1 Introduction

Total surface solar irradiance in the erythemally weighted ultraviolet (EUV), total ultraviolet (TUV), and global solar radiation (GS) bands is a key characteristic of solar radiative energy in the Earth climate system. It is an important component of studies of radiative transfer, biological effects, and solar energy utilization. The primary factors affecting ground-level solar irradiance are total-column ozone, solar elevation, clouds, and aerosols. Ozone strongly absorbs UV radiation at wavelengths below 320 nm, and solar elevation can be determined accurately by geometric equations. Previous studies have examined the effects of ozone on UV radiation (McKenzie et al. 2011; WMO 2011, 2014 and references therein). Clouds also influence the amount of solar radiation reaching the Earth’s surface. Atmospheric extinction by clouds reduces incoming solar energy at the ground level, particularly in the UV range, and absorption and backscattering by clouds modulates biologically effective radiation (e.g., Bais et al. 2018). As emission rates increase, aerosols have become a non-negligible factor in the attenuation of surface solar irradiance (SSI), particularly for UV radiation (e.g., Liu et al. 1991; Erlick and Frederick 1998; Eltbaakh et al. 2012; Kim et al. 2013). Top-of-atmosphere UV solar irradiance accounts for only 8.3% of the total solar radiation (e.g., Iqbal 1983). However, biological interactions are strongly affected by erythemal UV radiation (e.g., Sivamani et al. 2009), which in turn is affected primarily by ozone rather than clouds or aerosols. With increasing attention to climate change and biological effects, an interest in the evolution of SSI forcing factors (e.g., clouds, aerosols, and total-column ozone) has recently developed (e.g., Seckmeyer et al. 1996; Mateos et al. 2011; Kim et al. 2013, 2014; Chiodo et al. 2017; Schwarz et al. 2017; Bais et al. 2018; Rundel and Nachtwey 2018).

Cloud effects on surface solar radiation are difficult to estimate, and show large temporal and spatial variability (e.g., Calbó and González 2005; Mateos et al. 2011; Kim et al. 2014). A simple and robust method to evaluate the attenuation of radiation by clouds from available measurement datasets is thus desirable. Cloud transmission (cloud modification factor) is defined as the ratio of measured under cloud to clear-sky irradiance (e.g., Ilyas 1987; Lubin and Frederick 1991; Schafer et al. 1996). Early ground-based measurement studies utilized empirical relationship models applied to specific sites (e.g., Schafer et al. 1996; Calbó and González 2005). Empirical transmission models have been developed with second (Lubin and Frederick 1991; Kuchinke and Núñez 1999) and third order (Schafer et al. 1996; Calbó and González 2005) derived from long-term datasets. However, model accuracy and characteristics vary by region. Statistical models have also been used to predict the UV index (e.g., Lee et al. 2008).

Numerous studies have examined transmission by clouds (e.g., Frederick and Snell 1990; Krotkov et al. 1996; Seckmeyer et al. 1996; Frederick and Erlick 1997; Calbó and González 2005; López et al. 2009; Mateos et al. 2011; Wang et al. 2014; Serrano et al. 2015; Adam and Ahmed 2016). However, in this study we analyze statistical relationships between atmospheric transmission and associated meteorological parameters (e.g., clouds, aerosols, and total ozone amount) in Seoul, Korea. Seoul is a megacity with a population of more than 10 million, located at mid-latitudes in the Northern Hemisphere, with cloud conditions that are affected primarily by monsoons and the heavy air pollution of East Asia. Total ozone amount varies temporally, influenced by dynamical and photochemical processes in the atmosphere (Brasseur and Solomon 2005; Seinfeld and Pandis 2006). Thus, a reliable transmission model for Seoul will be useful to study the complex effects of these meteorological parameters. Here, measurements related to long-term EUV, TUV, and GS broadband transmission are investigated in combination with cloud, aerosol, and ozone data.

