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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:
(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).
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 calculationsand 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 to better approximate the Hartley and Huggins absorption bands. Equations S3 and S4show the empirical fits for ozone absorption cross sections at the Hartley() and Huggins () bands of ozone absorption, based on the original laboratory data from (Molina and Molina, 1986).
(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.
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.
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 is set to be as follows:
(S5)
finally, 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 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.
(S6)
Figure S6 shows the solution of equation S6 for five different air masses:
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°.
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