less than 1)-LEM: calculation of R/α-Determination of reflected direction3) MC simulation: Thermal emission events-LEM: Emissivity is determined by interpolating the LUT.
-Determination of reflected direction
Several test simulations were performed. The prescribed method had enough speed (using aPentium IV 3.6 GHz PC, we could achieve a reflectance simulation with over 6,000,000 events).
Figure 3.6.9 shows the validation results for the exact BRF and BRF calculated by the MC
method. The results had fairly good accordance.

106Fig. 3.6.9 Angular dependency of the BRF. The upper two examples are for a solar
zenith angle of 45° and the bottom two examples are for a solar zenith angle of 60°.
Left: exact simulation; right: MC simulation.

1073.7 Refraction
The refraction occurs at a boundary surface where different compositions and/or different
densities meet. The same phenomenon appears when light is transported in a medium in which
the refractive index changes continuously. Refraction is important for describing the reflection
at the ocean surface and light path bending in the atmosphere. For example, a mirage is theresult of refraction. These phenomena are basically explained by Snell’s law and the Fresnel
equations. In this section, algorithms for simulating non-polarized intensity with a MC modelare described.

3.7.1 Snell’s law and Fresnel equations
Figure 3.7.1 illustrates the transmission and reflection at a boundary surface. The incident vector
is Ω 0 with a zenith angle of θ0 and cosθ0 = µ0. The reflection and transmission vectors arerespectively expressed by Ω R and Ω T. Snell’s law is represented asm=n1 sin θ 0=
n0 sin θT(3.7.1)

where m is the ratio of refractive indexes. For the reflection and transmission vectors,

ΩR = 2 cosθ 0 N + Ω0 ,n
ΩT = 0 (Ω0 + cos θ 0 N) − cos θT N
nT

(3.7.2a)(3.7.2b)

All of the energy is reflected if the light is incident from a low density medium to a high density
medium, with no transmission, under the condition

Fig. 3.7.1 Definition of vectors and angles

108sin θ 0 > m(3.7.3)

The Fresnel reflectance RF for non-polarized incident light is given as

RF =

1 2 2
(r|| + r⊥ )2

(3.7.4)

The Fresnel coefficients modified to account for absorption are given as follows (Liou, 1992):[(n − k )cosθ −U ] + [2nk cosθ −V ]r =[(n − k )cosθ +U ] + [2nk cosθ +V ]
2

2
||

2

2

0

2

2

0

2

2

0

2

(3.7.5a)0cos θ 0 −U ) +V 2
(
r =
,
2
(cosθ 0 +U ) +V 2
2

2⊥

(3.7.5b)where n and k are respectively the real and imaginary parts of the refractive index:

m = n + ik
1U2 =G 2 + 4n 2 k 2 + G
21
V2 =G 2 + 4n 2 k 2 − G
2
G = n 2 − k 2 − sin 2 θ 0(((3.7.6)))(3.7.7a)(3.7.7b)(3.7.8c)

In a case with no absorption, (3.7.5) is simplified as

r|| =
r⊥ =

n 2 cosθ 0 − n 2 − sin 2 θ 0
n 2 cosθ 0 + n 2 − sin 2 θ 0,(3.7.9a)n 2 − sin 2 θ 0 − cosθ 0(3.7.9b)n 2 − sin 2 θ 0 + cosθ 0

2
2
2
r|| cosθ 0 n − sin θ 0 − sin θ 0=r⊥ cosθ 0 n 2 − sin 2 θ 0 + sin 2 θ 0(3.7.9c)

3.7.2 Treatment of refraction in atmosphere
The absorption can be neglected in the earth’s atmosphere (k = 0). The refractive index anomaly
from unity is proportional to the air density. The refractive index n is nearly 1, and n – 1 is in theorder of 0.0001, for the near-surface atmosphere. Continuous change in the refractive index canbe treated, but in this subsection, the atmosphere is divided into homogeneous layers, and therefraction is considered only at the layer boundaries. In the following, the previous equations
are modified so as to be suitable for the MC model. Similar equations are also described in
Marchuk et al. (1980).

109

From (3.7.2a), the reflection vector is given as⎛u ⎞ ⎛ u ⎞⎜ xR ⎟ ⎜ x0 ⎟⎜u yR ⎟ = ⎜ u y0 ⎟⎜ ⎟ ⎜⎟⎝u zR ⎠ ⎝−u z0 ⎠(3.7.10)

With the ratio of the refractive index

n=

n1
n0

(3.7.11)the transmission vector is given from (3.7.2b) as

⎛
⎞
⎛u ⎞
u x0
xT
⎜
⎟
⎜ ⎟ 1
⎜
⎟
u
=
u
⎜ yT ⎟y0n⎜⎟⎜ ⎟⎜ u 2 + n 2 −1 ⎟⎝u zT ⎠
⎝ z0⎠

(3.7.12)

If from (3.7.3),
2
u z0+ n 2 −1 < 0(3.7.13)then transmission is absent, with a reflectance of 1. If the light is incident to a dense layer froma thin layer (n > 1), the reflectance is derived from (3.7.9) and (3.7.12):⎛ u − B ⎞2 u 2 B 2 + A 2
z0
RF = ⎜ z 0⎟2⎝ u z 0 + B ⎠ (u z 0 B + A )(3.7.14)2
2
A = 1− u z0
= u 2x0 + u y0

(3.7.15)2
B = u z0
+ n 2 −1

In contrast, if the light is incident to a thin layer from a dense layer (n < 1),RF′ =n 2 −1+ RF
n2(3.7.16)

With attention to the fact that the refractive index is nearly unity, the anomaly from 1 ismore useful on the numerical side:n = 1+ Δn(3.7.17)

Then, the following are useful:1101≅ 1− Δn(1− Δn)n1≅ 1− Δn(2 − 3Δn)n2n 2 −1 = Δn(2 + Δn)These can be used in the calculations of (3.7.12–16).
In the MC model, the reflectance is first evaluated by (3.7.14) and (3.7.16). However, thereflectance is usually very small in the earth’s atmosphere. Then, a random number determines
whether the light transmits or reflects. The direction after the transmission or reflection is givenby (3.7.10) or (3.7.12).

3.7.3 Refractive index in earth’s atmosphere
The refractive index of the atmosphere is generally a function of atmospheric composition andnumber density of atmospheric molecules:

Δn ≡ n −1= (ns −1)NP Ts
= (ns −1)
Ns
Ps T

(3.7.18)

where N, P, and T are respectively the number density, pressure, and temperature of theatmosphere. The subscript s denotes the standard status of the atmosphere. Let us consider a dry
atmosphere. The number density is represented asNs =N A Ps⋅R* Ts(3.7.19)

where NA = 6.0221367 × 1023 mol–1 is Avogadro’s number and R* is the universal gas constant.
Empirical equations of the refractive index, based on experiments, are available for dry
standard atmosphere at wavelength λ (µm):
Edlén (1953):
294981025540( ns − 1) × 10 8 = 6432.8 + 146 − λ −2 + 41 − λ −2

(3.7.20)

Ps = 1013.25 hPa, Ts = 288.15 K, 300 ppm CO 2

Peck and Reeder (1972):
248099017455.7( ns − 1) × 10 8 = 8060.51 + 132.274 − λ −2 + 39.32957 − λ −2

(3.7.21)

Ps = 1013.25 hPa, Ts = 288.15 K, 300 (330?) ppm CO2

The correction formula for CO2 concentration (parts per volume, e.g., 360 × 10–6 for 360 ppm)
is given as (Edlén (1953))

111

(ns −1)CO= 1+ 0.54(CO2 − 0.0003)(ns −1) 300
2

Dry atmosphere is usually considered, neglecting the effect of water vapor.

112(3.7.22)

3.8 Calculation of irradiance and heating rate
The forward-propagating MC model can estimate subarea-averaged irradiances andvoxel-averaged heating rates at all regions in the domain, from a single simulation.

3.8.1 Definition of radiative quantities and sampling of quantities by MC method
Irradiance
The irradiance F (W m–2) at a location r on a plane with normal direction Ω c is the integral of
the radiance I (W m–2 sr–1) perpendicular component on the plane over the hemisphere:

F (Ωc , r ) = ∫ 2 π I (Ω, r ) Ω ⋅ Ωc dΩ

(3.8.1)

A special case of this is the irradiance on a horizontal plane:

F ± (r) = ∫ 2 π I ± (Ω, r) cos θ dΩ

(3.8.2)where θ is the zenith angle, and the plus/minus superscript denotes upward/downward.
Spheradiance
The spheradiance (W m–2) at a location r is defined as the integral of the radiance I (W m–2 sr–1)
over the sphere. It is the same as the so-called actinic flux or mean radiance. The spheradianceis more effective for the evaluation of the photosynthesis for vegetation and the ultraviolet
radiation on a creature’s body surface. The spheradiance is defined as

S (r) = ∫ 4 π I (Ω, r)dΩ

(3.8.3)A difference from (3.8.1) is that the spheradiance is not the integral of the perpendicularcomponent of radiance, but simply the integral of the radiance.
Radiative heating rate
The radiative heating rate H(r) (W m–3) is the convergence of the radiative energy, with anegative value for cooling.
Area-averaged irradiance and volume-averaged radiative heating rate
The area-averaged irradiance over an area RA with a finite section of A and thevolume-averaged heating rate over a region RV with a volume V are given, respectively, as

113

1F ± (r )dA∫
A
1
H (RV ) = ∫ H (r)dVV

F ± (R A ) =(3.8.4)(3.8.5)In the following, methods for calculating these quantities with the MC model are described.
Sampling of radiative quantities
When calculating the quantities of (3.8.4–5) by the MC model, estimates are integrated if thephoton location is included in the region RA or RV. The irradiance and heating rate are calculated
by integrating the contribution functions ψ and η:
α ( p)E1 N
F (R A ) = ∑ ∑ ψ p,s (Ω, r )A p=0 s=0±

(3.8.6)β ( p)E N
H (RV ) = 1 ∑ ∑η p,s (Ω, r ) ,V p=0 s=0

(3.8.7)

where N is the total number of model photons, α and β are the maximum number of samples,and E1 is the radiative energy (W) transported by a single model photon:

E1 =

Etot
N

(3.8.8)where Etot is the total energy emitted by all radiation sources. Several methods exist, withdifferent points of view on the conditions and target radiative quantities.
In the following, we limit our discussion to the irradiance on a horizontal plane and heatingrate. The irradiance on a non-horizontal plane and spheradiance can be calculated in a similarway.

3.8.2 Method I: Random sampling of transmittance
The optical thickness to a collision point is determined by a randomly determined transmittance.
If the collision is treated, including absorption and scattering, then the MC model interprets thetransmittance as 1 up to the collision point, and the transmittance changes suddenly to 0 fromthe point. In other words, the analytical transmittance

T (r0 , r1 ) = e −τ (r0 ,r1 )(3.8.10)is modified to⎪⎧1
T (r0 , r1 ) = ⎨⎪⎩0

for τ (r0 , r1 ) ≤ −ln ρfor τ (r0 , r1 ) > −ln ρ

114(3.8.11)

The absorption (heating) probability at the collision point is given as 1 – ω, where ω is the
single scattering albedo.
When calculating the irradiance and heating rate, one choice is to use the method based on
the randomly determined transmittance, as above. As for irradiance, if the model photon isincident at a sampling region RA, the model samples the following quantity:

ψ=w(3.8.12)

where the photon weight is taken at the initial point of the path. For the heating rate, if thecollision point is included in a region RV, the model samples

η = w(1− ω ) .

(3.8.13)

However, one might try to calculate heating rates for each component of mixed media (gases,
aerosols, and clouds). In this case, the weight of (3.8.13) is subdivided into components inproportion to their respective absorption coefficients, or a randomly chosen component can beused by using a random number.
This method is simple to implement. Its deficiency is that the heating is only at collision
points, and thus, the sampling frequency can be too low at optically thin regions. In such a case,the collision forcing method in 4.2 is useful.

Fig. 3.8.1 Schematic of sampling method I for irradiance and heating rate.

3.8.3 Method II: Analytical sampling of transmittanceOne can use just the analytical transmittance in (3.8.10) for sampling radiative quantities. Thismethod traces a virtual trajectory from the initial point r0 for an emission/scattering/reflection
event to the upper/lower boundary surface of the domain, sampling radiative quantities in thecourse of the ray path. The transmittance is the function of the optical thickness integrated fromthe initial point r0, and the contribution function of irradiance is given asψ (r) = wT (r) = w e −τ (r)(3.8.14)115

If the ray penetrates a sampling volume for the heating rate, from a location r to r , thecontribution function for heating rate is

η(r → r′) = w[T (r) − T (r′)][1− ω (r˜ )]= w e −τ (r) C (τ (r′) − τ (r))[1− ω (r˜ )](3.8.15)

where r˜ is the point of sampling the heating, and C(x) is the probability of the collision within apath segment to a point with optical thickness x:

C(x) = 1− e − x(3.8.16)

The sampling point r˜ is determined randomly:

C (τ (r˜ ) − τ (r)) = ρC (τ (r′) − τ (r))(3.8.17)

It should be noted that there is no need to determine this point if the single scattering albedo isconstant within the path segment between r and r .
The physical trajectory of the photon packet terminates at the randomly chosen collision
point, whereas the model should trace a virtual trajectory extended to the domain boundary (Fig.
3.8.2). Because with this method the model can trace the virtual trajectory up to the domainboundary, this method is useful for the frequent sampling of radiative quantities at points withvery large optical thicknesses compared to the emission point.
High accuracy can be expected for this method due to the analytical calculation of thesampled energy. On the other hand, this method requires time-consuming ray tracing and theevaluation of a numerically intensive function, the exponential function, many times. When
comparing its numerical efficiency with method I, which is better one depends on the problem,
extinction coefficient, inhomogeneity, geometrical complexity, and radiative quantity of theinterest. Method I samples energy only at the collision points, whereas method II samples many
points within the path. Method II is efficient for calculating heating rates in optically thin media.

Fig. 3.8.2 Schematic of sampling method II for irradiance and radiative heating
rate.

116

Method II requires the calculation of the geometrical and optical path lengths every time thephoton packet penetrates a voxel. Therefore, the MCS method (described in 4.1) cannot be usedwith method II. This is the reason that the method is inefficient in a case where the domain isdivided into many small volume elements.

3.8.4 Method III: Hybrid method
Another possible choice is a hybrid of methods I and II. The transmittance is treated analyticallyup to a threshold Tmin and by the MC method subsequently for larger optical thicknesses:⎧e −τ (r)⎪T (r) = ⎨e −τ max⎪0⎩for τ (r) ≤ τ max
for τ max < τ (r) ≤ τ max + τ freefor τ max + τ free < τ (r)(3.8.18)

where

Tmin = e −τ max
τ free = −ln ρ(3.8.19a)(3.8.19b)

Figure 3.8.3 shows the transmittance schematically. The model should always trace the photon
packet up to the point with an optical thickness of τmax. However, it is not necessary to trace thephoton packet up to the domain boundary.
Sampled contributions of irradiance and heating rate are respectively

ψ (r) = wT (r)η(r → r′) = w[T (r) − T (r′)][1− ω (r˜ )](3.8.20)(3.8.21)

The sampling point for the heating could be determined in a way similar to method II:

Fig. 3.8.3 Transmittance in method III

117

T (τ (r) ) − T (τ ( r˜ ) ) = ρ[T (τ (r) ) − T (τ ( r′) )](3.8.22)

There is no requirement to determine the point r˜ if the single scattering albedo is constant
between r and r .
Different sampling methods could be used for respective orders of scattering. For example,
a possible method would be to use the hybrid method for a direct beam and first-order scattering
light and subsequently use method I for multiple scattering light.

