A total of 40 weather stations of Japan Meteorological Agency, covering almost all over the country, were selected, where GSR Q [W·m⁻²] and the sunshine duration n [h·d⁻¹] had concurrently been monitored. The latitudes, longitudes, and elevation heights from land surface of the stations were tabulated in Table 1 . The measured values of Q [W·m⁻²] and n [h·d⁻¹] on daily basis were collected from the database that are opened to the public by Japan Meteorological Agency (2022) .

The Japan Meteorological Agency standardized the thermopile pyranometers used for measuring solar radiation at every meteorological station in the period between 1971 and 1974, after the uses of several types of pyranometer including Robitzsch-type pyranometers and Eppley-type pyranometers had mingled together. The measurement systems of sunshine duration had also been updated from Jordan sunshine recorders to rotating mirror sunshine recorders in the period from 1986 to 1990. Therefore, the measured data sets of GSR and sunshine duration obtained after these updates can be used for the analyses without any correction.

Values of daily precipitation, hourly-maximum precipitation, and cloud cover were also collected to characterize the interannual trends in GSR, sunshine duration, and the parameters of the atmospheric transmittance model. The stations where the cloud cover had been routinely measured throughout the study period were limited to 11 stations denoted in Table 1 .

Values of GSR Q [W·m⁻²] in relation to sunshine duration n [h·d⁻¹] have often been expressed by the Angstrom-Prescott type model:

$Q=Q_e\cdot \tau$ (1)

where Q_e [W·m⁻²] is the extraterrestrial solar radiation, and τ[-] is the atmospheric transmittance. Originally, the solar radiation expected in clear days Q_c [W·m⁻²] had been adopted in place of Q_e [W·m⁻²] ( Angstrom, 1924 ; Prescott, 1940 ), but, later, the original model was modified into Equation (1) by replacing Q_c [W·m⁻²] with Q_e [W·m⁻²] as seen on Doorenbos and Pruitt (1977) and Allen et al. (1998) .

In the Angstrom-Prescott type model, the atmospheric transmittance τ[-] is expressed as a function of sunshine duration as below:

$\tau =a_0+a_1\cdot n/N$ (2)

where N [h·d⁻¹] is the day length. The offset a₀[-] and the slope a₁[-] of this linear relation are likely to be site-specific. Even in cases of the original models, Angstrom (1924) showed the model with a₀ = 0.22[-] and a₁ = 0.54[-] while a₀ = 0.25[-] and a₁ = 0.54[-] were found on Prescott (1940) . Therefore, these two parameters are determined by fitting the model outputs to data sets of Q [W·m⁻²] and n [h·d⁻¹] obtained in a location of interest.

The day length N [h·d⁻¹] in Equation (2) was calculated as below:

$N=2h_{ahd}\cdot \left(24/2\pi \right)$ (3)

where h_ahd [rad] is the hour angle for half a day. The value of h_ahd [rad] was evaluated by using the following geometrically-derived equation:

$\cos h_{ahd}=-\tan \phi \cdot \tan \delta$ (4)

where φ [rad] is the latitude of a location of interest in radian unit, δ [rad] is the solar declination, which is an angle of elevation from the plane of the earth’s equator to the sun.

The solar declination δ [rad] was evaluated by using the following relation:

$\sin \delta =\sin {\delta }_{max}\cdot \sin \left(1.5\cdot \pi +f-f_{ws}\right)$ (5)

where δ_max [rad] is the angle of the tilt of the rotation axis of the earth from the direction normal to the earth’s orbital plane, being about 23.4393 in degree. f [rad] is the true anomaly of the earth for a given day of year (DOY). f_ws [rad] is the true anomaly of the earth on the day of the winter solstice in the northern hemisphere (DOY_ws).