The objectives of this study are to: 1) define the magnitude of and seasonal variations in SSI (EUV, TUV, and GS irradiance) along with clouds, aerosols, and ozone in Seoul climatologically; 2) develop semi-empirical relationships between transmission and clouds, aerosols, and ozone based on the Beer–Bouger–Lambert Law; and 3) investigate the wavelength dependence of transmission in the EUV, TUV, and GS spectral bands. The remainder of this paper is organized as follows: section 2 describes the study site and data, section 3 presents the transmission calculation methodology, section 4 presents the results, and section 5 provides a summary and conclusions.

2 Study Site and Dataset

The dataset used in this study was obtained for a site located in Yonsei University in Seoul (37.58°N, 126.99°E; 86 m SL). The north side of the site is surrounded by a small hill and the other sides are influenced by local pollution sources (Fig. 1). Daily measurements of total ozone (TO) have been made with a Dobson spectrophotometer (Beck #124; Komhyr 1980; Evans and Komhyr 2008) and a Brewer spectrophotometer (Brewer #148, MKIV), as used by Lee et al. (2018). The Dobson and Brewer spectrophotometers are described in detail by Cho et al. (2003a, b), Kim et al. (2007, 2013), and Kim et al. (2014). The ozone layer monitoring program at Yonsei University has been carried out using the Dobson spectrophotometer in the framework of the Global Ozone Observing System of the World Meteorological Organization (WMO/GAW/GO3OS, Station #252) since May 1984. The calibration history of the instrument is described by Cho et al. (2003a, b) and Kim et al. (2007, 2013). The Brewer #148 instrument has been in routine operation since October 1997 and measures direct UV radiation at five wavelengths (306.3, 310.1, 313.5, 316.7, and 320.1 nm) for ozone, sulfur dioxide, and aerosol (320.1 nm), through a window that tracks the sun. Global UV irradiance (290–363 nm with a 0.5 nm interval) was also measured, through a hemispheric dome and diffuser. The horizontal irradiance at each wavelength was integrated to produce the total UV irradiance (290–363 nm; TUV) and the erythemal UV-B value (290–325 nm; EUV) weighted by the Diffey action spectrum (McKinlay and Diffey 1987). The aerosol optical depth (AOD) at 320 nm was calculated as the residual optical depth after subtracting the total atmospheric optical depth due to molecular scattering, and absorption by ozone and sulfur dioxide from the direct solar irradiance (e.g., Meleti and Cappellani 2000; Gröbner et al. 2001; Cheymol and De Backer 2003). The Brewer #148 instrument was regularly recalibrated against the Brewer #017 instrument, the traveling standard of the International Ozone Service (Canada) between 2004 and 2013 at Yonsei University, Seoul (10–16 March 2004; 24–25 February 2006; 24 October–3 November 2007; 16 October 2009; 21 November 2011; 7 August 2013). Data for the cloud cover amount (CC) and total global solar irradiance (GS) are from the Seoul Meteorological Station of the Korean Meteorological Administration (KMA), located ~3 km east of the Yonsei site (Fig. 1). Both sites share the same meso- and local-scale conditions (Stewart and Oke 2012); thus, weather conditions were assumed to be the same at the two sites. The GS was measured by a Kipp & Zonen CM21 pyranometer with a wavelength range of 305–2800 nm. The CC was observed visually by researchers and is expressed as tenth of the sky covered by clouds according to WMO standards. Daily CC is obtained by averaging eight observations per day (03:00 to 24:00 LST). This study used daily data for GS, EUV, TUV, CC, AOD, and TO for the 11-year period between March 2004 and February 2015. Semi-empirical model was established using dataset from March 2004 to February 2013, and it was validated using independent measured dataset from March 2013 to February 2015.