3.8.5 Method IV: Use of equivalence theorem for absorption
As previously noted, it is possible to treat the scattering as a random process by the MC method
and to treat the absorption continuously by analytical expression. This is an application of theequivalence theorem. In this case, the collision point is solely determined by the distribution of
the scattering coefficient, and the photon weight is scaled by the transmittance due to thecontinuous absorption within the path the photon penetrated. The transmittance is solely afunction of the absorption optical thickness τa (see Fig. 3.8.4):⎧e −τ a (r)T (r) = ⎨⎩0for τ s (r) ≤ τ free.
for τ s (r) > τ free

(3.8.23)

The optical thicknesses for the absorption and scattering are respectivelyτ a (r) = ∫ βa (t)dt ; t = r′ − r0

(3.8.24a)τ s (r) = ∫ βs (t)dt ; t = r′ − r0

(3.8.24b)

r0 →r

r0 →r

The irradiance and heating rate are sampled at many points on the photon path for τ < τfree. Eachsampled contribution is given as follows:ψ (r) = wT (r)η(r → r′) = w[T (r) − T (r′)](3.8.25)(3.8.26)

This method has good efficiency for calculating the heating rate, compared with method I.
However, this method requires multiple calculations of the computationally intensive
exponential function, as in methods II and III, making the computation time longer than method
I. In addition, this method requires the calculation of the geometrical and optical path lengths
every time the photon packet penetrates a voxel, as in methods II and III. Therefore, the MCS
method (described in 4.1) cannot be used, which is a deficiency of method IV.
An advantage of this method is that simultaneous calculations for many different
wavelengths (with different absorption coefficients) are easy. If the scattering properties are thesame for different wavelengths, the photon path could be common for all wavelengths, with the
only difference being the transmittance of (3.8.23). The scattering properties of clouds, aerosols,

118and gases are usually functions that change slowly with wavelength and could be approximatedas constant within a narrow band. On the other hand, gaseous absorption coefficients can varysignificantly, even within a narrow band. If the photon weight is calculated according to (3.8.23),
simultaneous calculations for multiple wavelengths are possible and efficient. This method isefficient also when calculating band-averaged radiative quantities.

Fig. 3.8.4 Transmittance in method IV: Absorption is continuously treated, and thescattering event occurs at a point with τfree.

1193.9 LEM
The LEM (Marchuk et al., 1980; Evans and Marshak, 2005) calculates the analytical energy thatis scattered at the collision point and transmitted to a certain point of interest. The following
quantities could be calculated by the LEM, for example:-Area-averaged radiance in a specific direction on a plane with arbitrary orientation-Angular-averaged radiative quantities (radiance and irradiance) at a point

3.9.1 Formulas for local estimate
The LEM samples analytical contribution functions at each instance of source emission,scattering, or reflection, from the event point to the detector surface of the sensor of interest.
Area-averaged radiance on an arbitrary plane and angular-averaged radiance on a point, or
irradiance and spheradiance at a point, could be calculated by this method.
The method samples the contribution function of an event at a location r0:

ζ (Ω1 , r1 ) = wΨ (Ω0 ,Ω1 )T (r0 , r1 ) ,(3.9.1)

where w is the photon weight just after the emission/scattering/reflection event, Ψ is the PDF
for angular distribution, and T is the transmittance between the point r0 and the detector location
r1. The transmittance is a function of the optical thickness:

T (r0 , r1 ) = e −τ (r0 ,r1 ) ,τ (r0 , r1 ) =(3.9.2a)∫ β (t)dt; t = r′ − r .
e(3.9.2b)0

r0 →r1

Ray tracing is required to calculate the optical thickness in (3.9.2b), for the path between r0 and
r1. The PDF for the angular distribution is normalized as∫ Ψ (Ω ,Ω ) cosθ dΩ = 1 .
4π

0

1

(3.9.3)

1

The detection point r1 should be on the line oriented to the direction Ω 1 from the event point r0.
Thus,

Ω1 =

r1 − r0
r1 − r0(3.9.4)Generally, radiative quantities can be written in the form of the sum of the contributionfunctions for all events:120

N p N s ( p)Q = ∑ ∑ E1ψ p,s (Ω1 , r1 )Ξ p,s (Ω1 , r1 )(3.9.5a)p=0 s=0⎧1
Ξ p,s (Ω1 , r1 ) = ⎨⎩0

if Ω1 ∈ U and r1 ∈ Rotherwise(3.9.5b)where Np is the total number of model photons, Ns is the maximum order of scattering, U is thevolume of the direction vectors within the sensor field of view, and R is the volume of thelocation vectors for the incident points at the detector surface. E1 denotes the radiative energy(W) transported by a single model photon with a weight of 1:

E1 =

Etot
Np

(3.9.6)

where Etot is the total source energy.

Fig. 3.9.1 Schematic of sampling by local estimation.

Area-averaged radiance
The area-averaged radiance (W m–2 sr–1) for a direction Ω 1 in an arbitrary horizontal area R witha cross section A is defined as

I (Ω1 , R) =∫ I (Ω , r )dA = 1 I Ω , r dA
∫ ( )
A
dA
∫
R

1

1

R

1

1

(3.9.7)

R

Using the LEM, the function of (3.9.5) for this kind of averaged radiance is written asψ (Ω1 , r1 ) =ζ (Ω1 , r1 )A

(3.9.8)121

Local radiance averaged over solid angle
The local radiance (W m–2 sr–1) at a point r1 averaged over an angular region U with a solid
angle of ξ is defined as∫I˜ (U,r1 ) = U

I (Ω1 , r1 )dξ

∫ dξ

=

1
∫ I (Ω1, r1 )dξξ U(3.9.9)U

The small cross section dA at the point r0 is associated with a small solid angle dξ at anotherpoint r1 as follows:dξ = dA ⋅cosθ1r1 − r0(3.9.10)2

Using this, the function ψ for calculating the angular-averaged radiance by the LEM is derivedasψ (Ω1 , r1 ) =ζ (Ω1 , r1 ) cosθ12ξ
r1 − r0(3.9.11)MC integration of this type is difficult, because theoretically, the variance could become
infinite due to the dependence of ψ on the distance r1 − r0 . A large number of photons do not
help to get convergence. For example, a 100 times larger contribution is sampled if a scattering
event occurs at a point 10 times closer to the detector of interest. Variance reduction methods forthis kind of problem are discussed in Marchuk et al. (1980).
Local irradiance and spheradiance
The irradiance (W m–2) at a point r1 on a plane oriented in a direction Ω c integrated over anangular region U in the hemisphere is defined as

F (U,r1 ) = ∫ 2 π I (Ω1 , r1 ) Ω1 ⋅ Ωc dΩ1(3.9.12)The sampled contribution by the LEM is:ψ (Ω1 , r1 ) = ζ (Ω1 , r1 )cosθ1
r1 − r02

Ω1 ⋅ Ωc(3.9.13)

Radiative quantities integrated over an arbitrary angular region (not just for the hemisphere)
could be similarly derived.
A special case is the irradiance on a horizontal plane:122F (U,r1 ) = ∫ 2 π I (Ω1 , r1 ) cos θ1 dΩ1(3.9.14)

For this quantity, the contribution by the LEM becomesψ (Ω1 , r1 ) = ζ (Ω1 , r1 )

cosθ1

2

r1 − r02(3.9.15)The spheradiance (or actinic flux) (W m–2) at a point r1 is defined asF (r1 ) = ∫ 4 π I (Ω1 , r1 )dΩ1

(3.9.16)This quantity is defined as a simple integration of the radiance with an equal weight given to all
directions, being different from the projection to a specific plane as the irradiance in (3.9.12).
For example, the spheradiance is more important for the evaluation of plant photosynthesis or
the ultraviolet radiation absorbed by a living body. The contribution for this quantity when using
the LEM should beψ (Ω1 , r1 ) = ζ (Ω1 , r1 )cosθ1
r1 − r0(3.9.17)2

3.9.2 Definition of PDF for angular distribution
Scattering by particles in the atmosphereΨ (θ 0 , φ0 ,θ1 , φ1 ) =P (θ 0 , φ0 ,θ1 , φ1 )4 π cos θ1(3.9.18)

where P is the scattering phase function normalized as:

1∫ P(θ0 , φ0 ,θ1, φ1 )dΩ1 = 1
4π 4 π

(3.9.19)

For isotropic scattering, P = 1.
Reflection at surfaceΨ (θ 0 , φ0 ,θ1 , φ1 ) =R(θ 0 , φ0 ,θ1 , φ1 )α (θ 0 , φ 0 )(3.9.20)

where R is the BRDF, and α is the black-sky albedo. The BRDF is normalized as123∫ R(θ , φ ,θ , φ ) cosθ dΩ = α (θ , φ )
2π

0

0

1

1

1

0

0

(3.9.21)

For Lambertian reflection, R = α/π.
Emission of thermal radiation in atmosphereΨ (θ 0 , φ0 ,θ1 , φ1 ) =1
4 π cosθ1(3.9.22)

This is the same as the case for isotropic scattering.
Emission of thermal radiation from surfaceΨ (θ 0 , φ0 ,θ1 , φ1 ) =ε (θ 0 ),πε(3.9.23)

where ε is the emissivity, which is represented by the reflectance (albedo) α:ε (µ0 ) = 1− α (µ0 ) .

(3.9.24)

The mean reflectance and mean emittance are respectively

α = 2 ∫ 0 α (µ0 )µ0 dµ0
ε = 1− α1(3.9.25a)(3.9.25b)

For example, Ψ = 1/π for a Lambertian surface.
Isotropic source emission in conical angular region (as in solar incidence from above)⎧ 1⎪
Ψ (θ 0 , φ0 ,θ1 , φ1 ) = ⎨ υ (Θ, Δ)⎪0⎩

if Ω1 ⋅ Ω0 > cos Δ(3.9.26)

if Ω1 ⋅ Ω0 ≤ cos Δ

where Δ is the half cone angle for the angular region, and Θ is the zenith angle for the center
direction:υ (Θ, Δ) = ∫ 0

2π

∫

1

cosθ d cos θˆdφˆcosθ = sin Θsin θˆ cos φˆ − cosΘcosθˆ
cosΔ

If the angular region does not include the horizontal directions (cosΔ > sinΘ),υ (Θ, Δ) = π cosΘ sin 2 Δ .
124(3.9.27a)(3.9.27b)

In a special case where cosΘ = 1 and Δ = π/2,⎧1⎪
Ψ (θ 0 , φ0 ,θ1 , φ1 ) = ⎨ π⎪⎩0if Ω1 ⋅ Ω0 > 0(3.9.28)

if Ω1 ⋅ Ω0 ≤ 0which is the same as for a Lambertian surface. In general cases, the calculation method for thefunction in (3.9.27a) is described in 3.2.

125

3.10 Treatment of atmospheric sphericity
The radiative transfer in a planetary atmosphere should generally be treated as a spherical shellon a sphere or spheroid surface. Usually, the atmosphere could be approximated asplane-parallel. An exception is the case where the light travels in near-horizontal directions. For
example, the sphericity is very important when simulating light intensity at sunrise or sunset
with a low sun altitude, or when the instrument views a near-horizontal direction. The raybending due to refraction in the atmosphere is also important in these cases.
This section gives a correction of the atmospheric sphericity for calculating the radiances
using the LEM. In particular, a correction is discussed for the line of sight (from the scattering
event point to the detector surface).

3.10.1 Characteristics of sphericity effect
When calculating the radiance by the LEM in 3.9, the following contribution function is
sampled:ζ (Ω1 , r1 ) = wΨ (Ω0 ,Ω1 )T (r0 , r1 )(3.10.1)

where w is the photon weight just after the scattering (or reflection) event, Ψ is the normalized
PDF (sr–1) for the angular distribution, and T is the transmittance between the scattering point r0
and the detection point r1. The transmittance is a function of the optical thickness:

T (r0 , r1 ) = e −τ (r0 ,r1 ) .
τ (r0 , r1 ) =(3.10.2a)∫ β (t)dt; t = r′ − re(3.10.2b)0

r0 →r1

The function Ψ is given, for the scattering by a particle in the atmosphere, asΨ (Ω0 ,Ω1 ) =P(Ω0 ,Ω1 )4 π cos θ1(3.10.3)

where P is the scattering phase function. For surface reflection,

Ψ (Ω0 ,Ω1 ) =R(Ω0 ,Ω1 )α (Ω0 )

(3.10.4)

where R is the BRDF, and α is the black-sky albedo.
If the atmosphere is plane-parallel and the refraction is neglected, then the line of sight is astraight line. The scattering direction is equal to the direction at the detection point (left side ofFig. 3.10.1). For a spherical atmosphere with refraction, the light bends in the line of sight, andthe direction at the detection point differs from the scattering direction (right side of Fig. 3.10.1).
126

The function Ψ of (3.10.1) should be different between plane-parallel and spherical atmospheres.
In addition, the transmittance is also different because optical and geometrical path lengths aredifferent between plane-parallel and spherical atmospheres.

Fig. 3.10.1 Schematic of radiance sampling. Left: plane-parallel atmosphere, right:
spherical-shell atmosphere.

3.10.2 Correction of sphericity effect for line of sightLet us derive a path length a that corresponds to the height difference Δ, in the sphericalatmosphere, and the angle A between the path direction and the local normal vector (from thezero point at the center of the planet). The beginning and end points are respectively C and B,
and the corresponding distances from the planet center are b and c (see Fig. 3.10.2 fordefinitions). By representing the planet radius with Rp,

c = Rp + c′b = Rp + b′

(3.10.5a)(3.10.5b)

The height difference is given as

Δ = c − b = c′ − b′(3.10.6)

The light path has a cross point on the line of sight under the following condition:−Δ < b(1− sin C) = b

cos2 Ccos2 θ1= ( Rp + b′)
1+ sin C
1+ sin θ1(3.10.7)

If this condition is false, no light trajectory is possible to travel from point C to point B (in thespherical atmosphere). This could be possible if the ray is downward with a negative Δ.
From the law of cosines, the path length a is given by

a 2 = b 2 + c 2 − 2bc cos A(3.10.8)= 2(1− cos A)c(c − Δ) + Δ2

127

Fig. 3.10.2 Geometry in the spherical-shell atmosphere. Left: upward ray, rightdownward ray.

