a. Sun photometry

A conventional sun photometer, an instrument for measuring aerosol optical depth spectra, has several (usually less than 10) wavelength channels, defined with narrow-bandwidth interference filters [typically 5–10-nm full-width at half-maximum (FWHM)], and uses silicon detectors to measure the relative intensity at each wavelength (Volz 1974; Shaw 1983). The World Meteorological Organization recommends using wavelength channels at 368, 384, 500, 778, and 862 nm (Fröhlich 1977). These channels were selected at wavelengths that have minimal molecular absorption. To measure aerosol scattering, a sun photometer is calibrated from a Langley (semilog) plot of each channel versus air mass, which yields an AM0 intensity (Shaw 1983). Although conceptually they are very simple, history has shown that accurate and stable calibrations of sun photometers are problematic (Shaw 1983). In addition to aerosol optical depth measurements, precipitable water vapor measurements can be made with sun photometers that have a channel in one of the principal water vapor bands, usually 942–948 nm (Bird and Hulstrom 1982, 1983). Water vapor measurements are calibrated by comparing the 862- to 948-nm intensity ratio to a known water vapor amount, which is normally obtained from weather balloon data or a microwave radiometer. Unfortunately, even with this extra channel, no information about the other absorber amounts is available and a sun photometer is therefore not appropriate for reconstructing the spectral irradiance.

b. Spectroradiometry

A spectroradiometer, on the other hand, provides a direct measurement of spectral irradiance versus wavelength, but only over the wavelength range it is able to detect. For instruments that use silicon photodiode detectors, this range will be about 350 to 1100 nm, thus missing the ultraviolet (UV) and far-infrared (IR) regions. Assuming the instrument has a wavelength spacing of 1 or 2 nm, a silicon spectroradiometer will have several hundred discrete channels over this range. Accordingly, the method presented here, which we call spectroradiometric sun photometry (SSP), uses direct-beam solar spectral irradiance data as a multichannel sun photometer to obtain the parameters necessary for atmospheric transmittance calculations. These parameters can then be used to extrapolate direct spectral irradiance over wavelength ranges outside of the silicon detector region.

c. Atmospheric transmittance models

The Air Force Geophysical Laboratory has developed several comprehensive atmospheric transmittance models, especially MODTRAN and LOWTRAN (Kneizys et al. 1988, 1995). These names refer to the resolution of the models in wavenumbers, that is, moderate-resolution transmittance and low-resolution transmittance. LOWTRAN has a spectral resolution of 20 cm⁻¹; MODTRAN, 2 cm⁻¹ (0.5 and 0.05 nm at 500-nm wavelength), both of which are much higher compared to the spectroradiometer discussed above. The models can calculate transmittance along a path between any two arbitrary points in the atmosphere by modeling the atmosphere as a series of concentric shells that contain the temperature, pressure, and absorber (gas and aerosol) density as a function of altitude. These quantities are defined at the shell boundaries, and numerical integration between endpoints is then used for transmittance calculations. Because we assume that detailed vertical profile data are not available, and because of the wavelength resolution differences, we have adapted a simplified subset of the LOWTRAN 7 model to calculate atmospheric transmittance from spectral irradiance data.

d. Application and need

Development of these techniques was motivated by the need for spectral irradiance data over the 300–4000-nm range in primary photovoltaic reference calibrations (Osterwald et al. 1990; American Society for Testing and Materials 1998a). Such calibrations are performed under collimated sunlight from cloudless skies and compare the reference cell short-circuit current against an electrically self-calibrating, absolute cavity radiometer used for direct normal irradiance measurements. For a cavity radiometer traceable to the World Radiometric Reference in Davos, Switzerland, the total uncertainty in the irradiance measurement is less than 0.5%. The short-circuit current of a solar cell, which is directly proportional to the incident irradiance, is the current generated under illumination while the cell is operated at bias voltage of 0 V. A standard calibration is defined as the ratio of the short-circuit current to total irradiance (this ratio is termed the calibration constant, A m² W⁻¹) while illuminated by a reference solar spectral irradiance, such as American Society for Testing and Materials (1998b). Spectral corrections using numerical integrations of the product of the reference cell’s spectral responsivity and the incident spectral irradiance are used to correct the results to the reference spectral irradiance.

Minimization of error in these calibrations requires that the range of the spectral irradiance data (corresponding to the illumination conditions of the reference cell at the time of the calibration) match the wide, flat response range of the cavity radiometer and the wavelengths present in terrestrial sunlight. Thus, a spectroradiometer that cannot provide data over this range is inadequate (Field and Emery 1993), and the missing information must be obtained through other means (Osterwald et al. 1990). It should be noted that the techniques presented in Osterwald et al. (1990) predate those of this paper and are based on the earlier LOWTRAN 5 model (Kneizys et al. 1980).

a. Formulation

The direct-beam solar spectral irradiance can be expressed as the product of the extraterrestrial spectral irradiance E_o(λ) and the atmospheric transmittance τ(λ), or

EλE_oλDτλ(1)

where D is the earth–sun distance correction factor (Spencer 1971)

View Expanded

The day angle A_d is equal to

View Expanded

where J_d is the Julian day of the year integer.