Fig. 1
figure 1

Photograph of the Seoul area showing the two measurement sites: the global environment laboratory at Yonsei University (yellow triangle) and the Seoul meteorological station of the KMA (red circle). Image is courtesy of Google Earth digital globe

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3 Methodology

The atmospheric broadband transmission for clouds, aerosol, and ozone were normalized as the ratio of measured broadband SSI (EUV, TUV, and GS) for all-sky conditions (i.e. clear and cloudy) to a reference value for clear-sky irradiance (e.g., Frederick and Steele 1995; Schafer et al. 1996; Kuchinke and Núñez 1999; Josefsson and Landelius 2000). We define the atmospheric broadband transmission (T) as

$$ T\mathrm{i},\mathrm{j}=\frac{\mathrm{SSI}\ {\left(\mathrm{all}-\mathrm{sky}\right)}_{\mathrm{i},\mathrm{j}}}{\mathrm{SSI}\ {\left(\mathrm{clear}-\mathrm{sky}\right)}_{\mathrm{i}}}, $$
(1)

where the index i takes on three values to denote EUV, TUV, and GS; j is the associated parameter (CC, AOD, and TO); SSI (all-sky) is the measured SSI; and SSI (clear-sky) is a reference value for SSI. Thus, SSI (all-sky) can be normalized by the clear-sky value for transmission.

We calculated daily zenith-direction atmospheric transmission using the daily noon zenith angle (Z0; e.g., Iqbal 1983). Hereafter, atmospheric transmission is always in the zenith direction, T = T´1/m = e−τ for total optical thickness τ, where m is the relative optical air mass (m ≈ sec Z0).

3.1 Reference Value for Clear-Sky Irradiance in Transmission Calculations

When the sun is directly overhead (Z0), SSI = So. However, when zenith angle (Z) > 0, SSI is a function of Z. Thus, the SSI measured over a unit horizontal area is given by

$$ \mathrm{SSI}={\mathrm{S}}_{\mathrm{o}}\cdotp \cos Z $$
(2)

To estimate the SSI (clear-sky) reference values, a clear-sky model was developed based on the envelope of daily maximum SSI data that consists of a cosine function constructed as a function of time of year by harmonic analysis (Wilks 2011):

$$ {\mathrm{y}}_{\mathrm{r}}\ \left(\mathrm{clear}-\mathrm{sky}\right)=\overline{\mathrm{y}}+\mathrm{A}\cdot \cos \kern0.3em \left(\frac{2\uppi t}{\mathrm{n}}-{\upphi}_1\right), $$
(3)

where yr is the clear-sky SSI at time t in days, \( \overline{y} \)is the average value of the SSI (clear-sky) time series, n = 365, ϕ1 is the phase angle, and A is the amplitude. Figure 2 shows the monthly cycles of daily EUV, TUV, and GS measurements of SSI in Seoul between March 2004 and February 2015. Superimposed on each panel in the figure is the envelope cosine curve of monthly mean data obtained by harmonic analysis and used to estimate the maximum possible clear-sky irradiance employed as the reference value in Eq. (1). The yr indicated that the maximum possible irradiances (clear-sky) were 29.4 mW m−2, and 11.8 and 248 W m−2 for EUV, TUV, GS, respectively. The annual ranges of EUV, TUV, and GS irradiances, with minima in December and maxima in June, were from 8.0 to 51.1 mW m−2, 5.3 to 18.2 W m−2, and 132.4 to 364.4 W m−2, respectively. Areas above the envelope cosine curve correspond to partly cloudy conditions with relatively high sun that lead to enhanced scattering and multiple reflections at cloud edges (e.g., Madronich 1993; Mims III and Frederick 1994; Jung et al. 2011; Núñez et al. 2016). Thus, the modeled clear-sky (maximum possible) SSI was used as the reference value in Eq. (1) for calculations of atmospheric transmission.