The law of sines determines the angle A, using B = π – A – C

b sinC = csin B= csin(A + C)bsin(A + C) − sin C = 0csin(A + C) − (1− δ ) sin C = 0(3.10.9a)(3.10.9b)(3.10.9c)

This could be solved for A. In the above,

δ=

Δ
Δ
=
c Rp + c′(3.10.10)

Because the planet is large compared with the atmospheric depth, δ is usually very small. We
should exercise special caution on this point because of the numerical problem of calculating avery small A:

A = sin −1 ((1− δ ) sin C ) − C(3.10.11)

It is problematic to compute this formula with a computer because it rounds off. In practice,accurate calculation is possible using the following method.

a. Calculation of A by numerical method
Let us derive first the termsin −1 ((1− δ ) sin C )(3.10.12)

in (3.10.11). This is the solution of128

f (x) = sin x − (1− δ ) sin C = 0(3.10.13)

The derivative isf ′(x) = cos x(3.10.14)

Using the Newtonian method, a sequence of numbers could be derived:

xi+1 = xi − Δxi
f (xi )Δxi =f ′(xi )(3.10.15a)(3.10.15b)

With an initial estimate

x0 = C(3.10.16)

the solution of (3.10.9) is∞

x = C − ∑ Δxi(3.10.17)

i=0From (3.10.15b) and (3.10.16),

f (C)
sin C
=δ
f ′(C)
cosC

Δx0 =(3.10.18)

Finally, the angle A of (3.10.11) can be computed by∞

A = −∑ Δxi = −δi=0

sin C ∞− ∑ Δxi
cosC i=1(3.10.19)

The first-order approximation is as follows:A ≈ −δ

sinC
≥0
cosC

(3.10.20)

The calculation algorithm for (3.10.19) is as follows:
1) Compute the initial estimate A0:

A0 = −δsinC
cosC(3.10.21)

2) Compute the following:1
1 5
p = sinAi ≈ Ai − Ai3 +
Ai −
6
120
1
1 4
q = cos Ai ≈ 1− Ai2 +
Ai −
2
24
129

(3.10.22a)(3.10.22b)

r = 1− cos Ai =sin 2 Aip2=
1+ cos Ai 1+ q

(3.10.23c)

3) Compute Δxi:

f (C + Ai ) = sin(C + Ai ) − (1− δ ) sin C= p cosC − (r − δ )sinC

f ′(C + Ai ) = q cosC − p sinC
∴Δxi =(3.10.24a)(3.10.24b)f (C + Ai ) ( p cosC − r sin C ) + δ sin C=
f ′(C + Ai )q cosC − p sin C4) Ai+1 = Ai − Δxi(3.10.24c)(3.10.25)

5) If Δxi is small enough compared with Ai, then finish, else return to (2).
This iteration usually converges within two cycles.

b. Correction of geometrical and optical path lengths
The path length is derived substituting (3.10.22c) into (3.10.8)

a = 2rc(c − Δ) + Δ2

(3.10.26)

This equation gives a better result than using the law of sines. The corresponding path length in
the plane-parallel geometry is

Δ
.
−cosC

(3.10.27)

Therefore, by the spherical effect, the path length is factored by−a cosC
Δ(3.10.28)

c. Correction of local zenith angle
The local zenith angle for the line of sight rotates by the angle A up to the end of a path oflength a. Only the zenith angle changes, with the azimuth unaltered. With the zenith angle θ forthe ray,
Upward tracing:cosθ > 0
Downward tracing: cosθ < 0
In both cases, the local zenith angle at the terminal point is determined using (3.10.22a,b):

θ′ = θ − A(3.10.29a)130sin θ ′ = sin θ cos A − cosθ sin A = q sin θ − p cosθ
cosθ ′ = cos θ cos A + sin θ sin A = q cosθ + p sin θ(3.10.29b)(3.10.29c)

The angle A is always larger than 0, and the zenith angle should decrease, not depending on the
upward or downward direction.

d. Implementation algorithm
In practice, the refraction in the multilayered atmosphere should be included. For simplicity, therefraction occurs only at the layer boundaries. The sphericity correction, modification of thezenith angle by (3.10.29), is applied every time the photon packet penetrates a layer in the lineof sight. The direction is modified at layer boundaries due to refraction. The following is apossible algorithm:
1) First, one should know the direction Ω 1 at the last scattering point, when the directionof the radiance of interest is Ω 1. For this, the ray is traced in the backward trajectoryfrom the detector to the altitude of the scattering event at r0, and the direction Ω 1 isobtained. The refraction and sphericity should be included when tracing the backward
trajectory.
2) Determine the PDF Ψ(Ω 0, Ω 1) for the angular distribution at the scattering point r0.
3) Start the ray tracing from the scattering point to the detection point r1. The optical
thickness and geometrical path lengths at each layer are computed. The refraction andsphericity should be included in this step.
To accelerate the calculation of (1), an LUT would be useful. The LUT should tabulate thedirections at the initial points at various altitudes.
It should be noted that a light path may not always reach the destination altitude from anarbitrary altitude, if the refraction and sphericity are included.
Figure 3.10.3 shows the effects of refraction and sphericity for the radiance at the surface
and normalized average path length (air mass factor, AMF).

131Normalized Radiance, pure Rayleigh, 577 nm, SZA: 20˚
0.10

Normalized Radiance

0.08

0.06

0.04

0.02

0.00
1

2

3

4

5

6

7 8 9

2

10

3

4

5

6

7 8 9

Elevation Angle [deg.]
AMF Exercise 1, pure Rayleigh, 577 nm, SZA: 20˚ , elevation: 1˚

AMF Exercise 1, pure Rayleigh, 577 nm, SZA: 20˚ , elevation: 3˚

60

20
P-Parallel, Refraction: no
P-Parallel, Refraction: yes
Spherical, Refraction: no
Spherical, Refraction: yes

50

15

AMF

AMF

40
30

10

20
5
10
0

0
1

10

2

10

3

10

Altitude [km]

4

10

5

10

1

10

2

10

3

10

Altitude [km]410Fig. 3.10.3 Effects of refraction and sphericity for the radiance at the surface andnormalized average path length (air mass factor, AMF).

132

5

10

3.11 Spectral integration
Here, we describe an integration method for the physical quantity over a broad wavelengthdomain. In the MC method, it is possible to achieve the same computation time as for a
monochromatic wavelength. This is one of the advantages of the MC method.
In this section, two different meanings of “photon” are mixed: a light photon (or lightquantum) and the modeled photon as a sampling unit in the MC method. We should clearly
distinguish these two.

3.11.1 Extraterrestrial solar irradiance and photon flux density
Figure 3.11.1(a) shows a spectral solar irradiance at the top of the atmosphere Fλ (W m–2 µm–1).
The incoming solar irradiance with the solar zenith angle θs is Fλcos(θs). Figure 3.11.1(b) showsa spectral solar photon flux density Qλ (µmol m–2 s–1 µm–1). The photon flux density Qλ can bederived from the irradiance Fλ . The energy of the single light photon Ep is expressed by

E p = ch λ(3.11.1)

where c (= 2.997 × 108 m s–1) is the velocity of light and h (= 6.626 × 10–34) is Planck’s constant.
The irradiance Fλ can be converted to photon flux density N:

N = Fλ E(3.11.2)p

Hence, when (3.11.2) is rewritten using the mol unit (µmol m–2 s–1 µm–1), Qλ becomesQλ = N N =
A

1
λFλ
chN A

(3.11.3)

where NA is Avogadro’s number. The energy of a single light photon is inversely proportional to
the wavelength. Therefore, the energy of a single photon becomes smaller at longer wavelengths.
In other words, the number of photons increases with longer wavelengths at the same energy
level. When we compare the extraterrestrial spectral solar irradiance and photon flux density(Fig. 3.11.1 (c)), the photon flux density peak is located at a longer wavelength than the solar
irradiance.

133

Fig. 3.11.1 (a) Extraterrestrial spectral solar irradiance, (b) extraterrestrial spectral
solar photon flux density, and (c) comparison between irradiance and photon flux
(The values are normalized).

3.11.2 Calculation of spectral integration
In MC calculation, we divide the wavelength domain into subdomains, in which the opticalproperties in the atmosphere and plant canopy are assumed to be constant. The spectral
integration of the radiative quantity is derived by counting the sampling quantities in the
subdomains. In an actual simulation, the spectral integration is achieved by adding the spectral
integration loop as an outer loop (Fig. 3.11.2).

a. Approach to use same number of modeled photons in all spectral subdomains
Let us consider dividing the wavelength domain [λmin, λmax] into M subdomains. Then, spectral
integration is performed as a sum of the M subdomains. When the total number of modeled
photons is N and Ni (= N/M) modeled photons are used in each subdomain, the intensity (energy
or photon flux density) of a single modeled photon is expressed by

M
M
Fi or
Qi
N
N

(3.11.4)

134

Fig. 3.11.2 Flowchart of the spectral integration.

where Fi and Qi are the spectrally integrated incident irradiance and photon flux density within
the ith subdomain. F and Q can be calculated byM

F = ∑ Fi , Fi = ∫
i=1
M

λi
λi−1

Q = ∑ Qi , Qi = ∫
i=1

Fλ dλ

(3.11.5a)Qλ dλ(3.11.5b)λi
λi−1This is a simple, stable, and accurate method to calculate the average radiative quantity within
subdomains. However, it is necessary to allocate the same number of modeled photons as themost contributing spectral subdomain over all the subdomains. Therefore, this method requires avast computation time compared with a monochromatic wavelength simulation. The method
described in the next section has a better performance.

b. Approach using photon population proportional to incident solar radiation intensity
The accuracy of radiative quantities sampled by the MC method depends on the sampling
number. It is possible to simulate a physical quantity with the same accuracy as the
monochromatic wavelength simulation using the same number of modeled photons when themodeled photons are allocated to be proportional to the spectral solar radiation intensity. And, as
shown in Fig. 3.11.1, the spectral distributions of irradiance and photon flux density are similar.
Thus, the irradiance and photon flux density can be calculated simultaneously. When we use a
total of N modeled photons for the spectral integration in the domain [λmin, λmax], the weight of
the ith subdomain with respect to the incident spectral radiation intensity is

135

M

w i = Fi F , ∑ wi = 1(3.11.6)

i=1

where F and Fi are derived from (3.11.5a). The number of modeled photons Ni in the ith
subdomain is

N i = wiN(3.11.7)

In this case, the intensity of the ith subdomain is expressed in the irradiance unit:

ΔFi =

Fi F
=
Ni N

(3.11.8)

This is constant over all subdomains, when we use the photon flux density:

ΔQi =

Qi
Ni

(3.11.9)In the wavelength, where the thermal emission effect is not negligible, the total intensity isexpressed by

Fi = Fsolar,i + Fthermal,i(3.11.10)

Here, Fthermal,i is an average thermal irradiance, which is defined as the total thermal radiative
energy normalized by the total area in the simulation space.

1363.12 Canopy photosynthesis
We describe a method to calculate canopy photosynthesis using a radiative quantity derived
from MC simulation.

3.12.1 Single leaf photosynthesis rate
There are several biochemical models for the simulation of photosynthesis. Here, we briefly
introduce Farquhar’s model (Farquhar et al., 1980), which is widely used. A detailed description
is provided in (Farquhar et al., 1980; de Pury and Farquhar, 1997). When the photosynthetically
active radiation absorbed by a single leaf is defined by Il (µmol s–1 m–2), the absorbed radiationin photosynthesis II is expressed by
Ile = Il(1 – f)/2

(3.12.1)

where f is an absorbed energy other than chloroplast (Farquhar et al., 1980). Strictly speaking,the parameter f depends on the leaf thickness. de Pury and Farquhar (1997) proposed f = 0.15.
The electron transfer rate in a unit leaf area J can be derived as a solution of the quadratic
equationθl J2 – (Ile + Jm)J + IleJm=0(3.12.2)

where Jm and θl are maximum J and empirical parameters to determine the electron transfer
response to absorbed photosynthetically active radiation (APAR), respectively. In de Pury and
Farquhar (1997), θl = 0.7 is used. Jm is a function of temperature. Using J, the photosynthesis
rate Aj under the electron transfer limitation can be calculated by

Aj = Jpi − Γ*4( pi + 2Γ* )(3.12.3)

where Γ∗ and pi are a CO2 compensation point Γ (Pa) in the absence of mitochondrial respirationand partial CO2 pressure in the leaf.
The photosynthesis rate Av under the Rubisco-limited condition can be calculated bypi − Γ*Av = Vl,pi + 2K'(3.12.4)

where K′ is an effective Michaelis-Menten constant of Rubisco. Finally, the photosynthesis rate
Al can be derived by taking the minimum between Aj (3.12.3) and Av (3.12.4):
Al = min (Aj, Av) – Rl

(3.12.5)

where Rl is a leaf respiration rate.

1373.12.2 APAR calculation
In a 3-D canopy radiative transfer model, the following three radiative quantities can becalculated:- Total APAR in each voxel (µmol s–1 m–2 m–3) or (W m–2 m–3)
- Diffuse APAR in each voxel (µmol s–1 m–2 m–3) or (W m–2 m–3)
- Number of modeled photons of first-order scattering in each voxela. Calculation of sunlit leaf area in each voxel
The sunlit leaf area can be calculated by counting the number of modeled photons of first-orderscattering in the voxel. The number of incident modeled photons in a unit voxel Nd (m–2) at the
top of a canopy is
Nd = N/(xd yd)(3.12.6)

where N and xd, yd are the total modeled photons and the length of a voxel on the x and y-axes.
The projected sunlit leaf area in the unit voxel up (m2) is calculated by
up = (xd yd) Na/Nd(3.12.7)

where Na is the number of absorbed direct modeled photons in the voxel.
Since, as described in 2.3, the projected leaf area G toward Ω(θ,φ) is expressed byG(Ω) =π 2

1 2π∫ ∫ gL (θ L )Ω ⋅ ΩL dθ L dϕ L
2π 0 0(3.12.8)

the total sunlit leaf area can be calculated using G (3.12.8) and up (3.12.7):
us = up/G(3.12.9)b. Photosynthesis calculation for each voxel
The photosynthesis rate in each voxel can be calculated using physical quantities derived from
the MC radiative transfer simulation and single leaf photosynthesis model. We assume that thevoxel size is sufficiently small, the spatial heterogeneity of the diffuse PAR flux is negligible,
and the diffuse flux is incident from the upper hemisphere and is uniform over all directions
within the voxel. Equation (3.12.5) is a function of Il, which is defined as an APAR in a unit leaf
area. On the other hand, the APAR calculated from radiative transfer simulation is the APAR
normalized in a unit volume. The simulated APAR (unit volume) can be converted to the APAR
in a unit leaf area (Il,sun, Il,shade) using the following equations:138

Il,sun = Idir (Ωs ⋅ ΩL )(1− ω ) +Il,shade =APAR difu

APAR difu(3.12.10)(3.12.11)

where Idir is a direct PAR flux incident at the top of the canopy. Therefore, photosynthesis in avoxel can be calculated by

Avox =us(u − u s )Al (I l,sun )gL (Ω L )dΩ L +Al (I l,shade )gL (Ω L )dΩ L∫2π2π2π ∫2 π(3.12.12)

where the first term is a contribution from a sunlit leaf and the second term is a contribution
from a shaded leaf. It should be noted that (3.12.12) does not distinguish between the adaxial
and abaxial sides of a leaf. Generally speaking, the photosynthesis abilities for the adaxial andabaxial sides are not equal. Thus, it is necessary to consider the adaxial and abaxial sides of aleaf if it is necessary to perform a more detailed analysis, by changing the constant parametersin Farquhar’s model.

139

Chapter 4
Improvement of numerical efficiency

4.1 MCS method
The MC atmospheric radiative transfer model usually treats the atmosphere as divided into alarge number of voxels, representing the fine-scale structure of the atmosphere. Each voxel has
specific optical properties (extinction coefficient, single scattering albedo, and phase function).
Each voxel is homogeneous, but different voxels can have different properties. In thesecircumstances, photon tracing requires that every crossing point at each voxel boundary befound to compute the optical path length in the voxel (Fig. 4.1.1). This calculation can be verytime consuming if there are too many voxels. For example, the earth’s upper atmosphere is
optically thin. If the mean free path in the upper part is 100 km, and if there are 1,000 layers of
0.1 km thickness, then how many numerical instructions are required? Calculations for the
crossing points on the order of 1,000 are required to find the next collision point!
The MCS method (Marchuk et al., 1980) is useful to avoid this problem. This method isimplemented in many MC radiative transfer models.

4.1.1 Base of MCS method
Any inhomogeneous field with a varying extinction coefficient can be considered as equivalent
to a homogeneous field with a constant extinction coefficient that is the maximum βmax of theextinction coefficients in the domain, using the scaling transformation based on the similarity
relations (e.g., Liou, 1992). The medium in the voxel can be considered to be a mixed medium
of two components: one is the original, and the other is a medium that has pure forward
scattering with a scattering angle of 0 (equivalent to transmission). In other words, thetransmission in the original medium is treated as virtual forward scattering. By using the MCS
method, there are two kinds of collision processes:
1) Real collision: a physical collision as in the original2) Virtual collision: mathematical scattering due to the scaling transformation
By the MCS method, the extinction coefficient becomes as large as βmax, originally from βe.
The number of collisions should thus increase. However, the virtual scattering is relatively easy
to treat in the model, compared to the physical scattering. The reason is that no absorption is
included for the virtual scattering (ω = 1) and that a direction change never occurs in the virtual140

scattering. A possible algorithm for the MCS method is as follows:
1) Determination of the collision location, based on the maximum extinction coefficientβmax.
2) Determination of the kind of collision: “virtual” or “real”? The probability of the realscattering is βe β max , so that a random number determines the kind.
3) Iterate the above until the real scattering is detected. If virtual, return to 1.
4) If real scattering occurs, then the direction changes as usual.
Using this method, there is no requirement to determine the crossing points at every voxel
boundary, because the extinction coefficient is constant throughout the domain. The model
photon can thus jump to the net collision point from the current point. A schematic of photon
tracing using this method is shown in Fig. 4.1.1.
Table 4.1.1 Optical property changes due to the MCS method
Medium1
2
1+2

Extinction
coefficientβe
βmax – βe
βmaxSingle scattering albedo

Scattering phase function

ω
1

P(Θ)
2δ(Θ)

(ωβe + βmax − βe ) βmax
= 1− (1− ω ) βe β max

ωβe P(Θ) + 2(β max − βe )δ (Θ)ωβe + β max − βe

δ is Dirac’s delta function

Fig. 4.1.1 Schematic of photon tracing. Left: usual method, right: using the MCS
method.

