#### 3.1. Relationships between Monthly Average of Effective Day Length and Other Climatic Elements

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

N_{0} (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

N_{0}, at low light intensity of less than 10,000 lx, the following formula can be applied:

where

N_{0} 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)}.

#### 3.2. Consistency Check

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.