Mark Z. Jacobson Stanford University 100% Clean, Renewable Energy and Storage for Everything chap. 5

1. 100% Clean,
Renewable Energy
and Storage for
Everything
Mark Z. Jacobson
Stanford University

2. Chapter 5:
Photovoltaics and
Solar Radiation

3. How to PV Panels Work?

4. Photovoltaics (PVs)
• Defined: A material or device that converts photons of light
energy to electrical voltage and current
• A photon with short enough wavelength can cause an electron
in a PV material (semiconductor, such as Si) to break free of
the atom that holds it
• If a nearby electric field is provided, electrons can be swept
toward a metallic contact, where they can become part of an
electric current

5. First solar cell to generate electricity –
1883 (Charles Fritts, U.S.)
Pressed a thin film of selenium against a brass metal plate, then laid an
even thinner layer of gold on top. The gold layer was so thin that sunlight
could penetrate through it. Exposing the gold to sunlight resulted in
electricity generation.
Werner Siemens, who confirmed Fritts’ experiment, commented,
“In conclusion, I would say that however great the scientific importance of
this discovery may be, its practical value will be no less obvious when we
reflect that the supply of solar energy is both without limit and without cost,
and that it will continue to pour down upon us for countless ages after all the
coal deposits of the earth have been exhausted and forgotten.”

6. Photovoltaics (PVs)
• PVs are semiconductor materials that convert sunlight to electricity
• Main material is silicon (Si). Other materials: Germanium (Ge),
Gallium (Ga), and Arsenic (As)
• Si has 14 electrons, including 4 in outer shell, thus a +4 nucleus
charge. Pure Si crystals have Si bonded to each other with
covalent bonds.
• Boron (B) and Phosphorus (P) are added to Silicon to create
electric field in a PV cell.

7. Energy Bands – Photoelectric Effect
VB = valence band;
CB = conduction band
If an electron in the valence
band acquires energy Ebg, the
electron can jump to the
conduction band.

8. Photovoltaics (PVs)
• Si is a semiconductor
• Metals are conductors
• Metals, semiconductors, and insulators have energy bands:
 Valence band, forbidden band, and conduction band, forbidden
band
 Electrons in conduction band contribute to current flow
• With metals, P-N junction can’t form, so free electrons move too
randomly
• Insulators require too much energy to add electrons to conduction
band

9. Band Gap Energy
• Forbidden band
• Gap between conduction band and valence band
• Band-gap energy (Ebg)
• Energy needed for an electron to free itself from the electrostatic force
holding it to its own nucleus and jump from the valence band to the
conduction band
• Unit of band gap energy
• Electron-volt (eV) = energy an electron acquires when its voltage is
increased by 1 V (1 eV = 1.6x10-19 J)

10. Band Gap Energy
• Band gap energy for Si
• 1.12 eV
• When electron jumps, it leaves +4 nucleus with only 3
electrons  net + charge or hole.
• Unless electron swept away by current, the same one will
recombine to fill hole

11. Photovoltaics (PVs)
• When electron freed, other electrons in lattice may fill the hole,
thus moving the location of net positive charge.
• Band-gap energy Ebg (J) = hn = hc/lbg
• h=6.626 x 10-34 J-s; c=speed of light 3x108 m/s; n=frequency, Hz;
lbg=band-gap wavelength, m
•  lbg=hc/Ebg=1.11 mm when Ebg=1.12 eV
• As such, silicon solar cells produce electricity only for solar
wavelengths less than 1.11 mm

12. Photovoltaics (PVs)
Only wavelengths less than 1.11 mm result in electricity production from pure Si cells
Other band gaps (eV) and band-gap wavelengths (mm):
Si 1.12 1.11
a-Si 1.7 0.73
CdTe 1.49 0.83
CuInSe2 1.04 1.19
CuGaSe2 1.67 0.74
GaAs 1.43 0.87
Shorter band-gap wavelength higher band-gap energy

