Two- and Four-Stream Combination Approximations for ...qfu/Publications/jas.hua.2010.pdf · Two- and Four-Stream Combination Approximations for Computation of Diffuse Actinic Fluxes

Two- and Four-Stream Combination Approximations for Computationof Diffuse Actinic Fluxes

HUA ZHANG

National Climate Center, China Meteorological Administration, Beijing, China

FENG ZHANG

Chinese Academy of Meteorological Sciences, Beijing, China

QIANG FU

Department of Atmospheric Sciences, University of Washington, Seattle, Washington

ZHONGPING SHEN

Shanghai Climate Center, Shanghai, China

PENG LU

Chinese Academy of Meteorological Sciences, Beijing, China

(Manuscript received 3 November 2009, in final form 25 April 2010)

ABSTRACT

The d-two- and four-stream combination approximations, which use a source function from the two-stream

approximations and evaluate intensities in the four-stream directions, are formulated for the calculation of

diffuse actinic fluxes. The accuracy and efficiency of the three computational techniques—the d-two-stream

approximations, the d-two- and four-stream combination approximations based on various two-stream ap-

proaches, and the d-four-stream approximation—have been investigated. The diffuse actinic fluxes are ex-

amined by considering molecular, aerosol, haze, and cloud scattering over a wide range of solar zenith angles,

optical depths, and surface albedos. In view of the overall accuracy and computational efficiency, the d-two-

and four-stream combination method based on the quadrature scheme appears to be well suited to radiative

transfer calculations involving photodissociation processes.

1. Introduction

Photochemical processes in the atmosphere are driven

by solar radiation, which dissociates certain molecules into

reactive atoms or free radicals. Models that simulate the

chemistry of the atmosphere must accurately simulate

the radiation processes that initiate photodissociation. The

photodissociation rate is proportional to the actinic flux. In

addition, for some biological applications such as exposure

of small ‘‘bodies’’ suspended in air or in water (e.g., phy-

toplankton in the ocean), actinic fluxes are also used to

compute the dose rate and total dose (Kylling et al. 1995).

A variety of approximate techniques are now com-

monly used for the calculation of actinic fluxes, including

the variational method (Yung 1976), isotropic integra-

tion (Anderson and Meier 1979), the Isaksen–Luther

method (Isaksen et al. 1977; Luther 1980; Thompson

1984; Madronich 1987; Dvortsov et al. 1992), various

two-stream methods (Liou 1974: Coakley and Chylek

1975; Joseph et al. 1976; Meador and Weaver 1980; Toon

et al. 1989; Kylling et al. 1995; Qiu 1999; Lu et al. 2009),

four-stream methods (Liou et al. 1988; Li and Dobbie

1998; Li and Ramaswamy 1996), and discrete ordinates

methods (Stamnes et al. 1988).

Among these techniques, the variational method,

isotropic integration, and Isaksen–Luther method as-

sume that the phase function is isotropic or that the light

Corresponding author address: Prof. Hua Zhang, National Cli-

mate Center, Beijing 100081, China.

E-mail: [email protected]

3238 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67

DOI: 10.1175/2010JAS3370.1

� 2010 American Meteorological Society

scattered by an entire layer is isotropic. Such approxi-

mations cannot be applied to the presence of clouds and

aerosols, which have strong forward scattering. The

d-two-stream methods (Meador and Weaver 1980; Toon

et al. 1989; Kylling et al. 1995), which were developed

to calculate multiple scattering in aerosols and clouds,

are widely used to simulate diffuse actinic fluxes. The

d-four-stream method is more accurate but also more

computationally expensive. Even though computer speed

is rapidly improving these days, saving computational

cost is still very important in global climate models and

in some remote sensing operational applications. Here

we develop the d-two- and four-stream combination ap-

proximations for the calculation of diffuse actinic fluxes,

which are shown to be more accurate than the two-

stream methods and more efficient than the d-four-

stream method. Note that the photodissociation rate is

proportional to the total actinic flux that is summation

of the direct and diffuse radiation, and the former is

often dominated by the latter. Lary and Pyle (1991)

found that a correct treatment of the diffuse radiative

field is important in the modeling of ozone above

35 km. Considerable efforts were made to include the

effects of diffuse radiation in photochemical models

(Luther and Gelinas 1976; Fiocco 1979; Mugnai et al.

1979). In addition, Leighton (1961) showed that diffuse

radiation is more effective than direct radiation by

a factor of 2 cos(u0) (Madronich 1987).

The d-two- and four-stream combination approxima-

tions are developed based on the source function tech-

nique proposed by Davies (1980) and Toon et al. (1989).

Fu et al. (1997) showed that they are suitable for the

radiative flux and heating rate calculations in the

infrared, with an accuracy close to the d-four-stream

method but a computational efficiency only about 50%

more than the d-two-stream methods. Unfortunately,

for the solar radiation, when the single scattering albedo

is equal to 1, these approaches do not necessarily yield

conserved radiative fluxes (Toon et al. 1989). In the in-

frared and solar spectra, the approaches do yield a useful

approximation to intensities and can be used to obtain

quantities such as the geometric albedo that cannot be

found with various two-stream approximations (Toon

et al. 1989). However, little work has been done to apply

the d-two- and four-stream combination approximations

to the calculations of actinic fluxes.

In section 2, we briefly introduce the d-two-stream and

d-four-stream approximations and formulate the d-two-

and four-stream combination approximations based on

various two-stream approaches. In section 3, we examine

the accuracy and computational efficiency of these ap-

proximations for a wide range of cases. A summary and

conclusions are given in section 4.

