## 1. Introduction

The focus of this paper is to use spectroradiometric measurements to characterize atmospheric transmittance in a form that can then be used to calculate the entire terrestrial solar spectral irradiance from the airmass zero (AM0, or extraterrestrial) spectral irradiance at each wavelength. The direct-beam solar spectral irradiance at a point in the atmosphere is expressed as the product of the AM0 spectral irradiance and the optical transmittance through the atmosphere to that point. Atmospheric transmittance is a complex function of molecular and aerosol scattering, and molecular absorption in discrete bands (H_{2}O vapor, O_{2}, CO_{2}, and O_{3}), all of which vary with wavelength. A method for obtaining atmospheric transmittance parameters is therefore needed.

### 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 m^{2} 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).

## 2. Atmospheric optical depth measurements

### a. Formulation

*E*

_{o}(

*λ*) and the atmospheric transmittance

*τ*(

*λ*), or

*E*

*λ*

*E*

_{o}

*λ*

*Dτ*

*λ*

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

*A*

_{d}is equal to

*J*

_{d}is the Julian day of the year integer.

*κ*as (Iqbal 1983)

*τ*

*λ*

*κ*

*λ*

*κ*

*λ*

*k*

*λ*

*m.*

*R, a,*UVO

_{3}, IRO

_{3}, and

*C*refer to Rayleigh (molecular) scattering, aerosol scattering, ultraviolet and infrared O

_{3}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*

_{i}

*λ*

*C*

_{i}(

*λ*)

*X*

_{i}]

^{Ak}

*A*

_{k}absorber parameters are unity for all except O

_{2}, infrared O

_{3}, CO

_{2}, and H

_{2}O 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).

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.

## 3. Atmospheric model

### a. Vertical structure

*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 Δ

*s*

_{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.

*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

*ds*/

*dz*is now the absolute optical air mass

*m*

_{a}(Iqbal 1983), and is calculated with Kasten’s formula (Kasten 1966),

*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

*P*

_{z}

*T*

_{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:

*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}

*W*

_{CO2}

^{−3}for H

_{2}O vapor). For the uniformly mixed gases, which include O

_{2}and CO

_{2}, the densities are assumed to be the same as air. Therefore, the density functions for O

_{2}and CO

_{2}are simply the surface concentration times the relative air density. Although CO

_{2}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 CO

_{2}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

_{2}O vapor continuum optical depth is expressed as

*κ*

_{C}

*C*

_{s}

*λ*

*X*

_{s}

*C*

_{f}

*λ*

*X*

_{f}

*N*

_{L}is Loschmidt’s number (Avogadro’s number per unit cm

^{3}, or 2.68675 × 10

^{24}molecules cm

^{−2}km

^{−1}),

*R*

_{o}= 273.15 K/296 K, and

*C*= 3.3429 × 10

^{21}is noted in the LOWTRAN code as converting water vapor from g m

^{−3}to molecules cm

^{−2}km

^{−1}.

### d. Relative air density

*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];

*σ*

_{T}is positive because the normalized temperature is inversely proportional to the temperature):

*σ*

_{T}becomes negative, see Fig. 3). Table 2 lists the 0–10-km exponential slopes calculated from fits to the model atmospheres, along with the variations of these slopes. Note that these quantities do not change greatly for different atmospheres, as

*σ*

_{P}has a spread of about 13% and

*σ*

_{T}about 21%. Therefore, the temperature assumption should not have a large effect on the fitting results. Substituting Eqs. (25) and (26) into Eq. (23) and solving for

*σ*

_{N,M}gives

*σ*

_{N,M}

*N*

_{k}

*σ*

_{P}

*M*

_{k}

*σ*

_{T}

*σ*

_{air}equals

*σ*

_{N,M}when

*N*

_{k}and

*M*

_{k}are unity)

*σ*

_{air}

*σ*

_{P}

*σ*

_{T}

### e. Water vapor absorption

_{2}O 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 H

_{2}O vapor:

*W*

_{H2O}(

*z*)

*W*

_{H2O}

*h*

*σ*

_{H2O}

*z*

_{2}O vapor functions to be expressed in terms of a single parameter, the surface relative humidity. Because

