The experimental measurements of solar spectral irradiance for the validation process of the SSolar-GOA model were taken using two commercial spectroradiometers. The first spectroradiometer used was the LI-1800 model from Li-COR Biosciences (LI-COR, 1989), which covers the 300–1100 nm spectral range and is based on monochromator holographic grating of 800 grooves mm-1 with a nominal FWHM (full width at half-maximum) or spectral resolution of 6 nm (according to Vergaz et al., 2000, the FWHM measured at our Laboratory was 6.25 ± 0.07 at 632.8 nm He–Ne laser wavelength). The scanning system of LI-1800 takes about 40 s to measure a solar spectrum. The software of the system allows variable wavelength sampling, but currently 1 nm and also a programmable time are used for the measurement of global solar radiation spectra (or the direct component with a solar tracker). The LI-1800 is manufactured with a Remote Cosine Receptor for global solar irradiance measurements, but different fore-optic devices designed by the GOA Group allow for direct normal irradiance and reflected solar irradiance measurements (Durán, 1997).

The other spectroradiometer used was the FieldSpec Pro (hereafter, ASD), a general purpose portable spectroradiometer developed by ASD Inc. (ASD Full Range, Portable Spectrometers & Spectroradiometers | Malvern Panalytical, 2021; Milton et al., 2009; Goetz, 2012; Hannula et al., 2020). This spectroradiometer covers the 350–2500 nm shortwave range and is composed of three spectrometers: the VNIR from 350–1050 nm is composed of a 512-channel silicon photodiode (CCD) array overlaid with an order separation filter, a second scanning spectrometer (SWIR-1) from 1050 to 1800 nm, and a third scanning spectrometer (SWIR-2) to 2500 nm. Each SWIR consists of a concave holographic grating and a single thermoelectric cooled indium gallium arsenide (InGaAs) detector. Each grating is mounted on a common shaft which oscillates at 100 ms per scan, thus providing their spectra in a few seconds and the CCD array makes it simultaneously in the VNIR spectral range. The spectral resolution of the ASD is different for each of the three spectrometers: the VNIR has approximately 3 nm of spectral resolution at around 700 nm, and the SWIRs have about 10–12 nm.

The ASD system is provided with specific fore-optical accessories for field radiance and irradiance measurements of different FOVs (fields of view of 1, 3, and 8∘), and a remote cosine receptor is used for global irradiance and for measuring full-hemisphere albedo or reflectance spectra. Light collection is achieved through a bundle of optical fibre. For direct normal irradiance measurements, the earlier fore-optic accessories used for radiance measurements cannot be used. This is because each tube is provided with a lens which focuses the radiation over the optical fibre, and due to the high energy of the normal direct irradiance, this may damage the fibre. Thus, a new tube collimator was designed by the GOA Group which can be used with the ASD.

Calibration details, associated errors, and measurements of the LI-1800 for both direct and global irradiances were discussed in Vergaz (2001), Martínez-Lozano et al. (2003), Vergaz et al. (2005), and Estellés et al. (2006). As a general feature, the LI-1800 presents an experimental error of about 5 % in the 340–1100 nm spectral range while the instrument itself has proven to be durable and have a long-lasting calibration. The ASD solar irradiance measurements have similar errors to LI-1800. Despite the advantage of registering near-instantaneous spectra and automatic optimization, the latter results in variable integration times and gains for one measured spectrum over another, that being a drawback in the data processing. This requires special care and attention for field solar irradiance measurements, and normally post-processing is necessary since frequent saturation is observed in the spectra, which is not always avoidable. In both spectroradiometers, the LI-1800 and ASD diffuse solar irradiances are derived from the difference between near-simultaneous measured spectra of global and direct normal irradiances.

The global solar spectral irradiance GHI(SZA, λ) at ground level over a horizontal surface and for a given Sun position (specified by the solar zenith angle, SZA) can be expressed as the sum of its direct normal component (DNI(SZA, λ)) projected onto the horizontal surface (hence multiplied by cos (SZA)), plus the horizontal diffuse irradiance component DIF(SZA, λ), also dependent on the SZA. 1GHISZA,λ=DNISZA,λcos⁡SZA+DIFSZA,λ Although the wavelength is explicit in the above Eq. (1), it should be noted that it is valid for both spectral and integrated irradiance values (in this case removing λ and considering the integration over the entire solar range). These quantities are usually expressed in the units of W m-2 µm-1 (W m-2 nm-1) for spectral irradiance values, and W m-2 for integrated irradiance values. There is not a unified nomenclature to designate the three components of solar radiation at surface level (global or total horizontal solar irradiance (GHI) is also called shortwave downwelling solar irradiance (SWD), shortwave surface irradiance (SSI), surface total solar flux, etc.). Therefore, Eq. (1) incorporates the most recent and most widely used names in the solar energy community for these irradiances.