13. Maximum PV Efficiency
Of total solar spectrum, PV converts energy below max wavelength, and
for each wavelength only up to max band-gap energy (Ebg). Thus, for Si,
• 30.2% of solar spectrum unavailable below lbg=1.11 mm because
hn>Ebg
• 20.2% unavailable above lbg= 1.11 mm
• 49.6% available
• Another 7% loss due high temperatures (Stefan Boltzmann losses
sT4)
• Another 10% loss due to recombination of electrons
 33.7% max efficiency for single-junction PV cell=Shockley-Queisser
limit

14. Ideal PV Efficiency
• Lower Ebghigher band-gap wavelength/fewer losses above it
but more loss below it because more hn>Ebg
• Higher Ebglower band-gap wavelength/greater losses above
it but less loss below it because less hn>Ebg
 Greatest efficiency around Ebg = 1.2-1.6 eV

15. Temperature Effect on Band Gap
Energy
• Band gap energy T dependent
• Higher T  less energy needed to send electron into
conduction band Ebg decreases and lbg increases
• Lower T  Ebg increases and lbg decreases

16. Photovoltaics (PVs)
• PV panels contain a built-in electric field to prevent electrons
from recombining with Si by carrying those in conduction band
away
• To create electric field, contaminate each respective side of
PV cell with 1 atom per 1,000 of Si of
• Element with 5 electrons in valence band (e.g., P) from
Column V
• Element with 3 electrons in valence band (e.g, B) from
Column III

17. N-Type Material
• P has 5 electrons in valence
band, but there are only 4
electrons from surrounding Si
atoms to form covalent bonds
with, so 5th electron breaks free
and roams while P retains an
immobile positive charge
 n-type material since P donates
negatively-charged electron

18. P-Type Material
• B has 3 outer shell electrons, so
forms covalent bonds form with
only 3 Si atoms. So, B borrows
electron from nearby Si to form 4th
bond,  B has net negative
immobile charge. Si now has hole
filled by another electron  hole
elsewhere.  B is p-type since
creates roaming hole

19. Photovoltaics (PVs)

20. Types of PV Cells
• Si 2nd most abundant element = 20% of Earth’s crust.
• Si from high quality Silica or quartz (SiO2) from mines or sand.
• Single crystal Si (sc-Si) – uniform structure
• Polycrystalline Si (poly-Si) rock like chunks of a multifaceted metal
– less expensive and less efficient than sc-Si
• Amorphous Si (a-Si) made by vapor deposition of silane &
hydrogen gas – used in thin-film PV cells

21. Types of PV Cells
• First generation: 200 mm thick
 (sc-Si or poly-Si)
• Second generation: thin film (1-10 mm thick)
 (GaAs, CdTe, CIGS, or a-Si)
• Third generation: Multijunction tandem cells
 Single thin film with multiple materials or stack of different thin films
 One film: 33.7% max efficiency; two: 47%; three: 53%

22. PV Modules and Arrays
• One PV cell = ~0.5 V
• Module = 72, 96, or 128 pre-wired cells in a series in a package
• Array=modules wired in series to increase v or in parallel to
increase i.
• For array, must optimize modules in series or parallel for max
p=vi

23. PV Modules and Arrays
• Modules first stringed in series to increase v as much as safe,
then strings put in parallel to maximize power. This minimizes
i2Rw losses too.
• For series, total v is sum of individual module v’s and total i is
just the i in any one module.

24. PV Modules and Arrays
For strings in parallel, total current is sum of currents of each
string and total v is just the v of individual string.

25. Output of PV Panels

26. Solar PV Panel I-V and P-V Curves

27. Fill Factor and Panel Efficiency
Fill Factor
FF= pMPP / (iSCvOC)
Panel Efficiency
Epanel = pMPP,STC / (F1000 Apanel)
.