Considering the 300–4000-nm range, the total optical depth is simply a sum of the various scattering and absorber species

View Expanded

where the subscripts R, a, UVO₃, IRO₃, and C refer to Rayleigh (molecular) scattering, aerosol scattering, ultraviolet and infrared O₃ absorption, and the water vapor continuum. LOWTRAN expresses optical depth as a function of an absorption coefficient C_i(λ), an equivalent absorber amount X_i (which includes the optical pathlength) and an absorber parameter A_k:

κ_iλ[C_i(λ)X_i]^A_k(7)

In Eq. (7), the A_k absorber parameters are unity for all except O₂, infrared O₃, CO₂, and H₂O vapor. The absorption coefficients C_i(λ) are tabular values for each wavelength.

b. Instrumentation and smoothing

An accurate determination of the transmittance requires that the energy at the top of the atmosphere and at the calibration site be known. At the National Renewable Energy Laboratory, we use a Li-Cor model LI-1800 portable spectroradiometer fitted with a direct-beam collimating tube that matches the 5.0° field of view used on absolute cavity radiometers, and calibrated against a National Institute of Standards and Technology standard irradiance lamp. Advantages of the LI-1800 include portability, low cost, ease of calibration and use, and ease of integration with measurement systems. The LI-1800, which employs a silicon detector with an Instruments SA, Inc. model H-1061 compact, holographic, single-grating monochromator and an order-sorting filter wheel, measures the 300–1100-nm range with 2-nm resolution and has a monochromator slit width of either 0.5 or 1.0 mm. The H-1061 monochromator has a stray light rejection ratio of 10⁻⁵ at eight bandpasses from the 633-nm HeNe laser line. Because of decreasing signal-to-noise ratios at the ends of its full wavelength range, the LI-1800 has much higher uncertainty in these regions (Myers 1989). For this reason, we discard the data at the tails and only use the shorter range of 400 to 1050 nm.

The finite slit-width results in a range of wavelengths, centered at the current wavelength, which are able to pass through the monochromator, and thereby determines the amount of energy that reaches the detector. Generally speaking, the spectral bandwidth of the spectroradiometer will not be the same as that of the AM0 data. In the UV and visible regions, the spectral irradiance data of Wehrli (1985) have data every 0.5 nm. It is therefore necessary to smooth the AM0 data so that the data match the spectroradiometer bandwidth (influence of the AM0 data on the results is discussed later).

The first step in the smoothing process was to measure the spectroradiometer bandwidth by scanning a 633-nm HeNe laser line and to fit the results to a Gaussian distribution. For the LI-1800, the FWHM bandwidths were found to be 3.65 and 6.13 nm for the 0.5- and 1.0-mm slits. The measurement was repeated on a 488-nm Ar laser line and the bandwidth was found to be 3.4 nm for the narrow slits. In addition, several high-pressure lamp emission lines were scanned, and the results were Ar 912–4.1 nm, HgAr 405–4.3 nm, Ne 707–4.3 nm (these line sources are expected to have line widths wider than the laser sources). From these results, we concluded that the spectroradiometer bandwidth is independent of wavelength. Following these measurements, the AM0 data were smoothed at each of the LI-1800 wavelengths by convolution with a Gaussian of the correct FWHM bandwidth. The Gaussian bandwidth is related to the FWHM bandwidth by

View Expanded

Solving for the optical depth gives

View Expanded

Figure 1 is an example of spectral irradiance measured with an LI-1800 that has been processed to obtain the optical depth versus wavelength using Eq. (9).

After the optical depth versus wavelength is known, the data are then processed to obtain the individual optical depths in Eq. (6). This is done by selecting regions where only a few mechanisms dominate the optical depth. Because the optical depth is a simple linear summation, once the magnitude of an individual mechanism is known, it is easily removed from the data. Figure 2 illustrates the regions in the 300–1100-nm range where the scattering and absorbing mechanisms operate, and the wavelength regions selected for the optical depth calculations. Note that both aerosol and molecular scattering are continuous across this range.

c. Automatic error reduction

There is an inherent advantage to the optical depth calculation of Eq. (9) that needs to be emphasized. Any wavelength-independent, multiplicative calibration errors in either E(λ) or E_o(λ) are automatically reduced and become additive offsets. A 5% error, for example, is reduced to a vertical offset of ln(1.05) = 0.049 (cf. this value to the optical depth spectrum in Fig. 1). Aerosol scattering functions will be influenced by such offsets, but they will have no affect on the molecular absorber parameters obtained with the SSP method because the fitting procedures are sensitive only to wavelength-dependent changes. Wavelength-dependent errors in the spectroradiometer measurement and the AM0 are certainly possible, but they are also reduced in magnitude. However, to adversely affect the absorber parameter results, an error needs to vary with wavelength over the narrow ranges used for the parameter fits.

a. Vertical structure

As noted above, the atmospheric model used for the SSP method is a simplification of the LOWTRAN model, which calculates transmittance along a path between any two arbitrary points in the atmosphere. This is done by modeling the atmosphere as a series of concentric shells that contain the temperature, pressure, and absorber (gas and aerosol) density as a function of altitude. These quantities are defined at the shell boundaries, and between the boundaries the temperature is assumed to vary linearly, while the pressure and absorber densities are assumed to vary exponentially. Numerical integration between endpoints is used to calculate the total absorber amounts along the path (Kneizys et al. 1983):

View Expanded

where a and b are the starting and ending points of the path, the i subscripts refer to the various absorber species, z is the altitude, s is the length along the path, R_i is the LOWTRAN equivalent absorber density, and N is the number of layers traversed by the path (R_i and Δs_j are additional numerical integrations used internally to LOWTRAN that are not germane to the SSP method).

