For the purpose of the study, it is desirable to use daylight illuminance data measured in various fields around the World. However, the field measurement of daylight illuminance on a global scale has not yet been conducted. Therefore, we used solar radiation data as input data. Continual global solar radiation (Rs) observation data were gathered from the baseline surface radiation network (BSRN) of 18 stations located from 50°N to 50°S over a period of 4 years between 2004 and 2007 [14]. The site locations are listed in Table 1. In this study, only data from low- to mid-latitude sites were gathered, because data from high-latitude sites can interfere with the analysis because of the weak and long day light in summer. Solar radiation measurements were obtained from the BSRN system, which provides 1- or 3-min averages of the global solar radiation. In this study, only days for which 24-h observations were performed without missing any data were considered.

The solar radiation data were converted to light intensity data using a Igawa’s luminous efficiency model [15,16] that calculates the daylight intensity using solar radiation and solar altitude data. This model is well known for its validity. The luminous efficiency model is:

where Evg is light intensity (global horizontal illuminance; lx), Rs is global horizontal solar radiation (W m⁻²), h is solar zenith angle (rad).

The photoperiods exceeding each defined light intensity (1 lx, 300 lx, 500 lx, 1,000 lx, …, 20,000 lx) were integrated at daily intervals. Next monthly averages of these photoperiods were calculated. In this study, this photoperiod is defined as the effective day length [Ne_(lux)], where lux is the threshold of each light intensity indicated by the global horizontal illuminance.

The monthly average of daily extraterrestrial radiation Ra (MJ m⁻² day⁻¹) and possible sunshine duration N₀ (h day⁻¹) were calculated following the guidelines of Darula et al. [17], and the monthly average of daily amount of global solar radiation Rs (MJ m⁻²day⁻¹) was calculated using the BSRN data. In order to derive the model that estimates the effective day length at any light intensity, the relationships between the monthly averages of Ra, N, Rs, and Ne_(lux) (h day⁻¹) were investigated. In this case, only months when more than 20 days were observed without missing any data were considered. The numbers of analyzed months are shown in Table 1.

To determine the coefficients of the general regression equation, the following steps were taken. First, the regression coefficients between the monthly averages of Ra, N, Rs, and Ne_(lux) were calculated at the defined light intensity. Then, the trends of the coefficients for all the light intensities were investigated, and a generalized model that could be applied to any light intensity was derived.

Global solar radiation measurement data from Tateno (Japan), Chesapeake Light (USA), and Payerne (Switzerland) for 2008 were gathered from the baseline surface radiation network (BSRN), and were converted to light intensity data using an Igawa’s luminous efficiency model. We calculated monthly averages of effective day length from these data and used for a consistency check. Additionally, in order to verify the model, direct measurements of illuminance data from Vaulx-en-Velin (France) for 2008–2010 were obtained [18]. Monthly averages of the effective day length were calculated, and the estimated effective day length and the observed one were compared. Then, the root mean square error (RMSE) and mean absolute percentage error (MAPE) were calculated. The MAPE was calculated as follows:

The site locations used for the validation are listed in Table 2.

Statistical analyses were carried out using R version 2.7.1 [19].

Figure 1 shows the relationships between the monthly average of Ne₍₅₀₀₎ (h day⁻¹) and the monthly average of Rs (MJ m⁻² day⁻¹) and between the monthly average of Ne_(10000) and monthly average of Rs. At low light intensity (500 lx), the relationships were not clear. At low light intensity, the Ne_(lux) values were almost the same as the monthly averages of N₀ (h day⁻¹). This means effective day length at low light intensities will be free of the influence of any weather conditions. At high light intensity (10,000 lx), the relationships between Ne_(lux) and Rs were clear and a logarithmic curve was well fitted.

The trend of the coefficients was investigated for all light intensities (Figure 2), and a generalized formula that could be applied to any light intensity was derived:

Figure 3 shows the relationships between the monthly average of the effective day length (Ne_(lux)) and the light intensity at Tateno station in 2008. Because Ne_(lux) varies inversely with light intensity in relation to the monthly average of N₀, at low light intensity of less than 10,000 lx, the following formula can be applied:

where N₀ is the monthly average of possible sunshine duration (h day⁻¹). Equation (6) shows the linear interpolation between the possible sunshine duration and the monthly average of effective day length at light intensity of 10,000 lx. As we did not observe a trend for the change in the coefficients in Figure 3, we considered a simple linear relationship; however, the error caused by this is probably small. There was no significant relation between Ra, n, and Ne_(lux).