4.1.2 Discussion on efficiency of MCS method
The MCS method does not always improve the efficiency. What about with a significantly
larger maximum extinction coefficient than the average extinction coefficient? The efficiency
will be worse because too many virtual scattering events occur before reaching the real141

scattering point. Figure 4.1.2 shows an extreme case.
The MCS method does not work effectively if the average probability of a physical collision,βe β max , is low. Városi and Dwek (1999) discussed a similar point. The MCS method works
effectively under the following conditions:1) The size of the voxel is small compared with the average free path length: the extinction
coefficient is small.
2) The medium is relatively homogeneous: the inhomogeneity of the extinction coefficientis small.
The extinction coefficient is at a maximum in clouds and the value is relatively large. If the
maximum is used for the MCS method for the whole domain, the efficiency would be bad. The
MCS method is thus used for each of the subdomains (or super-voxel) that make up the entire
domain. Therefore, it is important to adaptively construct the super-voxels before the radiativetransfer calculation.
With thresholds related to the above two conditions, the voxels are hierarchically merged
into super-voxels in the pre-process until the following condition is broken:
1) The maximum extinction in the super-voxel is less than the average by a factor of A.
2) The inverse of the maximum extinction is larger than the voxel size by a factor of B.
The creation of super-voxels that are as large as possible under the above conditions would
improve the efficiency of the MCS method.

Fig. 4.1.2 Schematic of photon tracing in a case where the MCS method is notefficient: (Left) usual method, (right) using the MCS method.

142

4.2 Variance reduction methods
MC integration results in little variance (noise) if the number of samples is large. However, thecomputation time is generally proportional to the number of samples. Variance reduction
methods seek to obtain results with little noise in a limited computation time; that is, they are amethod for improving numerical efficiency. Several methods have already been described inprevious sections (e.g., the Russian roulette method). This section describes other methods forvariance reduction. Marchuk et al. (1980), Booth (1985), and Iwabuchi (2006) also presented
good methods for variance reduction. The MCML code of the Los Alamos National Laboratory
X-5 Monte Carlo Team also uses good methods.

4.2.1 Exponential transformation method
In the usual method for the determination of the collision point, the optical path length to thecollision point follows the exponential distribution with an average of 1. The transmittance is

T (τ ) = e − τ(4.2.1)

The randomly chosen optical path length to the collision point is determined by a uniform
random number ρ:∫ e dτ = ρ; τ = −ln(1− ρ )τ−τ(4.2.2)0

In this method, the photon weight is unchanged.
The PDF (transmittance) of the collision can be modified as

T ′(τ ) =1 −τ SeS(4.2.3)

with an average for the optical thickness of S. This is properly normalized to 1. If the PDF is
modified as this form, the random optical thickness to the collision point should be

1 τ ′ −τ S
∫ e dτ = ρS 0
τ ′ = −S ln(1− ρ ) = −S ln ρ˜(4.2.4a)(4.2.4b)

For energy conservation, the following equation should be true:

w′T ′(τ ′) = wT (τ ′) = w e −τ ′(4.2.5)Thus, the weight is given as143

w′ = w

⎡ ⎛ 1 ⎞⎤
S e −τ ′=wSexp⎢−τ ′⎜1− ⎟⎥ = wS exp[(S −1)ln ρ˜ ]e −τ ′ S⎣ ⎝ S ⎠⎦

(4.2.6)

By this modification, the average optical thickness is modified to an arbitrary value for S insteadof 1. When S > 1, then the photon will fly to a far point. In contrast, when S < 1, the collision
will occur at a closer point.
Using this method, the model photon is forced to escape from the domain even when too
many scattering events should usually occur, by setting S > 1. However, it should be noted thatthe photon weight varies due to (4.2.6). The Russian roulette method is easier to use for
reducing the multiple scattering. Another application is to force many scattering events whenthe media is too thin to sample enough scattering events, by setting S < 1 (4.2.4).

Fig. 4.2.1 Schematic of the exponential transformation: PDF of the collision(transmission function).

4.2.2 Modification of contribution function
The LEM samples the following contribution function for radiance:ζ (Ω1 , r1 ) = wΨ (Ω0 ,Ω1 )T (r0 , r1 )(4.2.7)

where w is the weight just after the scattering (or reflection), Ψ is the normalized PDF (sr–1), andT is the transmittance between the event point r0 and the detection point r1. The transmittance isthe function of the optical thickness:

T (r0 , r1 ) = e −τ (r0 ,r1 )τ (r0 , r1 ) =(4.2.8a)∫ β (t)dt; t = r′ − re(4.2.8b)0

r0 →r1

The function of (4.2.7) varies significantly sample-by-sample, especially when the phasefunction is anisotropic, as in cloud particles or ocean reflection. It is important to reduce thevariance of the contribution function ζ to reduce the radiance noise.

144

a. Modification of small ζ
The function ζ is frequently small. We are trying to present a method to modify ζ so as to keep
it larger than a threshold. If ζ < ζ min in the original form, a method such as the Russian roulette
method can be applied to modify (4.2.7). First, the following modification is valid:(4.2.9)Denoting

ζ min = wΨ e − τ max(4.2.10)the transmittance T is modified, using the random sampling method for the collision, to

T = e −τ = e − τ max e −(τ −τ max )⎧e − τ
if τ ≤ τ max⎪ −τ→ ⎨e max if τ > τ max and ρ < e − (τ −τ max )⎪0otherwise⎩(4.2.11)

where τ > τmax and ρ is a uniform random number. No sampling is needed if T = 0, of course.
Figure 4.2.2 shows schematically the transmittance modified in this method.
Combining (4.2.9) and (4.2.11), the following equation is derived (Iwabuchi, 2006):
When wΨ ≤ ζ min ,⎧ζif ρ1 ≤ wΨ ζ min and τ ≤ τ freeζ ′ = ⎨ minotherwise⎩0
When wΨ > ζ min ,⎧ζ
if τ ≤ τ max⎪
ζ ′ = ⎨ζ min if τ max < τ ≤ τ max + τ free⎪0
if τ > τ max + τ free⎩

(4.2.12a)(4.2.12b)

whereτ free = −ln ρ 2⎛ζ ⎞τ max = −ln⎜ min ⎟(4.2.13a)(4.2.13b)⎝ wΨ ⎠

Before the tracing between the scattering point and the detector, τfree and τmax are determined.
Tracing can terminate as soon as it is known that the photon energy is 0 (ζ′ = 0) for that sample,
which could be known before the tracing or at some point between the scattering point and thedetector.

145

This method has two advantages. One is a reduction of computation time because of the
reduction in the number of tracings. The other is a variance reduction in the contributionfunction due to the fact that the sampled functions are all larger than the threshold. Iwabuchi(2006) found that ζmin = 0.3 has the best performance for calculating reflectance from a clouddeck.

Fig. 4.2.2 Schematic of the transmittance modificationb. Use of MCS method
The analytical transmittance in (4.2.7) requires time-consuming ray tracing between the
scattering point r0 and the detection point r1, to calculate the optical thickness between the twopoints. The optical thickness integrating is a numerically heavy procedure if there are many
voxels in the atmosphere. Let us consider a use of the MCS method for calculating thetransmittance.
In (4.2.10), the threshold τmax is associated with ζmin. Similarly, introducing anotherparameter ζmax,

ζ max = wΨ e −τ max⎧ ⎛ ζ max ⎞⎪−ln⎜⎟ if wΨ > ζ max
τ max = ⎨ ⎝ wΨ ⎠⎪0otherwise⎩(4.2.14)(4.2.15)

Using these, we can derive the following:
When wΨ ≤ ζ min ,⎧ζ T ′(τ ) if ρ1 ≤ wΨ ζ minζ ′ = ⎨ minotherwise⎩0
When ζ min < wΨ ≤ ζ max ,ζ ′ = wΨT ′(τ )When wΨ > ζ max ,146(4.2.16a)(4.2.16b)⎧ζζ′ = ⎨⎩ζ maxT ′(τ − τ max )if τ ≤ τ max
otherwise(4.2.16c)

The modified transmittance is determined randomly:⎧1 if τ ≤ τ free = −ln ρ 2
T ′(τ ) = ⎨⎩0 otherwise(4.2.17)

This formulation is useful for applying the MCS method. Using the MCS method for calculatingT′, the transmittance is 1 if a physical collision does not occur between the two points. If aphysical collision does occur, the transmittance is 0, and the ray tracing can terminate
immediately.
Using this method, an algorithm follows:
1) If Ψ is small, the contribution function is modified as larger than ζmin, following (4.2.16a).
Else, τmax = 0.
2) Start the tracing from r0.
3) Analytical transmittance is calculated until τ is smaller than τmax. If the photon packet
reaches r1, then the transmittance is determined by the first equation of (4.2.16c) and thesampling for the local estimate finishes. Else, the MCS method is applied from the pointwith τ = τmax.
4) Randomly determined collision is due to mathematical scattering or physical collision. A
random number can determine the kind of collision.
5) If a physical collision is detected, then the transmittance becomes 0, and the tracing
terminates. Else, the photon reaches r1, and the transmittance with T′ = 1 is substituted into
(4.2.16a,b,c). Finally, the sampled contribution function should be⎧ζ min⎪ζ ′ = ⎨wΨ⎪ζ⎩ max

if wΨ ≤ ζ minif ζ min < wΨ ≤ ζ max
if wΨ > ζ max(4.2.18)

4.2.3 Truncation approximations for anisotropic scattering
The contribution function sampled by the LEM in (4.2.7) contains the PDF Ψ for angular
distribution, which is associated with the phase function for the scattering. If the phase function
contains sharp peaks, the contribution function sometimes becomes significantly large. Thus,
the convergence of the integral of (4.2.7) is very slow, with significant noise (Barker et al.,2003). The forward scattering peak for cloud particles and large aerosol particles is significantly
sharp, and the phase function is 104 or more at the peak.
Truncation approximations are effective for this problem (Iwabuchi. 2006). Commonly, themethod approximates the original phase function P by

P(Θ) ≈ 2 fδδ (Θ) + (1− fδ )Pˆ (Θ)(4.2.19)147

where Pˆ is the truncated phase function and fδ is the delta fraction. The forward scattering withzero scattering angle is equivalent to the transmission. By the similarity relations (e.g., Liou,
1992), the extinction coefficient and single scattering albedo are scaled asβˆ e = βe (1− fδω )1− fδωˆ = ω
1− fδω

(4.2.20)(4.2.21)

The radiative transfer can be calculated for a system with scaled properties βˆe , ωˆ , and Pˆ .
Various definitions of Pˆ can be possible (e.g., Nakajima and Tanaka, 1988; Antyufeev, 1996;
Thomas and Stamnes, 1999; Modest, 2003). All of the proposed methods are approximations
because the real forward peak is not exactly the same as the delta function. Bias is thus imposedon the calculation results. A method with as small a bias as possible is preferable.
Table 4.2.1 Changes in optical properties due to the truncation approximation.
Media
Original
ApproximatedExtinction
coefficientβe
βeSingle scattering
albedoωωTransformed

βˆ e

ωˆ

Scattering phase functionP(Θ)

2 fδδ (Θ) + (1− fδ )Pˆ (Θ)
Pˆ (Θ)

a. Delta-isotropic approximation
The delta-isotropic approximation (transport approximation) uses the isotropic phase function as
the truncated phase function:Pˆ (Θ) = 1(4.2.22)

Calculations become very simple. For accurately approximating the original radiative transfer, it
is important for the moments of the approximated phase function (right hand side of 4.2.19) tobe close to the original. The first-order moment (asymmetry factor) g1 is defined as

g1 =

1 π
∫ P(Θ)cosΘsin ΘdΘ
2 0(4.2.23)

Conservation of the first-order moment

g1 =

[

]

1 π
2 fδδ (Θ) + (1− fδ )Pˆ (Θ) cosΘsin ΘdΘ∫02

is the prerequisite. Then,148(4.2.24)

g1 = fδ ∫ δ (cosΘ −1)cosΘd cosΘ + (1− fδ )1

−1

= fδ + (1− fδ ) × 0∴ fδ = g1

1 1 ˆ∫ P(Θ)cosΘd cosΘ (4.2.25)
2 −1(4.2.26)

That is, the delta fraction is determined uniquely from this equation.
This method is not appropriate for radiance calculations because of very large bias. However,
the radiative flux and heating rate may be accurately calculated using this method, if the method
is applied only to multiple scattering. This method reduces the extinction coefficient and thenumber of scattering events, and the method is useful for optically-thick atmosphere. When g1 =0.86, for example, substituting (4.2.26) into (4.2.20), the extinction coefficient is decreased by afraction of 0.14. The number of scattering events is almost proportional to the extinction
coefficient, so that the computation time is significantly reduced.

b. Delta-Heyney-Greenstein approximation
In this method, the truncated phase function is expressed by the Heyney-Greenstein function.
For the first-order moment gˆ1 ,Pˆ (Θ) =1− gˆ12[1− 2 gˆ1 cosΘ + gˆ12 ](4.2.27)

3/2

As in (4.2.24), equating the first-order moment of the approximated function to the original,

g1 = fδ + (1− fδ )gˆ1
g −f
∴ gˆ1 = 1 δ
1− fδ(4.2.28)(4.2.29)

where(4.2.30)

Eq. (4.2.28) is true for an arbitrary fδ under the condition of (4.2.30), so that the user can freely
set fδ.
Although this method has better accuracy than the delta-isotropic approximation, it is notapplicable to calculating radiance (because of large bias).

c. Forward-end truncation approximation (FTA)Let us consider the truncated phase function as a geometrically truncated and renormalized one
without the forward peak part in the original:

149

⎧P(Θ) ftPˆ (Θ) = ⎨⎩0

for Θ ≥ Θffor Θ < Θf

(4.2.31)

where

ft =

1 π
∫ P(Θ)sin ΘdΘ
2 Θf(4.2.32)

The truncation angle Θf is determined from the conservation of the asymmetry factor (4.2.24):

1 π ˆ∫ P(Θ)cosΘsinΘdΘ
2 0
1 π
=
∫ P(Θ)cosΘsin ΘdΘ
2 ft Θfgˆ1 ≡(4.2.33)

We obtain

g1 = fδ + (1− fδ )gˆ1

g1 − fδ

(4.2.34)∫ P(Θ)cosΘsin ΘdΘ = 1− f ∫ P(Θ)sinΘdΘ
π

Θf

δ

π

Θf

(4.2.35)

The angle Θf can be determined by numerically solving the above equation. fδ is a freeparameter. In Iwabuchi (2006), the parameter is set so as to increase with the isotropy of thephoton packet (with multiple scattering).
This method is significantly better than the two methods previously described. The form of(4.2.31) is the same as the original function, except for the forward part, and this similarity ofthe phase function form is a key for achieving high accuracy. This method can be applicable toradiance calculations with small bias (usually less than 0.5%), except for the solar aureole
simulation.
The form of (4.2.31) has another merit for the MC model. LUTs for the original phasefunctions and associated cumulative distribution functions can be reused even for the truncatedphase functions. The LUT is usually tabulated for a very large number of grid points, requiring
the use of a large memory. The form of (4.2.31) does not require new tables for truncatedfunctions. This point is especially important when setting different approximation degrees withdifferent fδ values for each scattering order.

d. Dual-end truncation approximation
This method is similar to the previous one, but truncates both ends of the phase function. The
truncation of the backward part introduces another free parameter and results in better
calculation results in some situations. The truncated phase function is150

⎧P(Θ) ftPˆ (Θ) = ⎨⎩0

for Θ f ≤ Θ ≤ Θbfor Θ < Θf or Θ > Θb

(4.2.36)

where

ft =

1 Θb
∫ P(Θ)sin ΘdΘ
2 Θf(4.2.37)

For first and second moments,

1 π∫ P(Θ)cosΘsin ΘdΘ
2 0
1 π
g2 ≡ ∫ 0 P(Θ)cos 2 Θsin ΘdΘ2

g1 ≡(4.2.38a)(4.2.38b)

Similarly, for an approximated function,

1 π ˆ
1 Θb
P(Θ)cosΘsin ΘdΘ =∫∫ P(Θ)cosΘsin ΘdΘ
0
2
2 ft Θf
1 π
1 Θbgˆ 2 ≡ ∫ 0 Pˆ (Θ)cos 2 Θsin ΘdΘ =∫ P(Θ)cos2 Θsin ΘdΘ22 ft Θfgˆ1 ≡(4.2.39a)(4.2.39b)

Prerequisite conditions are the conservations of these two moments:

g1 = fδ + (1− fδ )gˆ1
g2 = fδ + (1− fδ )gˆ 2(4.2.40a)(4.2.40b)

By numerically solving the system of equations

g1 − fδ Θb∫ P(Θ)sin ΘdΘf
1− fδ ΘfΘb
g2 − fδ Θb∫Θf P(Θ)cos2 ΘsinΘdΘ = 1−
∫ P(Θ)sin ΘdΘ
fδ Θf

∫

Θb

P(Θ)cosΘsinΘdΘ =Θ

(4.2.41a)(4.2.41b)

the two angles Θf and Θb are determined.
The user can freely set the fraction fδ. If the parameter is set so as to increase with theisotropy of the photon packet (with multiple scattering), then higher accuracy is obtained(Iwabuchi, 2006). This method is slightly better than the previous one. The bias in radiance isless than 0.2%, except for the solar aureole. By using this method, the numerical efficiency forradiance calculation is significantly improved. In addition, as in the previous method, any LUT
related to the original phase functions can be used even for the truncated phase functions.