Because of the difficulties involved with obtaining real-time vertical profiles of the absorbing species at the location where the spectral irradiance measurements are made, it is assumed that atmospheric profile data are not available, and that it is not possible to use the LOWTRAN numerical integration methods for transmittance calculations. Therefore, the SSP method uses only the following surface data: 1) the direct-normal solar spectral irradiance, 2) the absolute barometric pressure, 3) the absolute ambient temperature, 4) the relative humidity, and 5) the solar geometric position in the sky.

To accomplish this, several simplifications are made. First, for the case of transmittance through the entire atmosphere to the surface, that is, for solar spectral irradiance, the optical path endpoints become the receiver site altitude h and infinity (top of the atmosphere), with the path direction defined by the solar zenith angle θ_z. Next, we assume that the local zenith does not vary along the path (true for zenith angles less than approximately 80°), which implies that ds/dz is constant. At latitudes of 40°, this assumption is valid from approximately 1 h after sunrise to 1 h before sunset. Therefore, Eq. (10) can be expressed for all absorber and scattering species as

View Expanded

Using this so-called secant approximation, ds/dz is now the absolute optical air mass m_a (Iqbal 1983), and is calculated with Kasten’s formula (Kasten 1966),

View Expanded

where the solar zenith angle θ_z can obtained from the site latitude, longitude, and time of day (see Walraven 1978; Wilkinson 1981), and P is the surface pressure in kilopascals. Note that if the pressure ratio is removed, Eq. (12) becomes the relative optical air mass m_r.

b. Absorber functions

LOWTRAN expresses the equivalent absorber densities as

View Expanded

Here, P_zT_z is called the relative air density and is a function of the absolute pressure and temperature, normalized to the standard sea level pressure and absolute temperature:

View Expanded

for units of kilopascals and kelvins. Figures 3 and 4 plot these functions for the U.S. Standard Atmosphere, 1976. The absorber parameters A_k [see Eq. (7)], N_k, and M_k are tabular values for individual absorption bands (provided in Table 1), and the k subscripts refer to the different bands. The W_O2, W_H2O, W_O3, and W_CO2 terms are the molecular densities of the different absorbers (in units of ppmv, or g m⁻³ for H₂O vapor). For the uniformly mixed gases, which include O₂ and CO₂, the densities are assumed to be the same as air. Therefore, the density functions for O₂ and CO₂ are simply the surface concentration times the relative air density. Although CO₂ may not be uniformly mixed in all environments, this assumption is made by LOWTRAN and we adopt it for SSP. However, any errors in the calculated CO₂ absorption band functions will be small compared to the total irradiance because these bands are all at wavelengths greater than 1400 nm.

c. Water vapor continuum

The water vapor continuum is divided into a self-broadened and a foreign-broadened portion. For the self-broadened portion, the LOWTRAN documentation indicates that the continuum has a strong temperature dependence, and an expression is given for the self-broadened continuum at 296 K plus a term that uses linear interpolation between 260 and 296 K (Kneizys et al. 1995). Unfortunately, the formulation of the temperature interpolation term prevented its inclusion in our model and we therefore neglect it. Thus, the H₂O vapor continuum optical depth is expressed as

κ_CC_sλX_sC_fλX_f(19)

and the absorber densities are

View Expanded

In Eqs. (20) and (21), N_L is Loschmidt’s number (Avogadro’s number per unit cm³, or 2.68675 × 10²⁴ molecules cm⁻² km⁻¹), R_o = 273.15 K/296 K, and C = 3.3429 × 10²¹ is noted in the LOWTRAN code as converting water vapor from g m⁻³ to molecules cm⁻² km⁻¹.

d. Relative air density

The pressure–temperature functions are approximated with

View Expanded

where σ_air is the exponential decay of the air density with altitude and σ_N,M is an exponential decay slope that depends on the individual absorption band. From Eqs. (17) and (18) P_h and T_h are the normalized pressure and temperature at the surface. For the uniformly mixed gases, the absorber amount calculations can then be expressed as [using Eqs. (11), (14), (22), and (23); W_i(h) are the surface molecular densities for these absorbers];

View Expanded

The absolute values of the negative exponential decay slopes are used to emphasize that the definite integral solutions are finite only if the exponential arguments are negative. Next, we assume the normalized pressure and temperature profiles are also simple exponential functions (σ_T is positive because the normalized temperature is inversely proportional to the temperature):

View Expanded

Because the absorber density functions are exponential decays from the actual surface values, the lower integration limit should be zero. Making the limit substitution and solving the definite integral in Eq. (24) results in

View Expanded

e. Water vapor absorption

Figure 4 shows the H₂O vapor profile from the U.S. Standard Atmosphere. For altitudes up to about 10 km, the water vapor density decreases in a manner that can be roughly approximated with an exponential decay. Therefore, in the absence of actual profile data, we also assume an exponential function for H₂O vapor:

W_H2O(z)W_H2Ohσ_H2Oz(30)