If we divide these irradiances by the irradiance at the top of the atmosphere (the extraterrestrial irradiance, Fo, multiplied by the corresponding correction of the Earth–Sun distance (D) projected over the horizontal plane, D⋅Focos⁡(SZA)), the corresponding atmospheric transmittances at surface level for the above three components are obtained: global transmittance TGHI(SZA, λ), normal direct transmittance TDNI(SZA, λ), and diffuse transmittance tDIF(SZA, λ). 2TGHISZA,λ=TDNISZA,λ+tDIFSZA,λ Here it must be noted that using the transmittances in Eq. (2), the horizontal global transmittance (TGHI) is given by the sum of the normal direct transmittance (TDNI) and the diffuse horizontal transmittance (tDIF). As can be seen, the explicit dependence on the cos(SZA) of Eq. (1) is removed in Eq. (2). The advantage of using transmittance functions instead of irradiance values is because in this way it works with normalized functions whose values are always equal to or less than 1. As well as this, Eq. (2) is also valid for integrated values of solar radiation, which translates to the definitions of the clearness indices Kn and KT for normal direct and global solar components, respectively, where Kn =TDNI and KT=TGHI, in this case referring to instantaneous and integrated values. Therefore, Eq. (2) is now written as Kn =KT(1-K), where K is the fraction of the diffuse radiation (K= DIF/GHI). These indices are widely used by the solar energy community and are the base of the so-called separation solar radiation models under all sky conditions (Gueymard and Ruiz-Arias, 2016; Yang and Boland, 2019).

The direct normal spectral solar component at any level of the atmosphere (expressed as radiance or irradiance quantities) is currently given by the Beer–Lambert–Bouguer (BLB) law. This law is a solution of the radiative transfer equation (RTE) when applied only to direct component. The simplicity of the resulting solution makes it possible to consider scattering by molecules and particles, and absorption by atmospheric gases as independent processes (non-interaction between them). This allows us to present TDNI as a product of independent transmittances of the different atmospheric constituents: ozone, water vapour, aerosols, molecules, etc. (see Sect. 3.1).

Therefore, it is standard in radiative transfer theory to separate the modelling of solar radiation into its two components, direct normal and diffuse, and solving the RTE for each component, considering a dispersive or scattering medium without absorption of atmospheric gases. However, solving the RTE for the diffuse component is not a straightforward task and different analytical and numerical methods have been developed depending on the approaches or the specific problem involved (see classical books on radiative transfer theory: Chandrasekhar, 1960; Sobolev, 1963; Kondratyev, 1969; Lenoble, 1985, 1993; Liou, 1992, 2002; Zdunkowski et al., 2007; Kokhanovsky, 2008; see also the different solvers used in libRadtran).

Since we are interested in solar spectral irradiances or fluxes and not radiances a more convenient approach for solving the RTE is addressed by those methods known as “two streams” or “two flux” which indistinctly solve the RTE for the diffuse component only or for the global component. The “two flux” methodology was extensively developed in the 70–80 s and presents numerous variants (Joseph et al., 1976; Meador and Weaver, 1980; Zdunkowski et al., 1980; King and Harshvardhan, 1986; Liou, 1992; Fouquart and Bonnel, 1980; Durán, 1997; Räisänen, 2002; Lin et al., 2019).

Although less frequent, another possible option is to consider other methods, such as the original method of “addition of layers” in a one-dimensional scattering medium, developed by Ambartsumian (Sobolev, 1963; Nikoghossian, 2009), which does not consider the RTE. The analytical expression obtained for the transmittance of the global solar irradiance in a scattering medium (without atmospheric gas absorption) composed of aerosols or molecules (or a mixture of both components) is the core of the SSolar-GOA model (see Sect. 3.2). After that, this transmittance is multiplied by the absorption transmittance of atmospheric gases giving the above spectral TGHI of Eq. (2) assuming non-interaction between scattering and gas absorption.

The libRadtran package is a software library for radiative transfer calculations of solar and thermal radiation (from 120 nm to 100 µm) in the Earth's atmosphere. The central part of the software package is an executable program called uvspec which was initially developed for UV radiation evaluation and which has undergone numerous extension and improvements to reach the current libRadtran estructure (Mayer and Kylling, 2005; Emde et al., 2016). It is freely available at the web page http://www.libradtran.org (last access: 23 May 2021), which contains all the available information about the program including the user's guides (libRadtran user's guide, 2015, 2020) and the software source code. The libRadtran package contains a complete treatment of the inputs, utilities, methods, and outputs to handle the complex structure that radiative transfer models have, allowing for the determination of the field radiation (radiances, irradiances, polarization, etc.) in the atmosphere. Therefore, libRadtran is a set of RT codes which serves as a reference tool which is widely used by the scientific community in different fields of study.

libRadtran requires detailed information specified in input files provided by the same package or constructed by the users, for example the Mie program (see libRadtran user's guide chap. 4) for the calculation of aerosol optical properties. For the irradiance values, the direct normal component is calculated based on the BLB in a similar way to SSolar-GOA (described in next Sect. 3.1). For our simulations, the algorithm for the spectral solar diffuse horizontal irradiance used the sdisort RTE solver with 10 streams. The global spectral radiation is constructed by the sum of direct horizontal plus diffuse horizontal components. All the atmospheric gases were considered in libRadtran for the simulations. To compare with the SSolar-GOA model, the adequate options of libRadtran are the “spectrally resolved calculation” for the UV and visible spectral range and the “pseudo-spectral” in the infrared solar region (i.e. water vapour, oxygen and carbon dioxide), represented by the band parameterization of the LOWTRAN7 code taken by the SBDART model and adopted in libRadtran (Kneizys, 1988; Mayer and Kylling, 2005). Therefore, the latter option was taken by us in accordance with the building of the SSolar-GOA model.