*X*

_{H2O}

_{2}O vapor absorption band. However, it will influence the H

_{2}O vapor continuum and the other H

_{2}O vapor absorption band results (this error is discussed further in section 6 below). Computing the definite integrals for the H

_{2}O vapor absorber densities then gives

_{2}O 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):

### f. Absorption coefficients

_{2}, H

_{2}O vapor, CO

_{2}, O

_{3}, and the H

_{2}O vapor continuum are tabular values at discrete wavelengths. These absorption coefficients are from the FORTRAN block data statements CPUMIX, CPH2O, CPO3, C4D, SF296, and BFH2O in the LOWTRAN computer code, and were processed with the following function (LOWTRAN stores the log

_{10}of the coefficients):

*C*

_{i}

*λ*

*C*

^{′}

_{i}

*λ*)

^{10Ak}

_{2}O vapor continuum coefficients in Clough et al. (1989) reflect these changes. These modifications are

*υ*is the wavenumber in inverse centimeters (

*υ*= 10

^{4}/

*λ,*for units in inverse centimeters and the wavelength

*λ*in micrometers). If the wavenumber is greater than 15725,

### g. Rayleigh scattering

### h. Aerosol scattering

*κ*

_{a}

*βλ*

^{−α}

*m*

_{a}

*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

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 O_{3} 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.

## 4. Optical depth processing

The details of how the optical depth data are processed to obtain the transmittance parameters are presented below in sequence.

### a. Oxygen absorption

Examination of the LOWTRAN absorption coefficients shows a strong, narrow O_{2} 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 O_{2} 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 H_{2}O vapor and the O_{3} 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 O_{2} absorber amount using the 762-nm band.

As the O_{2} 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 O_{2} absorber amount.

_{2}absorption band,

*κ*

^{′}

_{1}

*κ*

^{′}

_{2}

*X̃*

_{O2}

*κ*

_{1}

*κ*

_{2}

*C*

_{1}

*X*

_{O2}

*X*

_{j=1}

*X*

*X̃*

_{O2}

*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+1}

*X*

_{j}

*X*

*s*= +1 or −1 and

*G*

_{n}is the golden number

*X*is less than 0.5. The oxygen absorber amount is then calculated from the search result:

*X*

_{O2}

*X*

^{(1/AO2)}

*X*

^{1.7727}

### b. Rayleigh scattering

_{2}absorber amount from Eq. (56), the air exponential decay slope for the O

_{2}absorption band can be obtained from Eqs. (27)–(29):

_{2}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

*σ*

_{P}and

*σ*

_{T}known and using the uniformly mixed gas assumption noted above, the CO

_{2}absorber amount calculations are straightforward and only require the surface CO

_{2}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

*N*

_{k}and

*M*

_{k}differ, there is a corresponding absorber amount for each CO

_{2}absorption band. With the parameters for O

_{2}absorption, CO

_{2}absorption, and Rayleigh scattering known, the optical depth can now be corrected:

*κ*

*λ*

*κ*

*λ*

*C*

_{R}

*λ*

*X*

_{R}

*C*

_{O2}

*λ*

*X*

^{AO2}

_{O2}

*C*

_{CO2k}(

*λ*)

*X*

^{ACO2k}

_{CO2k}

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

### d. Short-wavelength aerosol scattering and UV ozone absorption

_{3}and the short-wavelength tail of the self-broadened H

_{2}O continuum band. Because the self-broadened continuum is four orders of magnitude less than the peak at 940 nm and is falling rapidly, we neglect its affects for the O

_{3}determination. Over this range, the optical depth can then be written as

*κ*

^{′}

_{SW}

*λ*

*C*

_{O3}

*λ*

*X*

_{UVO3}

*β*

_{SW}

*λ*

^{−αSW}

*m*

_{a}

*κ*

^{′}

_{SW}

*λ*

*C*

_{O3}(

*λ*)

*X*

_{UVO3}

*β*

_{SW}

*m*

_{a}

*α*

_{SW}

*λ*

Therefore, if the optical depth is corrected for UV O_{3} absorption, a log–log plot of optical depth versus wavelength should be a straight line. Figure 8 shows the O_{2} and Rayleigh scattering–corrected optical depth data from Fig. 7, curve 2, as the UV O_{3} 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 O_{3} 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.