Fig. 2
figure 2

Monthly variation in the extraterrestrial irradiance the all-sky average irradiance, and the maximum possible clear-sky irradiance (reference value) as a function of month for the (a) EUV, (b) TUV, and (c) GS bands from March 2004 to February 2013. The dashed, solid, and dotted lines indicate the envelope line of the data maxima for clear-sky, all-sky average, and extraterrestrial values, respectively

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3.2 SSI Data Corrections for Cloud, Aerosol, and Ozone Transmission Calculations

To separately calculate the transmission of clouds, aerosols, and ozone, the following procedure was applied assuming constant Z, AOD, and TO, and using cloud transmission (TCC) as an example. The SSI (all-sky) for a given cloud cover amount (CC) is affected by daily differences in Z, AOD, and TO on the days considered. Mean aerosol and ozone conditions were used for the TCC calculation shown in Eq. (4). For the cloud transmission at constant \( \overline{Z},\kern0.5em \overline{AOD} \), and\( \overline{TO} \), the SSI (all-sky) data were corrected for cloud coverage as follows:

$$ \mathrm{Corrected}\ \mathrm{SSI}\ \left(\mathrm{CC}\right)=\mathrm{f}\ \left(\mathrm{CC},\overline{\ Z},\overline{AOD},\overline{TO}\right). $$
(4)

In Eq. 4, other gases are considered to be constant.

All daily SSI (all-sky) data were corrected using the following procedure for the EUV, TUV, and GS bands for transmission by clouds, aerosol, and ozone. For cloud transmission under constant aerosol and ozone conditions, Eq. (4) takes the following forms for the three spectral bands:

$$ \mathrm{corrected}\ \mathrm{EUV}=\mathrm{EUV}\times \left(1+\frac{\mathrm{TO}-\overline{TO}}{\overline{TO}}\cdotp {TO}_{RAF}+\frac{\mathrm{AOD}-\overline{AOD}}{\overline{AOD}}\cdotp {AOD}_{RAF}\right), $$
(5a)
$$ \mathrm{corrected}\ \mathrm{TUV}=\mathrm{TUV}\times \left(1+\frac{\mathrm{TO}-\overline{TO}}{\overline{TO}}\cdotp {TO}_{RAF}+\frac{\mathrm{AOD}-\overline{AOD}}{\overline{AOD}}\cdotp {AOD}_{RAF}\right),\mathrm{and} $$
(5b)
$$ \mathrm{corrected}\ \mathrm{GS}=\mathrm{GS}\times \left(1+\frac{\mathrm{TO}-\overline{TO}}{\overline{TO}}\cdotp {TO}_{RAF}+\frac{\mathrm{AOD}-\overline{AOD}}{\overline{AOD}}\cdotp {AOD}_{RAF}\right). $$
(5c)

As part of this procedure, a radiation amplification factor (RAF; e.g., Madronich 1993; Booth and Madronich 1994; Kim et al. 2014) was introduced to account for the sensitivities of CC, AOD, and TO to EUV and TUV variability. The GS RAFs for CC, AOD, and TO (CCRAF, AODRAF, and TORAF, respectively) were calculated according to Kim et al. (2014), yielding values of −0.25, −0.16, and − 0.23, respectively. The EUV and TUV RAFs were also adopted from Kim et al. (2014). Also, TORAF was more sensitive to EUV than to TUV and GS. Therefore this RAFs were multiplied by the fractional difference between the TO (or AOD) of a particular day and the mean value, \( \overline{TO} \) (or \( \overline{AOD} \)), to yield the TO (or AOD) correction factors shown in Eq. (5a). Equations similar to Eq. (5b, c) were applied to SSI (all-sky) for calculations of aerosol (corrections to CC and TO) and ozone (corrections to CC and AOD) transmission.

3.3 Semi-Empirical Transmission Models

Atmospheric transmission can be described by the Beer–Bouguer–Lambert Law. A beam of radiation that passes through a medium is attenuated exponentially, where the extinction medium amount appears in the exponent. This law applies to monochromatic radiation, but in parameterizations the spectrally integrated transmission for an atmospheric constituent is needed to compute the broadband transmission of the atmosphere (Eq. 1; e.g., Iqbal 1983; Cho et al. 2003a, b; Fitzpatrick et al. 2004; Sedlar and Hock 2009).