2. Theory and method

a. Basic equations

To obtain the diffuse actinic fluxes, we begin with the

azimuthally averaged radiative transfer equation for the

diffuse solar intensity I(t, m) in plane-parallel atmo-

spheres (e.g., Liou et al. 1988):

mdI(t, m)

dt5 I(t, m)� -

4ppE

0e�t/m0 P(m, �m

0)

� -2

ð1

�1

I(t, m9)P(m, m9) dm9, (1)

TABLE 1. Summary of coefficients in the selected two-stream approximations. The Eddington and quadrature schemes are discussed

in detail by Meador and Weaver (1980). The hemispheric constant scheme is derived by assuming that the phase function P(m, m9) 5

1 1 3gmm9.

Method r1 r2 r3

Eddington 1/4[7 2 -(4 1 3g)] 21/4[1 2 -(4 2 3g)] 1/4(2 2 3gm0)

Quadrature (31/2/2)[2 2 -(1 1 g)] (31/2-/2)(1 2 g) 1/2(1 2 31/2gm0)

Hemispheric constant 2 2 -(1 1 3/4g) -(1 2 3/4g) 1/4(2 2 3gm0)

TABLE 2. Summary of coefficients in the two- and four-stream combination approximations. In the table, a(6m) 5Ð 1

0 P(6m, m9) dm9.

Method based on G(6m) H(6m) z(6m)

Eddington-4p

g1[2(1 1 R) 6 3gm(1� R)]-4p

g2[2(R 1 1) 6 3gm(R� 1)]-4p

[pE0P(6m, �m0) 1 2(Z1

1 Z�)

6 3gm(Z1

1 Z�)]

Quadrature31/2-g1

4pha(6m) 1 Rf[2� a(6m)]gi

31/2-g2

4p[Ra(6m) 1 2� a(6m)]

-4pfpE

0P(6m, �m

0) 1 31/2[Z

1a(6m)

1 2Z�� Z�a(6m)]g

Hemispheric constantg1-

2pha(6m) 1 Rf[2� a(6m)]gi

g2-

2p[Ra(6m) 1 2� a(6m)]

-4pfpE0P(6m, �m0) 1 2[Z

1a(6m)

1 2Z�� Z�a(6m)]g

OCTOBER 2010 Z H A N G E T A L . 3239

where m is the cosine of the zenith angle; t the optical

depth; - the single scattering albedo; P(m, m9) the

azimuthally averaged scattering phase function, defining

the light incidence at m9, which is scattered in the di-

rection m; pE0 the direct solar irradiance perpendicular

to the solar direct beam; and m0 the cosine of the solar

zenith angle. The quantity P(m, m9) can be written as

(e.g., Liou et al. 1988)

P(m, m9) 5 �N

l50-

lP

l(m)P

l(m9), (2)

where Pl(m) is the Legendre function and -l is the ex-

pansion coefficient of the scattering phase function in

terms of the Legendre polynomials (e.g., Liou et al. 1988);

-0 5 1 and -1 5 3g, where g is the asymmetry factor.

The diffuse actinic fluxes and irradiance fluxes in the

upward and downward directions are

F 5 2p

ð1

�1

I(t, m) dm, (3)

E6 5 2p

ð1

0

I(t, 6m)m dm. (4)

Next, we introduce various approximations for com-

putation of the diffuse actinic fluxes.

b. Two-stream approximations

Two-stream approximations have been widely used

for many years to rapidly solve radiative transfer prob-

lems. They avoid the complex and lengthy algorithms

necessary for numerical solutions of the radiative transfer

equation by yielding analytical solutions that are relatively

easy to implement. Meador and Weaver (1980) have

shown that various two-stream schemes can generally

be expressed in the form

›E 11

›t5 r

1E 1

1 � r2E�1 � r

3pE

0-e�t/m0 , (5)

›E�1›t

5 r2E 1

1 � r1E�1 1 (1� r

3)pE

0-e�t/m0 , (6)

where r1, r2, and r3 are coefficients that depend on the

particular form of the two-stream schemes. Table 1 pres-

ents the values of r1, r2, and r3 for some commonly used

two-stream approximations (Meador and Weaver 1980;

Thomas and Stamnes 2002).

The solution for Eqs. (5) and (6) can be written as

TABLE 3a. Diffuse actinic fluxes at the layer top for conservative Rayleigh layer. The d-two-stream approximation and d-two- and four-

stream combination approximation are based on the Eddington scheme.

F(0)

R m0 t1 Exact Eddington Err (%) SFE4 Err (%) Four Err (%)