28. I-V Curves For Series or Parallel
Panels

29. PV Losses
PAC=PDC,STC x Derate Factor x Ctemp
Derate factor (range)
PV module nameplate DC rating 0.98 (0.9-1.05)
Inverter DC to AC efficiency 0.98 (0.97-0.99)
Diodes and connections 0.995 (0.99-0.997)
DC wiring 0.98 (0.97-0.99)
AC wiring 0.99 (0.98-0.993)
Soiling 0.98 (0.7-0.995)
System availability 0.99 (0.7-1)
Age 0.98 (0.7-1) (0.5% per year)
Shading 0.97 (0.7-1)
Total derate factor 0.864 (0.2-1)

30. PV Output Correction for Cell
Temperature
Ctemp = 1 – brefmax(min(Tc-Tref,55),0)
bref = Temperature coefficient (0.0011-0.0063 K-1) (e.g., 0.0025
K-1)
Tc = Ta+0.32Ftot/(8.91+2w) = PV cell temperature (K)
Tref = reference temperature (298.15 K)
w = ambient wind speed (m/s)
Ftot = solar flux (W/m2) normal to a panel

31. Shading Derate Factor
• Ground Cover Ratio (GCR)=Collector Area (AC)/Total Ground Area (AG)
• AC= panel height x width
• AG= (panel height x cos(tilt angle)+distance between panels) x width
• For tilt angle=30o, panel width=1.2 m, panel height=0.6 m, d= 0.76 m,
• GCR = 0.6 x 1.2 / [(0.6 cos(30o) + 0.76) x 1.2] = 0.47.
• This gives a derate factor for 30o fixed tilt of 0.975 (next slide)

32. Shading Derate Factor

33. Solar Output With Tilted and Tracked
PV Panels

34. Geometry For Calculating Solar Zenith
Angle on a Sphere

35. Solar Zenith Angle
• Cosine of solar zenith angle
• Solar declination angle (d)
 Angle between the equator and the north or south latitude of the point
the point at which the sun is directly overhead
• Local hour angle (Ha)
 Angle, measured westward, between longitude of the point at which
which the sun is directly overhead and longitude of location of
interest.
cos qs = sinjsind + cos jcos dcos Ha

36. Solar Declination Angle
• Solar declination angle (angle between equator and point at which sun is
overhead)
• Obliquity of the ecliptic [Angle between the plane of the Earth's equator and the
plane of the Earth's orbit around the Sun (ecliptic)].
• Number of days since 12 PM GMT, January 1, 2000
NJD = 364.5 + 365 (Y – 2001) + DL + DJ
eob = 23
o
.439 - 0
o
.0000004NJD
d = sin-1
sineob sinlec
( )

37. Solar Declination Angle Terms
• Ecliptic longitude of the sun
Angular distance along the plane of Earth’s orbit around the sun (ecliptic) between a
line from the sun to the current position of the Earth and a reference line found when
the sun passes closest to the Earth (perihelion) during the NH spring equinox on a
specific date
• Mean longitude of the sun. Same as ecliptic longitude, but assuming a circular orbit.
• Mean anomaly of the sun. Angular distance of Earth at its perihelion with elliptical
versus circular orbit
LM = 280
o
.460 +0
o
.9856474NJD
gM = 357
o
.528 + 0
o
.9856003NJD
lec = LM +1
o
.915singM +0
o
.020sin2gM

38. Solar Zenith Angle
• Local hour angle (longitude angle between point of interest and overhead
sun; ts=# seconds past local noon)
• Example:
 At noon, when sun is directly overhead, Ha = 0 --->
 When the sun is over the equator, d = 0 --->
Ha =
2pts
86,400
cos qs = sinjsind + cos jcos d
cos qs = cosjcos Ha
cos qs = sinjsind + cos jcos dcos Ha

39. Solar Zenith Angle
Example:
1:00 p.m., PST, Feb. 27, 2018,  = 35 oN
---> NJD = 6,631.4
---> gM = 6893.4o (mean anomaly of sun)
---> Lm = 6816.7o (mean longitude of sun)
---> lec = 6818.2o (ecliptic longitude of sun)
---> ob = 23.436o (obliquity of ecliptic)
---> d = -8.489o (solar declination)
---> Ha = 15.0o (hour angle)
---> s = arccos[sin(35o)sin(-8.489o)+cos(35o)cos(-
8.489o)cos(15o)]=45.7o