3.3.1 Total Transmission

The total transmission, T, is the product of the transmission by clouds, aerosols, and ozone:

$$ {T}_i={T}_{\mathrm{CC}}\cdotp {T}_{\mathrm{AOD}}\cdotp {T}_{TO}\cdotp {T}_{\mathrm{others}} $$
(6)

where TCC, TAOD, TTO, and Tothers are the transmission by clouds, aerosols, ozone, and other variables, respectively. Consequently, the logarithmic daily total transmission can be derived from Eq. (6) using an approach based on Beer’s Law and employing multiple linear regression, as follows:

$$ T=\exp\ \left(a\cdotp \mathrm{CC}+b\cdotp \mathrm{AOD}+c\cdotp \mathrm{TO}+{d}_{others}\right)\ \mathrm{or} $$
(7a)
$$ lnT=\left(a\cdotp \mathrm{CC}+b\cdotp \mathrm{AOD}+c\cdotp \mathrm{TO}+{d}_{others}\right), $$
(7b)

where a, b, c, and d are regression coefficients from a multiple linear regression model.

3.3.2 Individual Transmission

In this study, four semi-empirical transmission models for clouds, aerosols, and ozone in the EUV, TUV, and GS bands are considered: an exponential relationship model (Eq. 8a), a simple linear regression model (Eq. 8b), a second-order (quadratic) polynomial model (Eq. 8c), and a third-degree (cubic) polynomial model (Eq. 8d). For convenience, we write the transmission T as Y and the extinction optical thickness as a·X in the following.

  1. (a)

    Exponential relationship model

$$ Y=\exp\ \left(a\cdotp X+d\right)\ \mathrm{or} $$
(8a)
$$ lnY=a\cdotp X+d, $$
  1. (b)

    Simple linear model

$$ Y=1+a\cdotp X, $$
(8b)
  1. (c)

    Second- order polynomial model

$$ Y=1+a\cdotp X+b\cdotp {X}^2,\mathrm{and} $$
(8c)
  1. (d)

    Third- order polynomial model,

$$ Y=1+a\cdotp x+b\cdotp {X}^2+c\cdotp {X}^3. $$
(8d)

The constants a, b, and c were obtained by regression analyses of data from 2004 to 2013.

3.4 Clearness index

The clearness index, KT, is useful in characterizing sky conditions over a particular location (Liu and Jordan 1960; Iqbal 1983; Bano et al. 2013; Adam 2014; Wang et al. 2014) and is calculated using SSI (all-sky) and extraterrestrial radiation (Ho) as follows:

$$ KT=\mathrm{SSI}\ \left(\mathrm{all}-\mathrm{sky}\right)/{\mathrm{H}}_{\mathrm{O}}, $$
(9)

where the HO (Fig. 2, dotted line) irradiance is calculated according to Iqbal (1983).

4 Results and Discussion

4.1 Climatology of SSI and Associated Parameters

Monthly average SSI (EUV, TUV, and GS) for a horizontal surface, along with associated meteorological parameters (CC, AOD at 320.1 nm, and TO) under all-sky conditions, are listed in Table 1 (spring: MAM; summer: JJA; autumn: SOM; winter: DJF) from March 2004 to February 2013. The EUV irradiance varied range from 8.0 ± 1.5 (winter) to 33.7 ± 7.6 mW m−2 (summer), with an annual average value of 21.1 ± 4.6 mW m−2. The TUV value varied in the range from 4.5 ± 0.8 (winter) to 10.4 ± 2.3 W m−2 (summer) with an average annual average of 7.7 ± 1.6 W m−2. Also the GS varied between 105.8 ± 21.0 (winter) and 194.4 ± 48.3 W m−2 (summer) with an annual average value of 159.1 ± 37.9 W m−2. The contribution of the TUV to the GS (TUV/GS) varied from 4.1 to 5.7% (annual average 4.8%) on a monthly basis, while that of EUV (EUV/GS) ranged from 6.7 × 10−3% to 19.5 × 10−3% (annual average 12.3 × 10−3%). The CC varied between 3.1 ± 2.1 (winter) and 5.5 ± 2.1 tenth (summer) with an annual average of 4.2 ± 2.2. The AOD varied in the range from 0.9 ± 0.4 (autumn) to 1.5 ± 0.7 (summer) with an annual average of 1.2 ± 0.6. The TO varied from 294.9 ± 14.5 (fall) to 363.1 ± 30.8 DU (spring) with an annual average value of 329.5 ± 21.9 DU.

