## Calculation formula for analysis

We calculated absolute humidity (Contrary to relative humidity, it indicates the amount of water vapor(g) contained per 1m^{3}) using the following approximation equation.

The amount of water vapor(g/m^{3})=217×(6.11×10^{(7.5×Ta/(Ta+237.3))})/(Ta+273.15)×RH/100

(Ta: Average temperature(℃), RH: Average relative humidity(%))

Why is absolute humidity used even though relative humidity is used in the weather forecast? The reason is that the correlation between relative humidity and the heatstroke patients is too weak. In our opinion, relative humidity shows the ratio of water vapor to amount of saturated water vapor. This means, the higher the temperature is, the lower the humidity becomes when the same amount of water vapor might make the correlation weaker. Therefore, we decided to use absolute humidity.

Since the actual measurements of WBGT are obtained only in some prefectures, we estimated the daily average WBGT for each prefecture. The following formula is used for WBGT published on the Ministry of the Environment’s website.

WBGT=0.7×Tw+0.2×Tg+0.1×Ta

(Tw: Wet-bulb temperature (℃), Tg: Globe temperature (℃), Ta: Average temperature (℃))

We estimated the globe temperature in reference to the paper that is “Parameter adjustment and application to an extension area of Okada and Kusaka's formula for the black globe temperature”.

Tg =(S_{0}-38.5)/(0.0217×S_{0}+4.35×U+23.5)＋Ta

(Tg: Globe temperature (℃), Ta: Average temperature (℃), S_{0}: The amount of global solar radiation(W/m^{2}), U: Wind velocity(m/s))

Because some prefectures have no data about declination of the sun for use in calculating the globe temperature, we estimated by referring to the paper of “太陽の位置、可照時間、大気外水平面日射量” “アメダス地点における全天日射量(日平均)の推定法”.

⑴Horizontal insolation intensity of upper atmosphere (Sd_{0}(W/m^{2}))

Sd_{0}=(1367/π)×((d_{0}/d)^{2})×(hsinφsinδ+sinhcosφcosδ)

h=cos^{-1}(-tanφtanδ)

(d_{0}/d)^{2}=1.00011+0.034221cosη +0.00128sinη+0.000719cos2η+0.000077sin2η

η=(2π/365)×DOY

δ=23.5cos(0.01689×(DOY-173))×(π/180)

(1367: Solar constant, h: Angle of sunset, φ: Latitude, δ: Declination of the sun, DOY: Day of year)

⑵Possible sunshine duration(N0(h))

N0=2H/0.2618

2H=4asin((A/(cosφcosδ)) ^{0.5})

A=sin(π/4+(φ-δ+ 0.01)/2)×sin(π/4-(φ-δ-0.01)/2)

(φ: Latitude, δ: Declination of the sun)

⑶Day average declination of the sun(Sd(W/m^{2}))

Sd=(0.244+0.511(N/N_{0}))×Sd0 (N≠0)

Sd=0.118×Sd0 (N=0)

(N: Hours of daylight(h))