A midlatitude summer atmospheric profile with a default of an aerosol profile in the summer season was chosen, but the contribution of aerosols was constructed on the alpha and beta Ångström turbidity parameters (they are also represented in the text by the symbols α and β, respectively; see next sections). It must be noted that under clear skies the aerosol contribution is the most important factor for solar irradiance, and the spectral behaviour of AOD(λ) is given by the alpha parameter. The spectral AOD(λ) is the most relevant input for a proper comparison between libRadtran and our model since it determines the curvature shape and height of the transmittance of the direct normal component. The other two aerosol parameters, the asymmetry parameter (g) and single scattering albedo (SSA), are of secondary importance and are taken as fixed values (not wavelength dependent) in libRadtran and SSolar-GOA models.

Assuming the validity of the BLB law, the spectral irradiance of the direct normal component of solar radiation, DNI (SZA, z), at any time (given by the SZA) and at any vertical altitude z of the atmosphere, is given by 3DNISZA,z,λ=DFoλTDNISZA,z,λ=DFoexp⁡-τz,λm, where τ(z,λ) is the spectral atmospheric optical thickness at the level z, or the altitude in the atmosphere which accounts for scattering by molecules and particles as well as absorption by atmospheric gases. Fo(λ) is the spectral irradiance at the top of atmosphere (extraterrestrial spectrum) and D is the correction factor of the Earth–Sun distance.

Considering the ground surface level z=0, τ(z=0,λ)=τ(λ) represents the total spectral optical thickness of the atmosphere at the site. m is the relative optical air mass giving the slant path of the Sun's rays relative to the zenith, which is given by m=1/cos⁡(SZA) for a plane-parallel atmosphere. For a spherical atmosphere, more accurate expressions for m are necessary when the SZA is greater than 70∘; in order to account for the curvature of the atmosphere and the refraction effects, various expressions were developed for each atmospheric component (Kasten and Young, 1989; Gueymard, 1995, 2005; Tomasi et al., 1998; Chiron de la Casinière and Cachorro Revilla, 2008; Rapp-Arrarás and Domingo-Santos, 2008).

As mentioned, the advantage of solving the RTE only for the direct normal component is that TDNI can be calculated as a product of transmittances due to the different processes of attenuation due to the different atmospheric components, where the non-interaction between these processes is implicitly assumed. 4TDNISZA,λ=TRSZA,λTaSZA,λTgas,iSZA,λ In Eq. (4), the different transmittances are given by exponential functions of the optical thickness of each process: the scattering by molecules or Rayleigh scattering, τR (λ); scattering by aerosols, τa(λ) or AOD(λ); and the absorption by atmospheric gases, τgas,i (λ) (subscript i refers to different selected gases), multiplied by the corresponding relative air mass. 5TDNISZA,λ=exp⁡-τλm=exp⁡-τRλmRexp⁡-τaλmaexp-τgas,iλmgas,i The BLB law of Eq. (3) is also valid for integrated irradiance values (i.e. removing the wavelength dependency), where τ represents the integrated total optical thickness of the atmosphere, but Eqs. (4)–(5) are only valid for spectral values (Cachorro et al., 2000b, a; Utrillas et al., 2000). Despite this, Eq. (4) is taken as a good approach for the “broadband solar models” (Gueymard, 2008; Ruiz-Arias and Gueymard, 2018) assuming that the transmittance of each atmospheric component is an integrated value over the entire solar range. According to Eq. (5), it follows that 6τλm=τRλmR+τaλma+τgas,iλmgas,i. Therefore, the total optical thickness of the atmosphere is given by the sum of the different optical thicknesses due to the different attenuation processes of solar radiation assuming the same relative optical air mass. According to Eqs. (5) and (6) we can use either transmittances or optical thickness. Observe that optical thickness is a dimensionless parameter like the relative optical air mass. Although it is usual to consider m=mR=ma, for atmospheric gases it is more convenient to use different expressions specifically determined for each absorbing gas (Gueymard, 1995; Tomasi et al., 1998).