0 0.10 0.02 0.0454 0.0181 260.13 0.0286 237.00 0.0287 236.78

0.25 0.1698 0.0973 242.70 0.1422 216.25 0.1477 213.02

1.00 0.2116 0.1343 236.53 0.1808 214.56 0.1915 29.50

0.4 0.02 0.0471 0.0195 258.60 0.0299 236.52 0.0307 234.82

0.25 0.2837 0.1889 233.42 0.2626 27.44 0.2667 25.99

1.00 0.5342 0.4268 220.11 0.5116 24.23 0.5276 21.24

0.92 0.02 0.0395 0.0198 249.87 0.0266 232.66 0.0282 228.61

0.25 0.2793 0.2205 221.05 0.2724 22.47 0.2768 20.90

1.00 0.6907 0.6562 24.99 0.7053 2.11 0.7188 4.07

0.25 0.10 0.02 0.089 0.0631 229.10 0.0730 217.98 0.0730 217.98

0.25 0.1891 0.1198 236.65 0.1627 213.96 0.1672 211.58

1.00 0.2196 0.1448 234.06 0.1898 213.57 0.1993 29.24

0.4 0.02 0.2345 0.2125 29.38 0.2210 25.76 0.2208 25.84

0.25 0.4023 0.3229 219.74 0.3840 24.55 0.3863 23.98

1.00 0.5837 0.4865 216.65 0.5624 23.65 0.5762 21.28

0.92 0.02 0.4765 0.4698 21.41 0.4723 20.88 0.4717 21.01

0.25 0.5972 0.5754 23.65 0.5984 0.20 0.5945 20.45

1.00 0.8505 0.8456 20.58 0.8697 2.26 0.8772 3.14

0.8 0.10 0.02 0.1869 0.1632 212.68 0.1722 27.87 0.1721 27.92

0.25 0.2388 0.1765 226.09 0.2146 210.13 0.2174 28.96

1.00 0.2471 0.1800 227.16 0.2203 210.85 0.2262 28.46

0.4 0.02 0.6531 0.6420 21.70 0.6464 21.03 0.6459 21.10

0.25 0.707 0.6601 26.63 0.6935 21.91 0.6953 21.65

1.00 0.7781 0.6864 211.79 0.7349 25.55 0.7432 24.49

0.92 0.02 1.453 1.4716 1.28 1.4644 0.78 1.4633 0.71

0.25 1.404 1.4693 4.65 1.4221 1.29 1.4157 0.83

1.00 1.398 1.4797 5.84 1.4250 1.93 1.4211 1.65


E 11 (t) 5 g

1e�k(t1�t) 1 g

2Re�kt 1 Z

1e�t/m0 , (7)

E�1 (t) 5 g1Re�k(t1�t) 1 g

2e�kt 1 Z�e�t/m0 , (8)

where

k 5 (r21 � r2

2)1/2, (9)

R 5r

1� k

r2

5r

2

r1

1 k, (10)

Z1

5-pE

0[(r

1� 1/m

0)r

31 r

2(1� r

3)]

k2 � 1/m20

, (11)

Z�5-pE

0[(r

11 1/m

0)(1� r

3) 1 r

2r

3]

k2 � 1/m20

, (12)

and the coefficients g1,2 are to be determined from

boundary conditions.

In the two-stream approximations, the diffuse actinic

flux is given by

F1(t) 5 2p

ð1

�1

I(t, m) dm 5 [E11(t) 1 E�1 (t)]/m1 , (13)

where

m1 5 (1� -)/(r1� r

2). (14)

The parameter m1 is a constant such as ½ orffiffiffi3p

rather

than a frequency-dependent property (Toon et al. 1989).

c. Four-stream approximation

The four-stream approximation is based on the gen-

eral solution to the discrete ordinates method for radi-

ative transfer. It has been discussed in detail by Liou

et al. (1988) and Fu and Liou (1993). Using Gaussian

quadrature and the phase function expansion in Eq. (2),

the four-stream approximation may be written as

mi

dI(t,mi)

dt5I(t, m

i)�-

2�

3

l50-

lP

l(m

i)

3 �2

j5�2I(t, m

j)P

l(m

j)a

j� -

4ppE

0

3 �3

l50-

lP

l(m

i)P

l(�m

0)e�t/m0 , i561,62,

(15)

where the quadrature point m2j 5 2mj, j 5 1, 2, and the

weight a2j 5 aj, �2

j5�2a� j 5 2. The ‘‘double Gauss’’

TABLE 3b. As in Table 3a, but for the diffuse actinic fluxes at the layer base.

F(t1)

R m0 t1 Exact Eddington Err (%) SFE4 Err (%) Four Err (%)