40. Optimal Tilt Angles by
Country/Latitude
Jacobson and Jadhav (2018)
Northern Hemisphere Southern
Hemisphere
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
0 8 16 24 32 40 48 56 64
C09: 2.14+ j0.764
T11: 7.203+ j0.6804
This study: 1.3793+ j(1.2011+ j(-0.014404
+j0.000080509)) (R=0.96)
Optimal
Tilt
Angle
(degrees)
Latitude (degrees)
Northern Hemisphere
-50
-40
-30
-20
-10
0
10
-50
-40
-30
-20
-10
0
10
-50 -40 -30 -20 -10 0
C09: -2.14+ j0.764
T11: -7.203+ j0.6804
This study:-0.41657+ j(1.4216+ j(0.024051
+j0.00021828)) (R=0.97)
Optimal
Tilt
Angle
(degrees)
Latitude (degrees)
Southern Hemisphere

41. Solar Zenith Angles in a Vacuum For
Tilting/Tracking
Horizontal cosz=sin sin(d) + cos cos(d) cosH
Optimal tilt cosz=sin sin(d + b) + coscos(d + b)cosH
1-Axis vertical tracking cosz=sin2 + cos2 cosH
1-Axis horizontal tracking cosz=sin sin(d + b) + cos cos(d + b)
2-Axis tracking cosz=sin2 + cos2 z=1
 = latitude
d = solar declination
H = hour angle
b = optimal tilt angle

42. Solar Zenith in Air for Tilting/Tracking
z,air = arcsin(sinz/rair) for z ≤ p /2
z,air = z + z,crit - p/2 for z > p/2
Critical zenith angle
z,crit = arcsin(1/rair) = 88.649o
Refractive index of air at 550 nm
rair = 1.000278

43. Solar Flux Normal to a Panel
Solar flux (W/m2/mm) normal to a panel at wavelength l
Ftot,l = Fdiffuse,l + cosz,airFdirect,l
Fdiffuse,l = diffuse irradiance normal to a panel at wavelength l
Fdirect,l = direct irradiance parallel to solar beam at
wavelength l
z,air = solar zenith angle in air

44. 0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
-80 -40 0 40 80
Latitude (degrees)
Annual average
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
-80 -40 0 40 80
GATOR-GCMOM 2050 2o
x2.5o
(global: 4.48)
ESS data 1983-2005 1
o
x1
o
(global: 4.53)
Global
horizontal
radiation
(kWh/m
2
/day)
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
-80 -40 0 40 80
Optimal tilt
1-Axis vertical tracking
1-Axis horizontal tracking
2-Axis tracking
Ratio
diffuse+direct
radiation
Latitude (degrees)
Annual average
normal
to
tilted
or
tracked
panel
versus
horizontal
panel
0
2
4
6
8
0
2
4
6
8
-80 -40 0 40 80
Horizontal
Optimal tilt
1-Axis vertical tracking
1-Axis horizontal tracking
2-axis tracking
(d) Latitude (degrees)
Annual average
Direct+diffuse
radiation
(kWh/m
2
/day)
Ann. Avg. Model v. SSE Data
Direct + Diffuse Solar Radiation
to Horizontal PV panel
Modeled Ratio of Radiation to
Tilted or Tracked PV Panel to
Horizontal PV Panel
Modeled Direct + Diffuse Solar
Radiation Reaching Tilted or
Tracked PV Panel
JacobsonandJadhav(2018)

45. Annual Average Incident Direct+Diffuse
Solar Radiation to Horizontal PV Panel
Incident solar, horizontal panel (kWh/m2/day) (g:4.57; l:4.48; o:4.60)
2
3
4
5
6
-180 -90 0 90 180
-90
0
90
JacobsonandJadhav(2018)

46. Incident solar, horizontal panel (kWh/m2/day) (g:4.57; l:4.48; o:4.60)
2
3
4
5
6
-180 -90 0 90 180
-90
0
90
Incident solar ratio, 2-axis track:flat panel (gl:1.39; l:1.50; o:1.34
1
1.5
2
2.5
-180 -90 0 90 180
-90
0
90
Annual Avg Direct+Diffuse Solar Radiation
Reaching Horizontal PV Panel
Ratio of Solar Radiation
Reaching 2-Axis Tracked v.
Horizontal PV Panel
JacobsonandJadhav(2018)