There are different parameterized expressions to evaluate the Rayleigh optical thickness (Teillet, 1990; Gueymard, 1995; Bodhaine et al., 1999; Tomasi et al., 2005) with insignificant differences for our purpose, and hence the Gueymard (1995) formula was taken for the SSolar-GOA model. 7τRλ=1117.2594λ4-1.3215λ2+0.00032-0.000076λ-2 Since this expression is evaluated at sea level, it is necessary to multiply by the factors P and Po, where P and Po are the pressure at the site (or altitude) and the sea-level pressure, respectively. The transmittance of scattering by aerosols Ta (SZA) is accounted for by a simple approach for the aerosol optical depth given by the Ångström formula (Ångström, 1929, 1930, 1961, 1964). This is an empirical expression extensively used in the field of aerosol studies and in solar radiation applications (Cachorro et al., 1987b, c, 2000a, b) and is expressed as follows: 8τaλ=βλ-α, where α (alpha) and β (beta) are the Ångström turbidity parameters. The α parameter, also called Ångström exponent (the symbol AE is now more commonly used in place of α), is related to the bulk size of the particles, and the β parameter is the aerosol optical thickness at 1 µm wavelength. Bear in mind that this is an empirical expression which may be applied to a given extended spectral range. Frequently, two wavelengths can be selected and hence Eq. (9) allows for the determination of the α parameter: 9τaλ1τaλ2=λ1λ2-α. When several wavelengths are available, such as in Sun photometers or spectroradiometers, the α–β parameters can be determined simultaneously by a linear regression of log[τa(λ)] versus log[λ] (Cachorro et al., 1987b, c, 1989, 2000b; Martìnez-Lozano et al., 1998). In this case, different values of the α–β pair are obtained depending on the selected spectral range (or wavelengths). Therefore, some solar radiation models use more than one pair of α–β values to cover the entire solar spectral range (Gueymard and Myers, 2008). However, in the SSolar model, only a pair of values is taken. Despite its simplicity, the Ångström formula has proven to be an excellent approach for modelling the spectral behaviour of the aerosol optical depth, AOD(λ). In RT studies, the aerosol optical thickness and other optical properties are determined by the Mie scattering theory (Bohren and Huffman, 1998; Cachorro and Salcedo, 1991). In Cachorro et al. (2000a), experimental direct normal irradiance measurements of the LI-1800 together with rigorous Mie scattering expressions were used to determine the distribution of aerosol particle size and other aerosol parameters.

The absorption processes by atmospheric gases must be accounted for and are given by different transmittances. In the solar spectral range, the SSolar-GOA model uses tabulated absorption coefficients of water vapour (H2O(v)), ozone (O3), oxygen (O2), nitrogen dioxide (NO2), and carbon dioxide (CO2), but the current version only considers water vapour, ozone, and oxygen because of the low absorption features of the other two components and the necessity of a rapid running of the model. These absorbing gases are represented by the product of the different transmittances. 10Tgas,iSZA,λ=TH2OSZA,λTO3SZA,λTO2SZA,λ The selective line absorption of these molecular gases is treated under the “band model approach” method mentioned above. This results in parameterized expressions which are adequate for models of low-median spectral resolution, as explained below. The transmittance of ozone absorption is given by the following expression: 11TO3=exp⁡-τO3λmO3=exp⁡-CO3λLO3mO3, where τO3(λ) is the spectral ozone optical thickness. CO3(λ) refers to the ozone absorption coefficients (or cross section, depending on the units taken), which carry the wavelength dependence, and LO3 is the columnar ozone content. Usually, LO3 is given in Dobson units, DU (1 DU = 1 cm-atm ×10-3), and thus the absorption coefficients are given in (cm-atm)-1. The relative optical air mass, mO3, is given by the expression from Komhyr (1980). Ozone in the region of 280–350 nm corresponds to the Hartley (200–310 nm) and Huggins (300–350 nm) bands, and the Chappuis band in the visible range (400–650 nm). The cross sections taken in our model for the UV region are those from Bass and Paur (1985). The original values are given with a spectral resolution of 0.05 nm, so they were convoluted with a triangular slit function of 7 nm of FWHM and evaluated or interpolated in 1 nm steps. The cross sections were also provided for three different temperatures, and 226 K was selected for our model. For the visible Chappuis band, the CO3 values were taken from Amoruso et al. (1990), Anderson and Mauersberger (1992), and Brion et al. (1998). These values were also changed to the spectral resolution as before. These CO3(λ) values are sufficient to predict solar irradiance values (Redondas et al., 2014; Orphal et al., 2016).

The transmittance of water vapour is given by the parameterized expression of Pierluissi et al. (1989): 12TH2Oλ=exp⁡-τH2OλmH2O=exp⁡-CH2OλWmH2Oa, where CH2O (λ) refers to the absorption coefficient of water vapour which was taken from LOWTRAN7 with a spectral resolution of 20 cm-1 in steps of 5 cm-1. These coefficients were accommodated as before at a spectral resolution of 7 nm and step of 1 nm. The parameter “a” presents a smooth dependence on wavelength and is given by Pierluissi et al. (1989) for each water absorption band. W is the equivalent absorber amount over the vertical which is related to the amount of absorber, U, or precipitable water vapour, PWV, expressed in cm or g cm-2 by 13W=PePonToTemU. The above expression applied the Curtis–Godson approximation to the whole single layer of the atmosphere for our model, where Pe and Te are the effective pressure and temperature of the atmosphere, respectively (we take those of the standard atmosphere), and To and Po are the values at standard conditions. The parameters n and m are also given by Pierluissi et al. (1989) for each band of water vapour. An integration is used to model several atmospheric layers, where Pe and Te are substituted for the values P(z) and T(z), and U by dU=ρv(z)dz, where ρv(z) is the profile of water vapour density.