0 0.10 0.02 0.0451 0.0181 259.87 0.0285 236.81 0.0284 237.03

0.25 0.1091 0.0863 220.90 0.1164 6.69 0.1124 3.02

1.00 0.0566 0.0657 16.08 0.0626 10.60 0.0543 24.06

0.4 0.02 0.0471 0.0195 258.60 0.0299 236.52 0.0294 237.58

0.25 0.2521 0.1829 227.45 0.2486 21.39 0.2477 21.75

1.00 0.3026 0.3076 1.65 0.3148 4.03 0.3023 20.10

0.92 0.02 0.0395 0.0198 249.87 0.0266 232.66 0.0254 235.70

0.25 0.2663 0.2174 218.36 0.2661 20.08 0.2685 0.83

1.00 0.5435 0.5633 3.64 0.5708 5.02 0.5672 4.36

0.25 0.10 0.02 0.0928 0.0644 230.60 0.0751 219.07 0.0754 218.75

0.25 0.1417 0.1173 217.22 0.1488 5.01 0.1448 2.19

1.00 0.0818 0.0920 12.47 0.0905 10.64 0.0792 23.18

0.4 0.02 0.2517 0.2183 213.27 0.2300 28.62 0.2312 28.14

0.25 0.4523 0.3670 218.86 0.4414 22.41 0.4475 21.06

1.00 0.4577 0.4568 20.20 0.4720 3.12 0.4572 20.11

0.92 0.02 0.5168 0.4833 26.48 0.4936 24.49 0.4960 24.02

0.25 0.7987 0.7054 211.68 0.7787 22.50 0.7993 0.08

1.00 1.047 1.0368 20.97 1.0683 2.03 1.0721 2.40

0.8 0.10 0.02 0.1992 0.1675 215.91 0.1789 210.19 0.1805 29.39

0.25 0.2254 0.1952 213.40 0.2304 2.22 0.2288 1.51

1.00 0.1683 0.1800 6.95 0.1832 8.85 0.1648 22.08

0.4 0.02 0.7099 0.6607 26.93 0.6755 24.85 0.6823 23.89

0.25 0.9668 0.8307 214.08 0.9269 24.13 0.9636 20.33

1.00 0.989 0.9566 23.28 0.9966 0.77 0.9895 0.05

0.92 0.02 1.583 1.5152 24.28 1.5329 23.16 1.5480 22.21

0.25 2.166 1.9344 210.69 2.0674 24.55 2.1706 0.21

1.00 2.772 2.6221 25.41 2.7292 21.54 2.8059 1.22


quadrature formula (Sykes 1951) is used, which leads to

m1 5 0.211 324 8, m2 5 0.788 675 2, a1 5 0.5, and a2 5

0.5. As for various two-stream approximations, an ana-

lytic solution for the four-stream approximation can be

derived explicitly (Liou et al. 1988).

The diffuse actinic fluxes and irradiance fluxes in the

upward and downward directions are defined as

F2(t) 5 2p �

2

i51a

iI(t, m

i) 1 �

2

i51a

iI(t, �m

i)

24

35

, (16)

E62 (t) 5 2p �

2

i51a

im

iI(t, 6m

i). (17)

d. Two- and four-stream combinationapproximations

For a homogeneous layer with an optical depth t1,

Eq. (1) is formally solved to obtain (m . 0)

I(0, m) 5 I(t1, m)e�t1/m 1

ðt1

0

M(t9, m)e�t9/m dt9/m, (18)

I(t1, �m) 5 I(0, �m)e�t1/m

1

ðt1

0

M(t9, �m)e�(t1�t9)/mdt9/m, (19)

where I(t1, m) and I(0, 2m) are, respectively, the in-

ward intensities at the bottom and top surfaces. In Eqs.

(18) and (19), M(t9, 6m) is the source function asso-

ciated with multiple scattering and single scattering. It

can be written as

M(t, 6m) 5-4p

pE0e�t/m0 P(6m, �m

0)

1-2

ð1

�1

I(t, m9)P(6m, m9) dm9. (20)

In the two- and four-stream combination methods, we

first solve the source function using various two-stream

schemes. Then, we use Eqs. (18) and (19) to evaluate

intensities in the four-stream directions. Here, the double

Gauss points and weights are used in the four-stream in-

tensity and flux calculations.

Using various two-stream schemes, the source func-

tion is given by

TABLE 4a. As in Table 3a, but for the d-two stream approximation and d-two- and four-stream combination approximation based on

the quadrature scheme.

F(0)

R m0 t1 Exact Quadrature Err (%) SFQ4 Err (%) Four Err (%)

0 0.10 0.02 0.0454 0.0157 265.42 0.0286 237.00 0.0287 236.78

0.25 0.1698 0.0849 250.00 0.1436 215.43 0.1477 213.02

1.00 0.2116 0.1188 243.86 0.1860 212.10 0.1915 29.50

0.4 0.02 0.0471 0.0169 264.12 0.0299 236.52 0.0307 234.82

0.25 0.2837 0.1640 242.19 0.2634 27.16 0.2667 25.99

1.00 0.5342 0.3739 230.01 0.5196 22.73 0.5276 21.24

0.92 0.02 0.0395 0.0171 256.71 0.0266 232.66 0.0282 228.61

0.25 0.2793 0.1911 231.58 0.2721 22.58 0.2768 20.90

1.00 0.6907 0.5716 217.24 0.7082 2.53 0.7188 4.07

0.25 0.10 0.02 0.089 0.0546 238.65 0.0729 218.09 0.0730 217.98

0.25 0.1891 0.1039 245.06 0.1629 213.86 0.1672 211.58

1.00 0.2196 0.1270 242.17 0.1939 211.70 0.1993 29.24

0.4 0.02 0.2345 0.1837 221.66 0.2204 26.01 0.2208 25.84

0.25 0.4023 0.2777 230.97 0.3799 25.57 0.3863 23.98

1.00 0.5837 0.4222 227.67 0.5652 23.17 0.5762 21.28

0.92 0.02 0.4765 0.4062 214.75 0.4709 21.18 0.4717 21.01

0.25 0.5972 0.4927 217.50 0.5866 21.77 0.5945 20.45

1.00 0.8505 0.7265 214.58 0.8589 0.99 0.8772 3.14

0.8 0.10 0.02 0.1869 0.1413 224.40 0.1719 28.03 0.1721 27.92

0.25 0.2388 0.1526 236.10 0.2133 210.68 0.2174 28.96

1.00 0.2471 0.1559 236.91 0.2216 210.32 0.2262 28.46

0.4 0.02 0.6531 0.5558 214.90 0.6453 21.19 0.6459 21.10

0.25 0.707 0.5695 219.45 0.6842 23.22 0.6953 21.65

1.00 0.7781 0.5914 223.99 0.7284 26.39 0.7432 24.49

0.92 0.02 1.453 1.2739 212.33 1.4618 0.61 1.4633 0.71

0.25 1.404 1.2664 29.80 1.3983 20.41 1.4157 0.83

1.00 1.398 1.2684 29.27 1.3935 20.32 1.4211 1.65


M(t, 6m) 5 G(6m)e�k(t1�t) 1 H(6m)e�kt

1 z(6m)e�t/m0 . (21)