151

4

10

f" = 0.56
3Phase Function P(!)

10

2

10

ff = 0.57
110ft = 0.43

010

1 – ff – ft = 0.0005

-1

10

-2

10

0

30
!f = 1360
90
120
Scattering Angle !

150

180
!b = 176

Fig. 4.2.3 Example of the dual-end truncation approximation (DTA). Effective
radius of 10 µm, wavelength at 0.67 µm.

Fig. 4.2.4 Radiance images computed without DTA (scheme S) and with DTA(scheme T). A thousand photons are incident per pixel. Right images show noise
error. Note that a large Fmax corresponds to a large delta fraction.

152

4.2.4 Collision-forcing method
Collisions rarely occur in optically thin media. This causes rare sampling of heating rate andradiance calculated using the LEM, both of which are sampled at collision events. To improve
the numerical efficiency for these quantities, collision-forcing methods are useful. Threemethods are introduced in the following.

a. Method using scaling transformation
The scaling transformation based on the similarity relations (e.g., Liou, 1992) can easily
increase the extinction coefficient (Iwabuchi, 2006). Increasing the extinction coefficient by afactor of 1/fe,

βe′ =

βe
fe

(4.2.42)

The transformed single scattering albedo and scattering phase function are as follows:ω′ = 1− (1− ω ) feP′(Θ) = 2 fdδ (Θ) + (1− fd )P(Θ)(4.2.43)(4.2.44)1− fe= (1− fe ) ω′
1− (1− ω ) fe

(4.2.45)

where

fd =

In other words, the transmission in the original medium is treated as forward scattering with ascattering angle of zero. The same theory is also used in the MCS method. The extinction
coefficient increases, and the amount of scattering increases with a single scattering albedo
close to 1 (weak absorption) and a phase function with the forward scattering peak. Opticalproperties are summarized in Table 4.2.2. The original medium (1) is equivalent to the mixed
media (1 + 2). This transformation is mathematically correct and results in no bias.

Table 4.2.2 Changes in optical properties due to the collision-forcing method
based on the scaling transformation
Medium1
2
1+2

Extinction
coefficient
βe

βe
− βe
fe
β
βe′ = efeSingle scattering albedo

Scattering phase function

ω
1

P(Θ)
2δ(Θ)

ω′ = 1− (1− ω ) feP′(Θ) =
2 fdδ (Θ) + (1− fd )P(Θ)

153

Calculation using this method is done with parameters given in (4.2.42–44). The phasefunction for the local estimate should be (1− fd )P(Θ) from (4.2.44). When determining thescattering direction, a random number first determines whether the event is mathematicalscattering (due to the forward scattering) with a probability of fd or the physical scattering. The
photon direction is not altered for the mathematical scattering.
Using this method, the extinction coefficient increases to an arbitrary (possibly large) value,and collisions should occur frequently enough. This method is the simplest and most efficientfor forcing collisions.

b. Forced sampling of collision in path segments
As mentioned in 3.4, virtual collisions can be sampled in arbitrary path segment(s). The
probability of a collision in a path segment with an optical thickness of Δτ isC(Δτ ) = 1− e −Δτ

(4.2.46)

To determine a random collision point in this path segment, the optical thickness up to the point
is given as

C(τ )
=ρ
C(Δτ )∴e −τ = 1− ρ (1− e −Δτ )(4.2.47)(4.2.48)

The photon weight is w0 at the initial point of this path segment, and the single scattering albedo
at the collision point is ω(τ). The photon weights for absorption and scattering are respectively

wabs = w0 (1− e −Δτ )[1− ω (τ )]wsca = w0 (1− e−Δτ(4.2.49a))ω(τ )(4.2.49b)

These weights can be used for sampling the heating rate and radiance. A random number
determines whether the photon collides at the point or transmits up to the terminal point of thepath segment. As noted in 3.4 (sampling absorption based on path length), one can force tosample only absorption or scattering.
This method is useful for sampling frequent collisions at specific volume elements ofinterest.

c. Exponential transformation method
As described in 4.2.1, collisions occur frequently with S < 1. However, with such a method, thephoton weight can vary significantly between 0 and (possibly) infinity. This causes large noisein calculated radiative quantities. Therefore, the authors do not recommend this method.

154

4.2.5 Numerical diffusion
The usual MC models sample the radiative flux, heating rate, and radiance at local points. If
these local samples are distributed in non-local regions (area or volume), then sampling noise
could be reduced. Each sample is subdivided into subsamples and distributed in a horizontal
area. The photon trajectory is not altered, but at each sampling process, the photon energy is
divided and replaced in subpixels. The domain average of the calculated quantity has no bias if
each sampling conserves the total of the original contribution.
For example, Iwabuchi (2006) proposed a method to distribute the contribution in arectangular region (diffusion area) with uniform spatial distribution. The size of the region wasdetermined according to the number of photon packets incident to the domain, diffusivity
diagnostics of the photon packet, and sampled contribution function.
This method is purely technical, and the optimal redistribution region size cannot be
determined theoretically. The optical size should be determined by tests. The noise should be
reduced with a wide diffusion area, and the spatial distribution of the calculated quantity should
become smooth. However, artifacts will increase if the diffusion area is too large, and there will
also be an increase in the computation time that is not negligible.

Fig. 4.2.5 Examples of computed radiance without using the numerical diffusion(scheme T) and with (scheme V). 1000 photon packets were incident. Right images
show the error.

4.2.6 Semi-analytical calculation of direct-beam and single-scattering components
It is important for noise reduction with MC integration to use analytical calculation for the part
where this is possible. As for solar radiative transfer, the direct-beam and single-scattering
components can be calculated semi-analytically. The direct beam flux can be determined at local
points by tracing the direct-beam ray from each point of interest for obtaining the optical155

thickness for the beam. The direct beam flux should be exp(–τ) multiplied by the solar incident
flux. However, spatial averaging is required for obtaining the area-averaged irradiance andvolume-averaged heating rate (due to the initial scattering). This averaging requires numericalintegration because the optical thickness for the direct beam is not given in analytical forms for
3-D inhomogeneous fields. Therefore, the area-averaged direct-beam irradiance and first-orderscattering heating rate should be calculated semi-analytically.
Two methods can be considered. One is to trace the backward trajectories from the target
region of interest. The other is to trace the forward trajectories from the radiation source. The
forward-type algorithm is better when one tries to calculate radiative quantities at many regions.
Several options could be used for spatial integration. The simplest one is to distribute uniformly
many rays at the top of the atmosphere. This seems very similar to the MC algorithm. However,they are different in two points: 1) The MC method has a random distribution of photon packets
at the top of the atmosphere, but the semi-analytical algorithm has uniform distribution, and thesame number of rays penetrates areas with constant cross sections. 2) The MC method
determines a random initial collision point, but the semi-analytical method always traces the ray
to the bottom of the atmosphere. Therefore, the semi-analytical method has better accuracy thanthe MC method. The number of rays determines the trade-off between computation time andaccuracy. Methods of spatial interpolation and integration are important for achieving better
accuracy with a small number of rays. Calculation algorithms for calculating the direct beamirradiance and first-order scattering heating rate are similar to those described in 3.8.3.
The ray tracing is relatively time consuming, especially if the atmosphere contains a large
number of cells. The analytical calculation method cannot use the tricks for increasing
efficiency, such as the MCS method in 4.1. Several techniques may improve the calculation
efficiency. For example, if the integrated direct-beam optical thickness is large enough, it isefficient to switch to MC sampling of the random collision point, as described in 3.8.2, and toterminate the ray tracing at the initial collision point. In addition, if the ray penetrates many
optically-thin, small cells, the MC method with the MCS method can be used to accelerate thecalculation.
Another advantage of the semi-analytical calculation is that radiative quantities for all
spectral wavelength (or bands) of interest could be obtained from one ray tracing. If there aremany target wavelengths, the method is useful.
The first-order scattering radiance, too, can be calculated semi-analytically. There are also
forward and backward-type algorithms for the method.

156

4.3 Parallelization
Since the MC method traces many photons, it takes a large amount of computation time.
Parallelization, which divides the simulation tasks into several CPUs, is an effective tool for MC
simulation. We next describe a parallelization method for MC radiative transfer simulation.

4.3.1 Parallel programming
So far, several hardware and software architectures have been proposed. With respect to thedependency between the memory and processor, two types of architecture exist: the distributed
memory type and shared memory type. The shared memory type consists of memories andprocessors directly connected with each other in a computer network. On the other hand,processors have their own memories in the distributed memory type.
The message passing method is widely employed in the distributed memory type for sharing
and exchanging data. A message passing interface (MPI) is commonly used to achieve the data
sharing and exchange.
There are two approaches to designing the parallel programming. The data parallel approach
distributes the same data source to all processors to execute the same type of calculations. The
controlled parallel approach divides the simulation task between the processors. Each processor
executes a different type of simulation.
Hereafter, we will mainly describe MPI-based programming. In particular, the data parallel
approach has the advantage of easy programming and expansion from single processor code.
The details of parallel programming can be found in Pacheco, (1997) etc.

4.3.2 Parallel computing by MC method I: Photon split method
Let us consider that the MC simulation is performed using N modeled photons with Nprc
processors. When the MC simulation is conducted in the Nprc independent processors,
independent random number sequences are required for each processor. This can be achieved by
preparing independent initial seeds for the random number sequences for each processor.
In the basic parallel programming, the MC simulation is first performed using N/Npr photons.
Then, the results obtained in each processor are assembled in one processor and the physicalquantities are calculated. The following is a detailed procedure for the photon split method.
i) Determination of the initial random number seed and photon number for each processor(= N/Nprc)ii) Execution of the MC simulationiii) Wait until all processors finish their simulationsiv) Gather the results from all processors and calculate the physical quantities
In this approach, data are exchanged only at the beginning and end of the simulation. The

157

amount of exchanged data is small. When the number of photons is sufficiently large, thecomputation time becomes almost equal for all processors. Therefore, it is efficient. Thismethod is very simple, and it is easy to modify the program with less work. On the other hand,since all the processors have all of the data required in the simulation, it consumes a large
amount of memory.
One idea to reduce the required memory resources is to consolidate the 3-D data for the
optical properties in one or a few parent processors, which work as a data transfer system. Child
processors perform the simulation, and if it is necessary to use or save the data, the childprocessor requests that the parent processors exchange the data. The information exchanged
between the parent and child processors would include the extinction coefficient, single
scattering albedo, phase function, and so on. In this idea, it is necessary to exchange the data forevery scattering event. The maximum cross section method described in 4.1 will help to reducethe data exchange.

4.3.3 Parallel computing by MC method II: Space split method
Another method for efficient parallel programming is a space split method, which divides thesimulation space so that each processor performs a simulation in their allocated space. Thismethod reduces the amount of memory used. Therefore, it enables a simulation using a large
number of simulation grids, compared with the single processor simulation. This means that it ispossible to conduct a larger spatial-scale simulation or finer spatial resolution simulation.
On the other hand, this approach is rather complicated. Assuming that the modeled photon
enters the (i, j) space, and emits to the (i+1,j) space after the photon tracing, it is necessary totransfer the photon information such as position, weight, and direction. The processor
simulation for the (i, j) space may receive the modeled photon from the nearest space. The
number of modeled photons used in the MC simulation is very large. Therefore, thecomputation for the simulation requires frequent exchanges of data between processors.
To reduce the photon exchanges among processors, it is necessary to set the space as large
as possible. It is not efficient to exchange small amounts of data many times due to the latency
issue (Pacheco, 1997). Therefore, it is preferable to exchange the data when some amount ofphoton data, for photons emitted from the current space, are stored in the current space. In thismethod, photon data are stored in the local memories and the next simulation then starts. This
procedure is continued until the stored data reaches some predetermined volume.
Another problem with this method is that if a simulated space, which is composed of asevere heterogeneous medium, is divided with the same spatial scale between all the processors,
it cause a computational imbalance between processors: some processors are required to handle
many photon tracings, while others handle only a few. To prevent this imbalance, it is necessary
to achieve equal computation time for all the processors. For example, the simulation space isfirst divided into many small spatial areas. Then, the simulation space for each processor is
randomly determined, or if some processors finish their simulation, they help with thesimulation of another processor (though it may be difficult to create programming for this).
The method described above is an example for parallel computing. An efficient methodshould be designed by considering the spatial homogeneity/heterogeneity of the optical158

properties.

Fig. 4.3.1

Data exchange in the simulation space.

159

Chapter 5 Introduction of radiative transfer
models and examples of applications

5.1 Simple code for MC radiation
The radiative transfer in a simple system can be easily modeled in code. For example, consider
the code for the MC radiative transfer model for a plane-parallel atmosphere on a flat
Lambertian surface. The atmosphere is plane-parallel, it is not necessary to calculate thephoton’s horizontal location coordinates (x and y), which makes the code simple. Isotropic
scattering is assumed. The code computes reflectance, transmittance, and absorptance, in
addition to the nadir-looking radiance at the top of the atmosphere, by using the LEM. The
Russian roulette method is applied each time after a scattering or reflection event.
Although the code shown here is very short, with this alone, we can accurately calculate
radiative flux and radiance under arbitrary conditions for the solar zenith angle, optical
thickness, single scattering albedo, and surface reflectance. Using a modern personal computer,
200,000 photon packets can be simulated in a second for an optical thickness of 10. The
accuracy of such a simulation would be approximately 0.3%. In addition, the code can be easily
modified as follows:-to compute separately the direct-beam and diffuse components for irradiance and heating
rate, respectively-to compute the radiance for an arbitrary direction-to model vertically inhomogeneous layers-to treat scattering with an arbitrary phase function-to treat the medium as a mixture of gases, aerosols, and cloud particles and sample theheating rates separately for each component
The function rand() can usually be used in a UNIX/Linux system, although the function can be
replaced by any good random number generator. Table 5.1.1 shows the results for various
optical thicknesses.