Even though the actual water vapor profile is not exponential, this assumption allows all three H₂O vapor functions to be expressed in terms of a single parameter, the surface relative humidity. Because X_H2O is obtained from a fit of the 868–936-nm optical depth data, this assumption does not introduce error in the 867–1040-nm H₂O vapor absorption band. However, it will influence the H₂O vapor continuum and the other H₂O vapor absorption band results (this error is discussed further in section 6 below). Computing the definite integrals for the H₂O vapor absorber densities then gives

View Expanded

Assuming that the density of dry air is equal to the density of dry air at saturation, the saturation density of H₂O vapor at the surface and at ambient temperature can be calculated from the relative humidity H_r using the following empirical expression (Kneizys et al. 1980):

View Expanded

with units of grams per cubic meter. This assumption, valid for temperatures from −50° to +50°C, results in a 0.26% error for 293-K air temperature, 80-kPa surface pressure, and 10% relative humidity.

f. Absorption coefficients

It should be noted that the LOWTRAN continuum coefficients are modified in the code by a hyperbolic tangent function and that the H₂O vapor continuum coefficients in Clough et al. (1989) reflect these changes. These modifications are

View Expanded

where

View Expanded

and υ is the wavenumber in inverse centimeters (υ = 10⁴/λ, for units in inverse centimeters and the wavelength λ in micrometers). If the wavenumber is greater than 15725,

View Expanded

otherwise

View Expanded

g. Rayleigh scattering

The optical depth due to Rayleigh (molecular) scattering can be expressed as (Kneizys et al. 1995)

View Expanded

for wavelength units of micrometers. Integrating Eq. (11) with Eq. (13) gives

View Expanded

h. Aerosol scattering

Because the detailed vertical aerosol data in LOWTRAN are not available, a substitute aerosol scattering model is needed. We therefore assume that aerosol scattering can be described by Ångström’s turbidity formula (Iqbal 1983):

κ_aβλ^−αm_a(46)

where α and β are the Ångström turbidity parameters. When expressed with wavelength units of micrometers, β is commonly called the turbidity. If nanometers are used instead, turbidity can be calculated from

View Expanded

Aerosol optical depth spectra in the visible wavelength region generally follow Eq. (46) closely (see Shaw 1983). In the near-IR region, it is possible for deviations to appear, especially for conditions of low aerosol optical depth. Because of the locations of the principal absorption bands in the 350–1100-nm range (see Fig. 2), it was found to be convenient to obtain an aerosol function over the 400–560-nm range using Eq. (46) at the same time a fit is made that determines the O₃ absorber amount. While this aerosol function could be used for the entire 300–4000-nm range, it was deemed desirable to not discard any available aerosol information in the near-IR region. Accordingly, an aerosol function is also calculated in this region, which results in a two-segment overall aerosol function.

a. Oxygen absorption

Examination of the LOWTRAN absorption coefficients shows a strong, narrow O₂ absorption band centered at 762 nm (see Fig. 2). If it is assumed that scattering is continuous and varies slowly over this short wavelength band, it should be possible to iteratively adjust the O₂ absorber amount to minimize the variation in optical depth. Fig. 5 plots the nonzero absorption coefficients in the 750–778-nm range, and of these coefficients, shows that the H₂O vapor and the O₃ coefficients have significant structure. The water vapor is at a minimum between two adjacent weak bands that peak at three orders of magnitude higher than at 762 nm, whereas the ozone is tailing off from a broad, weak absorption peak at 600 nm that is about one order of magnitude greater. For these reasons, it was decided that the first step following the optical depth calculation would be to determine the O₂ absorber amount using the 762-nm band.

As the O₂ absorber amount is varied and the corresponding optical depths are subtracted from the total optical depth, a series of curves is obtained. Using the optical depth data in Fig. 1, Fig. 6 shows these curves approaching a line before becoming negative. If a linear least squares fit is performed to the optical depth versus wavelength (Natrella 1966), it will be seen that the standard deviation of the fit goes through a minimum as the absorber amount is varied. This minimum represents a best estimate of the O₂ absorber amount.

To determine the minimum standard deviation of the fit, the following iterative search procedure is used. At any two wavelengths in the O₂ absorption band,

View Expanded

Assuming the absorption coefficient at the second wavelength is approximately zero, the absorber parameter is unity, and the optical depths due to other mechanisms at both wavelengths are nearly the same, that is, κ^′₁ ≈ κ^′₂, these equations can be combined as

X̃_O2κ₁κ₂C₁(50)

which is used as an initial estimate of X_O2. For wavelengths 1 and 2, we use 762 and 756 nm to calculate the estimate (see Fig. 6). The search is initiated with the following:

X_j=1XX̃_O2(51)

where the j subscript refers to iteration steps. While monitoring the standard deviation of the linear fit over the 752–774-nm range, the absorber amount is increased by

X_j+1X_jX(52)

until the standard deviation increases, at which point the linear search has gone through the minimum. To isolate the minimum, a golden search procedure is then performed (Harkins 1964):

View Expanded

where the sign s = +1 or −1 and G_n is the golden number

View Expanded

b. Rayleigh scattering

Using the O₂ absorber amount from Eq. (56), the air exponential decay slope for the O₂ absorption band can be obtained from Eqs. (27)–(29):