The transmittance of oxygen is treated with an expression similar to that of water vapour (Pierluissi and Maragoudakis, 1986). In contrast to the variability of water vapour, oxygen is constant in the atmosphere. The value used in our model for the equivalent vertical oxygen content was 87068.53 cm-atm, corresponding to the midlatitude summer atmosphere. This value does not differ substantially for other atmospheres, and therefore, no variation in transmittance was observed. In all the absorbing gas transmittances, the amount of absorbing gas is given in units of cm-atm or in g cm-2 and hence the absorption coefficients have the inverse units. As can be seen, the procedure followed for the absorption gas transmittances in our model is equivalent to the “pseudo-spectral calculations” according to libRadtran.

The simplicity of the SSolar_GOA model is based on the parameterized Eq. (14) to calculate the total scattering transmittance TMix (SZA) for a mixed layer of aerosols and molecules, considering the interaction between the two scattering processes. These Eqs. (14)–(16) are obtained by the original method of “addition of layers” in a one-dimensional medium developed by Ambartsumian (Sobolev, 1963; Nikoghossian, 2009). For simplicity, the wavelength is removed in some of the next Eqs. (14)–(16), but the generic parameters τ(λ), ω(λ), and g(λ) carry this wavelength dependence, and the following subscripts are R for the molecules, a for the aerosols, and Mix for the mixture. Equations (14)–(16) are as follows: 14TMixSZA=1-ro2exp⁡-kτtm1-ro2exp⁡-2kτtm, where τt (λ) is the total scattering optical depth (τt=τa+τR), and m is the relative optical air mass. The parameters ro and k are given by 15ro=k-1+ωMixk+1-ωMixk=1-ωMix1-ωMixgMix, where ωMix and gMix are the single scattering albedo and the asymmetry parameter of the mixed layer of aerosols and molecules defined by the corresponding parameters of individual molecules (R) and aerosols (a). They are given by the following expressions: 16ωMix=τR+ωaτaτt,withωa≠1,gMix=gaτaτt. However, we must state that the transmittance of Eq. (14) may be also used to evaluate an isolated aerosol layer, represented by the scattering aerosol transmittance, Ta. In this case, we need an expression for the scattering transmittance for an isolated pure Rayleigh atmosphere, TR, for example that given in Vermote and Tanré (1992). The total transmittance of the scattering atmosphere of the aerosol and molecular mixed layer is obtained as the product Ta-R=TaTR, where it is implicitly assumed that there is no interaction between the molecules and aerosols. Therefore, Ta-R is equivalent to TMix, but not the same. No significant differences have been found between these two approaches for moderate atmospheric aerosol loads. Scattering and gas absorption are applied to a single atmospheric homogeneous layer in the SSolar-GOA under the consideration of non-interaction of both processes, which simplify considerably the formulation of the model.

The above expressions were derived assuming a zero reflectance or albedo of the underlying surface (considered as a black body), so its influence must be taken into account by the contribution of the multiple reflections between it and the atmosphere. For this effect, we have followed the formulation of Lenoble (1998), where an amplification factor independent of the SZA is defined as 17f_amp(λ)=1/(1-ρSλ), where ρ is the surface albedo taken in the model as a constant value and considered Lambertian. S(λ) is the spectral atmospheric albedo of the mixed Rayleigh–aerosol layer, given by the sum of both scattering components (Tanré et al., 1986; Vermote and Tanré, 1992). 18Sλ=SRλ+Saλ19SRλ=τR2+τR1-exp⁡-2τR20Saλ=g′τa2+g′τa1-exp⁡-g′τawithg′=ωa(1-ga) As previously mentioned, all Eqs. (14)–(20) are wavelength dependent by means of the corresponding parameters. However, due to the difficulty of providing accurate spectral values for the aerosol single scattering albedo ωa (or SSA) and the asymmetry parameter ga, these two parameters are taken as constant values in the SSolar-GOA model. These values for the different types of aerosols are given in different publications (Dubovik et al., 2002; Hamill et al., 2016). Finally, we call attention to the total number of expressions/formulas which define the SSolar-GOA model in comparison with other spectral models of similar characteristics given in the bibliography (e.g. Bird, 1984; Gueymard, 1995, 2005; Xie and Sengupta, 2018), which results in a major complexity and computational cost.