160

Sample code, mcpp.f
program mcpp!************************************************************************! Demo program of Monte Carlo radiative transfer model for! plane-parallel homogeneous atmosphere on Lambertian surface!****
implicit none
! Input
integer iseed, nph, nsmax
real the0, tau, omg, alb, dz, wrr! Workinteger iph, isreal ext, afp, uz0, w, z, uz, ftau, znew, rad, pi
real rand
parameter (pi = 3.141592653, rad = pi/180.0)! Work/Output
double precision r, t, a, eu! User variables
iseed = 11nph = 100000
nsmax = 1000
the0 = 60.0*rad
tau = 2.0
omg = 0.99
alb = 0.2
dz = 1000.0
wrr = 0.5
! Initializations
w = rand(iseed)ext = tau/dz
afp = 1.0/ext
uz0 = -cos(the0)r = 0.0
t = 0.0
a = 0.0
eu = 0.0! initial seed for random number generator! # of photons! max # of collisions! solar zenith angle! optical thickness of the atmosphere! single scattering albedo! surface albedo! geometrical thickness (m) of the atmosphere! dummy for random number initialization! extinction coefficient (/m)! average free path (m)! downward incidence! Monte Carlo simulation
do iph = 1, nph
w = 1.0z = dz! source from top
uz = uz0
do is = 0, nsmax
ftau = -log(max(1.0e-35, rand(0))) ! free optical thickness
znew = z + uz * ftau * afp
if (znew .ge. dz) then! escaper = r + w
exit
else if (znew .le. 0.0) then ! Lambertian reflectiont = t + w
z = 0.0
uz = sqrt(rand(0))
w = w * alb
eu = eu + w * exp(-dz*ext)else! isotropic scatteringa = a + w * (1.0 - omg)z = znewuz = 1.0 - 2.0 * rand(0)w = w * omg
eu = eu + w / 4.0 * exp(-(dz - z)*ext)
end ifif (w .lt. wrr*0.5) then
! Russian roulette
if (w .gt. wrr*rand(0)) thenw = wrrelse
w = 0.0
exit
end ifend ifend doend do161

! Results (normalized quantities)r = r/real(nph)! TOA upward fluxt = t/real(nph)! BOA downward flux
a = a/real(nph)! Absorptanceeu = eu/real(nph)! pi*(TOA nadir radiance)
write (*,*) r, t, a, eu
end

Table 5.1.1 Simulation results using the mcpp code

τ
0.1
0.5
1
2
4

R
0.25890.42670.5451
0.66140.7511T
0.92320.70090.53870.36950.2184A
0.00253
0.01267
0.023960.04295
0.07415

162

Nadir BRF
0.2267
0.33690.44110.56470.67295.2 Introduction of atmospheric radiative transfer model
5.2.1 Radiative transfer model MCARaTS
The radiative transfer model MCARaTS (Monte Carlo atmospheric radiative transfer simulator)
is a multipurpose model to treat the 3-D radiative transfer in a system composed of theatmosphere, ocean, and land. The model can simulate detailed spatial and angular distributionsof the radiative energy in a cloudy atmosphere. Such a model is important for realisticallysimulating the 3-D distribution of radiative heating in cloud-resolving model simulations. For
remote sensing purposes, too, the model is useful for relating the observable spectral radiances,which could be available from satellite observations, to the parameters for clouds, aerosols,
gases, and land.
The forward-type MC method is used for the radiative transfer calculations. Many model
photons are emitted from the radiation sources, scattering in the atmosphere and surface
reflection are simulated by using random numbers, and the simulation for each single model
photon is continued until the model photon’s energy is completely absorbed in the system or
escapes from the system (to space). In addition to several standard variance reduction methods
such as the Russian roulette method, the truncation approximation of Iwabuchi (2006), thecollision-forcing method, the MCS method, the numerical diffusion, and the methods describedin this report are implemented in the model. This model was originally developed for the
purpose of studying 3-D radiative effects in a field with inhomogeneous clouds. Especially, theproblems of satellite remote sensing of the physical properties of boundary layer clouds havebeen studied (Iwabuchi and Hayasaka, 2002, 2003). Efforts now continue for model
development and improvements for the purpose of online coupling with the cloud/large eddy
resolving model.
The 3-D atmosphere is defined in the Cartesian coordinate system, and the 3-D distributionof clouds, aerosols, and gases (scattering and absorption) is taken into account. Several typicalmodels for the aerosol mixture, such as Hess et al. (1998), can be selected by the user. The
surface is defined in two dimensions, and its BRDF can vary spatially. Semi-empirical BRDF
models of water and land are incorporated. BRDF models can be selected from the Lambertian,
DSM, RPV, and LSTR models. At present, the surface is macroscopically flat, whereas thewater surface model has microscopic roughness due to wind velocity. The applicable spectral
bands include solar spectra, infrared, and microwave in a wavelength range of 0.2–2000 µm.
The radiation source can be the solar incidence, thermal emission, or artificial light.
The model’s outputs are the irradiance (averaged over pixel cross section, local irradianceon a horizontal plane, or local spheradiance), radiative heating rate, radiance (averaged overarea or angular region with arbitrary solid angle), and average photon path length distribution.
The direct beam and total irradiance are separately computed. Because the model employs theforward-type MC algorithm, from a single simulation, one can derive 3-D distributions ofirradiance, heating rate, and radiance. The radiance can be computed at arbitrary locations andfor arbitrary directions. It could be applied to the generation of photo-realistic computer
graphics (Fig. 5.2.4). Lidar signal simulation is another application.

163

Three solver modes are implemented in the model, for the purpose of studying the 3-D
radiative transfer effects. One is used to solve fully the 3-D transfer (F3D). Another is theindependent column approximation (ICA, Cahalan et al., 1994b), which applies the 1-D transfer
independently to every atmospheric column. The ICA is used in the current climate/weather
prediction models and in the remote sensing of the atmosphere and land. The other computation
mode is the partly-3-D (P3D) scheme, which uses the 3-D transfer for low-order scattering and
1-D for higher-order scattering.
The model’s code is written in Fortran 77 and runs on many common computers under theUNIX/Linux OS. The radiative transfer code can be parallized with an MPI, and the code canrun on large-scale parallel computers. The code is open and distributed for free (GPL license),with
associated
code
and
some
examplefiles,
ontheWeb:
http://www.geocities.co.jp/null2unity/mcarats/.
This model has been used in several model comparison projects such as I3RC (Cahalan etal., 2005), Wagner et al. (2006), and Ishida (2006), and several reasonable agreements havebeen obtained.
Figure 5.2.2 shows the algorithm flowchart. Due to the MCS method, a collision iscategorized as a mathematical collision or a physical one, and the model photon moves only
when a physical collision occurs. The photon tracing for a model photon terminates if thephoton weight is zero or the photon escapes from the system into space. The irradiance andheating rate are calculated by method I in 3.8 (random sampling of the transmittance). The
radiance and local flux are calculated by the LEM in 3.9, being sampled at every scattering
event.
Figure 5.2.3 shows the grid system for the model. The atmosphere is divided into
horizontally homogeneous and inhomogeneous layers. This is a numerically efficient way torealize the atmosphere by treating the cloudy layers as 3-D inhomogeneous layers and the clear
layers as possibly homogeneous layers. The optical properties (extinction coefficient, single
scattering albedo, and phase function) are constant within each cell (voxel). Cyclic boundary
conditions are assumed for the horizontal domain boundary. The radiative quantities arecomputed at layer boundary surfaces, except for the heating rate, which is a cell-averaged
quantity. The domain subdivision method is used for the photon tracing, and the model photon
is tracked, traveling voxel-by-voxel.

z (m)

2000
1500
1000
500
0
0

1000

2000

3000
x (m)

4000

5000

6000

Fig. 5.2.1 Examples of photon trajectories in the MC simulation.

164

StartInitialize weight w, position (x, y, z), direction,order of scattering n, and free optical path !", etc

Search for an intersection
point with objects

Scattering?

no

Motion to the boundary

yes
Determine type of scattering(mathmatical/gas/aerosols/
hydrometeors)

Integrate flux
yes
Left TOA?
no

Increase free
optical path !"noPhysical scattering?

End

yes
Transfer photon packet(modify x, y, z)

Hit surface?

Integrate flux
Increase n, modify w

noyes
Integrate flux

Integrate radiance and heating rate
Russian roulette

Determine new direction (#, $)and free optical path !"now = 0?
yesEnd

Fig. 5.2.2 Algorithm flowchart for the MCARaTS model. The algorithm for asingle photon packet is illustrated.

Fig. 5.2.3 Grid system used in the MCARaTS model165

5.2.2 Calculation examples using MCARaTS
Calculation examples for the MCARaTS are shown in the following. Figures 5.2.4–5 show theangular distribution of computed radiances at a local point at the surface or above the clouds,
using a 3-D distribution of cloud water simulated by large eddy simulation (LES). The surface isassumed to be an ocean with a wind velocity of 7 m s–1. The model successfully simulated thereflection at the ocean surface, cloud shadows, and multiple scattering among broken clouds.

Fig. 5.2.4 Photorealistic computer graphics generated by using the MCARaTS
model. Local radiances were plotted in the hemispheric angular region, and atrue-color RGB composite was made. The 3-D distribution of the amount of cloudwater was obtained from the LES simulation.

166

Fig. 5.2.5 Photorealistic computer graphics generated by using the MCARaTS
model for sunset scenes over an ocean. A true-color RGB composite was made. The
3-D distribution of the amount of cloud water was obtained from the LES
simulation. Left images show moderate amounts of aerosol (moderately clean
atmosphere), and right images show large amounts of aerosol (realizing a dirty
atmosphere).

167

Figure 5.2.6 shows horizontal-vertical cross sections of irradiance and radiance in a scene
with two cumulus-type clouds. The forward-type MC algorithm makes it easy to estimate thespatial distribution of radiative energy from one simulation.

(a)

(b)

(c)

(d)

(e)

Fig. 5.2.6 Calculation example for a wavelength of 500 nm, in a case with two
cumulus clouds (x-z cross section): (a) extinction coefficient; (b) downward
irradiance; (c) upward irradiance; (d) downward radiance for µ1 = –0.95, φ1 = 0˚;and (e) upward radiance with µ1 = 0.95, φ1 = 0˚. The solar zenith angle is 53°, andthe sun is on the left side.

168

Figure 5.2.7 shows simulation results for radiance, which could be obtained from satellite
observation. Figure 5.2.8 shows simulation results for the multiple-scattering lidar signals. The
radiation model can be used as a tool for the remote sensing of cloud properties.

Fig. 5.2.7 Simulation results of satellite observation radiance: (a) BRF for nadir
looking, (b) backward viewing (geometry of the backscattering), and (c) forward
viewing (geometry of the forward scattering). These are from case 6 of I3RC phase
III.

Fig. 5.2.8 Simulation results of the multiple-scattering lidar: 2-D plot of irradiancefrequency as a function of flight path length in the cloud (bottom axis) and beamemission direction (right axis). The laser beam light is emitted, scattered in thecloud many times, and detected at the receiver near the emitter. (a) Infinitely thicklayer with isotropic scattering, (b) as (a) but with Heyney-Greenstein phasefunction, and (c) thickness of 300 m with Mie scattering phase function. These arefrom case 7 of I3RC phase III.

169

5.3 Introduction of plant canopy radiative transfer model
A plant canopy radiative transfer model is important for evaluating forest and grassland lightenvironments. The light environment is one of the primary environmental parameters for plant
photosynthesis. Plant canopy photosynthesis can be simulated using the method described in
3.12 when the spatial distribution of the plant canopy light environment, such as the APAR, andincident irradiance at the top of the understory layer from radiative transfer simulation isavailable. Also, the simulated spectral reflectance at the top of a plant canopy is helpful inanalyzing the relationship between the observed spectral reflectance and forest structures.
We introduce the plant canopy radiative transfer model based on the method described in
Chapter 2 to Chapter 4. Then, some examples of the simulation are shown.

5.3.1 Plant canopy radiative transfer modela. Simulation scene
It is necessary to prepare the simulation scene for radiative transfer calculation. Figure 5.3.1
shows the forest scene and individual tree and forest floor conditions. As described in 2.3,
several representations of the forest scene can potentially be used for the radiative transfer
simulation. We use the geometrical representation of individual trees (see center image in Fig.
2.3.4 in 2.3).
Each tree canopy is modeled as an ellipsoid (Fig. 5.3.1(a)). The leaves are distributed
uniformly within this geometric object (leaf angle distribution = constant). Stems are modeled
as cylinders. A photon does not enter the stem. The photon is reflected at the surface of the stem.
The forest floor is assumed to be a plane-parallel layer (see left image in Fig. 2.3.4 in 2.3), andunder the forest floor vegetation, a soil layer exists.
Trees are distributed in a 30 × 30 m simulation scene. Figures 5.3.1(b) and (c) are thesimulation scenes used in this example. A total of 100 trees are randomly positioned in this 30 ×
30 m area. The heights of the trees, lengths of the canopies, and radii of the canopies are
determined to be in the range of 15.0 m ± 7.5 m, 5.0 m ± 1.2 m, 1.5 m ± 0.3 m, respectively.
In the plant canopy radiative transfer model, a simulation is conducted in this defined scene
by tracing the photon and sampling the radiative quantities.

b. Preparation of optical properties of plant canopy element
The optical properties of the forest elements are required to perform the radiative transfer
simulation. The parameters required in the simulation are the reflectance and transmittance ofthe forest elements (canopy leaf, forest floor plant leaf, stem, soil), canopy leaf area index,
forest floor leaf area index, and leaf angle distribution of the canopy and forest floor.
As described in 2.3, the reflectance and transmittance for a single leaf are calculated fromthe leaf radiative transfer model or they can be obtained from measured data. As an example, we

170

use the parameters summarized in Table 5.3.1 and Table 5.3.2.

Fig 5.3.1 Forest canopy scene used in the simulation: (a) Modeled single tree, (b)nadir view of the modeled forest scene, and (c) modeled forest scene.

Table 5.3.1 Optical properties of canopy and forest floor elements.
Canopy leaf reflectance
Canopy leaf transmittance
Floor leaf reflectance
Floor leaf transmittance
Stem
Soil

450 nm
0.038
0.017
0.050
0.136
0.142
0.085

550 nm
0.110
0.038
0.186
0.272
0.175
0.115

650 nm
0.046
0.021
0.075
0.131
0.247
0.149

800 nm
0.502
0.190
0.420
0.560
0.428
0.268

PAR
0.065
0.025
0.100
0.180
0.190
0.120

Table 5.3.2 Other parameters.
Solar zenith angle
Canopy leaf area density
Canopy branch area density
Floor leaf area index

40o0.5, 2.00.5
0.5

c. MC simulation in forest scene
Figure 5.3.2 shows a flowchart for the canopy radiative transfer model. In the forest scene, thismodel traces the photon trajectory, which is input at the top of the canopy with an initial weightw0. Then, scattering events are generated within the canopy and for forest floor objects or on thestem and ground surfaces. The non-analog method (see 3.5) is employed to evaluate the171radiative quantities from sampled photons. The scattering direction is determined using the LUT
method (see 3.5). The photon tracing will be stopped when the photon exits the simulation sceneor the weight of the photon w becomes 0 after the Russian roulette. Then, the next photon
tracing will start.
There are no universal criteria to determine the threshold ε of the Russian roulette. The
accuracy becomes worse when we use larger values for ε, and the computation time becomes
longer when we use smaller values for ε. In plant canopies, leaf reflectance and transmittancehave lower values in the visible region and higher values in the near-infrared region, as shownin 2.3. Therefore, ε should be determined empirically by trial and error. Here, we use thefollowing threshold ε., which is defined using the single scattering albedo ω:ε = 0.1ω 2(5.3.1)

By using this equation, the Russian roulette is executed after third-order scattering on average inthe visible region (ω ~ 0.1). In the near-infrared region (ω ~ 0.9), the Russian roulette is
executed after twentieth-order scattering on average if all scattering occurs in the leaf surface.
However, in an actual forest scene, the Russian roulette is executed after fifth- to tenth-orderscattering because of the existence of the stem and soil, the single scattering albedo of which isless than that of a leaf. We employ the spectral integration method described in 3.11 forintegrating the radiative quantities for all of the solar radiation and the photosynthetically active
radiation.

Fig. 5.3.2 Flowchart for the plant canopy radiative transfer model.