View Expanded

using a value of 20.95% for the volume mixing ratio of O₂ to air. Unfortunately, this is one equation with two unknowns, so σ_P and σ_T cannot be isolated. A solution is to select an appropriate temperature slope σ_T from the model atmospheres in Table 2 and then solve for σ_P in Eq. (57). The choice of σ_T was influenced by the fact that numerically, σ_T is about an order of magnitude smaller than σ_P, so the error contribution due to this choice should be somewhat smaller. Once σ_P and σ_T are known, all other decay slopes can be calculated using Eqs. (27) and (28).

c. Carbon dioxide absorption

With σ_P and σ_T known and using the uniformly mixed gas assumption noted above, the CO₂ absorber amount calculations are straightforward and only require the surface CO₂ mixing ratio. Iqbal (1983) lists a value of 333 ppmv, but more recent publications recommend 355–360 ppmv for years after 1994 (Anderson et al. 1996). From Eq. (29), the absorber amounts are then

View Expanded

Figure 7, curve 2, shows the corrected optical depth data from Fig. 1.

d. Short-wavelength aerosol scattering and UV ozone absorption

Therefore, if the optical depth is corrected for UV O₃ absorption, a log–log plot of optical depth versus wavelength should be a straight line. Figure 8 shows the O₂ and Rayleigh scattering–corrected optical depth data from Fig. 7, curve 2, as the UV O₃ absorber amount is varied and the resulting optical depth subtracted from κ′. The correlation coefficient of a linear least squares fit over the 400–560-nm range therefore goes through a maximum, and a search for this maximum gives both the UV O₃ absorber amount and the aerosol scattering function in this region.

During development of the SSP method, we tried to use wavelengths shorter than 400 nm, hoping that the additional wavelength points would improve the linear fitting. Instead, it was found that these points tended to not fall on the same line as the longer wavelengths and resulted in a reduction of the correlation coefficient. Two possible causes for this behavior are offered. First, this is a region where the spectral sensitivity of the silicon detector in the spectroradiometer is falling, which results in an decreasing signal-to-noise ratio (Myers 1989). Second, almost all of the variability in the AM0 spectral irradiance occurs below 400 nm, especially during cycles of maximum solar activity (Woods et al. 1996). Because SSP uses a constant AM0 spectral irradiance to calculate optical depth, solar activity could account for some of these deviations.

Because the fit uses the logarithm of κ", the search will fail if the UV O₃ absorber amount becomes greater than this critical amount. Clearly, the search needs to be constructed to avoid this problem, which can be accomplished by limiting the UV O₃ absorber amount value to less than

View Expanded

The search amount is obtained from the critical UV O₃ absorber amount as

View Expanded

Because the correlation coefficient maximum can be close to the critical amount, a small initial search step is used to reduce the possibility of the search failing if the linear portion steps over the maximum and goes beyond the critical amount. With these values, a combined linear-golden search, identical to the one used for the O₂ absorber amount, is used to obtain X_UVO3 [see Eqs. (52)–(55)]. The golden search is terminated when ΔX is less than 1 × 10⁻⁵, and the slope and intercept results of the last linear fit during the search provide α_SW and β_SW.

Next, the optical depth is corrected by subtracting the O₃ optical depth at each wavelength using Eq. (62). Example results of this correction are shown in Fig. 7, curve 3, and the best fit shown in Fig. 8 is very similar to the aerosol optical depth in Fig. 3a of Shaw (1983) for central Alaska in 1977.

e. Infrared ozone absorption

The O₃ absorption search procedure produces an absorber amount that would be obtained by integration of Eq. (11) with Eq. (15), without knowledge of the O₃ vertical profile. There are also some shallow O₃ absorption bands beyond 3000 nm that are expressed in the same band-model form as O₂ and CO₂ [see Eq. (14)]. Thus, the UV O₃ amount cannot be used for these bands, and some knowledge of the O₃ vertical profile is needed if Eqs. (11) and (14) is to be integrated. Unfortunately, the O₃ density is highly variable and does not vary in an exponential manner, as shown in Fig. 9. Starting at the surface, the density is nearly constant up to 6–8 km, where it increases to a maximum at 15–20 km. Above the maximum, the density decays roughly exponentially.

Over the 0–80-km altitude range, the average decay constant was calculated to be −0.06928 km⁻¹. Integrating Eqs. (11) and (15) and solving for the surface concentration (in ppmv) gives

View Expanded

It should be emphasized that the O₃ concentration obtained by Eq. (67) is not the actual surface value (in fact, it is probably about an order of magnitude smaller);rather, it is a value that gives the corresponding absorber amount for an exponential profile. Using this result with integration of Eqs. (11) and (14), the IR O₃ amounts can be expressed as

View Expanded

Because these bands are outside the spectroradiometer response range, correction of the overall optical depth is not necessary.

f. Water vapor absorption

Determination of the H₂O vapor absorption is complicated by the fact that H₂O vapor absorption is not a single parameter but is actually three parameters: H₂O vapor band absorption and the self- and foreign-broadened portions of the H₂O vapor continuum. We need a way of relating these functions to a single parameter that can be adjusted for a best fit. Equations (31)–(33) indicate that two parameters are common to all three: the surface H₂O vapor saturation density W_H2O(h) and the H₂O vapor exponential decay slope |σ_H2O| Of these two, W_H2O(h) is calculated from the relative humidity and is therefore known, whereas no information is available about the vertical water vapor distribution. Thus, it was decided to fit the absorption by varying |σ_H2O|.