The coefficients G(6m), H(6m), and z(6m) are deter-

mined for each two- and four-stream combination ap-

proximation. Table 2 gives the values of G(6m), H(6m),

and z(6m). Replacing M(t, 6m) in Eqs. (18) and (19) by

M(t, 6m) from Eq. (21), we may write the intensity at

a given m as follows (Toon et al. 1989):

I(0, m) 5 I(t1, m)e�t1/m 1

G(m)

km� 1(e�t1/m � e�kt1 )

1H(m)

km 1 1[1� e�t1(k11/m)]

1z(m)m

0

m 1 m0

[1� e�t1(1/m011/m)], (22)

I(t1, �m) 5 I(0, �m)e�t1/m

1G(�m)

km 1 1[1� e�t1(k11/m)] 1

H(�m)

km� 1(e�t1/m � e�kt1 )

1 z(�m)m

0

m0� m

(e�t1/m0 � e�t1/m). (23)

The upward and downward diffuse fluxes at a given

level t for this method are defined by

E63 (t) 5 2p �

2

i51a

im

iI(t, 6m

i), (24)

where m1 5 0.211 324 8, m2 5 0.788 675 2, a1 5 0.5,

and a2 5 0.5 in the two- and four-stream combination

approximations.

The two- and four-stream combination approxima-

tions are significantly better at representing the angular

characteristics of the radiance field than the two-stream

approximations (Davies 1980). However, the methods

are somewhat unsatisfactory, because where - 5 1, they

do not necessarily yield flux conservation at solar wave-

lengths (Toon et al. 1989). We therefore extend it by

defining

I9(t, 6m) 5 I(t, 6 m)E61 /E6

3 , (25)

where I(t, 6 m) is taken from Eqs. (22) and (23). In

addition, E11, E1

2, and E36 are calculated from Eqs. (7),

(8), and (24), respectively. Note that I9(t, 6m) has the

same angular characteristics as I(t, 6 m) but retains flux


F(t1)

R m0 t1 Exact Quadrature Err (%) SFQ4 Err (%) Four Err (%)

0 0.10 0.02 0.0451 0.0157 265.19 0.0285 236.81 0.0284 237.03

0.25 0.1091 0.0741 232.08 0.1155 5.87 0.1124 3.02

1.00 0.0566 0.0544 23.89 0.0591 4.42 0.0543 24.06

0.4 0.02 0.0471 0.0169 264.12 0.0299 236.52 0.0294 237.58

0.25 0.2521 0.1580 237.33 0.2482 21.55 0.2477 21.75

1.00 0.3026 0.2621 213.38 0.3081 1.82 0.3023 20.10

0.92 0.02 0.0395 0.0171 256.71 0.0266 232.66 0.0254 235.70

0.25 0.2663 0.1881 229.37 0.2652 20.41 0.2685 0.83

1.00 0.5435 0.4845 210.86 0.5638 3.74 0.5672 4.36

0.25 0.10 0.02 0.0928 0.0559 239.76 0.0753 218.86 0.0754 218.75

0.25 0.1417 0.1014 228.44 0.1486 4.87 0.1448 2.19

1.00 0.0818 0.0770 25.87 0.0866 5.87 0.0792 23.18

0.4 0.02 0.2517 0.1895 224.71 0.2308 28.30 0.2312 28.14

0.25 0.4523 0.3210 229.03 0.4463 21.33 0.4475 21.06

1.00 0.4577 0.3941 213.90 0.4684 2.34 0.4572 20.11

0.92 0.02 0.5168 0.4196 218.81 0.4951 24.20 0.4960 24.02

0.25 0.7987 0.6203 222.34 0.7905 21.03 0.7993 0.08

1.00 1.047 0.9076 213.31 1.0762 2.79 1.0721 2.40

0.8 0.10 0.02 0.1992 0.1456 226.91 0.1798 29.74 0.1805 29.39

0.25 0.2254 0.1712 224.05 0.2335 3.59 0.2288 1.51

1.00 0.1683 0.1559 27.37 0.1824 8.38 0.1648 22.08

0.4 0.02 0.7099 0.5744 219.09 0.6789 24.37 0.6823 23.89

0.25 0.9668 0.7392 223.54 0.9545 21.27 0.9636 20.33

1.00 0.989 0.8562 213.43 1.0289 4.03 0.9895 0.05

0.92 0.02 1.583 1.3174 216.78 1.5402 22.70 1.5480 22.21

0.25 2.166 1.7290 220.18 2.1379 21.30 2.1706 0.21

1.00 2.772 2.3882 213.85 2.8698 3.53 2.8059 1.22


conservation when - 5 1. Using I9(t, 6m) from Eq. (25),

the diffuse actinic flux is given by

F3(t) 5 2p �

2

i51a

iI9(t, m

i) 1 �

2

i51a

iI9(t, �m

i)

24

35

. (26)

The two- and four-stream combination methods are

appealing because they combine the advantages of the

speed of the two-stream approximations and the accu-

racy of the four-stream approximation.

In the solar radiative transfer, it is important to per-

form the delta adjustment to account for the forward-

scattering peak (Joseph et al. 1976; Liou et al. 1988; Fu

et al. 1997; Cuzzi et al. 1982). If a fraction of the scat-

tering energy f is considered to be in the forward peak,

the above solution can still be used, as long as the fol-

lowing transformations are applied to the optical prop-

erties:

t9 5 t(1� f-), (27)

-9 5 (1� f )-/(1� f-), (28)

-9l5 [-

l� f (2l 1 1)]/(1� f ). (29)

For the d-two-stream and d-two- and four-stream

combination approximations, l 5 0, 1 and f 5 -2/5;

for the d-four-stream approximation, l 5 0, 1, 2, 3 and

f 5 -4/9. The use of function adjustment would enhance

the accuracy of approximate treatments of multiple

scattering.