_{3}optical depth is increasing. Thus, there is a value of the UV O

_{3}absorber amount that will result in a corrected optical depth

*κ*" of zero at the end of this range, 560 nm. From the left-hand side of Eq. (61),

*κ*" is

*κ*

*λ*

*κ*

*λ*

*C*

_{O3}(

*λ*)

*X*

_{UVO3}|

_{λ=560nm}

*κ*", the search will fail if the UV O

_{3}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 O

_{3}absorber amount value to less than

_{3}absorption has dropped to zero by 424 nm, the initial estimates of

*α*

_{SW}and

*β*

_{SW}are obtained from a linear least squares fit of ln[

*κ*′(

*λ*)] versus ln(

*λ*) over the 400–424-nm range. The slope of the fit

*m*is

*α̃*

_{SW}

*β̃*

_{SW}

*b*:

*β̃*

_{SW}

*b*

_{3}absorber amount as

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 O_{2} absorber amount, is used to obtain *X*_{UVO3}*X* 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 O_{3} 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 O_{3} absorption search procedure produces an absorber amount that would be obtained by integration of Eq. (11) with Eq. (15), without knowledge of the O_{3} vertical profile. There are also some shallow O_{3} absorption bands beyond 3000 nm that are expressed in the same band-model form as O_{2} and CO_{2} [see Eq. (14)]. Thus, the UV O_{3} amount cannot be used for these bands, and some knowledge of the O_{3} vertical profile is needed if Eqs. (11) and (14) is to be integrated. Unfortunately, the O_{3} 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.

_{3}absorption bands entirely, the following procedure is used to calculate these absorber amounts. This is done by assuming an exponential profile using a decay constant that is an average of an exponential least squares fit to the six model atmospheres:

*W*

_{O3}(

*z*)

*W*

_{O3}(

*h*)

*σ*

_{O3}|

*z*)

^{−1}. Integrating Eqs. (11) and (15) and solving for the surface concentration (in ppmv) gives

_{3}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 O

_{3}amounts can be expressed as

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 H_{2}O vapor absorption is complicated by the fact that H_{2}O vapor absorption is not a single parameter but is actually three parameters: H_{2}O vapor band absorption and the self- and foreign-broadened portions of the H_{2}O 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 H_{2}O vapor saturation density *W*_{H2O}(*h*)_{2}O vapor exponential decay slope |*σ*_{H2O}*W*_{H2O}(*h*)*σ*_{H2O}

*σ*

_{H2O}

_{2}O vapor band:

The search therefore starts well above the minimum using a small absorber amount and moves downward. At each iteration, the absorber amounts for the H_{2}O 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 Δ*X* is less than 5 × 10^{−5}.

_{2}O vapor optical depths are subtracted from the total optical depth using the resulting absorber amounts:

*κ*

*λ*

*κ*

*λ*

*C*

_{H2O}

*λ*

*X*

^{AH2Ok}

_{H2Ok}

*C*

_{s}

*λ*

*X*

_{s}

*C*

_{f}

*λ*

*X*

_{f}

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 H_{2}O continuum) in the visible and IR regions complicates calculation of the aerosol-scattering optical depth. Figure 11 is a log–log plot of selected O_{3} 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 O_{2} 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

*κ*

_{SW}=

*κ*

_{LW}, it is possible to solve for the wavelength at which Eqs. (76) and (77) are equal:

*κ*

_{c}

*β*

_{c}

*λ*

^{−αc}

*m*

_{a}

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 *λ*_{1} and *λ*_{2}. 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

_{3}absorber amount used in Eq. (61) is already in the correct units and needs only to be converted to the vertical amount, which can be done with the relative optical air mass:

*l*

*X*

_{O3}

*m*

_{r}

*A*

_{k},

*N*

_{k}, and

*M*

_{k}. Precipitable water

*w*is defined as the total amount of H

_{2}O vapor at standard temperature and pressure in the zenith direction, and Eq. (30) gives the H

_{2}O vapor density as a function of altitude in grams per cubic meter. Therefore, a vertical integration of Eq. (30) should give

*w*:

^{−3}to g cm

^{−2}and the pressure–temperature reduction is from Paltridge and Platt (1976) [note that 1 g cm

^{−2}of H

_{2}O 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.