The true anomaly f [rad] is the angle between the direction of the perihelion and that of the earth’s position as seen from the sun and, thus, is the function of DOY. In this study, f [rad] was calculated in the following steps: 1) a mean anomaly M [rad] of the earth’s position was determined for a given DOY; 2) an eccentric anomaly E [rad] of the earth’s orbital position was derived from M [rad]; and 3) f [rad] was determined by E [rad]. In the step 1), the following relation was adopted:

$M=2\pi \cdot \left(\text{DOY}-{\text{DOY}}_{\text{ph}}\right)/T_{ty}$ (6)

where DOY_ph is the day of the year on which the earth passes the perihelion in a year, T_ty [d] is the period of a tropical year (365.2422 days). In the step 2), M [rad] was converted into E [rad] by solving the Kepler’s equation:

$0=M-E+e_o\cdot \sin E$ (7)

where e_o[-] (≒0.01671) is the eccentricity of the earth’s orbit. The Newton-Raphson iteration was adopted to solve this equation. In the step 3), f [rad] was calculated from E [rad] by using the following geometrical relation:

$f=\arctan \left(\sqrt{1-e_o^2}\cdot \sin E/\left(\cos E-e_o\right)\right)$ (8)

Then, both f [rad] and f_ws [rad] were determined by applying Equation (6) through (8) for DOY and DOY_ws.

The values of N [h·d⁻¹] for the 40 stations were calculated for any date by making the series of Equations (3) through (8) work as a function of φ [rad] and DOY. Their annual averages for each year are tabulated in Table 1 . The resultant values ranged from 12.03 to 12.06 [h·d⁻¹] with the mean of 12.04 [h·d⁻¹] among the 40 stations.

The extraterrestrial solar radiation Q_e [W·m⁻²] is a fraction of the solar radiation reaching the outer surface of the earth’s atmosphere, namely the solar constant S_c [W·m⁻²]. The solar constant S_c [W·m⁻²] is likely to be about 1366 [W·m⁻²] in a large time scale ( Frohlich, 2009 ; Passos et al., 2007 ), though it is not a constant quantity. The difference between S_c [W·m⁻²] and Q_e [W·m⁻²] owes to the facts that the angle of incidence of the sun ray from the sky to the land surface varies with time in a day, and that the sun ray does not reach the land surface after dark in the day. Therefore, Q_e [W·m⁻²] can be calculated by multiplying S_c [W·m⁻²] with the integral of the sine of the solar elevation angle h_e [rad] with respect to the hour angle h_a [rad] in the domain of the day length (−h_ahd ≦ h_a ≦ h_ahd) divided by the hour angle for the period of one day (equal to 2π). In addition, a coefficient should also be multiplied as the correction factor for the varying distance between the sun and the earth as the earth orbits. The sine of h_e [rad] is related to the hour angle h_a [rad] by the following relation:

$\sin h_e=\sin \phi \cdot \sin \delta +\cos \phi \cdot \cos \delta \cdot \cos h_a$ (9)

Therefore, the integration and the multiplications mentioned above leads to:

$\{\begin{array}{l}{Q}_e=\frac{S_c}{\pi }\cdot D\cdot \left(h_{ahd}\cdot \sin \phi \cdot \sin \delta +\cos \phi \cdot \cos \delta \cdot \sin h_{ahd}\right)\\ D=1+2\cdot e_o\cdot \cos M\end{array}$ (10)

The values of Q_e [W·m⁻²] for the 40 stations were calculated for any date from Equation (10) with the aids of Equations (4) through (8). Thus, the input variables for determining Q_e [W·m⁻²] were the same as for determining N [h·d⁻¹], φ [rad] and DOY. Their annual averages were tabulated in Table 1 . The obtained values ranged from 309.5 to 385.3 [W·m⁻²] with the mean of 349.8 [W·m⁻²] among the 40 stations.

The pair of the parameters a₀[-] and a₁[-] in Equation (2) was determined for every year at every station by minimizing the sizes of least squared errors between values of daily-averaged Q [W·m⁻²] evaluated by using Equation (1) and those measured daily in the year at the station. In order for clarifying interannual trends in Q [W·m⁻²] and n [h·d⁻¹], annually-mean values of these two quantities were determined for every station and every year in the study period.