3. Computational results and discussion

Here we examine the accuracy of the d-two-stream,

d-two- and four-stream combination, and d-four-stream

approximations by comparing results with the exact values

taken from Yung (1976) or computed from the discrete

ordinates method (Stamnes et al. 1988).

The impact of Rayleigh scattering on photochemical

processes in the stratosphere is important. We first com-

pare the diffuse actinic fluxes at the top [labeled F(0)] and

the bottom [labeled F(t1)] in a conservative Rayleigh

scattering atmosphere. Tables 3–5 show diffuse actinic

fluxes using a solar flux (pE0) of 1, with different op-

tical depths, surface albedos, and solar zenith angles,

TABLE 5a. As in Table 3a, but for the d-two-stream approximation and d-two- and four-stream combination approximation based on

the hemispheric constant scheme.

F(0)

R m0 t1 Exact Hemispheric constant Err (%) SFH4 Err (%) Four Err (%)

0 0.10 0.02 0.0454 0.0181 260.13 0.0286 237.00 0.0287 236.78

0.25 0.1698 0.0987 241.87 0.1443 215.02 0.1477 213.02

1.00 0.2116 0.1400 233.84 0.1882 211.06 0.1915 29.50

0.4 0.02 0.0471 0.0195 258.60 0.0299 236.52 0.0307 234.82

0.25 0.2837 0.1897 233.13 0.2637 27.05 0.2667 25.99

1.00 0.5342 0.4367 218.25 0.5222 22.25 0.5276 21.24

0.92 0.02 0.0395 0.0198 249.87 0.0266 232.66 0.0282 228.61

0.25 0.2793 0.2209 220.91 0.2730 22.26 0.2768 20.90

1.00 0.6907 0.6639 23.88 0.7133 3.27 0.7188 4.07

0.25 0.10 0.02 0.089 0.0629 229.33 0.0728 218.20 0.0730 217.98

0.25 0.1891 0.1201 236.49 0.1631 213.75 0.1672 211.58

1.00 0.2196 0.1486 232.33 0.1949 211.25 0.1993 29.24

0.4 0.02 0.2345 0.2118 29.68 0.2202 26.10 0.2208 25.84

0.25 0.4023 0.3182 220.91 0.3782 25.99 0.3863 23.98

1.00 0.5837 0.4886 216.29 0.5639 23.39 0.5762 21.28

0.92 0.02 0.4765 0.4681 21.76 0.4706 21.24 0.4717 21.01

0.25 0.5972 0.5618 25.93 0.5837 22.26 0.5945 20.45

1.00 0.8505 0.8319 22.19 0.8544 0.46 0.8772 3.14

0.8 0.10 0.02 0.1869 0.1631 212.73 0.1720 27.97 0.1721 27.92

0.25 0.2388 0.1759 226.34 0.2133 210.68 0.2174 28.96

1.00 0.2471 0.1800 227.16 0.2201 210.93 0.2262 28.46

0.4 0.02 0.6531 0.6414 21.79 0.6457 21.13 0.6459 21.10

0.25 0.707 0.6547 27.40 0.6854 23.06 0.6953 21.65

1.00 0.7781 0.6789 212.75 0.7220 27.21 0.7432 24.49

0.92 0.02 1.453 1.4702 1.18 1.4630 0.69 1.4633 0.71

0.25 1.404 1.4545 3.60 1.4047 0.05 1.4157 0.83

1.00 1.398 1.4480 3.58 1.3830 21.07 1.4211 1.65


for each of the computational techniques. In these ta-

bles, the d-four-stream approximation is labeled Four

and the d-two-and four-stream combination approxima-

tions based on the Eddington, quadrature, and hemi-

spheric constant schemes are labeled SFE4, SFQ4, and

SFH4, respectively. In addition, the d-two-stream methods

based on the Eddington, quadrature, and hemispheric

constant schemes are respectively labeled Eddington,

Quadrature, and Hemispheric constant. The exact values

are taken from Yung (1976). Relative errors (labeled Err)

between these approximations and exact values are also

given.

Results from the present d-two-stream method based

on quadrature approximation in Tables 4a and 4b agree

with those of the d-two-stream method given by Toon

et al. (1989, Table 6 in their paper) and Kylling et al.

(1995, Table III in their paper). Since the present d-two-

stream method based on quadrature is the same as the

method used by Toon et al. (1989) and Kylling et al.

(1995), the slight differences between them are due to

numerical rounding errors.

For F(0) and F(t1), no significant differences are ob-

served among the d-two- and four-stream combination

approximations based on the Eddington, quadrature,

and hemispheric constant schemes. The maximum rel-

ative errors in both F(0) and F(t1) of the d-two- and

four-stream combination approximations are ;37%,

which are similar to these of the d-four-stream ap-

proximation (;37%) but much smaller than these of

the two-stream approximations (;65%). For both

F(0) and F(t1) when the optical depth t1 $ 0.25, the

d-two-stream, d-two- and four-stream combination,

and d-four-stream approximations have errors less than

;43%, ;16%, and ;13% respectively. In general, for a

conservative Rayleigh scattering atmosphere, the accu-

racy of the d-two- and four-stream combination ap-

proximations based on various two-stream approaches

is close to that of the d-four-stream approximation and

much more accurate than that of the d-two-stream

approximations.