## 5. Spectral irradiance extrapolation

### 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 *μ*m, 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 H_{2}O 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.

## 6. Limitations

### 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 O_{3}, O_{2}, and the 867–1040-nm H_{2}O 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 H_{2}O vapor bands, the H_{2}O vapor continuum, and the CO_{2} and infrared O_{3} 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 O_{3} 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 H_{2}O 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.

## 7. Summary

Motivated by a need for 300–4000-nm direct-normal spectral irradiance data, we presented a method for calculating atmospheric transmittance parameters from 300– 1100-nm spectral irradiance measurements. By making simple exponential assumptions for vertical molecular distributions, analytical solutions for the LOWTRAN 7 numerical integrations were obtained, resulting in lumped parameter absorber amounts. Numerical procedures were developed that search for best fits of these parameters in measured atmospheric optical depth data. Using these results, we showed how the absorber amounts can be used to extrapolate the spectral irradiance over the wider wavelength range. For a large number of measurements over a 10-yr period, extrapolated spectral irradiances were compared with absolute cavity radiometer data and the ratios were found to be within ±0.032 of unity (two standard deviations). After the spectroradiometer used for this study was equipped with improved temperature controls, the ratios were within ±0.008 of unity. Lastly, we discussed limitations of the SSP method.

## Acknowledgments

We acknowledge the efforts of several coworkers at the National Renewable Energy Laboratory. Many useful discussions and assistance about LOWTRAN and spectral irradiance models were provided by Carol Riordan, and Daryl Myers gave invaluable help with spectroradiometer calibrations and issues. Halden Field and Don Dunlavy performed many of the spectral irradiance measurements, and David Clair performed the temperature–sensitivity analysis of the LI-1800 spectroradiometer.

This work was supported by the U.S. Department of Energy under Contract DE-AC36-83CH20093.

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A graphical illustration of the individual absorption bands in the wavelength region of a spectroradiometer that uses a silicon photodiode detector. Also shown are the wavelengths used for the absorber parameter determinations

Citation: Journal of Atmospheric and Oceanic Technology 17, 9; 10.1175/1520-0426(2000)017<1171:SSP>2.0.CO;2

A graphical illustration of the individual absorption bands in the wavelength region of a spectroradiometer that uses a silicon photodiode detector. Also shown are the wavelengths used for the absorber parameter determinations

Citation: Journal of Atmospheric and Oceanic Technology 17, 9; 10.1175/1520-0426(2000)017<1171:SSP>2.0.CO;2

A graphical illustration of the individual absorption bands in the wavelength region of a spectroradiometer that uses a silicon photodiode detector. Also shown are the wavelengths used for the absorber parameter determinations

Citation: Journal of Atmospheric and Oceanic Technology 17, 9; 10.1175/1520-0426(2000)017<1171:SSP>2.0.CO;2

Normalized air pressure (*P*_{z}) and air temperature (*T*_{z}) profiles for the *U.S. Standard Atmosphere, 1976* (Kneizys et al. 1980)

Normalized air pressure (*P*_{z}) and air temperature (*T*_{z}) profiles for the *U.S. Standard Atmosphere, 1976* (Kneizys et al. 1980)

Normalized air pressure (*P*_{z}) and air temperature (*T*_{z}) profiles for the *U.S. Standard Atmosphere, 1976* (Kneizys et al. 1980)

Water vapor density (g m^{−3}) and LOWTRAN relative air density (*P*_{z}*T*_{z}) profiles for the *U.S. Standard Atmosphere, 1976* (Kneizys et al. 1980)

Water vapor density (g m^{−3}) and LOWTRAN relative air density (*P*_{z}*T*_{z}) profiles for the *U.S. Standard Atmosphere, 1976* (Kneizys et al. 1980)