172

d. Sampling of radiative quantities
The radiative quantities are sampled by tracing photons in the forest. In remote sensing andterrestrial carbon cycle studies, the following five parameters are important:
1. Bidirectional reflectance factor (BRF)
2. Downward irradiance at the forest floor
3. Total APAR in the voxel (µmol s–1 m–2 m–3) or (W m–2 m–3)
4. Diffuse APAR in the voxel (µmol s–1 m–2 m–3) or (W m–2 m–3)
5. Number of direct photons first scattered in the voxel
We employ the LEM for BRF sampling. The downward irradiance at the forest floor is an
important parameter to evaluate the light environment for the forest floor vegetation. We sample
it by counting the weight crossing over the horizontal plane at the forest floor. The parametersfrom 3 to 5 are required to calculate the canopy photosynthesis. The parameters 3 and 4 areequivalent to the heating rate in the atmospheric radiative transfer model. These parameters canbe sampled by counting the absorbed energy wi(1 – ω) after the scattering events.
In the springtime, deciduous forests have little leaf canopy (low leaf area density), causing adifficulty in sampling the APAR accurately, due to few scattering events. In this case, the scaling
method described in 4.2 (collision-forcing method) is effective to improve the computation timeand accuracy.

5.3.2 Simulated results
Figure 5.3.3 shows the simulated BRF in the blue, green, red, and near infrared spectral regions.
The x-axis indicates the observation angle over the principal plane. The negative angle indicatesthe solar direction (backscattering direction). The BRF in the backscattering direction is larger
than that in forward scattering. In the forward scattering direction, the BRF gradually decreases
with an increase in the observation angle. In the solar direction (backscattering direction), strong
BRF peaks are found. This is called the hotspot or opposition effect, which occurs when all of
the shadows are hidden from the observation view.
Figure 5.3.4 shows the nadir view simulated RGB images in a forest landscape. The left
image shows the lower leaf area index case and the right image is the higher leaf area index case.
When the leaf coverage in the canopy is low, the forest floor can be identified. Also, we canclearly identify the dark shadows caused by the stems and solar geometry (solar radiation isincident from the right side of these images).
Figure 5.3.5 shows the results of the 3-D distribution of the APAR at three different canopy
height levels. As shown on the right side of Fig. 5.3.5, we can identify the effect of the stem
shadow at a low canopy level. Figure 5.3.6 shows the spatial distribution of the downward
irradiance at the forest floor. Even if the canopy leaf area index is low (leaf area index = 0.5),
the irradiance in the shaded area is only 20% of the sunlit area.

173

Fig. 5.3.3 Bidirectional reflectance factor at the top of the canopy (principal
plane): (a) 450 nm, 550 nm, 650 nm and (b) 800 nm.

Fig. 5.3.4 Nadir view of the simulated forest landscape. Left: leaf area index = 0.5;
right: leaf area index = 2.0.

174

(µmol m–2 s–1)
Fig. 5.3.5 3-D APAR distribution in the forest canopy (leaf area index = 0.5). Left:average for 19–20 m height, middle: average for 14–15 m height, right: average for
9–10 m.

(µmol m–2 s–1)
Fig. 5.3.6 Downward PAR irradiance at the forest floor (leaf area index = 0.5).

175Chapter 6 Conclusion
This report described three-dimensional (3-D) radiative transfer models based on Monte Carlo(MC) methods. The fundamental theory of the model and the physical and optical properties of
the atmosphere and plant canopy were described, and details of the algorithms used for themodel were also described with significant emphasis. Recently, computing power usinglarge-scale computers has been rapidly growing, at a factor of 400 per 10 years. This tendency
can be expected to continue in the future (at least the near future). With the aid of thiscomputing power, the realism of the models used for scientific studies will improve, accurately
depicting physical processes that could not be treated explicitly in the past. As for radiation
models, more realistic processes could be included in the model in the future. As described inthis report, the MC methods enable the accurate and efficient calculation of 3-D radiativetransfer in a cloudy atmosphere and plant canopy. The model can precisely treat complicatedmixed media and a bidirectional reflectance distribution function, compute the radiative quantity
for each scattering order, and treat arbitrary objects with complicated geometrical shapes in anarbitrary coordinate system. In addition, the MC model is very efficient for calculating theradiative quantities averaged over a spectral band.
Although in this report, discussions were limited to a cloudy atmosphere and plant canopy,
the algorithms described in this report are not limited to these problems. For example, they canbe applied to the radiative transfer in city canopies, mountains (rough surfaces), or astrophysics,
as well as in medical sensing or mechanical engineering. The authors will be pleased if thesealgorithms help the reader to develop new models when studying new problems.
Marchuk et al. (1980) described very good calculation methods for atmospheric optics using
the MC methods. Even though 25 years have passed since these were published, they remain thebest for an MC radiative transfer model. For a plant canopy, too, the algorithms of Marchuk et al.
would be useful, because the theory behind the problem is very similar.

176

Acknowledgements
The authors are grateful to Prof. K. Tsuboki of Nagoya University and Dr. A. T. Noda of
FRCGC, JAMSTEC, for providing the large eddy simulation data for boundary layer clouds
used in 5.2.
This work was partly supported by KAKENHI (Wakate-Kenkyu B, No. 18710020) of the
Ministry of Education, Culture, Sports, Science and Technology (MEXT).
This work was supported by Japan EOS (Earth Observation System) Promotion Program of the
Ministry of Education, Culture, Sports, Science and Technology (MEXT).

177

Appendix

A1 Analytical functions for size distribution
Size distribution for cloud and aerosol particles is often expressed in analytical form. In thefollowing, common distribution functions are introduced, along with the relationships between
their parameters and M, V, and N. See also 2.1 for the representations of parameters.

A1.1 Power-law distribution
The size distribution of ice particles could be expressed as a combination of the power-law
functions (OPAC, Hess et al., 1998; Heymsfield and Platte, 1984):⎧0⎪n(r) = NMf × ⎨a1r b1
⎪ b2
⎩a2 r

for r < r0 or r2 < r
for r0 < r < r1

(A1.1)

for r1 < r < r2

Eight parameters other than M and N are required.

A1.2 Gamma distribution
The gamma distribution is often used for approximating the cloud particle size distribution:n(r) = N

⎛ r⎞
rα −1
exp⎜ − ⎟αβ Γ(α )⎝ β⎠

(A1.2)

where α > 0 is the scale parameter. The mode radius rmod isrmod = (α −1)β , α ≥ 1(A1.3)The moment (about zero) of the pth-order is given byβ pΓ(α + p)Γ(α )(A1.4)

Using this, various definitions of average radius and variance could be derived:

rgeo = βα(A1.5a)

178

2
rsec
= β 2α (α +1)
3rvol= β 3α (α +1)(α + 2)
vgeo = β 2α

(A1.5b)(A1.5c)(A1.5d)where vgeo is the variance. The standard deviation normalized by the average is

σ˜ ≡

vgeo
rgeo

=

1
α

(A1.6)The effective radius and variance are respectivelyrr
reff = (α + 2)β = (α + 2) mod = geo
α −1 1− 2veff
1veff =
α+2(A1.7a)(A1.7b)

The parameter κ of (2.2.23) is derived from (A1.5c) and (A1.7a) asκ=(α +1)α(α + 2)2(A1.8)

By representing β with M and N, the following equation is derived:

⎡
3
M⎤
β =⎢
⋅
⎥
⎣ 4 π (α + 2)(α +1)α ρN ⎦1/ 3

(A1.9)

By substituting this to (A1.7a), the effective radius can be written as a function of M (or V) andN.
To determine the parameters α and β from the effective radius and the volume mean radius,
the following can be used:r ⎡3⎤
β = eff ⎢3 − 1+ 8(rvol reff ) ⎥⎦4 ⎣r
α = eff − 2 .
β(A1.10a)(A1.10b)

Using rgeo or rmod for representing the size distribution, instead of using β,⎛ α ⎞α r α −1
⎛ α ⎞n(r) = N ⎜⎜ ⎟⎟exp⎜⎜−r ⎟⎟⎝ rgeo ⎠ Γ(α )⎝ rgeo ⎠⎛ α −1 ⎞α r α −1
⎛ α −1 ⎞n(r) = N ⎜
exp⎜−
r⎟
⎟
⎝ rmod ⎠ Γ(α )⎝ rmod ⎠179(A1.11a)(A1.11b)

Note that the mode radius is defined only when α > 1.

A1.3 Modified/generalized gamma distribution
The modified/generalized gamma distribution is also frequently used for expressing the cloudparticle size distribution:⎡ ⎛ r ⎞γ ⎤r αγ −1n(r) = N αγ
exp⎢−⎜ ⎟ ⎥β Γ(α )⎢⎣ ⎝ β ⎠ ⎥⎦(A1.12)where α, β, and γ are parameters. The mode radius is1γ⎛ αγ −1 ⎞rmod = β ⎜⎟⎝ γ ⎠for α >

1γ

(A1.13)The moment of pth order is given as⎛p⎞
β pΓ⎜α + ⎟
γ⎠
⎝
Γ(α )

(A1.14)

As for the gamma distribution, this can be used to derive various average radii:⎛
1⎞
rgeo = β Γ⎜α + ⎟ Γ(α )
γ⎠
⎝
⎛
2⎞
2
rsec
= β 2 Γ⎜α + ⎟ Γ(α )
γ⎠
⎝
⎛
3⎞
3
rvol
= β 3 Γ⎜α + ⎟ Γ(α )
γ⎠
⎝
⎛
⎛
3⎞
2⎞
reff = β Γ⎜α + ⎟ Γ⎜α + ⎟
γ⎠ ⎝
γ⎠
⎝

(A1.15a)(A1.15b)(A1.15c)(A1.15d)

The size distribution of (A1.12) could be expressed using rgeo or rmod, instead of β:Xαexp(−X )rΓ(α )⎛ Γ(α +1 γ ) r ⎞γ αγ −1 ⎛ r ⎞γ
X = ⎜⎜⎟⎟ =⎜⎟γ ⎝ rmod ⎠⎝ Γ(α ) rgeo ⎠n(r) = N

where the mode radius exists only when α > 1/γ.

180(A1.16a)(A1.16b)

A1.4 Lognormal distribution (number density)
Size distribution of number density for cloud and aerosol particles is often expressed by thelognormal distribution (Nakajima and King, 1990; Hess et al., 1998):⎡ 1 ⎛ ln r − ln r ⎞2 ⎤1
modn(r) = Nexp⎢− ⎜
⎟⎥
⎝
⎠⎦
2
ln
s
2 π r ln s⎣

(A1.17)

where rmod corresponds to the mode in the logarithmic scale (the mode of dN/dlnr). The momentof pth-order is⎛1 2 2 ⎞p ln s ⎟⎝2⎠(ln rmod ) exp⎜p(A1.18)Using this, the following are derived:⎛1⎞
rgeo = rmod exp⎜ ln 2 s ⎟⎝2⎠2
rsec = rmod exp(ln s)⎛3⎞rvol = rmod exp⎜ ln 2 s ⎟⎝2⎠[(A1.19a)(A1.19b)(A1.19c)]2vgeo = rmod
exp(ln 2 s) exp(ln 2 s) −1⎛5⎞
reff = rmod exp⎜ ln 2 s ⎟⎝2⎠2veff = exp(ln s) −1(A1.19d)(A1.19e)(A1.19f)

κ = exp(−3ln 2 s)

(A1.19g)

By substituting (A1.19g) into (2.2.25), the effective radius and the mode radius are written asfunctions of M and N:⎡ 3M ⎤
2
reff = ⎢⎥ exp(ln s)⎣ 4 πNρ ⎦1/ 3⎡ 3M ⎤⎛ 3 2 ⎞
rmod = ⎢⎥ exp⎜− ln s ⎟⎝ 2⎠⎣ 4 πNρ ⎦1/ 3(A1.20a)(A1.20b)

Additionally, the mode radius and the standard deviation (s) are expressed as follows:⎛ r ⎞3/2
rmod = rvol ⎜⎜ vol ⎟⎟ = rvol κ⎝ reff ⎠⎛r ⎞1
ln 2 s = ln⎜ eff ⎟ = − ln κ3
⎝ rvol ⎠(A1.21a)(A1.21b)181

A1.5 Lognormal distribution (volume density)
The volume size distribution could be assumed to be expressed by the lognormal function(Hayasaka et al., 1994; Higurashi and Nakajima, 1999):⎡ 1 ⎛ ln r − ln rmod ⎞ 2 ⎤dV
1v(r) ≡=V
exp ⎢− ⎜
⎟ ⎥
⎠ ⎥⎦
dr
ln s
2π r ln s⎢⎣ 2 ⎝(A1.22)In this case, the number density is written asn(r) =dV3
×
dr 4 πr 33⎡ 1 ⎛ ln r − ln r ⎞2 ⎤
rvol
mod
=N
exp
⎢− ⎜
⎟⎥
4
⎝
⎠⎦
ln s
2 π r ln s
⎣ 2

(A1.23)

Various radius averages are as follows:⎛ 5⎞rgeo = rmod exp⎜ − ln 2 s⎟
⎝ 2⎠

(A1.24a)rsec = rmod exp(−2ln 2 s )(A1.24c)⎛ 3⎞rvol = rmod exp⎜ − ln 2 s⎟
⎝ 2
⎠
⎛ 1
⎞
reff = rmod exp⎜ − ln 2 s⎟
⎝ 2⎠

(A1.24c)(A1.24d)κ = exp(−3ln 2 s ) .

(A1.24e)

By substituting (A1.24e) into (2.2.25), the effective radius and the mode radius are written withM (or V) and N:⎡ 3M ⎤
2
reff = ⎢⎥ exp(ln s)
4
π
N
ρ
⎣
⎦
1/ 3
⎡ 3M ⎤⎛1 2 ⎞
rmod = ⎢⎥ exp⎜ ln s ⎟⎝2⎠⎣ 4 πNρ ⎦1/ 3(A1.25a)(A1.25b)

The mode radius and the standard deviation (s) are expressed as⎛ reff ⎞rmod = rvol ⎜ ⎟⎝ rvol ⎠

3/ 2

rvol
κ

(A1.26a)⎛r ⎞ln κln 2 s = ln⎜ eff ⎟ = −3⎝ rvol ⎠(A1.26b)=182

A2 Optical properties of mixed media
Let us consider a mixed medium with several kinds of aerosol, cloud water, and ice particle
dispersions. The optical and physical properties of each component are already known. If theabsolute amounts (N(i) or V(i) or q(i) for ith component) of the respective components are
known, the optical properties of the mixed medium are calculated, for a wet status withhygroscopic aerosols, as follows:wet
βe = ∑σ e (i)N(i) = ∑ Qe (i)π (rsec(i)) N(i)
2

i

i

wet
β s = ∑σ s (i)N(i) = ∑ Qs (i)π (rsec(i)) N(i)
2

i

ω=

βs
βe

(A2.1a)i(A2.1c)∑σ (i)N (i)P(i,Θ)
s

P (Θ) =

(A2.1a)i∑σ (i)N(i)
s

=

2
1
wet
Qs (i)π (rsec(i)) N(i)P(i,Θ)∑βs i(A2.1d)i

Another case is where the absolute amounts are unknown but the mixing ratio is known.
With the number density mixing ratio αN,∑α (i) =1(A2.2a)Ni

α N (i) =N (i)N(A2.2b)where the total number density of the mixed medium is

N = ∑ N(i)(A2.3)i

The cross-section-mean radius, the volume-mean radius, and the internal mass density of thewet/dry polydispersion are given as⎡⎤
2dry/ wetr= ⎢∑ (rsec(i)) α N (i)⎥
⎣i
⎦
1/ 3
⎡
⎤
3
dry/ wetdry/ wetrvol= ⎢∑ (rvol (i)) α N (i)⎥
⎣i
⎦
1/2

dry/ wetsec∑ (rρ dry/ wet = idry/ wet
vol(A2.4a)(A2.4b)(i)) ρ dry/wet (i)α N (i)3∑ (rvoldry/ wet (i)) α N (i)
3

i

183

(A2.4c)The optical properties of wet polydispersion are calculated by∑Q (i)π (r (i)) α (i)

Qe =

i

2

wet
sec

e

N

(A2.5a)∑ π (r (i)) α (i)
2

wet
sec

N

i

∑Q (i)π (r (i)) α (i)Qs = i

2

wet
sec

s

N

(A2.5b)∑ π (rsecwet (i)) α N (i)
2

i

Q
ω= s
Qe

(A2.5c)∑Q (i)π (r (i)) α (i)P(i,Θ)

P (Θ) = i

2

wet
sec

s

N

(A2.5d)∑Qs (i)π (rsecwet (i)) α N (i)2iUsing these formulas, the extinction and scattering coefficients are calculated from the total
number density:

βe = N × Qe (rsecwet ) , β s = N × Qs (rsecwet )2

2(A2.6)

Instead of the number density mixing ratios, the volume density mixing ratios at the dry/wet
status or the mass mixing ratios could be the known parameters. In such a case, the number
mixing ratios can be converted from the known ratios if the volume-mean radius and internalmass density of each component are known. For various mixing ratios,∑α (i) =∑α (i) =∑α (i) =∑α (i) =1
dry
V

i

dry
M

i

wet
V

wet
M

i

(A2.7)i

The relationship with the number mixing ratio is as follows:

αVdry/ wet (i) ∝ α N (i) × (rvoldry/ wet (i))

3

(A2.8a)α Mdry/ wet (i) ∝ α N (i) × (rvoldry/ wet (i)) ρ dry/ wet (i)3(A2.8b)Using this and (A2.7), any mixing ratios can be converted to number mixing ratios. Table A1
lists the optical properties of typical aerosol mixtures, calculated according to Hess et al. (1998).