47. Calculation Solar Radiation Reaching
the Surface of the Earth

48. Solar Radiation at Surface Depends on
Cloudiness
MODIS Cloud Fraction 2001-5 average (0.693)
0
0.5
1
-180 -90 0 90 180
-90
0
Incident solar, horizontal panel (kWh/m2/day) (g:4.57; l:4.48; o:4.60)
2
3
4
5
6
-180 -90 0 90 180
-90
0
90

49. kWh/m2/day)

50. World Solar Resource
(TW, Annually Averaged)
Land Ocean Global
Solar hitting ground 27,800 69,200 97,000
Max electricity 5,600 13,800 19,400
Practical electricity 1,300 50 1,350

51. Radiation Spectrum
10
-28
10
-20
10
-12
10
-4
104
10
12
1020
10
28
1036
10-6
10-4
10-2
100
102
104
106
108
1010
1012
10
-6
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
10
10
10
12
Radiation
intensity
(W
m
-2
mm
-1
)
Wavelength (mm)
6000
K
15
million
K
Gamma
X
UV
Visible
Infrared
Short
radio
AM
radio
Long
radio
Television
&
FM
radio
300
K
1
K

52. Radiation Spectra of Earth and Sun
10
-4
10
-2
10
0
10
2
10
4
0.01 0.1 1 10 100
Radiation
intensity
(W
m
-2
mm
-1
)
Wavelength (mm)
Sun
Earth
Ultraviolet
Visible
Infrared

53. Stefan-Boltzmann Law
Integrate spectral irradiance over all wavelengths
Stefan-Boltzmann law (W m-2)
Fb = sBT4
Stefan-Boltzmann constant
sB = 5.67 x 10-8 W/m2-K4
Example
T = 5800 K ---> Fb = 64 million W/m2
T = 288 K ---> Fb = 390 W/m2

54. Wien’s Law
100
102
104
106
108
0.01 0.1 1 10 100
Spectral
irradiance
(W
m
-2
mm
-1
)
Wavelength (mm)
6000 K
4000 K
2000 K
1000 K
300 K
0.5 mm
10 mm
Radiation
Intensity
(W/m
2
/mm)

55. Wien’s Displacement Law
Differentiate Planck's law with respect to wavelength at constant
temperature and set result to zero
Peak wavelength of emissions from blackbody
Sun’s photosphere lp = 2897/5800 K = 0.5 mm
Earth’s surface lp = 2897/288 K = 10.1 mm
lp mm
( ) »
2897
T K
( )

56. Earth-Sun Distance Varies Over Year

57. Solar Flux to Earth
Solar flux at top of Earth’s atmosphere (W/m2)
Averages 1,365 W/m2 over a year but varies daily
Res = Earth-sun distance (Astronomical Units, AU)×149,598,000 km/AU
Rp = radius of sun’s photosphere = 693,600 km
Fp = solar flux at photosphere = 63.5 million W/m2
Fe = Fp
Rp
Res
æ
è
ç
ç
ö
ø
÷
÷
2

58. Earth-Sun Distance
Earth-sun dist. (Astronomical Units, AU) Varies
0.984 to 1.018 (3.4%)
Res,AUs = 1.0014-0.01671cos(gM)-
0.00014cos(2gM)
Radiation intensity proportional to 1/Res
2 – varies
0.97 to 1.04 (by 7%)
Mean anomaly of the sun gM=357o.528 +
0o.9856003NJD
NJD=Number days (including leap days) from 12
PM GMT Jan 1, 2000. NJD=1 at 12 PM January
2, 2000;
NJD=366 at 12 PM January 1, 2001

59. Solid Angle
Radiance emitted from point (O) passes through
incremental area dAs at distance rs from the point.
Incremental surface area
Incremental solid angle (sr)
Steradians analogous to radians
Solid angle around a sphere
dAs = rsdq
( ) rs sinqdf
( )= rs
2
sinqdqdf
dWa =
dAs
rs
2
= sinqdqdf
Wa = dWa
Wa
ò = sinqdqdf
0
p
ò
0
2p
ò = 4p

