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

1.0 Introduction

SECTION 2.0

DIRECT NORMAL RADIANCE

Minor modifications have been made to the methods we reported (Bird) for calculating direct normal irradiance. The changes include the addition of an earth-sun distance factor, the use of Leckner's water vapor transmittance expression with some modification of Leckner's absorption coefficients, and the use of Robinson's ozone mass expression as given by Iqbal. These changes and other minor adjustments are described in this section.

The direct irradiance on a surface normal to the direction of the sun at ground level for wavelength is given by

                                .                             (2-1)

The parameter is the extraterrestrial irradiance at the mean earth-sun distance for wavelength ; D is the correction factor for the earth-sun distance; and , , , , and and are the transmittance functions of the atmosphere at wavelength for molecular (Rayleigh) scattering, aerosol attenuation, water vapor absorption, ozone absorption, and uniformly mixed gas absorption, respectively. The direct irradiance on a horizontal surface is obtained by multiplying Eq. 2-1 by cos Z, where Z is the solar zenith angle.

The extraterrestrial spectral irradiance used here was obtained from Fröhlich and Wehrli of the World Radiation Center. A major segment of this spectrum that is of interest here was taken from the revised Neckel and Labs spectrum. A 10-nm-resolution version of this spectrum is shown in Table 2-1 for the 122 wavelengths used in this model.

The earth-sun distance factor as given by Spencer is

                                          (2-2)

The day angle in radians is represented by

                                                                     (2-3)

where d is the day number of a year (1-365).

2.1 RAYLEIGH SCATTERING

                                         (2-4)

where M' is the pressure-corrected air mass. The relative air mass as given by Kasten is

                                                  (2-5)

where Z is the apparent solar zenith angle. The pressure-corrected air mass is M' = MP/P_o, where P_o = 1013 mb and P is measured surface pressure in mb.

2.2 AEROSOL SCATTERING AND ABSORPTION

                                                                     (2-6)

Values for and were derived using a rural aerosol model. Two values were used for this aerosol model: ₁ = 1.0274 for wavelengths <0.5 µm, and ₂ = 1.2060 for wavelengths 0.5 µm. The value of _n was chosen appropriately for each wavelength interval to produce accurate turbidity values (aerosol optical depth in a vertical path) at 0.5 µm wavelength. The turbidity in Eq. 2-6 is represented by the Angstrom formula, namely,

                                                                            (2-7)

For some types of aerosols, it may be important to separate the aerosol extinction into two or more segments, as we have done here for the rural aerosol model. The form of Eq. 2-6 allows the turbidity versus the wavelengths on a log-log plot to be nonlinear, which often occurs in the real atmosphere, as shown by King and Herman. However, for the rural aerosol model, this does not appear to significantly improve the accuracy of the modeled results since the function is approximately linear. Also, the approximate nature of this simple model approach sometimes masks the effect of refinements such as this. When a single value of is used to represent the rural aerosol model, the value should be = 1.140.

2.3 WATER VAPOR ABSORPTION

                                (2-8)

where W is the precipitable water vapor (cm) in a vertical path and , is the water vapor absorption coefficient as a function of wavelength. The water vapor amount W is not temperature- or pressure-corrected because this has been accounted for in the form of Eq. 2-8. We modified Leckner's values of somewhat and added several values to achieve better agreement with experimental data. The coefficients are given in Table 2-1. In our previous model, we used a misprinted version of Leckner's expression, which necessitated modifications to the expression and to the absorption coefficients to obtain reasonable agreement with rigorous model results. The correct form, shown in Eq. 2-8, gives better results.

2.4 OZONE AND UNIFORMLY MIXED GAS ABSORPTION

                                                                 (2-9)

where is the ozone absorption coefficient, O₃ is the ozone amount (atm-cm), and M_o is the ozone mass. We used Leckner's ozone absorption coefficients shown in Table 2-1. The ozone mass expression of Robinson as given by Iqbal has been adopted. The ozone mass is given by

                                          (2-10)

The parameter h_o is the height of maximum ozone concentration, which is approximately 22 km. The ozone height varies with latitude and time of year. If one does not have ozone measurements available, the ozone amount can be estimated using the expression of Van Heuklon. Since the total ozone amount is an approximation, using O₃M_o rather than O₃M may not be an improvement.

Leckner's expression for uniformly mixed gas transmittance is used, and it is expressed as

                                      (2-11)

where is the combination of an absorption coefficient and gaseous amount. We used Leckner's values of shown in Table 2-1 with a few additions and modifications. Final adjustments were made in the gaseous absorption coefficients by comparing the modeled data with measured data, as described in Section 4.0.

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