Radiative transfer through ozone layer around 35 km

is also important (Lary and Pyle 1991). Because of ab-

sorption by ozone, the single scattering albedo is not

equal to 1 in the ultraviolet. The diffuse actinic fluxes in

the two test cases with v 5 0.7 and v 5 0.9 for non-

conservative Rayleigh scattering atmosphere are shown

in Fig. 1. It shows that the accuracy of the d-two- and

four-stream combination approximations based on various


F(t1)

R m0 t1 Exact Hemispheric constant Err (%) SFH4 Err (%) Four Err (%)

0 0.10 0.02 0.0451 0.0181 259.87 0.0285 236.81 0.0284 237.03

0.25 0.1091 0.0848 222.27 0.1144 4.86 0.1124 3.02

1.00 0.0566 0.0600 6.01 0.0567 0.18 0.0543 24.06

0.4 0.02 0.0471 0.0195 258.60 0.0299 236.52 0.0294 237.58

0.25 0.2521 0.1821 227.77 0.2475 21.82 0.2477 21.75

1.00 0.3026 0.2976 21.65 0.3035 0.30 0.3023 20.10

0.92 0.02 0.0395 0.0198 249.87 0.0266 232.66 0.0254 235.70

0.25 0.2663 0.2169 218.55 0.2657 20.23 0.2685 0.83

1.00 0.5435 0.5556 2.23 0.5623 3.46 0.5672 4.36

0.25 0.10 0.02 0.0928 0.0647 230.28 0.0755 218.64 0.0754 218.75

0.25 0.1417 0.1168 217.57 0.1482 4.59 0.1448 2.19

1.00 0.0818 0.0857 4.77 0.0838 2.45 0.0792 23.18

0.4 0.02 0.2517 0.2194 212.83 0.2317 27.95 0.2312 28.14

0.25 0.4523 0.3748 217.14 0.4514 20.20 0.4475 21.06

1.00 0.4577 0.4533 20.96 0.4675 2.14 0.4572 20.11

0.92 0.02 0.5168 0.4860 25.96 0.4972 23.79 0.4960 24.02

0.25 0.7987 0.7282 28.83 0.8066 0.99 0.7993 0.08

1.00 1.047 1.0596 1.20 1.0924 4.34 1.0721 2.40

0.8 0.10 0.02 0.1992 0.1688 215.26 0.1806 29.34 0.1805 29.39

0.25 0.2254 0.2006 211.00 0.2368 5.06 0.2288 1.51

1.00 0.1683 0.1800 6.95 0.1831 8.79 0.1648 22.08

0.4 0.02 0.7099 0.6663 26.14 0.6827 23.83 0.6823 23.89

0.25 0.9668 0.8795 29.03 0.9855 1.93 0.9636 20.33

1.00 0.989 1.0242 3.56 1.0687 8.06 0.9895 0.05

0.92 0.02 1.583 1.5282 23.46 1.5490 22.15 1.5480 22.21

0.25 2.166 2.0673 24.56 2.2235 2.65 2.1706 0.21

1.00 2.772 2.9077 4.90 3.0373 9.57 2.8059 1.22


two-stream approaches is close to that of the d-four-

stream approximation and much more accurate than

that of the d-two-stream approximations. It should be

noted that it is necessary to integrate over part of the

solar spectrum in the computations of photolysis rates.

The actual errors may be significantly smaller than those

in the monochromatic radiative transfer computations

because of the compensating errors.

FIG. 1. Diffuse actinic flux F for a nonconservative Rayleigh scattering atmosphere vs the optical depth t : v 5 (left)

0.7 and (right) 0.9. Here, the surface albedo R equals 0, the solar flux (pE0) equals 1, the optical depth t1 5 1, and the

cosine of the solar zenith angle m0 5 0.5. The d-four-stream approximation is labeled Four; the d-128-stream method

is denoted Exact. The d-two- and four-stream combination approximations based on the Eddington, quadrature, and

hemispheric constant schemes are SFE4, SFQ4, and SFH4, respectively. In addition, the d-two-stream methods based

on the Eddington, quadrature, and hemispheric constant schemes are labeled as Eddington, Quadrature, and

Hemispheric constant, respectively.


We also calculated the diffuse actinic flux by consid-

ering nonabsorbing aerosols. The asymmetry factor of

aerosols is used to represent the phase function through

the Henyey–Greenstein function. Figures 2–4 show the

diffuse actinic fluxes calculated by the d-two-stream,

d-two- and four-stream combination, d-four-stream, and

d-128-stream approximations. The latter is labeled Ex-

act. Figures 2–4 show that the d-two- and four-stream

combination approximations based on the Eddington,

quadrature, and hemispheric constant schemes are much

more accurate than the d-two-stream approximations

based on the Eddington, quadrature, and hemispheric

FIG. 2. Diffuse actinic flux F for the conservative scattering aerosol layer vs the optical depth t : t1 5 (top) (left)

0.02 and (right) 0.25; (middle) (left) 0.50 and (right) 1.00; and (bottom) (left) 2.50 and (right) 5.00. Here, the surface

albedo R is equal to 0.25, the phase function is the Henyey–Greenstein with g 5 0.65, the solar flux (pE0) is equal to 1,

and the cosine of the solar zenith angle m0 5 0.5. The d-four-stream approximate is denoted Four; the d-128-stream

method is denoted Exact. In addition, the d-two- and four-stream combination and d-two-stream approximations

based on the Eddington scheme are denoted SFE4 and Eddington, respectively.


constant schemes for all cases. When the optical depth

t1 is equal to 0.02, 2.5, and 5, the accuracy of the d-two-

and four-stream combination approximation based on

the quadrature scheme is similar to that of the d-four-

stream approximation. In other cases, the d-four-stream

approximation performs better. Hence, the d-two- and

four-stream combination approximation based on the

quadrature scheme is better than those based on the

Eddington and hemispheric constant schemes.