60. Spectral Irradiance
Flux of radiant energy propagating across a flat surface
Incremental spectral irradiance
Integral of dFl over the hemisphere above the x-y plane
Isotropic spectral irradiance
dFl = Il cosqdWa
Fl = dFl
Wa
ò = Il cosqdWa
Wa
ò = Il cosqsinqdqdf
0
p 2
ò
0
2p
ò
F
l = Il cosq sinqdqdf
0
p 2
ò
0
2p
ò = pIl

61. Extinction Coefficient
Loss of radiation through the atmosphere per unit distance
Total extinction coefficient (km-1)
ss,g = due to scattering by gases
sag = due to absorption by gases
ssp = due to scattering by aerosol and cloud particles
sap = due to absorption by aerosol and cloud particles
l = wavelength
sl = ss,g,l +sa,g,l +ss,p,l +sa,p,l

62. Particle Scattering
Combination of reflection, refraction, and diffraction

63. Optical Depth
Dimensionless = extinction coefficient
multiplied by distance thru air
dSb = incremental distance along solar
beam
dt = -sld = zincremental optical depth
dz = incremental altitude distance
m = cosine of solar zenith angle
tl = sl dz
¥
z
ò = slms dSb
¥
Sb
ò
dz = cosqsdSb =msdSb

64. Radiative Transfer Equation
Change in radiance / irradiance along a beam of interest
Change in radiance along incremental path length
Scattering of radiation out of the beam
Absorption of radiation along the beam
dIl = -dIso,l -dIao,l +dIsi,l +dISi,l
dIso,l = Ilss,ldSb
dIao,l = Ilsa,ldSb

65. Scattering Angles
Single scattering of direct solar
radiation and multiple
scattering of diffuse radiation.

66. Radiative Transfer Equation
Multiple scattering of diffuse radiation into the beam
Single scattering of direct solar radiation into beam
dIsi,l =
ss,k,l
4p
Il, ¢
m , ¢
f Ps,k,l,m, ¢
m ,f, ¢
f d ¢
m
-1
1
ò d ¢
f
0
2p
ò
æ
è
ç
ç
ö
ø
÷
÷
k
å
é
ë
ê
ê
ù
û
ú
ú
dSb
dISi,l =
ss,k,l
4p
Ps,k,lm,-ms ,f,fs
æ
è
ç
ç
ö
ø
÷
÷
k
å
é
ë
ê
ê
ù
û
ú
ú
Fs,le
-tl ms
dSb

67. Extinction Coefficients
Extinction due to total scattering
Extinction due to total absorption
Extinction due to total scattering plus absorption
ss,l = ssg,l +ssp,l
sa,l = sag,l +sap,l
sl = ss,l + sa,l

68. Scattering Phase Function
Gives angular distribution of scattered energy vs. direction
Scattering phase function for diffuse radiation
redirects diffuse radiation from m’, ’ to m, 
Scattering phase function for direct radiation
redirects direct solar radiation from -ms,
s to m, 
P
s,k,l,m, ¢
m ,f, ¢
f
P
s,k,l,m,-ms,f,fs

69. Scattering Phase Function
Scattering phase function defined such that
 = angle between directions m’, ’ and m, 
Substitute -->
1
4p
P
s,k,l Q
( )
4p
ò dWa =1
dWa =sin QdQdf
1
4p
P
s,k,l Q
( )sinQdQdf
0
p
ò
0
2p
ò = 1

70. Scattering Phase Function
Phase function for isotropic scattering
Phase function for Rayleigh scattering
Ps,k,l Q
( )=1
Ps,k,l Q
( )=
3
4
1+cos2
Q
( )

71. Scattering Phase Function
Scattering phase functions for (a) isotropic and (b) Rayleigh
scattering
-1.5
-1
-0.5
0
0.5
1
1.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Q
(Q)
Ps