In order to check the model, global solar radiation measurements data from Tateno (Japan), Chesapeake Light (USA), and Payerne (Switzerland) for 2008 were converted to light intensity data using a luminous efficiency model and monthly average of effective day length were derived and used for a consistency check. Figure 4 shows the seasonal transition of observed and estimated monthly mean effective day length [Ne_(lux) h day⁻¹] at 3,000 lx (upper) and 20,000 lx (lower) at Tateno (a), Chesapeake Light (b) and Payerne (c). At 3000 lx, the estimation was carried using Equations (3)–(6); 20,000 lx, Equations (3)–(5). The RMSE calculated through the year was 0.44 h day⁻¹ (MAPE: 3.9%) at 3,000 lx and 0.29 h day⁻¹ (MAPE: 3.7%) at 20,000 lx in Tateno. Likewise, the RMSE was 0.36 h day⁻¹ (MAPE: 3.3%) at 3,000 lx and 0.47 h day⁻¹ (MAPE: 5.7%) at 20,000 lx in Chesapeake Light, and the RMSE was 0.51 h day⁻¹ (MAPE: 5.0%) at 3,000 lx and 0.44 h day⁻¹ (MAPE: 5.9%) at 20,000 lx in Payerne.

Figure 5 shows the transition of RMSE and MAPE at all the light intensities for the monthly base estimation at Tateno station in 2008. The RMSE and MAPE increased at around 10,000 lx, with a similar trend in Chesapeake Light and Payerne. This is because, at low light intensities, the effective day length is almost the same as the possible sunshine duration. Hence, at high light intensities, the logarithmic curve fitted well. The maximum value of RMSE at all the light intensities was 0.56 h day⁻¹, and the maximum value of MAPE was 5.2% at Tateno.

In Figure 5, the transitions of RMSE and MAPE are discontinuous around 10,000 lx. This is because a different equation was used to estimate the effective day length. We decided to set a limit on the choice of equations from the fitting conditions of the equation or the results of validation, but this sometimes causes a discontinuity in the estimation. The maximum value of RMSE at all the light intensities was 0.53 h day⁻¹ and 0.82 h day⁻¹ in Chesapeake Light and Payerne stations, respectively. The maximum value of MAPE was 5.7% and 6.8% for Chesapeake Light and Payerne stations, respectively.

Figure 6 shows the seasonal transition of observed and estimated monthly mean effective day length [Ne_(lux) h day⁻¹] at 3,000 lx (upper) and 20,000 lx (lower) at Vaulx-en-Velin (France) station in 2008 (a), 2009 (b) and 2010 (c).

At 3000 lx, the estimation was carried using Equations (3)–(6); 20,000 lx, Equations (3)–(5). At 3,000 lx, the estimated values slightly exceeded the observed values throughout the year in all years. This was probably due to the linear interpolation of Equation (6). The RMSE calculated through the year was 0.43 h day⁻¹ (MAPE: 4.4%) at 3,000 lx and 0.42 h day⁻¹ (MAPE: 9.3%) at 20,000 lx in 2008. Likewise, the RMSE was 0.57 h day⁻¹ (MAPE: 5.5%) at 3,000 lx and 0.52 h day⁻¹ (MAPE: 10.1%) at 20,000 lx in 2009, and the RMSE was 0.57 h day⁻¹ (MAPE: 6.0%) at 3,000 lx and 0.45 h day⁻¹ (MAPE: 9.9%) at 20,000 lx in 2010.

Figure 7 shows the transition of RMSE and MAPE calculated through the year at all the light intensities for the monthly base estimation at Vaulx-en-Velin station in 2008. The RMSE increased at around 10,000 lx, with a similar trend in the consistency check. The maximum value of RMSE at all the light intensities was 0.54 h day⁻¹ (at 10,000 lx), and the maximum value of MAPE was 9.3% (at 20,000 lx) in 2008. The MAPE increased at high light intensities, as the actual effective day length was so short. The maximum value of RMSE at all the light intensities was 0.65 h day⁻¹ and 0.52 h day⁻¹ in 2009 and 2010, respectively. The maximum value of MAPE was 10.1% and 9.9% in 2009 and 2010, respectively.

As a result, the effective day length at any light intensity could be estimated with an accuracy of less than 0.9 h day⁻¹ of RMSE (11% of MAPE) in the estimation of the monthly base photoperiod.