Table 1 Seasonal variations in CC, AOD, TO, and SSI (GS, TUV, and EUV irradiances) over the period March 2004 to February 2013. Numbers in parentheses indicate one standard deviation
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4.2 Total Transmission and Clearness Index

The division of all-sky measurements and clear-sky values calculated using the clear-sky model (Eq. 3) provides an estimation of total transmission for CC, AOD, and TOC. Table 2 lists the seasonal and annual measured total transmissions. Total transmissions varied in the range 71 (summer)-85% (winter) with annual average of 78% for EUV, in the range of 62 (summer)-83% (winter) with annual average of 74% for TUV, in the range of 58 (summer)-83% (winter) with annual average of 72% for GS. Thus, the seasonal and annual average total transmission increases from the EUV to TUV and then the GS bands, except for spring.

Table 2 Seasonal and annual average total measured transmissions (%). Numbers in parentheses indicate slant transmissions
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The clearness index, KT, which characterizes the sky conditions of a particular location, was estimated from Eq. (9) for Seoul. The monthly KT was calculated using SSI (EUV, TUV, and GS) and extraterrestrial radiation. Jung et al. (2016) studied the spatio-temporal characteristics of KT using daily GS measurements (2000–2014) at 21 sites in Korea, yielding an annual average of 46% for all the sites. The annual average KT values were found to be 1%, 45% and 48% for the EUV, TUV, and GS bands, respectively. This spectral dependence appeared because of the combined effects of the variability in clouds, aerosols, and ozone throughout the year. The ratio of KT to T which indicated as SSI (clear-sky) to the HO was 1%, 69%, and 74% for EUV, TUV, and GS irradiance, respectively, and was calculated as follows:

$$ \frac{K_T}{T}=\frac{Clear- sky\ solar\ irradiance}{Extraterrestrial\ solar\ irraidnace} $$

The low value of KT/T for EUV is due to strong absorption in the EUV band by the ozone layer. Figure 3 presents a schematic depiction of atmospheric transmission, clearness index, and the ratio of the clearness index to atmospheric transmission, (KT/T), for the three spectral bands.

Fig. 3
figure 3

Schematic depiction of atmospheric transmission (T), clearness index (KT), and the ratio of clearness index to transmission (KT/T) for the EUV, TUV, and GS bands from March 2004 to February 2013

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4.3 Multiple Linear Regression Models for Atmospheric Transmission

A multiple linear regression model was derived for the calculation of total transmission from Eqs. (7a, b) for the EUV, TUV, and GS bands. The regression model used daily measurements of the SSI (for the EUV, TUV, and GS bands) and SSI forcing factors (CC, AOD, and TO) from March 2004 to February 2013, with 1440 total measurements. Table 3 lists the regression and correlation coefficients for the EUV, TUV, and GS bands. The correlation coefficients (R2) of SSI explain 56%, 48%, and 57% and are statistically significant at the P < 0.001 level. Cho et al. (2008) investigated the contributions of air temperature, clouds, and humidity to variations in downward longwave radiation using a multiple regression model. In this study, the correlations of SSI forcing factors with SSI are used to evaluate the partial contribution of each SSI forcing factor to SSI. The partial contributions of each parameter are listed in Table 4. Among the R2 of the 56% in the EUV band, the partial contributions of CC (in tenth), AOD (unitless), and TO (in cm) to EUV were estimated to be 15%, 29%, and 12%, respectively. Similarly, R2 of the 48% to total contribution in the TUV band, the contributions of CC, AOD, and TO were estimated to be 20%, 27%, and 1%, respectively. R2 of the 57% to in the GS band, the contributions of CC, AOD, and TO were estimated to be 41%, 15%, and 0.3%, respectively. Thus, the largest contribution in the EUV and TUV bands was from aerosols. However, the GS band had a large contribution from CC. The contribution of ozone in the TUV and GS bands was negligible, as expected from the characteristics of ozone absorption at these wavelengths.