According to the above expressions, the input parameters for the SSolar-GOA model are as follows:

the solar zenith angle, SZA (degrees);

Hence, a total of 10 input parameters are required. N is the Julian day (from 1 to 365) which is required as an input to correct the Earth–Sun distance, D, which multiplies the extraterrestrial irradiance spectrum. The pressure, P, at the site (altitude of the bottom surface) is required to account for the correction of altitude in the Rayleigh scattering optical thickness. The α and β Ångström coefficients build the aerosol optical thickness, AOD (λ), for the whole solar spectral range. Observe that another possible option for the spectral construction of the modelled AOD is to take 4–6 values of the spectral AOD provided by AERONET. As mentioned, the single scattering albedo ωa (or SSA) and the asymmetry parameter ga are taken as constant values because these two parameters are of the second order of importance in relation to the contribution of the aerosol optical depth. As well as this, the Julian day is also required if GMT (Greenwich Mean time, a.k.a. UTC) is used as input instead of the SZA, but in this case it is also necessary to add the latitude and longitude of the site in order to calculate the SZA, and hence a total of 12 parameters must be entered into the model. GMT (or also local time) is currently used when building a set of spectra or when daily solar irradiance values are calculated, since the SSolar-GOA model may also calculate instantaneous integrated irradiance values.

Under clear-sky conditions, AOD and water vapour content are the two atmospheric parameters of major importance for irradiance values, and ozone and oxygen absorption are also considered because of their strong spectral absorbing features. Other minor absorbing gases, such as CO2 and NO2, are included in the file of absorbing coefficients but are neglected in the running of the current version of the SSolar-GOA model, partially due to their low contribution, but mainly for simplicity and calculation speed.

Generally, the spectral resolution of the model is given by the spectral resolution taken for the spectrum of the extraterrestrial solar irradiance according to that of the absorption coefficients of atmospheric gases. In our model, we can select three different extraterrestrial work files, given by Wehrli (1985), Kurucz (1992), and Gueymard (2004), as they appear in Fig. 1 together with the default input parameters above described.

The comparison between the SSolar-GOA model and libRadtran is carried out as a theoretical exercise, given the latter as a framework reference. For the comparison with experimental spectral irradiance data, the SSolar-GOA model is fed with measured values of the required atmospheric input parameters. Figure 2 shows a set of simulated solar irradiance spectra at sea level by the SSolar-GOA model at three SZAs: 6, 30, and 60∘, with typical values of the input parameters (given at the top of the figure) under clear-sky conditions. This figure illustrates the main characteristics of solar radiation components: horizontal irradiances of direct, diffuse, and global components and the direct normal irradiance. Irradiance values of direct-normal and global solar components show the well-known spectral distribution of solar radiation and their behaviour on the wavelength due to the absorption by atmospheric gases. They increase quickly from near zero at 300 nm to the maximum at visible wavelengths around 500 nm (reaching ∼ 1800, ∼ 1600, and ∼ 800 W m-2 nm-1 at SZA = 6, 30, and 60∘, respectively, for the global irradiance) and decreasing very slowly along the wavelengths of infrared range. Moreover, the features of water vapour and oxygen band absorptions are the most evident. Overall, the prevalence of global irradiance can be highlighted for low SZA values, but the inverse situation happens when the SZA increases. In this case, direct normal irradiance prevails, starting in the infrared wavelength region and then spreading throughout the whole spectral range, with a greater separation of the spectra of both components.

As typical characteristics, the direct normal irradiance shows a less pointed shape than the global component and a smoother curvature at peak values (from 470 to 700 nm) for increasing SZAs and also the strong variation of these solar irradiances with the SZA in the first part of the spectrum (440 to 1100 nm) in relation to this last part of the spectrum (1000–2500 nm). On the other hand, the low values of the diffuse irradiance in relation to the other components under clear skies present the particularity of their minor variations with the SZA and their maximum at the UV region. Thinking about solar radiation as an energy source, it must be noted that irradiance values for SZA = 6∘ are only frequent in sites near the tropics where these low SZAs are reached, while SZAs from 30 to 60∘ are most frequent in midlatitudes and high latitudes where the influence of cos (SZA) on direct horizontal irradiance values are very important and hence greatly influence the global component.

Figure 3 shows the comparison between both models for the direct normal irradiance at SZA = 6, 30, and 60∘ from top to bottom, respectively, with typical values of the input parameters (shown at the top of the figure) corresponding to middle-latitude sites. For better visualization, we have selected the 300–1100 nm spectral range. As can be seen, the results of both models are nearly identical, with relative differences ((libRadtran - SSolar-GOA) / libRadtran) around 0.5 % or less than 1 % in the non-band absorption regions throughout the entire solar spectral range and covering this large range of SZAs. However, high relative differences with strong and rapid variations, going from positive to negative values (about ±30 %), are shown in the regions of the absorption bands of water vapour and oxygen (mainly at 940 nm for water vapour and the oxygen A band (759–771 nm)). This behaviour must be due to the different spectral resolution in this regions of strong absorption, where the cause could be the slightly different values of the absorption coefficients of each model. Minor differences are found in the visible region due to the smooth and low absorption of the ozone absorption band (400–650 nm) and due to the very low absorption of the water vapour bands.