Water vapor density (g m^{−3}) and LOWTRAN relative air density (*P*_{z}*T*_{z}) profiles for the *U.S. Standard Atmosphere, 1976* (Kneizys et al. 1980)

The LOWTRAN 7 absorption coefficients (various units) in the wavelength region used to determine the O_{2} absorber amount

The LOWTRAN 7 absorption coefficients (various units) in the wavelength region used to determine the O_{2} absorber amount

The LOWTRAN 7 absorption coefficients (various units) in the wavelength region used to determine the O_{2} absorber amount

Net optical depth for the data in Fig. 1 over the 762-nm band as the O_{2} absorber amount is varied from 0 to 800 g cm^{−2}

Net optical depth for the data in Fig. 1 over the 762-nm band as the O_{2} absorber amount is varied from 0 to 800 g cm^{−2}

Net optical depth for the data in Fig. 1 over the 762-nm band as the O_{2} absorber amount is varied from 0 to 800 g cm^{−2}

The optical depth data from Fig. 1 (curve 1) after successive corrections for O_{2} absorption, CO_{2} absorption, and Rayleigh scattering (curve 2); O_{3} absorption (curve 3); H_{2}O absorption (curve 4); and aerosol scattering (curve 5)

The optical depth data from Fig. 1 (curve 1) after successive corrections for O_{2} absorption, CO_{2} absorption, and Rayleigh scattering (curve 2); O_{3} absorption (curve 3); H_{2}O absorption (curve 4); and aerosol scattering (curve 5)

The optical depth data from Fig. 1 (curve 1) after successive corrections for O_{2} absorption, CO_{2} absorption, and Rayleigh scattering (curve 2); O_{3} absorption (curve 3); H_{2}O absorption (curve 4); and aerosol scattering (curve 5)

A log–log plot of optical depth vs wavelength for the O_{2} absorption and Rayleigh scattering-corrected data in Fig. 7 over the 400–564-nm range as the UV O_{3} absorber amount is varied from 0.0 to 0.8 atm-cm. The solid line shows the best linear least squares fit to the data after the correlation coefficient has been maximized and represents a best estimate of the short-wavelength aerosol function

A log–log plot of optical depth vs wavelength for the O_{2} absorption and Rayleigh scattering-corrected data in Fig. 7 over the 400–564-nm range as the UV O_{3} absorber amount is varied from 0.0 to 0.8 atm-cm. The solid line shows the best linear least squares fit to the data after the correlation coefficient has been maximized and represents a best estimate of the short-wavelength aerosol function

A log–log plot of optical depth vs wavelength for the O_{2} absorption and Rayleigh scattering-corrected data in Fig. 7 over the 400–564-nm range as the UV O_{3} absorber amount is varied from 0.0 to 0.8 atm-cm. The solid line shows the best linear least squares fit to the data after the correlation coefficient has been maximized and represents a best estimate of the short-wavelength aerosol function

*U.S. Standard Atmosphere, 1976* O_{3} density profile (g m^{−3}) (Kneizys et al. 1980)

*U.S. Standard Atmosphere, 1976* O_{3} density profile (g m^{−3}) (Kneizys et al. 1980)

*U.S. Standard Atmosphere, 1976* O_{3} density profile (g m^{−3}) (Kneizys et al. 1980)

Optical depth vs wavelength for the O_{3} absorption–corrected data in Fig. 7 (curve 3) over the 868–936-nm range as the H_{2}O vapor exponential decay slope is varied from 0.2 to 3.2 km^{−1}

Optical depth vs wavelength for the O_{3} absorption–corrected data in Fig. 7 (curve 3) over the 868–936-nm range as the H_{2}O vapor exponential decay slope is varied from 0.2 to 3.2 km^{−1}

Optical depth vs wavelength for the O_{3} absorption–corrected data in Fig. 7 (curve 3) over the 868–936-nm range as the H_{2}O vapor exponential decay slope is varied from 0.2 to 3.2 km^{−1}

A log–log plot of optical depth vs wavelength for selected wavelengths from the O_{3} absorption–corrected data in Fig. 7, curve 3, over the 652–1044-nm range. The criteria used to select these wavelengths are listed in the text. The solid line shows the long-wavelength aerosol function determined for the example spectrum, and the dashed line is the overall function