Several schemes were attempted for the fit, and the following produced the best results. Looking at Fig. 7, curve 3, it is seen that outside the absorption bands, for the example data, the optical depth is nearly flat over this region (the data have been corrected for molecular scattering). Also, because the optical depth is not large, small errors at single wavelengths can cause the corrected optical depth to be negative. These complications make a linear least squares fit problematic, and we finally chose a simple minimization of the population standard deviation of the net optical depth:

View Expanded

Figure 10 shows the effects of varying |σ_H2O| over the 868–936-nm range, which has little absorption at 868 nm, and extends to the peak of the large absorption band at 936 nm. This scheme gives a sharp minimum and the search for this minimum is initiated with an estimate of the absorber amount for the 867–1040-nm H₂O vapor band:

View Expanded

Using this quantity in Eq. (31) gives an initial estimate of |σ_H2O|:

View Expanded

As before, a combined linear-golden search is employed to find the minimum, using the following initial conditions:

View Expanded

The search therefore starts well above the minimum using a small absorber amount and moves downward. At each iteration, the absorber amounts for the H₂O vapor bands and the continuum are calculated, and the corresponding optical depths are subtracted from κ"(λ). The population standard deviation S is then calculated over the 868–936-nm range using Eq. (69), and this quantity is minimized with the same search described by Eqs. (52)–(55). The golden search portion is terminated when ΔX is less than 5 × 10⁻⁵.

Results of the optical depth correction are shown in Fig. 7, curve 4.

g. Long-wavelength aerosol scattering

The lack of wavelengths free of molecular absorption (including the H₂O continuum) in the visible and IR regions complicates calculation of the aerosol-scattering optical depth. Figure 11 is a log–log plot of selected O₃ absorption–corrected optical depth data from Fig. 7, curve 4. Over the 652–1044-nm range, these are the wavelengths where the water vapor absorption coefficients are less than 2 × 10⁻², the O₂ absorption coefficients are less than 1 × 10⁻⁴, and the optical depths are greater than 0.001. These criteria were chosen to avoid the peaks of the absorption bands and to discard wavelengths where the optical depth is very small (the data have been corrected for molecular scattering). Figure 11 shows that it is not possible to obtain something better than just a rough determination of the aerosol scattering function in this region. A linear least squares fit of ln[κ"'(λ)] versus ln(λ) at these wavelengths is used to calculate α_LW and β_LW, and the results of this fit for the data from Fig. 7, curve 4, are plotted in Fig. 11.

A problem with the long-wavelength aerosol fit needs to be noted. In a handful of cases where the LI-1800 data were noisier than usual, the aerosol linear fit produced a negative value of α_LW. This problem, which seems to occur when the number of wavelengths that pass the criteria above is small (<10), results in an unrealistic function where the aerosol optical depth is increasing with wavelength in the infrared. We deal with the problem in the section below.

h. Overall aerosol optical depth

Two aerosol functions have now been obtained, one for the 400–560-nm region and one for the 652–1044-nm range:

View Expanded

A transition between these two functions is needed. Setting κ_SW = κ_LW, it is possible to solve for the wavelength at which Eqs. (76) and (77) are equal:

View Expanded

To smooth the transition between the two functions, a crossover aerosol function is constructed from the optical depths at two wavelengths 50 nm above and below the crossover wavelength:

View Expanded

Dividing Eq. (81) by Eq. (82) and solving for α_c gives

View Expanded

Here, β_c is obtained from either Eq. (81) or (82):

View Expanded

If the crossover wavelength calculated in Eq. (78) is not within 100 of 600 nm, the calculated wavelength is discarded, and λ_c is arbitrarily set to 600 nm instead. This avoids having Eq. (76) or (77) represent the optical depth in the wrong region, which can produce unnatural aerosol functions and errors in the calculated optical depth.

The overall aerosol optical depth is thus defined by Eqs. (76), (77), and (85), in the corresponding regions defined by wavelengths λ₁ and λ₂. Figure 7, curve 5, shows the aerosol scattering optical depth correction. The overall function is displayed in Fig. 11 as a log– log graph, and illustrates the crossover function.

For the rare cases where α_LW is negative, the long-wavelength and crossover aerosol functions are discarded, and the short-wavelength aerosol function is used for the entire wavelength range.

i. Meteorological parameters

The precipitable water vapor calculation is more involved because the units are different and because of the absorber parameters A_k, N_k, and M_k. Precipitable water w is defined as the total amount of H₂O vapor at standard temperature and pressure in the zenith direction, and Eq. (30) gives the H₂O vapor density as a function of altitude in grams per cubic meter. Therefore, a vertical integration of Eq. (30) should give w:

View Expanded

where the constant 0.1 converts from g km m⁻³ to g cm⁻² and the pressure–temperature reduction is from Paltridge and Platt (1976) [note that 1 g cm⁻² of H₂O vapor corresponds to a height of 1 cm (Iqbal 1983)]. It should be noted that the same caution about the simple exponential water vapor profile assumption discussed in section 3 is applicable here.

