Supplementary Materials:
Radiation Calculations
All spectra were calculated using the NREL Simple Model of the Atmospheric Radiative
Transfer Sunshine (SMARTS) model. Surface level irradiance was calculated using the
default values, i.e. a U.S. Standard Atmosphere (“USSA”) at 1.5 air masses, with one
exception: the CO2 concentration was changed from 370 to the values at the time of
calculation –395 ppm (NOAA 2014). Global horizontal values were taken, as canopies
should be considered flat, instead of tilted, collectors (red lines in Figure 1). This spectrum
was then multiplied by the reflectance spectrum of the individual plant to determine the
reflected irradiance at the surface of the plant, I s ( ) . The theoretical surface-level
irradiance spectra of a plant engineered to maximal reflectance outside the PAR, I s ( ) was
calculated in an identical manner, by using a reflectance spectrum of unity over all
wavelengths. The marginal increase in radiative forcing at the surface of the plant, in a
spectral region bound by wavelengths λ1 and λ2 is then
M S (1 , 2 )  Fs(1 , 2 )  Fs (1 , 2 ) (S1)
where
Page 1
2
Fs(1 , 2 )   I s ( )d
1
2
(S2)
Fs (1 , 2 )   I s ( )d
1
These numbers, as calculated numerically using trapezoidal integration, are reported in the
“at surface” columns of Table 1.
To determine the amount of light that makes it back out to space, plant surfaces were
modeled as Lambertian reflecters, i.e. the surface-level reflected irradiation was emitted
isotropically in all directions. The light reflected back out to space, then, is
I bo ( )  I s( ) ( )
(S3)
where η(λ) is the average absorption of the atmosphere over the upward-facing hemisphere,

2
 ( ) 
1
2
2
  A( , ) sin dd
0 0

(S4)
2
  A( ,  ) sin d
0
where A(λ,θ) is the straight-line absorption spectrum through the atmosphere at that polar
angle. Because the absorption depends on the path-length (i.e. number of air masses),
which is symmetric around zenith, A(λ,θ) does not depend on azimuthal angle (Φ).
Page 2
A(λ,θ) can be approximated by inverting the problem: the absorption and scattering of a ray
emanating from the plant surface outward at an angle θ from zenith will be similar to that
scattered and absorbed from a ray emanating from a sun at an angle θ from zenith as
detected by a tracking detector.
A( , ) 
I gt ( , L( ))
I etr ( )
(S5)
where Igt(λ,L(θ)) is the global-tilted irradiance at air mass L(θ) (representing a tracking
detector), and Ietr(λ) is the extraterrestrial spectrum, both of which were calculated using
SMARTS2 (Gueymard 1995), using the empirically-derived relationship between air mass
and zenith angle (Kasten & Young 1989),

180


L( )  cos( 180
  )  0.15 93.885   
1.253
 (S6)
η(λ) was therefore calculated by trapezoidal integration of equation S4 and is plotted as the
black line in Figure S1. A(λ,θ) is also plotted for θ = 0 (blue), π/4 (orange), and π/2 (red).
The marginal increase in radiative forcing back out to space, then, was calculated in an
identical manner to equation S1, using the spectra of light reflected back out to space,
 ( ) instead of that at the surface of the plant. These numbers are given in
I bo ( ) and I bo
the “back to space” columns of Table 1.
Page 3
Radiative Forcing / Water Use Calculations
The fraction increase in radiation at the canopy surface and back out to space, as calculated
in Table 1, is formally given by
Fs(non  PAR )  Fs (non  PAR )
Fs ( total )
F  (non  PAR )  Fbo (non  PAR )
 bo  bo
Fs ( total )
s 
(S7)
where Fs(total) is the surface-level insolation (“global horizontal”) over all wavelengths
(equation S2), Fs(non-PAR) is that over wavelengths outside the PAR, primed variables
indicate those arising from plants of perfect reflectivity, and “bo” subscripts indicate
irradiance back out to space.
Table 1 indicates that κs = 0.3-0.4 and κbo = 0.2-0.3. Note that because the fractional
increases in reflection were both calculated on the basis of surface-level incident radiation
(“Total Incident” in Table 1), Fs appears in the denominator of both equations above (this
was the precise reason for calculating the marginal gains on the basis of total incident
horizontal insolation at the plant surface).The reflected irradiance at the plant surface and
back out to space, respectively, is then trivially given by
Fs  Fs   s
Fbo  Fs   bo
(S8)
Page 4
where we have used the convention that Fs  Fs ( total ) . We use the insolations for a given
crop producing region (Table 3, row 2, Table 1, row 4) for these values. Note that these are
averages over the growing season, approximated as the two months before, three months
after, and the month of highest insolation. Thus the average annual radiative forcing offset
was approximated as half the (six-month) growing season value
Fs , a  12 Fs (S9)
These numbers are given in row 3 of Table 3.
The global radiative forcing decrease (considering reflection back out to space) was
converted to an effective carbon offset by the equation
C  C0 1  e

 Fbo,a 

 (S10)

where C0 is the reference concentration (278 ppm CO2) (Butler & Montzka 2013), α=5.35
W/m2 is the empirically derived constant relating CO2 concentration and insolation, ΔC is
the projected change in concentration (Myhre et al. 1998), and χ is the fraction of the
Earth’s surface taken up by that crop (Food and Agriculture Organization of the United
Nations).
Page 5
Water use, in inches/month was calculated by equating the decrease in local, surface level
radiative forcing with the heat needed to vaporize water.
dWlocal 
sec in  Fs MWH 2O
 1.4  108

dt
mo  m  H vap , H 2 O  H 2O

(S11)
where MWH 2O = 18 g/mol, H vap ,H 2O = 41 kJ/mol,  H 2O = 1000 kg/m3 are the molecular
weight, enthalpy of vaporization, and density of water, respectively. These numbers were
scaled to global values by dividing by 2 (to get average monthly savings over the year,
assumes a six-month growing season) and multiplying by the area of crop production for
that crop.
For “all crops,” the global average surface exergy (Hermann 2006) of 168 W/m2 was used
for Fs.
Page 6
Fig. S1. Attenuation of radiation through the atmosphere.
Simulated absorption spectra for various zenith angles. The
averaged spectra is the area integral of all absorption spectra
over the upward facing half-sphere as described by equation
S4. Data simulated from the SMARTS2 program (Gueymard
1995).
Fig. S2. The effect of temperature on plant transpiration.
Simulated rates of leaf transpiration as a function of leaf
temperature, for various light levels (fraction of full-sunlight,
Sffs), stomatal resistances to water vapor diffusion (SR, in
s/m×10-2), and boundary layer thicknesses (δ, in mm). All data
taken from Figure 1 of Smith (1980) (Smith & Geller 1980).
Page 7