A log–log plot of optical depth vs wavelength for selected wavelengths from the O_{3} absorption–corrected data in Fig. 7, curve 3, over the 652–1044-nm range. The criteria used to select these wavelengths are listed in the text. The solid line shows the long-wavelength aerosol function determined for the example spectrum, and the dashed line is the overall function

A log–log plot of optical depth vs wavelength for selected wavelengths from the O_{3} absorption–corrected data in Fig. 7, curve 3, over the 652–1044-nm range. The criteria used to select these wavelengths are listed in the text. The solid line shows the long-wavelength aerosol function determined for the example spectrum, and the dashed line is the overall function

The spectral irradiance extrapolated from the parameters obtained with SSP for the optical depth data of Fig. 1, plotted vs wavelength, along with the original spectroradiometer data used for the fitting

The spectral irradiance extrapolated from the parameters obtained with SSP for the optical depth data of Fig. 1, plotted vs wavelength, along with the original spectroradiometer data used for the fitting

The spectral irradiance extrapolated from the parameters obtained with SSP for the optical depth data of Fig. 1, plotted vs wavelength, along with the original spectroradiometer data used for the fitting

The ratio of the integrated extrapolated spectral irradiance to the direct irradiance measured with an absolute cavity radiometer vs measurement number. Measurement numbers were assigned sequentially with time, and Table 3 lists the dates on which these data were obtained

The ratio of the integrated extrapolated spectral irradiance to the direct irradiance measured with an absolute cavity radiometer vs measurement number. Measurement numbers were assigned sequentially with time, and Table 3 lists the dates on which these data were obtained

The ratio of the integrated extrapolated spectral irradiance to the direct irradiance measured with an absolute cavity radiometer vs measurement number. Measurement numbers were assigned sequentially with time, and Table 3 lists the dates on which these data were obtained

A histogram of the irradiance ratios from Fig. 13: 91.6% of the ratios are in the 0.98–1.03 range, and 88.4% are in the 0.99– 1.03 range

A histogram of the irradiance ratios from Fig. 13: 91.6% of the ratios are in the 0.98–1.03 range, and 88.4% are in the 0.99– 1.03 range

A histogram of the irradiance ratios from Fig. 13: 91.6% of the ratios are in the 0.98–1.03 range, and 88.4% are in the 0.99– 1.03 range

A histogram of the 1998 irradiance ratios from Fig. 13. For these measurements, the LI-1800 spectroradiometer was equipped with a thermoelectric temperature controller

A histogram of the 1998 irradiance ratios from Fig. 13. For these measurements, the LI-1800 spectroradiometer was equipped with a thermoelectric temperature controller

A histogram of the 1998 irradiance ratios from Fig. 13. For these measurements, the LI-1800 spectroradiometer was equipped with a thermoelectric temperature controller

The irradiance ratios from Fig. 15 plotted vs the precipitable water vapor amount calculated using Eq. (87)

The irradiance ratios from Fig. 15 plotted vs the precipitable water vapor amount calculated using Eq. (87)

The irradiance ratios from Fig. 15 plotted vs the precipitable water vapor amount calculated using Eq. (87)

The change in total irradiance over 1-min periods (measured with an absolute cavity radiometer) vs time of day for clear-sky conditions in Golden, CO. The date was 26 Sep 1998

The change in total irradiance over 1-min periods (measured with an absolute cavity radiometer) vs time of day for clear-sky conditions in Golden, CO. The date was 26 Sep 1998

The change in total irradiance over 1-min periods (measured with an absolute cavity radiometer) vs time of day for clear-sky conditions in Golden, CO. The date was 26 Sep 1998

LOWTRAN absorber parameters for the 300–4000-nm wavelength range and the individual bands considered (Kneizys et al. 1995)

Normalized pressure and normalized temperature exponential decay slopes for the LOWTRAN model atmospheres (Kneizys et al. 1995), and the same slopes normalized by column averages

The dates on which the spectral irradiances represented in Fig. 13 were obtained. For each date, the number of spectra, the average irradiance ratio, and the standard deviation for the average irradiance ratio are also listed