a. Extrapolation

With the individual absorber and scattering parameters known, it is then possible to calculate the spectral irradiance over the entire 300–4000-nm range using Eq. (1). This can be done with the AM0 solar spectrum of Wehrli (1985), which contains 920 data points from 200 nm to 10 μm, with wavelength resolutions that vary from 1 nm in the UV to about 200 nm in the far IR. On the other hand, the simple model of Bird and Riordan (1986) provides only 122 points and gives much less resolution. Because of the application considered, that is, numerical convolution of the resultant spectral irradiances with solar cell spectral responses, it was decided that neither spectrum met these needs. A medium-resolution spectrum that preserves as much detail of the Wehrli (1985) spectrum and the LOWTRAN absorption coefficients as possible was therefore developed. This spectrum has 446 wavelength points, variable resolution, and a minimum resolution of 0.5 nm.

Next, the AM0 spectral irradiance and absorption coefficients corresponding to the selected wavelengths were needed. These were obtained in a manner similar to the one described above for the LI-1800 spectroradiometer, with the exception that a variable, rather than fixed, bandwidth was used for the Gaussian smoothing. At any wavelength, the FWHM bandwidth was equal to the distance to the nearest adjacent wavelength.

Figure 12 shows the results of reconstructing and extrapolating the spectral irradiance of Fig. 1 using the parameters obtained from the optical depth calculation procedures. This figure, which shows that the agreement with the measured spectral irradiance is very good, is typical of the results obtained with the method. Note that in the UV–visible region below 600 nm the two curves are quite close, and the same is true of the 867– 1040-nm H₂O vapor absorption band.

b. Comparison with absolute cavity radiometer data

As a gauge of the quality, the extrapolated spectrum is then integrated and compared to the total irradiance measured by the absolute cavity radiometer at the same time (to minimize possible errors in the reference cell calibration constants caused by different limits of integration in the spectral correction factor, we then normalize the extrapolated spectral irradiance by this ratio so that the integrated irradiance matches the value measured with the cavity radiometer). For the data in Fig. 12, the ratio of the integrated spectral irradiance to the total irradiance was 0.9908, which again shows that the agreement is excellent.

c. History

Since 1990, we have accumulated a large amount of data using SSP. Figure 13 summarizes these results and shows the irradiance ratios for over 2300 spectra. The overall average of these ratios is 1.006, with a standard deviation of 0.016. Table 3 lists the dates on which these spectra were obtained, along with the daily averages and standard deviations of the ratios. A histogram of the ratios appears in Fig. 14, which shows a large peak very close to 1.00 and a broad, smaller peak at about 1.02.

Daily variations of the solar constant cannot explain the variations seen in Fig. 13 because over a 20-yr period, measurements of the extraterrestrial solar constant have a total range of 1363–1368 W m⁻² (0.37%) and a standard deviation of just 0.58 W m⁻² (Fröhlich and Lean 1998).

d. Spectroradiometer temperature sensitivity

Temperature sensitivity of the LI-1800 silicon detector and analog signal processing electronics is a known problem. The instrument used to obtain the data in Fig. 13 was fitted with a resistive heater and a thermostat to hold the internal detector at 40° ± 1°C (the heater also affected the electronics and monochromator). Unfortunately, this scheme cannot cool the detector if the temperature rises above 40°C, which can easily occur when the instrument is exposed to direct sunlight. This problem was alleviated somewhat with a reflective foil shield (Osterwald et al. 1988), but it is believed that heating was responsible for most of the variation in Fig. 13 (also note that the maximum operating temperature of the detector is 40°C).

A more recent study of the LI-1800 temperature sensitivity showed that 1) the silicon detector response changes with temperature, and 2) the silicon detector, the voltage amplifier, and the low-pass filters have DC offsets that vary with temperature (D. Clair 1997, personal communication). Following this study, for the 1998 calibrations, the LI-1800 instrument used for the NREL reference cell calibrations was fitted with a thermoelectric temperature controller set to hold the internal detector at 20° ± 0.02°C. The effect of this modification is evident in Fig. 13, where the 1998 data are seen to have much reduced variability and the ratio is much closer to unity. Figure 15 is a histogram of the irradiance ratios for just the 1998 data, which have an overall average ratio of 0.999 and a standard deviation of 0.004. When compared with an absolute cavity radiometer, then, the data show a lower uncertainty for direct normal irradiance measurements than a normal-incidence pyrheliometer, a thermopile instrument that has an uncertainty of about 2%–4%.

In an attempt to determine if the irradiance ratio is dependent on atmospheric conditions, we plotted the 1998 ratios versus the direct irradiance, the relative optical air mass, and the precipitable water vapor amount calculated using Eq. (87). The results indicated there have very little dependence on the direct irradiance or the air mass. Figure 16 shows the irradiance ratios plotted versus the water vapor amount, and a slope of about −0.006 cm ⁻¹ is evident. This slight dependence is probably responsible for most of the lower, secondary peak in Fig. 15. A possible cause of this dependency could be the inaccuracies in the calculated water vapor continuum absorber amounts previously discussed.

a. General

Several limitations to the SSP method can be listed. First, the atmospheric transmittance model only applies to cloudless skies, and it has no provisions for transmittance through liquid water or ice crystals. The results are therefore suspect if, for example, cirrus clouds are present in the direct beam. In such cases, however, the ratio of the integrated spectral irradiance produced by the extrapolation to the direct total irradiance will probably not match those under better conditions.

