Simple Solar Spectral Model for Direct and Diffuse Irradiance on Horizontal and Tilted Planes at the Earth's Surface for Cloudless Atmospheres

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2.0 Direct Normal Irradiance



The diffuse irradiance is difficult to determine accurately with the simple parameterization methods that were used to calculate direct normal irradiance in the previous section. We used tabulated correction factors in our previous research (Bird) to make the simple formulation for the diffuse irradiance of Brine and Iqbal match the results from a rigorous radiative transfer code. Justus and Paris changed the diffuse formulation somewhat and obtained reasonable agreement with rigorous code results without using tabulated correction factors. We examined this new formulation and made some minor adjustments, which we believe improve its accuracy. The correction table approach is still valid and may be the most accurate approach; however, this new formulation is more flexible and is easier to implement.

In addition, we examined different simple formulations for producing spectra on inclined surfaces. We obtained reasonable success with this effort and report our results here.


The diffuse irradiance on a horizontal surface is divided into three components: (1) the Rayleigh scattering component , (2) the aerosol scattering component , and (3) the component that accounts for multiple reflection of irradiance between the ground and the air . The total scattered irradiance is then given by the sum

                                        .                                 (3-1)

If we consider the Rayleigh and aerosol scattering to be independent of each other, the following expressions would be approximately correct:

                          .                  (3-2)

                        .                (3-3)

In these formulas, we have assumed that half of the Rayleigh scatter is downward regardless of the zenith angle of the sun, and that a fraction . of the aerosol scatter is downward and can be a function of the solar zenith angle. The transmittance terms and are for aerosol absorption and aerosol scattering, respectively. In our previous model, (Bird) we used the assumption of independent scattering, and Eq. 3-3 has the following form:

                        .                  (3-4)

where is the aerosol single scattering albedo at one wavelength and is the aerosol forward scattering fraction which is independent of the sun position. We found that this formula significantly underestimated the scattered irradiance for Z > 60°.

In the new simple spectral model reported here, we used modifications of the Justus and Paris expressions for diffuse irradiance. Comparisons with diffuse irradiance calculated using a rigorous radiative transfer code (BRITE) indicated a tendency for the Justus and Paris model to overestimate the energy in the UV and visible portions of the spectrum. This overestimation increased as the turbidity and air mass increased. By slightly modifying the expression, we were able to obtain closer agreement with BRITE results. Table 3-1 gives examples of the results of these comparisons. The modified expressions are

                       .              (3-5)

                     .               (3-6)

                        .                (3-7)

                  .          (3-8)

                                  .                                (3-9)

                               .                             (3-10)

                        .                       (3-11)

                      .                  (3-12)

                     .                 (3-13)

                                    .                                (3-14)

                          .                      (3-15)

                             .                         (3-16)


The parameter is the ground albedo as a function of wavelength, is the sky reflectivity, and the primed transmittance terms are the regular atmospheric transmittance terms evaluated at M = 1.8. is the aerosol single scattering albedo as a function of wavelength, is the single scattering albedo at 0.4 Ám wavelength, is the wavelength variation factor, and is the aerosol asymmetry factor. For the rural aerosol model, = 0.945, = 0.095, and = 0.65. The equations for and ensure that is equal to and that the wavelength-dependent single scattering albedo is correctly defined by . The parameters and are the aerosol scattering and absorption coefficients, respectively. In a homogeneous medium, the optical depth is related to these coefficients by , where L is the path length in the medium.

The only adjustments that we made to the Justus and Paris model were to take to the 0.95 power instead of to the 1.0 power in Eq. 3-5, to take to the 1.5 power instead of to the 1.0 power in Eq. 3-6, and to multiply by Cs in Eqs. 3-5, 3-6, and 3-7. As mentioned previously, we also changed several absorption coefficients.

It is important to note that the parameters used in the simple spectral model for the comparisons in Table 3-1 were selected to match the atmospheric conditions used in the BRITE code. This includes the use of the rural aerosol model and parameters that represent it. Since the rural aerosol model was used in a rigorous fashion in the BRITE code, our modifications to the Justus and Paris model are based on realistic aerosol data as well as other realistic atmospheric conditions. Some model comparisons could be misleading if sufficient attention is not given to the details of the parameters used. This could be the case for comparisons with the Dave aerosol models. A constant complex index of refraction for all wavelengths is used in these models, which is not representative of real aerosols and has an effect on the single scattering albedo as a function of wavelength.

It should also be noted that is difficult to determine and is quite variable in the real world. Justus has derived an expression for for the urban aerosol model as a function of relative humidity. This parameter affects only the diffuse component, so the global radiation at the ground should not be overly sensitive to the values used. This is not the case when the upwelling radiation at the top of the atmosphere is calculated as Justus and Paris did.


There have been several algorithms produced [5, pp. 19-22] that convert the broadband global horizontal irradiance to the broadband global irradiance on a tilted surface. Most of these conversion algorithms require the direct normal and the diffuse on a horizontal surface as input. Several algorithms have been evaluated with measured data [23, 24, 25, and 26], in recent years. Some of them appear to be quite accurate for broadband applications for east-, west-, and south-facing surfaces. Perez et al. found that the algorithms were somewhat inadequate for north-facing surfaces. This is partially because there is less irradiance on north-facing slopes.

We used three of these simple conversion algorithms to produce spectral irradiance on tilted surfaces by using the spectral direct and diffuse irradiance calculations of the previous section as inputs to the conversion algorithm. We obtained the best agreement with rigorous modeled data for clear-sky conditions using the Hay and Davies algorithm. This was somewhat surprising because the way in which the algorithms were formulated would favor the Temps and Coulson algorithms over the Hay and Davies and Klucher algorithms for clear-sky applications. The Hay and Davies algorithm is presented in this section and the results of comparisons with rigorous code results and measured data are presented in the next section.

The spectral global irradiance on an inclined surface is represented by

                        .                       (3-18)

where is the angle of incidence of the direct beam on the tilted surface and t is the tilt angle of the inclined surface. The tilt angle is zero for a horizontal surface and 90° for a vertical surface. The following relationship holds for the spectral global irradiance on a horizontal surface:

                                        .                                   (3-19)

The first term in Eq. 3-18 is the direct component on the inclined surface. The second and third terms account for the circumsolar or aureole component and the diffuse skylight component. The last term in Eq. 3-18 represents the isotropically reflected radiation from the ground. A component that is missing from this model is the horizon-brightening radiation. There are arguments that could be made as to why this algorithm should not be accurate, but the fact that it is reasonably accurate for the cases that we have checked cannot be ignored. It is somewhat surprising that a broadband model can be used for spectral data.

4.0 Comparisons of the New Simple Model with Rigorous Models and Measurements

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