Table 3 Multiple linear regression coefficients (a, b, c, and d) and correlation coefficients for SSI (GS, TUV, and EUV irradiances)
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Table 4 Partial and total contributions (%) from the multiple regression analysis. Partial contributions are the product of individual correlation coefficients and beta coefficients
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The SSI predictions using monthly averaged data from March 2013 to February 2015 were compared with the results of multiple linear regression models using all-sky measurements (Table 5). The modeled total transmission was 75%, 71%, and 67% for the EUV, TUV, and GS bands, respectively. The average ratios of the modeled (M) to measured (M0) total transmission were 1.09, 1.05 and 1.04 for SSI in the EUV, TUV, and GS bands, respectively. The annual average transmissions tend to decrease from the GS to the TUV and then the EUV bands (Table 5). For modeled AOD and ozone transmission the physical cause of this trend is unknown. The modeled transmission by clouds was found to be wavelength dependent, ranging from 87% for EUV to 85% for TUV and 75% for GS (e.g., Krotkov et al. 1996; Seckmeyer et al. 1996). In contrast, for cloud transmission the modeled aerosol transmission in the EUV, TUV, and GS bands was 83%, 85%, and 89%, respectively. Thus, aerosol transmission increases with wavelength. Transmission by ozone was 39%, 77%, and 79%, for the EUV, TUV, and GS bands, respectively. It should be noted that the EUV is strongly attenuated by ozone absorption. The wavelength dependence of aerosol and ozone transmission is the inverse of that of clouds using the multiple linear regression model. Results for both clouds and aerosols are consistent with the findings of Erlick and Frederick (1998) when clouds and aerosols are mixed (Frederick and Erlick 1997).

Table 5 Comparison of measured and modeled transmissions (%) computed from multiple linear regression. M and M0 indicate the modeled and measured transmission, respectively over the period the March 2013 to February 2015. The ratio is M divided by M0
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4.4 Semi-Empirical Models of Individual Transmission

To develop semi-empirical models of the individual transmission of clouds, aerosols, and ozone in the EUV, TUV, and GS bands, the following procedures were performed using Eqs. (8a, b, c and d). We applied four different semi-empirical models (March 2004 to February 2013) for the regression analyses: an exponential relation model (Eq. 8a; exponential Beer’s Law), a simple linear model (Eq. 8b), a second-order (quadratic) polynomial model (Eq. 8c), and a third-order (cubic) polynomial model (Eq. 8d) for cloud transmission under constant aerosol and ozone conditions, for aerosol transmission under constant cloud and ozone conditions, and for ozone transmission under constant cloud and aerosol conditions in the EUV, TUV, and GS bands. An ideal clear-sky dataset would produce transmission close to unity (i.e., T = 1). Therefore, all regressions were forced to a transmission of 1.0 for cloudless conditions (CC = 0) except for the exponential relation model (Eq. 8a). The cloud transmission models obtained from the regression analyses, along with the aerosol and ozone models, are summarized in Table 6. The R2 of the regression models varied between 0.03 to 0.50 depending on the cloud, aerosol, and ozone transmission models, with statistical significance at the p = 0.000 level, except for the simple linear regression model for ozone transmission in all three spectral regions.

Table 6 Summary of individual semi-empirical transmission models for clouds, aerosols, and ozone determined from March 2004 to February 2013. The dependent variable (Y) is transmission and the independent variables (X) are cloud cover (tenth), aerosol optical depth (320.1 nm), and total ozone (cm). a, b, and c are regression coefficients, and SE is the standard error of the regression models
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The second-order polynomial (quadratic) model in this study takes the form Y (TUV) = 1–0.102·X + 0.005·X2. However, Lubin and Frederick (1991) developed a quadratic model of the form Y (345 nm) = 1–0.002·X − 0.004·X2, and Kuchinke and Núñez (1999) derived a equation for zenith angles in the range 35°–59° of the form Y = 1 + 0.027·X − 0.013·X2 with X (CC) in oktas cloud cover amount. The empirical third-order polynomial (cubic) model takes the form Y (TUV) = 1–0.169·X + 0.029·X2–0.002·X3 for the TUV band in this study. However, Schafer et al. (1996) derived the equation Y = 1–0.131·X + 0.029·X2 − 0.002·X3 for z = 50°. Thus, four previous cloud models have somewhat different coefficients depending on the locations and data periods. Previous studies of empirical aerosol and ozone transmission have not been investigated.

