Appendix I
Table 1 in the article contains expressions for individual transmittances required for the
calculation of the DGR. Some of these expressions are further simplifications of more
complex tables and algorithms used by the SMARTS software; these have been created for
the specific purposes of the present work. Shown in this appendix here are comparisons of the
proposed simplified expressions against SMARTS outputs and data. Note that the
supplementary figures and equations, presented in this appendix, are numbered with the
prefix S before the figure or equation number.
Simplification of the Rayleigh scattering transmittance equation
For the purposes of this work, a simplified approximation to Rayleigh scattering as
calculated by SMARTS is provided in the following equation:
𝑇𝑅𝜆 = √exp (−
((3604 )∗(M))
𝜆4
)
(S1)
The following figures (Fig. S1 and S2) show the Rayleigh transmittance for different
SZAs, as calculated by Eq. (S1) (Figure S1) and a comparison against the calculations of
SMARTS for an atmospheric pressure of 1atm (Figure S2).
2
Figure S1 shows the solution of Eq. 1 for different SZA (0 to 90 degrees). This simplified
approximation closely matches the calculations proposed by Gueymard, as shown in figure
S2.
Figure S2 Showing the linear relationship between Gueymard’s Rayleigh
transmittance calculations and the approach proposed in the present work.
Ozone absorption and the effective ozone absorption of the diffuse radiation
For the specific purposes of this work, simple empirical fits are also proposed to the
ozone absorption coefficients provided by Molina and Molina (1986). Two fits were needed
3
to better approximate the Hartley and Huggins absorption bands. Equations S3 and S4 show
the empirical fits for ozone absorption cross sections at the Hartley(A𝑂𝜆𝐻𝑎𝑟𝑡𝑙𝑒𝑦 ) and Huggins
(A𝑂𝜆𝐻𝑢𝑔𝑔𝑖𝑛𝑠 ) bands of ozone absorption, based on the original laboratory data from (Molina
and Molina, 1986).
A𝑂𝜆𝐻𝑎𝑟𝑡𝑙𝑒𝑦 = 1140 exp(−1.5E − 3 ∗ (𝜆 − 253.65)2 ))
1
A𝑂𝜆𝐻𝑢𝑔𝑔𝑖𝑛𝑠 = A𝑂𝜆𝐻𝑎𝑟𝑡𝑙𝑒𝑦 ∗ 2 exp((2.7E − 4) ∗ (𝜆 − 253.65)2 ))
(S3)
(S4)
Note that the constants 1140 and 253.65 in Eq. S3 correspond to the wavelength with
maximum absorption (253.65) and its absorption coefficient according to the reported values
of Molina and Molina (1986).
Equation S4 returned good estimates of the spectral ozone absorption cross sections at
wavelengths near 253.65 nm (Hartley band) (see Figure S3). However, as shown in Figure
S4, the Huggins absorption band (λ > 320 nm) was strongly underestimated.
Figure S3 Showing the empirical fit (Eq. S3) to the ozone absorption data of Molina
and Molina (1985). Dots are the original laboratory data; the line represents the solution
to the empirical fit.
4
Figure S4 A detailed section of the empirical fit to ozone absorption in wavelengths
from 320 to 350 nm. Illustrating the associated error of Equation S3.
In Figure S4 the fit to the Huggins band (Eq. S4) was used to better approximate
absorption cross sections at larger wavelengths in the ultraviolet range:
Figure S5 The Huggins band fit (Eq S4) to the ozone absorption cross section of
Molina and Molina.
5
Equation S3 better approximated absorptions at wavelengths below 320nm and
underestimated them above this limit. Equation S4 showed the inverse pattern. For this
reason, a final function for the ozone absorption coefficients A𝑂𝜆 is set to be as follows:
A𝑂𝜆 = max(A𝑂𝜆𝐻𝑎𝑟𝑡𝑙𝑒𝑦 , A𝑂𝜆𝐻𝑢𝑔𝑔𝑖𝑛𝑠 )
(S5)
finally, A𝑂𝜆 can then substituted into Eq S2.
Effective to standard ozone transmittance
Gueymard (1995 p 28) presents a figure showing that the ratio of effective-to-standard
ozone transmittance
Г𝑜𝜆
𝑇𝑜𝜆
is approximately a logarithmic linear function of the ozone optical
thickness and air mass. Equation S6 represents an approximation to Gueymard’s figure for
the ratio of effective-to-standard ozone transmittance.
Г𝑜𝜆
𝑇𝑜𝜆
≈ 10
O3𝑜𝑡 (
3
1
1
−
)
cos(Z) cos(35)
(S6)
Figure S6 shows the solution of equation S6 for five different air masses:
6
Figure S6 The solution of Eq. S6 for values of ozone optical thickness of 0 to 50
and 5 different air masses, i.e. 1, 1.22, 1.55, 2 and 5.7, corresponding respectively to
SZAs of 0°,35°, 50°, 60°, 80°.
7