As already indicated, although both models employed the LOWTRAN7 band model parameterization and similar original coefficients for absorption in the infrared, it seems that the absorption coefficients have undergone a slightly different mathematical handling related to the convolution and interpolation processes. The original LOWTRAN7 absorption coefficients are given in wavenumber (in cm-1) and not in wavelength (in nm). Therefore, the transformation from “cm-1” to “nm” gives rise to an inhomogeneous spectral interval of the model, requiring a subsequent interpolation and smoothing (or convolution with a given FWHM) to have a constant step interval, which also depends on the spectral resolution chosen for the model. For example, at the wavelength λ=1 µm (1000 nm), a spectral resolution of 20 cm-1 corresponds to 2 nm, but at λ=0.5 µm (500 nm) the spectral resolution is 0.5 nm. Observing the irradiance values of the spectra and the shape features of these two absorption bands for both models, it is evident that libRadtran presents a slightly higher spectral resolution than the SSolar-GOA model in the regions of gas absorption. For both models, the extraterrestrial spectrum (Kurucz, 1992) is taken with a spectral resolution of 1 nm. Therefore, in the intervals of non-absorption both models present the same spectral resolution and hence they show an exact coincidence for each nm, since the transmittance of scattering processes have a smooth behaviour, and hence the spectral resolution is given by the extraterrestrial spectrum.

For a better visualization of these differences and to confirm the above reasoning, Fig. 4 shows in detail the comparison of both models in the region of the 940 nm water vapour absorption band for direct normal (a) and global irradiances (b) at SZA = 30∘. A perfect spectral correspondence (point to point) can be seen in the region of 840–890 nm (just before the “940 nm absorption band” begins) due to the absorption coefficients being zero. On the other hand, one can observe the lower spectral resolution of the SSolar model with a slight smoother behaviour than libRadtran into the “940 nm absorption band”. This is because the absorption coefficients of the SSolar-GOA model were convoluted with a slit function of FWHM equal to 7 nm.

Although the relative differences in the regions of high absorption by water vapour and oxygen may seem very high, this is the typical behaviour when the spectral resolution of two models is not the same. This is also evident when observing the sharp shape of the A band of oxygen in the libRadtran package with respect to the SSolar-GOA model. Other minor differences between both models are due to the fact that by default the libRadtran considers the complete list of absorption atmospheric gases (such as NO2, CO2, minor gases, etc.) and SSolar-GOA only considers ozone, water vapour, and oxygen.

Figure 5 shows the comparison for the global spectral component with the same input parameters and SZAs as Fig. 3. The global irradiance differences at SZA = 30∘ show a slight increase of 1 %–2 % compared with the 0.5 % given by the direct-normal irradiance from 330 nm to nearly 700 nm and decreasing for longer wavelengths. The relative differences also decrease with the SZA, and at SZA= 60∘ the relative differences are less than 1 %, which is lower than those at 30 and 6∘. Relative differences in the regions of the water vapour and oxygen absorption bands show the same variations or features as before. However, a different behaviour in the region of UV ozone absorption band, between 300–320 nm, is observed. The increasing differences range from 0.5 % at 330 nm to -25 % at 300 nm for SZAs of 6 and 30∘, but this trend decreases to 10 % for SZA = 60∘. This feature does not appear in normal direct irradiance values where a very good agreement was observed, and only the difference very close to 300 nm increases slightly to 10 % for SZA = 60∘ (Fig. 3). This problem in the ozone UV absorption band (around 300 nm) will be discussed later.

Like Figs. 3 and 5, Fig. 6 shows the comparison for the spectral diffuse component. As mentioned, the diffuse component is obtained as the difference between the global and direct components according to Eq. (1). Here, the relative differences in the region of non-gas absorption also increase to reach a maximum of 9 % in the visible and near-infrared range, but this behaviour also decreases at longer wavelengths and with increasing SZA values, with the relative differences ranging from 0 %–2 % at SZA = 60∘. In general, our model underestimates the diffuse irradiance values for low SZA values in comparison with libRadtran, but there is a good correspondence for the SZA between 40–60∘, and thus a better agreement for midlatitudes to low latitudes where these angles are most frequent. The differences are reasonable due to the low diffuse irradiance values under clear-sky conditions, which accentuates the relative differences. Moreover, it should be emphasized that there are the different physical approaches used by each model for the determination of the diffuse component. LibRadran directly obtains the diffuse irradiance by solving the RT Equation (DISORT solver), while in our model the diffuse component is obtained via the difference between the global and direct-horizontal irradiances.

The problem of the absorption for global irradiance and consequently for diffuse irradiance in the SSolar model in the ozone UV Huggins band may be due to the different treatment of the interaction between scattering and gas absorption. Apart from the above-mentioned procedures for solving the scattering problem (discrete-ordinate/Ambartsumian) libRadtran performed an adequate treatment of the absorption-scattering interaction for the diffuse component (see libRadtran user's guide, 2015, 2020), while our model only performs a simple multiplication of absorption and scattering transmittances. Furthermore, the multilayer approach may have also played an important role in this case. Other possible factors, such as the influence of temperature on the ozone absorption coefficients, seem to have had a minor impact because the direct-normal irradiance does not show these high relative differences. On the other hand, this problem does not appear in the absorption bands of other atmospheric gases or in the Chappuis band of ozone because of the lesser absorption of these bands and their rapid saturation compared to the strong absorption of ozone in the Huggins band. However, considering the strong fall that UV irradiances present close to 300 nm (over 3 orders of magnitude) and the low irradiance values, the increase to -30 % of the relative differences (the SSolar model overestimates UV values) is not so big and in part is enhanced with the artifice due to the different spectral resolution of the absorption coefficients of the two models (this always happens in the regions of strong absorption, as observed).