Both cloud and haze affect the radiation field. For

overcast and hazy atmospheres, the error in photolysis

rates incurred by the use of the two-stream approach

becomes larger than that of clear-sky situations (Kylling

et al. 1995). The diffuse actinic fluxes of two test cases,

one from the Haze L scattering model (case 1 in Table 21

of Garcia and Siewert 1985) and the other from the Cloud

C1 scattering model (case 4 in Table 21 of Garcia and

Siewert 1985), are shown in Fig. 5. The phase function

FIG. 3. As in Fig. 2 but for the d-two- and four-stream combination and d-two-stream approximations based on the

quadrature scheme, denoted SFQ4 and Quadrature, respectively.


of Cloud C1 and Haze L are specified by Garcia and

Siewert (1985). The two cases are summarized in our

Table 6. Figure 5 shows that the d-two- and four-stream

combination approximation based on the quadrature

scheme is much better than d-two-stream approxima-

tions based on the quadrature scheme. Since the d-two-

stream approximation based on the quadrature scheme

is the same as the d-two-stream method used by Kylling

et al. (1995), the accuracy of the d-two- and four-stream

combination approximations based on the quadrature

scheme is better than the method reported by Kylling

et al. (1995). For both Haze L and Cloud C1, the accuracy

of the d-two- and four-stream combination approxima-

tions based on the Eddington and hemispheric constant

scheme is slightly better than or similar to that of the

d-two-stream approximation based on the Eddington and

hemispheric constant schemes. Hence, the d-two-and

four-stream combination approximation based on the

FIG. 4. As in Fig. 2 but for the d-two- and four-stream combination and d-two-stream approximations based on the

hemispheric constant scheme, denoted SFH4 and Hemispheric constant, respectively.


FIG. 5. Diffuse actinic flux F for the haze and cloud vs the optical depth t : t1 5 (left) 1.00 and (right) 64.0; (top)

quadrature and SFQ4, (middle) Eddington and SFQ4, and (bottom) Hemispheric constant and SFH4. Four and

Exact are for comparison references to the left and right. Here, the left of the figure is about Haze L, and the

right of it is about Cloud C1. The d-four-stream approximation is labeled Four; the d-two-and four-stream

combination approximations based on the Eddington, quadrature, and hemispheric constant schemes are la-

beled SFE4, SFQ4, and SFH4, respectively. In addition, the d-two-stream methods based on Eddington,

quadrature, and hemispheric constant schemes are labeled Eddington, Quadrature, and Hemispheric constant,

respectively.


quadrature scheme is better than those based on the

Eddington and hemispheric constant schemes.

In conclusion, the d-two- and four-stream combina-

tion approximation based on the quadrature scheme

may be used for the calculation of diffuse actinic fluxes

when accuracy greater than that of the d-two-stream

approximations is required. In some cases, such as

Rayleigh scattering, its accuracy is similar to that of the

d-four-stream approximation.

For applications to three-dimensional atmospheric

chemistry modeling, radiative computations of diffuse

actinic flux are required. Thus, it is important to examine

both the accuracy and efficiency of the radiative transfer

parameterization. Table 7 shows the calculation times

using various approximations. For these comparisons,

the atmosphere was divided into 1000 layers, and the

computing time was normalized by that of the d-two-

stream approximations.

The d-two-stream approximations are the most com-

putationally efficient. However, as demonstrated in Ta-

bles 3–5 and Figs. 1–5, they produce significant errors

in the diffuse actinic flux calculations. High accuracy in

diffuse actinic fluxes can be obtained using the d-four-

stream approximation. However, its computation time

is 6.49 times more than that of the d-two-stream ap-

proximations (see Table 7).

As shown in Tables 3–5 and Figs. 1–5, the accuracy of

the d-two- and four-stream combination approximations

based on the quadrature scheme is close to that of the

d-four-stream approximation, but their computational

time is less than half that of the d-four-stream approxi-

mation (see Table 7).

4. Summary and conclusions

We formulated d-two- and four-stream combination

approximations for the calculation of diffuse actinic

fluxes. We investigated the accuracy and efficiency of

these methods and compared them to the d-two-stream

and d-four-stream approximations. A wide range of solar

zenith angles, optical depths, and surface albedos were

considered for considering molecular, aerosol, haze, and

cloud scattering.

We found that for the calculation of diffuse actinic

fluxes, the errors in the d-two-stream methods were

largest. For the d-four-stream approximation, reliable

results were obtained under all conditions. The accuracy

of the d-two- and four-stream combination approxima-

tions based on the quadrature scheme was close to that

of the d-four-stream approximation.

In view of their accuracy and computational effi-

ciency, the d-two- and four-stream combination method

based on the quadrature scheme is well suited to diffuse

actinic flux calculations.

Acknowledgments. This work is financially supported

by the National Natural Science Foundation of China

(Grant 40775006), the National Basic Research Program

of China (Grant 2006CB403707), the Public Meteorology

Special Foundation of MOST (Grant GYHY200706036),

and the National Key Technology R and D Program

(Grants 2007BAC03A01 and 2008BAC40B02). QF is in

part supported by DOE Grant DE-FG02-09ER64769.

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