72. Asymmetry Factor
First moment of phase function -- relative direction of scattering
ga,k,l
> 0 forward (Mie) scattering
= 0 isotropic or Rayleigh scattering
< 0 backward scattering
ì
í
ï
î
ï
ga,k,l =
1
4p
Ps,k,l Q
( )cosQsinQdQdf
0
p
ò
0
2p
ò

73. Radiative Transfer Equation
Single scattering albedo
dIl,m,f
dSb
= -Il,m,f ss,l +sa,l
( )+
ss,k,l
4p
Il, ¢
m , ¢
f Ps,k,l,m, ¢
m ,f, ¢
f d ¢
m
-1
1
ò d ¢
f
0
2p
ò
æ
è
ç
ç
ö
ø
÷
÷
k
å
+Fs,le
-tl ms
ss,k,l
4p
Ps,k,l,m,-ms ,f,fs
æ
è
ç
ç
ö
ø
÷
÷
k
å
ss,l
sl
=
ss,g,l +ss,p,l
ss,g,l +sa,g,l +ss,p,l +sa,p,l
= ws,l

74. Rewrite Radiative Transfer Equation
where
m
dIl,m,f
dtl
= Il,m,f - Jl,m,f
diffuse
- Jl,m,f
direct
Jl,m,f
diffuse
=
1
4p
ss,k,l
sl
Il, ¢
m , ¢
f Ps,k,l,m, ¢
m ,f, ¢
f d ¢
m
-1
1
ò d ¢
f
0
2p
ò
æ
è
ç
ç
ö
ø
÷
÷
k
å
Jl,m,f
direct
=
1
4p
Fs,le
-tl ms
ss,k,l
sl
Ps,k,l,m,-ms ,f,fs
æ
è
ç
ç
ö
ø
÷
÷
k
å

75. Two-Stream Method
Divide phase function into upward (+) and downward
component
1
4p
Il, ¢
m , ¢
f Ps,k,l,m, ¢
m ,f, ¢
f d ¢
m
-1
1
ò d ¢
f
0
2p
ò »
1+ ga,k,l
( )
2
I ­+
1- ga,k,l
( )
2
I ¯ upward
1+ ga,k,l
( )
2
I ¯+
1- ga,k,l
( )
2
I ­ downward
ì
í
ï
ï
î
ï
ï
Substitute this equation into Jdiffuse
1
4p
ss,k,l
sl
Il, ¢
m , ¢
f Ps,k,l,m, ¢
m ,f, ¢
f d ¢
m
-1
1
ò d ¢
f
0
2p
ò
æ
è
ç
ç
ö
ø
÷
÷
k
å »
ws,l 1- bl
( )I ­+ws,lblI ¯
ws,l 1- bl
( )I ¯+ws,lblI ­
ì
í
ï
î
ï

76. Two-Stream Method
Integrated fraction of forward scattered energy
Integrated fraction of backscattered energy
1 - bl =
1+ ga,l
2
Effective asymmetry parameter
bl =
1- ga,l
2
ga,l =
ss,a,lga,p,l
ss,g,l +ss,p,l

77. Two-Stream Approximation
Upward radiance equation
Downward radiance equation
Irradiances in terms of radiance for two-stream approximation
m1
dI -
dt
= I --ws 1- b
( )I --wsbI ¯-
ws
4p
1-3gam1ms
( )Fse
-t ms
-m1
dI ¯
dt
= I ¯-ws 1- b
( )I ¯-wsbI --
ws
4p
1+3gam1ms
( )Fse
-t ms
F ¯= 2pm1I ¯
F -= 2pm1I -

78. Two-Stream Approximation
Substitute irradiances and generalize for different phase function
approximations
Surface boundary condition
dF -
dt
= g1F --g2F ¯-g3wsFse
-t ms
dF ¯
dt
= -g1F ¯+g2F -+ 1- g3
( )wsFse
-t ms
F -B= AeF ¯B +AemsFse
-tNL+1 2
ms

79. Two-Stream Approximation
Coefficients for two stream approximations using two
techniques
g1 =
1-ws 1+ ga
( ) 2
m1
g2 =
ws 1- ga
( )
2m1
g3 =
1-3gam1ms
2