Table 7 shows the parameters obtained for the mean bias error (MBE) for all-sky conditions in a comparison of the modeled and measured transmissions for the March 2013 to February 2015 dataset. The MBE varied from −2.4% to 4.3% for all models. The model products overestimate measurement values. It should be noted that the exponential model of AOD and ozone (Eq. 8a) results were closest to the measured data according to R2 at the p = 0.000 significance level, and this model yielded the lowest MBE for the EUV, TUV, and GS bands. The exponential models underestimated the measurements by 0.6% on average, with MBEs of −1.9%, −1.9%, and − 2.4% for the cloud models; 0.8%, 0.9%, and − 0.1% for the aerosol models; and 1.0%, −0.3%, and − 1.2% for the ozone models for the EUV, TUV, and GS bands, respectively.

Table 7 Comparison of modeled (M) and measured (M0) transmissions (%) over the period March 2013 to February 2015
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5 Summary and Conclusions

This study developed semi-empirical models for atmospheric broadband transmission in the EUV, TUV, and GS spectral regions using daily average CC, AOD, and TO measurements for Seoul (March 2004 to February 2015).

The climatology in SSI for Seoul is briefly summarized as follows. The monthly average EUV irradiance ranged from 5.8 (December) to 34.9 mW m−2 (July), with an annual average of 21.1 mW m−2. The TUV irradiance ranged from 3.6 (December) to 11.3 W m−2 (June) with an annual average of 7.7 W m−2, whereas GS irradiance ranged from 85.9 (December) to 232.7 W m−2 (May), with an annual average of 159.1 W m−2. The fraction of EUV and TUV to GS showed an annual average of 12.3 × 10−3% and 4.8%, respectively. The annual clear-sky averages were 248.1 and 11.8 Wm−2 for GS and TUV irradiance, respectively, compared with 29.4 mW m−2 for EUV irradiance.

Atmospheric transmission was calculated as the ratio of SSI (all-sky) to calculated SSI (clear-sky) using a harmonic analysis model. The measured total transmissions had annual averages of 78%, 74%, and 72% in the SSI bands, under all-sky conditions. We constructed multiple linear regression models for total transmissions under all-sky conditions that explain 56%, 48%, and 57% of the total variance (R2) in the EUV, TUV, and GS bands, respectively. The average ratios of measured to modeled total transmission were 1.09, 1.05, and 1.04 for the EUV, TUV, and GS bands, respectively, with the measured values being higher than modeled values for all three spectral regions. The annual average clearness index (KT) values, which are also used to characterize atmospheric transmission, were 1%, 45%, and 48% for the EUV, TUV, and GS bands, respectively. The ratio of clear-sky irradiance to extraterrestrial irradiance (KT/T) was 1%, 69%, and 74% for the EUV, TUV, and GS bands, respectively. The small KT/T for the EUV band is caused by strong absorption in the ozone layer. We developed four models (exponential, simple linear, second-order polynomial, and third-order polynomial) for transmission by clouds, aerosols, and ozone in the EUV, TUV, and GS bands. Results of a regression analysis showed that the exponential model for clouds, aerosols, and ozone was the closest to measured values, with the lowest MBE and the highest R2 (statistically significant at the p = 0.000 level), as expected from the fact it is based on Beer’s Law. The exponential models underestimate measurements by 0.6% on average. The annual average transmission of clouds, aerosols, ozone, and a combination of the three, decreases from the EUV to the TUV and then the GS spectral bands.