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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3.0 Diffuse Irradiance



Comparisons of the new, simple spectral model with results of rigorous radiative transfer codes and with measured data are given in this section. These initial comparisons give the reader some measure of the accuracy of the simple model. Additional high-quality spectral measurements, including an error analysis of the measured data, are required for a quantitative analysis of the performance of the model. These measured data are necessary to obtain statistics on the accuracy of the model for different regions of the spectrum and for a range of atmospheric condiotns and collector orientations/modes.


Dave produced several data sets using the Spherical Harmonics method of solving the radiative transfer equation. One of the data sets was for a Rayleigh atmosphere (no aerosols) with molecular absorption. The atmospheric model that was used (the Midlatitude Summer model) contained 2.93 cm of precipitable water and 0.31 atm-cm of ozone. A comparison of Dave's results with the results of the model presented here (SPCTRAL2) is shown in Tables 4-1 and 4-2 at a few wavelengths throughout the spectrum. Table 4-1 compares the direct normal irradiance for three solar zenith angles, and Table 4-2 compares the diffuse irradiance for the same solar zenith angles. The direct normal irradiance was produced using Eq. 2-1 with = 1.0 and D = 1.0. The diffuse horizontal irradiance was found using Eq. 3-5 with D = 1.0 and = 1.0.


Examples of comparisons between BRITE code results and results of the SPCTRAL2 code are presented in Figures 4-1 through 4-4 for global irradiance. Figure 4-1 is for a horizontal surface with zenith angle (Z) = 0.0°, ozone = 0.344 atm-cm, water vapor = 1.42 cm, ground albedo = 0.2, surface pressure = 1013 mb, and a turbidity at 0.5 Ám of 0.1. The only differences for the horizontal spectra shown in Figure 4-2 are that Z = 80.0° and a turbidity of 0.51 was used.

Figure 4-3 is a comparison of BRITE and SPCTRAL2 results for a surface titlted 30° from the horizontal for Z = 48.19° and turbidity at 0.5 Ám of 0.27. The spectra compared in Figure 4-4 were produced with the same parameters as those of Figure 4-3, except Z = 37° and the surface is tilted 60°.


The measured data used to make final adjustments to gaseous absorption coefficients in the model were taken with a unique spectroradiometer [27,28] and an automatic sun photometer [29, 30]. Several comparisons between the spectra produced with the new simple spectral model and measured data are presented here to indicate the extent of agreement.

The first comparison was made with a global horizontal spectrum taken on 5 August 1981 in Golden, Colorado. This site is at 39.75° north latitude and 105.156° west longitude. The spectral measurement was made at 15:09 mountain standard time (MST), and the sun photometer measurements were made at five-minute intervals throughout the day, as illustrated in Figures 4-5 and 4-6. The following parameters were determined:

Solar zenith angle      = 44.8°
Turbidity at 0.368 Ám  = 0.39
Turbidity at 0.500 Ám  = 0.28
Turbidit- at 0.862 Ám  = 0.13
Precipitible water      = 2.25 cm
Surface pressure       = 829.6 mb.

The rural aerosol model used in the BRITE code for a turbidity of 0.27 at 0.500 Ám wavelength produced turbidities of 0.37 at 0.368 Ám wavelength and 0.14 at 0.862 Ám wavelength. From this we can infer that the aerosol present during the 5 August measurement was nearly identical to that in the rural aerosol model, which adds validity to this particular comparison.

The ozone amount was assumed to be 0.31 atm-cm. The results of this comparison are shown in Figure 4-7 and the agreement between experiment and model is extraordinary. There is a slight wavelength calibration error evident in the measured data at the infrared end of the spectrum. This calibration error is due to the linear wavelength calibration procedure, which requires that a slope and intercept be determined. The spectroradiometer system determines both slope and intercept in real time in the visible wavelengths. In the infrared wavelengths, only the intercept is calibrated in real time; the slope appears to have changed slightly between laboratory calibration and the time the measurement was taken.

Several comparisons were also made on 19 August 1981. A global horizontal spectrum was measured at 10:44 MST, a direct normal spectrum was measured at 10:56 MST, and a global spectrum on a 40° south tilt was measured at 13:42 MST. The atmospheric pressure was 832 mb for these measurements. The meteorological and geometrical parameters are shown in Table 4-3 for these measurements.

Results of these comparisons are shown in Figures 4-8 through 4-10. The agreement between the modeled and measured data is very good for these data sets. One has to keep in mind that the circumsolar scattered radiation within a 6° field-of-view (FOV) is included in the direct normal measurements. This could add 1%-5% to the irradiance in the 0.5 Ám region and could explain why the measured direct normal irradiance is larger. Differences similar in magnitude but in the opposite direction have been observed in measured and modeled diffuse radiation. The circumsolar radiation is missing in the diffuse measurement, which causes the opposite effect.

Another comparison (Figure 4-11) was made on 18 August 1981 at 13:22 hours for the global horizontal mode. The parameters for this measurement are as follows:

Solar zenith angle      = 30.11°
Turbidity at 0.368 Ám  = 0.320
Turbidity at 0.500 Ám  = 0.225
Turbidity at 0.862 Ám  = 0.069
Precipitable water      = 1.97 cm
Surface pressure       = 830 mb.

The agreement between modeled and measured data is not as good for this set of data. The reason for the disagreement is unknown, but possibly indicates the accuracy limitations of the modeled and the measured results. Additional measured data will be gathered in the future to verify the model and to assess the accuracy of modeled versus measured data comparisons. Justus and Paris have shown that the use of urban rather than rural aerosol parameters can account for differences of the magnitude and type shown in Figure 4-11. It is not known whether or not urban aerosols from nearby Denver, Colorado, were present during these measurements.

5.0 Examples of the Application of the New Simple Spectral Model

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