These figures are only a visual snapshot of the extensive comparison between SSolar-GOA and libRadtran where hundreds of spectra were compared covering a wide range of SZA values under the varied atmospheric conditions.

A more quantitative evaluation of this comparison was carried out applying linear regression and the RMSE % (root mean square error, in percentage) statistical indicator using the earlier solar irradiance values and relative differences. The SSolar-GOA model takes the interval 300 to 2600 nm as the entire solar range, so this interval was used to apply the linear regression in the comparison between libRadtran and SSolar-GOA for the three SZAs of the earlier figures, as can be seen in Table 1. As well as this, Table 2 collects the values of the RMSE % applied to different spectral ranges: UV(300–400 nm), VIS(401–700 nm), NIR(701–1100 nm), and the full shown range (300–1100 nm). The reason for analysing these four spectral ranges is due to their different behaviour in the comparison and the fact that not taking the entire range 300–2600 is due to the number of zero values (infinite relative differences) of the irradiance in the water vapour bands existing in the last part of the solar range.

Table 1 shows the slope (a), intercept (b), and correlation coefficient (r2) for the three SZAs and the three components of solar radiation (it was also added the results for the comparison of measured–modelled data evaluated in the next section). The values of r2 are always higher than 0.99, the slope varies from 1.01 to 1.06, and the intercept from 0.0005 to 0.013 W m-2 nm-1 (or 0.5 to 13 mW m-2 nm-1) for the three SZAs, thus resulting in general a very good agreement. Direct normal and global components have similar slopes, near 1, for the three SZAs, but diffuse components have a worse value (a=1.06) for SZA of 6 and 30∘ but improve to 60∘ with a slope of 1.01, a value similar to the other two components. Intercept (b) values of each case of Table 1 are very low, and they reflect for a given spectrum of a solar component the constant value that the SSolar-GOA model always underestimates the irradiances in relation to libRadtran, but this value of intercept refers to all sets of wavelengths of the whole spectrum. The equivalent information refers to the slope; therefore linear regression is only a relatively good method to know the agreement between the SSolar and libRadtran and more when the r2 reaches high values near 1.

Table 2 gives the values of RMSE % for the three solar components evaluated for the three SZAs and the four intervals in the solar range (for consistency, it was also added to the results for the comparison of measured–modelled data which will be analysed in the next section). For the visualized range 300–1100 nm and considering the three SZAs, the values are very similar for direct and global components, varying from 5 % to 8 %, but the diffuse component is more stable for the three angles at about 9 %–10 %. Very different values are observed for the other intervals with a clear behaviour for lower (3 and 30∘) or greater SZA (60∘). The VIS range stands out for its low values and low variation of the RMSE % (0.8 %–1.6 %) for direct and global components for the three SZAs, increasing for the diffuse component to about 6 %–8 % for the lower SZA but also decreasing to 2 % for SZA = 60∘. The UV interval presents higher to lower RMSE % for increasing SZAs, from 6 to 60∘, with very different values between the three components: very low values for direct component (0.8 % to 2.5 %), high values for the diffuse (11 % to 4.5 %), and intermediate values for the global component (6.8 %–3.6 %), with a substantial improvement in the comparison for SZA around 60∘ for global and diffuse components. Finally, the NIR = 701–1100 nm interval shows in general higher values of RMSE % than the other intervals, in this case always increasing with the SZA for the three components. Global and direct components present similar values (from 6 % to 11 %) and the diffuse component less variation, from 9.7 % to 13 %. Summarizing these results, global and direct components present similar numbers for the RMSE % for the four intervals and a similar behaviour with the SZA but for the UV interval direct normal component present significant lower values. Diffuse component present the highest RMSE % values, but with a substantial improvement for high SZA, mainly in the UV and VIS ranges. In spite of the valuable information provided by the parameters of the linear regression and the RMSE %, they do not give detailed information on which wavelengths fail in the compared irradiance spectrum. Therefore, relative differences evaluated across the spectrum together with the evaluation of a large number of spectra are necessary in order to improve the estimated irradiances of the SSolar-GOA model.

To validate the SSolar-GOA model, we selected specific well-suited spectra from our irradiance solar databank. Thousands of solar irradiance spectra have been measured over the past 25 years by GOA for different research activities, most of them focused on atmospheric studies for the determination of atmospheric components (Cachorro et al., 1987b, 1996, 1998, 2000b, a; Vergaz et al., 2005) and modelling of solar spectral radiation. In Cachorro et al. (1985, 1987a, c), one of the first comparisons between field experimental spectral solar irradiance measurements and their modelling with simple spectral solar radiation models can be seen. Detailed radiative transfer models have been also used and compared with experimental spectral solar irradiance data (Cachorro et al., 1997; Durán, 1997; Utrillas et al., 2000; García et al., 2016).