Table 2 shows the annual mean values of GSR (Q_a [W·m⁻²]) measured in each of the 40 stations. The series of Q_a [W·m⁻²] for each station was averaged and named m_Qa [W·m⁻²]. The obtained m_Qa [W·m⁻²] tended to take a higher value for a station with lower latitude, ranging from the minimum of 130.4 [W·m⁻²] for Abashiri in the northern most island Hokkaido to the maximum of 190.1 [W·m⁻²] for Minami-Daito Island in the south-western sub-tropical region.

The interannual trend in Q_a [W·m⁻²] was mostly upward in the 22-year period. When the values of Q_a [W·m⁻²] for each of the stations were plotted against dominical years Y_d, the linear regression procedure applied to the Y_d - Q_a relation gave the rates dQ_a/dY_d [W·m⁻² y⁻¹] as tabulated on Table 2 . Counting the number of dQ_a/dY_d [W·m⁻²·y⁻¹] with positive values indicated that Q_a [W·m⁻²] had increased for the study period in 33 out of the 40 stations. Averaging the 40 values of dQ_a/dY_d [W·m⁻²·y⁻¹] resulted in 0.252 [W·m⁻²·y⁻¹], which is equivalent to the increase in Q_a [W·m⁻²] by 5.5 [W·m⁻²] in the 22-year period, representing the upward trend in GSR over the country.

Many of the stations with negative dQ_a/dY_d [W·m⁻² y⁻¹] were found in the south-western region of the country with latitudes lower than 32[˚]. Since the values of GSR in that region are ordinarily in the possible highest levels, it can be difficult to expect additional increases in them, suggesting that the interannual trends in GSR found in that region might not have been downward but been effectively flat.

Values of annually-averaged sunshine durations n_a [h·d⁻¹] were determined for each station ( Table 3 ). The temporal means in the 22-year period m_na [h·d⁻¹] ranged from 4.07 to 6.17 [h·d⁻¹] among the values for the 40 stations, and had the spatially averaged value of 5.21 [h·d⁻¹]. The values of n_a [h·d⁻¹] also have increased in most of the stations. The rates of increase dn_a/dY_d [h·d⁻¹·y⁻¹] were 0.015 [h·d⁻¹·y⁻¹] in average and was 0.036 [h·d⁻¹·y⁻¹] at maximum. The value 0.015 [h·d⁻¹·y⁻¹] in dn_a/dY_d [h·d⁻¹·y⁻¹] implied that n_a [h·d⁻¹] had increased by 0.33 [h·d⁻¹] in the 22-year period.

A few negative dn_a/dY_d [h·d⁻¹·y⁻¹] were obtained mainly in the south-western region with low R² values, as some negative dQ_a/dY_d [W·m⁻²·y⁻¹] ( Table 2 ). These apparent coincidences found among Q_a [W·m⁻²] and n_a [h·d⁻¹] induced us to estimate the extent to which the trend in n_a [h·d⁻¹] can explain that in the measured Q_a [W·m⁻²]. Estimating this extent requires the values of the parameter a₁[-] in Equation (2). Thus, the relation between Q_a [W·m⁻²] and n_a [h·d⁻¹] was analyzed in terms of Equation (2) in the next section.

The values of a₀[-] in Equation (2) were determined for each of the stations in the series of years ( Table 4 ). m_a0[-] denotes the interannual average of a₀[-] obtained for each of the stations. The values of m_a0[-] ranged from 0.163 to 0.239 with their mean of 0.201. σ_a0[-] denotes the standard deviation of a₀[-] associated with m_a0[-], whose values ranged from 0.006 to 0.017, and with the nation-wide value of 0.004. No obvious interannual trend was found among the series of a₀[-] values. The signs of da₀/dY_d [y⁻¹] varied from positive to negative throughout the 40 stations, resulting in the mean of da₀/dY_d [y⁻¹] being 0.00006. And their sizes were less than 0.001 in most cases. These features suggested that a₀[-] had been practically constant for the study period.