184

Table A1 Optical properties of typical aerosol mixtures: wavelength of 0.55 µm,
relative humidity of 80%, asymmetry parameter g.
Aerosol typeContinental clean
Continental average
Continental polluted
Urban
DesertMaritime clean
Maritime polluted
Maritime tropical
ArcticAntarctic
Mineral transported
Yellow dust
Stratosphere
Volcanic ash

reff (µm)
0.23
0.21
0.18
0.18
1.28
1.07
0.69
1.13
0.29
0.46
2.37
2.19
0.44
0.44

ρ (102 kg m–3)
15.1
14.9
14.3
14.6
25.5
12.2
12.2
12.2
12.4
11.2
26.0
20.0
11.0
10.0

Qe
0.941
0.893
0.853
0.796
1.92
1.94
1.49
2.02
1.00
2.33
2.34
2.43
2.33
2.50

ω
0.973
0.930
0.899
0.828
0.889
0.997
0.976
0.998
0.893
1.000
0.820
0.871
1.000
0.930

g
0.710
0.706
0.700
0.692
0.729
0.768
0.753
0.770
0.721
0.776
0.782
0.738
0.776
0.707

A3 Spectrally averaged optical properties
Optical properties spectrally averaged over a spectral band are required for the radiative transfer
calculation for the band. The spectral averaging is usually performed with weighting by thespectral source flux S(λ) (solar incident flux or the Planck function for the thermal emission):Q˜ e =Q˜ s =Q˜ a =

∫

λ max
λ min

∫

∫

Qe (λ )S(λ )dλ
λ max

λ min
λ max

Qs (λ )S(λ )dλ

λ min

∫

∫

λ max

λ min

(A3.1b)S(λ )dλ

λ min
λ max

Qa (λ )S(λ )dλ

∫
∫

λ max

λ min
λ max

P˜ (Θ) =(A3.1a)S(λ )dλλ min∫(A3.1c)S(λ )dλP(Θ, λ )Qs (λ )S(λ )dλ
λ max
λ min

Qs (λ )S(λ )dλ(A3.1d)These could result in an inconsistent balance between the scattering and absorption efficiency
factors:

Q˜ e ≠ Q˜ s + Q˜ a

(A3.2)

185

This causes many possibilities to define the spectrally averaged single scattering albedo:ω (0) =

∫

λ max
λ min

∫

ω (λ )S(λ )dλ

λ max

λ min

(A3.3a)S(λ )dλω (1) = Qs Qe
ω (2) = (Qe − Qa ) Qeω (3) = Qs (Qs + Qa )1ω (4) = (ω (1) + ω (2))
2

(A3.3b)(A3.3c)(A3.3d)(A3.3e)There are also other definitions (e.g., Fu, 1996; Nakajima et al., 2000). The differences between
these definitions are especially large near a wavelength where the absorption coefficient varies
rapidly with wavelength. The best choice for the definition would depend on the problem: thestrength of the absorption and shape of the source flux spectrum.

A4 Vertical profile of aerosols
The vertical profile of aerosols is often expressed, as a first-order approximation, by an
exponential function. If the compositions of the medium, size distributions, mixing ratios, and(for hygroscopic aerosols) the relative humidity are constant for a vertical column, then theaerosol extinction coefficient should solely follow the profile of the number density. If therelative humidity is not constant, then the proportionality becomes invalid.
Let us assume an exponential vertical profile for the number density:

N(z) = N (0)exp(−sz), s =1.
Zs(A4.1)

With bottom and top altitudes (Zmin and Zmax, respectively), the vertically integrated number
concentration per unit cross section of the column is

N col = ∫ Z

Z max

N (z)dzmin[exp(−sZ min ) − exp(−sZ max )]= N (0)(A4.2a)s

If s is nearly equal to 0,

N col ≈ N (0)( Z max − Z min )(A4.2b)From (A4.2a) or (A4.2b), the vertical profile is uniquely determined by N(0) or Ncol. Eq. (A4.1)
becomes186

N(z) = N cols exp(−sz)
exp(−sZ min ) − exp(−sZ max )(A4.3)The vertical profile of the extinction coefficient is

βe (z) = N (z)σ e (z)2= Qe (z)πrsec(z)N cols exp(−sz)
exp(−sZ min ) − exp(−sZ max )(A4.4)

where the average extinction efficiency Qe(z) should be calculated for the wet status of themixed medium, using the method described in the previous subsection. The average extinction
efficiency is constant within the layer and could be written as Qe (j). The optical thickness forthe vertical column (integral of the extinction coefficient) is given asτ = ∫ZZ maxminβe (z)dz= N col

Z maxs2
Qe (z)πrsec(z)exp(−sz)dz∫
exp(−sZ min ) − exp(−sZ max ) Z min= N colzjs⎡⎤2
Qe ( j)πrsec( j) ∫ exp(−sz)dz⎥
∑
⎢
z
⎦
j−1
exp(−sZ min ) − exp(−sZ max ) j ⎣= N col

1∑ Qe ( j)πrsec2 ( j) exp(−sz j−1 ) − exp(−sz j )exp(−sZ min ) − exp(−sZ max ) j

{

[

]}

(A4.5)
From this, Ncol can be calculated if τ is given. In addition, using the volume-mean radiusaveraged over all components, V(z) can be calculated from N(z). The vertically integrated
volume density (m3 m–2) is

4 3
N(z) πrvol
(z)dz
min
min
3
⎧4 3
⎫
1
= N col⎨ πrvol ( j) exp(−sz j−1 ) − exp(−sz j ) ⎬∑⎭exp(−sZ min ) − exp(−sZ max ) j ⎩ 3

Vcol = ∫ Z

Z max

V (z)dz = ∫ Z

Z max

[

]

(A4.6)
Thus, Ncol can be derived from Vcol. Similarly, Ncol can be derived from the vertically integrated
mass density Mcol (kg m–2). Furthermore, the volume and mass could be formulated for both thedry and wet statuses. Finally, a single parameter out of τ, Vcol, and Mcol at the dry/wet status isenough to derive all of the other parameters.
Let us assume that the mixing ratios of aerosol species and relative humidity are all constant.
Eqs. (A4.5–6) can be reduced todry/ wetτ dry/ wet = N colQedry/ wet π (rsec)

2(A4.7a)187

4 π dry/ wet 3(rvol )
34 π dry/ wet 3 dry/ wetdry/ wet
M col= N col(rvol ) ρ3

Vcoldry/ wet = N col(A4.7b)(A4.7c)Observations using an instrument such as lidar show that the vertical profile of theextinction coefficient is roughly exponential and that the extinction coefficient is somewhat
large in the planetary boundary layer. If a more complicated profile shape is needed, acombination of several functions (e.g., the gamma distribution) would be a choice.

A5 Representation of photon transport by direction vector
The transport direction of the photon can be expressed by a unit vector with (x, y, z) components.
For the zenith angle θ and azimuth angle φ, the direction vector is⎛u ⎞ ⎛sin θ cos φ ⎞⎜ x⎟ ⎜⎟
Ω = ⎜u y ⎟ = ⎜sin θ sin φ ⎟⎟⎜ ⎟ ⎜cos θ⎠⎝u z ⎠ ⎝(A5.1)

u x2 + u y2 + u z2 = 1(A5.2)sin θ = u 2x + u 2y = 1− u z2

(A5.3)

where

A5.1 Movement of photon
When the photon travels in the direction Ω by a path length l, the coordinates of the photon
become⎛ x ′⎞ ⎛ x ⎞⎜ ⎟ ⎜ ⎟⎜ y′⎟ = ⎜ y ⎟ + lΩ
⎜ ⎟ ⎜ ⎟⎝ z′ ⎠ ⎝ z ⎠(A5.4)

A5.2 Rotation of direction due to scattering
When scattering occurs, the photon transport direction rotates. The unit vector Ω rotates by anangle Θ to increase the zenith angle and subsequently rotates by an azimuth angle Φ about theoriginal vector Ω. The new direction is188⎡⎛u x u z sin θ ⎞⎛−u y sin θ ⎞⎤
⎢
⎜
⎟
⎜
⎟⎥
Ω′ = cosΘΩ + sin Θ⎢cosΦ⎜u yu z sin θ ⎟ + sin Φ⎜u x sin θ ⎟⎥
⎜
⎟
⎜0
⎟⎥
⎢
⎝−sin θ⎠⎝⎠⎦⎣(A5.5)The numerical evaluation of this formula fails if sin 2 θ = u x2 + u y2 = 1− u z2 ≈ 0 . In such a case, thenew direction can be calculated by⎛sin ΘcosΦ⎞⎜⎟Ω′ = sign(u z )⎜ sin Θsin Φ ⎟⎜⎟⎝ cosΘ ⎠(A5.6)

where sign(.) is a function to return +1 or –1 depending on the sign of the argument.

A5.3 Angle between two vectors
For the angle between two vectors Ω 1 and Ω 2,cos α = Ω1 ⋅ Ω2 .

(A5.7)

The angle can be derived from this formula. However, it is numerically difficult if α is nearly 0or π, because of the round-off error. This problem is especially important for the simulation of alidar signal, for example. The following gives an algorithm for accurate calculation of the angle
even when α is nearly 0 or π.
First, let us derive sinα for a small α. For the narrow triangle with the two vectors Ω 1 andΩ 2, the length of the other side isa 2 = Ω1 − Ω2

2

(A5.8)

From the law of cosines,

a 2 +1− 2a cos β = 1(A5.9)∴sin β = 1− a 4(A5.10)2

From this and the law of sines,

sin α = a sin β = a 1− a 2 4(A5.11)

Next, fromy = sin α − C = 0(A5.12)and using the first-order Newtonian formula,189

α =C−y(α = C )sin C − C=C−y′(α = C )cosC(A5.13)

Expanding sinC and cosC by the Taylor series, truncating at the second term, and substituting itto the above formula, we get

α =C

6 − 2C 2.
6 − 3C 2(A5.14)The solution is obtained by substituting C = sinα into this equation.

A6

Determination of random point in a circle: Polar coordinate

methodAt a scattering or reflection event, a random number and the sine and cosine of a random
azimuth angle are needed. A possible method to get the random azimuth is to computeΦ = 2 πρ Φ

(A6.1)

using a random number uniformly distributed in [0, 1]. However, the required values are its sineand cosine, both of which are numerically heavy functions. A better method is the polar
coordinate method, which directly determines ρ, cosΦ,sin Φ (Fushimi, 1989). From two
random numbers, ρ1 and ρ2,

W1 = 1− 2 ρ1 ,W2 = 1− 2 ρ 2(A6.2)

r 2 = W12 +W22
are computed. This is iterated until

10 −12 ≤ r 2 ≤ 1(A6.3)Such an iteration converges fast because the probability that (A6.3) is true is large as π/4 ~
0.785, for one iteration cycle. Finally, random points in a circle in two dimensions are
determined, and such points should be distributed uniformly within the circle. The coordinatesof the points are given by W1 and W2. Then,ρ = r2cosΦ = W1 rsinΦ = W2 r

(A6.4)

This method does not require the evaluation of the sine and cosine!

190

For example, let us consider the Lambertian reflection. The azimuth angle after thereflection should be distributed isotropically and cos2θ should be uniform, so that the following
is derived using (A6.4):

cosθ = ρ = r
cos φ = W1 r
sin φ = W2 r

(A6.5)The reflection vector becomes⎛W 1−r 2 r ⎞
⎜ 1
⎟
2
⎜
Ω = W2 1−r r ⎟
⎜
⎟
r
⎜
⎟
⎝
⎠

(A6.6)

A7 Mathematical functions
A7.1 cos–1x
The solution of this function should be in a range between 0 and π . This function is numericallyvery heavy, so that an LUT is useful for –0.99 < x < 0.99. If x is nearly 1, then the following
approximate calculation is useful.
Let us try to solve the following equation:y = cosα − x = 0(A7.1)The initial guess by first-order approximation isα ≈ 2(1− x)(A7.2)

Applying the Newtonian method and using the Taylor series expansion,

α = 2(1− x) −
= 2δ

(
)
y′(α = 2(1− x) )y α = 2(1− x)(A7.3)

60 −15δ + 2δ60 − 20δ + 2δ 2
2where δ = 1 – x. If x is nearly equal to –1, then the solution is α = π – cos–1(–x).

191

A7.2 exp(–x)
This function is frequently used for the radiative transfer for calculating the transmittance,
usually with an optical thickness x between 0 and xmax = 80. The transmittance exp(–x) for an x
larger than 80 is almost equal to 0, with single precision. This function is tabulated in an LUT
with about 10000 grid points. For evaluating the function, the following are computed:⎛ N⎞
i = int⎜x + 0.5 ⎟⎝ xmax
⎠
xmax
δ= x−
i.
N

(A7.4a)(A7.4b)Using the Taylor series expansion,exp(−x) ≈ LUT(i)[1− δ (1− δ / 2)](A7.5)

With N = 10000, and xmax = 80, the accuracy of the above approximation is 0.00001% or better.
If x < 0.01, more simply

exp(−x) ≈ 1− x(1− x / 2)

(A7.6)

can be used.

A7.3 1 – exp(–x)
This function is used for calculating the collision probability and heating rate. When x is large(0.03 or more), the above function exp(–x) could be used. However, caution should be paid for asmall x because of the round-off error. Using the Taylor series expansion,⎡⎛ 1 x ⎞⎤
1− exp(−x) ≈ x⎢1− x⎜ − ⎟⎥⎝ 2 6 ⎠⎦⎣

(A7.7)The maximum error for this is less than 0.0001% for x < 0.03.

A7.4 sin(x) and cos(x)These functions are numerically time-consuming. Let us assume that the argument x is between
0 and 2π. An LUT of about 10,000 grid points is useful. For evaluating the sine and cosine,⎛N⎞i = int⎜x + 0.5 ⎟⎝ 2π⎠(A7.8)

192

δ= x−2πiN(A7.9)are computed. Representing the sine and cosine LUT values S and C, respectively, using the
Taylor series expansion, we getsin(x) ≈ S(i) + δC(i)cos(x) ≈ C(i) − δS(i)(A7.10a)(A7.10b)

If N = 10000, the error of the above is less than 0.00001%.

193

References-in JapaneseAraya, S., 2003: 明解 3 次元コンピュータグラフィックス. 共立出版 (Kyoritsu-shuppan),
pp. 191 (in Japanese).
Shibata, K., 1999: 光の気象学, 朝倉書店 (Asakura-shoten), pp. 182 (in Japanese).
Taniguchi, H., W-J. Yang, K. Kudoh, A. Kuroda, A. Mochida, 1994: パソコン活用のモンテカ
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Fushimi, M., 1989: 乱数. UP 応用数学選書 12 (UP Applied Mathematics Series 12), 東京大学
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d’Almeida, G. A., P. Koepke, and E. P. Shettle, 1991: Atmospheric Aerosols. Global
Climatology and Radiative Characteristics. A. Deepak Publishing, pp. 561.
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canopies, Remote Sens. Environ., 31, 183.4.23.2091.
Antyufeev, V. S., 1996: Solution of the generalized transport equation with a peak-shaped
indicatrix by the Monte Carlo method. Russ. J. Numer. Anal. Math. Modeling, 11,
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