Also, SSP assumes that the atmospheric conditions do not change over the time period needed for the spectral irradiance measurement. As noted previously, the measurement time of the LI-1800 spectroradiometer is about 30 s. To ensure stable conditions during this time, we digitize the absolute cavity radiometer output every 0.5 s, along with the short-circuit currents of several photovoltaic reference cells. If any of these vary by more than 1%, the spectral irradiance measurement is discarded.

The measurement time limitation also determines how close to sunrise and sunset the method can be used. This is caused by the varying rate of change of the optical air mass over the course of a day. Figure 17 shows that to avoid high rates of change with instruments such as the LI- 1800, SSP should be not be used within about an hour after sunrise and before sunset. This restriction is coincidentally very similar to that imposed by the local zenith angle assumption used to obtain Eq. (11).

b. Exponential profile assumptions

The SSP method relies on obtaining the X_i absorber amounts in Eq. (7) for each of individual absorption bands in the 300–4000-nm range. As a consequence of the optical depth fitting procedures, for the major absorption bands in the spectroradiometer range, that is, UV O₃, O₂, and the 867–1040-nm H₂O band, the corresponding absorber amounts are determined without the need for any knowledge of the vertical profiles of the absorber densities. The only exponential profile assumptions needed to obtain these parameters are the pressure and temperature functions, Eqs. (25) and (26). Figures 3 and 4 show that these are reasonable assumptions for pressure up to at least 100 km, but break down for temperature above about 10 km, where the tropopause ends (note that all of the outdoor data used to develop the SSP method were obtained at a site altitude of 1.7 km). Therefore, the majority of the error in these absorption functions will be due to E(λ) and E_o(λ), and the LOWTRAN absorption coefficients. For the remaining H₂O vapor bands, the H₂O vapor continuum, and the CO₂ and infrared O₃ bands, the lack of actual density profiles will increase the optical depth errors.

c. LOWTRAN accuracy

The accuracy of this model depends on the errors in the LOWTRAN 7 formulation, especially Eq. (7) MODTRAN and LOWTRAN are classified as band model radiative transfer codes, which have computational speed advantages over much more rigorous and very high resolution line-by-line (LBL) algorithms [an example of an LBL is the HITRAN (High-resolution Transmittance) spectroscopic database, from which MODTRAN is derived; Anderson et al. (1996)]. For the earlier LOWTRAN 5 model, the authors stated the overall accuracy for transmittance was better than 10%, and that “the largest errors may occur in the distant wings of strongly absorbing bands in regions which such bands overlap appreciably” (Kneizys et al. 1980). The transmittance functions were upgraded for LOWTRAN 7, and a discussion of how the LOWTRAN 7 band model was developed is summarized in Kneizys et al. (1995), and quoting from this report: “Calculations using the band-model parameters, agreed within a mean (over all wavenumbers and gases) rms transmittance difference of 2.0% with the degraded LBL data used in their determination.” The band model parameters were then validated against laboratory and field measurements.

d. Ångstrom turbidity assumption

Short wavelength aerosol scattering is determined from a linear fit of the natural logarithm of optical depth versus the natural logarithm of wavelength as the ozone absorber amount is varied (see section 4). Because we assume an Ångstrom relationship for aerosol scattering [Eq. (46)], using the SSP method in aerosol conditions where Eqs. (46) and (61) are invalid will result in an incorrect value for the O₃ absorber amount, as well as the aerosol optical depth.

e. AM0 accuracy

Finally, the accuracy of the transmittance calculated with Eq. (1) depends on the accuracy of the extraterrestrial solar spectral irradiance, and finding estimates of error in the literature is difficult. The Wehrli (1985) AM0 spectrum, which does not include an error estimation in the supporting text, was scaled so that the total integrated irradiance (i.e., the solar constant) equals 1367 W m⁻², which is very close to the mean value of 1366.1 W m⁻² of the solar constant measured in earth orbit since 1978 (Fröhlich and Lean 1998). However, because this AM0 spectrum is composed of a number of different measurements that span only portions of the entire wavelength range, localized errors of up to several percent in magnitude should be expected (Colina et al. 1996).

According to Colina et al. (1996), the spectrum used by LOWTRAN 7 is based on the Wehrli (1985) spectrum, which in turn is mostly based on the Neckel and Labs (1984) spectrum (up to 870 nm), and the Arvesen et al. (1969) spectrum from 870 to 2500 nm. In this region, the Arvesen et al. (1969) spectrum apparently shows anomalous absorption features, especially between 870 and 970 nm where differences up to 5% have been noted. MODTRAN required a much higher resolution AM0 spectrum (1 cm⁻¹) than the Wehrli (1985) spectrum, so a very high-resolution spectrum calculated from a sophisticated solar irradiance model produced by the Harvard–Smithsonian Astrophysics Observatory was used (Kurucz 1993). Preliminary results with the SSP method show that using the Kurucz (1993) spectrum improves the 867–1040-nm H₂O band fit, and results in a slight improvement to the ratios of Fig. 13. This would appear to illustrate how multiplicative errors are reduced by the logarithmic processing of the measured transmittance data, as discussed above in section 2.