Solar radiation data are essential for the design of very many energy systems. These data are needed for obtaining solar energy resource assessment, its transmission and also to obtain the efficiency of energy delivery. A few examples are solar water heating, and space PV systems, daylighting, building air conditioning load and solar-driven ventilation. The starting point for the above computational chain is almost always global and diffuse horizontal radiation. Usually, the computations are carried out using hourly or sub-hourly data.

Note that not always it is possible to obtain a long-term series of hourly or sub-hourly data for the above parameters.

The most commonly measured solar data are global irradiation and these are available for a limited number of stations within any given country at an hourly, daily or monthly frequency. For example within the UK and Spain a historical records of hourly data are available for 71 and 31 stations, respectively.

Of these stations due to higher operational costs associated with diffuse radiation measurements the respective meteorological offices tend to record the latter variable at much fewer locations. For example, since the year 2002 within the UK the diffuse radiation is recorded at only two locations, at North latitudes of Camborne (50.21°) and Lerwick (60.80°).

On the contrary, through the work of NASA ( it is now possible to obtain daily-averaged irradiation data for virtually any location in the world. A sample table of climatic data for Easthampstead (Bracknell) is provided in Table 1.

Table 1

Climatic data for Easthampstead (Bracknell) with the NASA reported irradiation data and averaged measured data for the period 1981-1983

UnitClimate data location
Heating design temperature°C–1,74
Cooling design temperature°C22,96
Earth temperature amplitude°C14,35
Frost days at siteday37,00
MonthAir temperatureRelative humidityDaily solar radiation-horizontalAtmospheric pressureWind speedEarth temperatureHeating degree-daysCooling degree-daysAverage measured radiation
Measured at (m)10,00,0

This information was downloaded from the above-mentioned NASA website. The NASA reported irradiation data were compared by the present research team against averaged measured data for one UK location for the period 1981–1983 (three complete years) (see Fig. 1). The statistics within the latter figure shows that there is a close concordance between the satellite-based NASA irradiation and the UK Meteorological Office measured data set.

Fig. 1 

Comparison between NASA reported irradiation data and ground-based averaged measured data for Bracknell, kWh/m2-day

Following the original work of Liu and Jordan (Liu & Jordan ) a great many number of research teams from around the world have produced regressions relating diffuse ratio (k) and clearness index (kt) regressions at an hourly, daily, monthly and annual frequency. Each of the above four category of regression is unique and statistically different as shown in the work of Muneer (Muneer ) and Saluja et al. (Saluja et al. ).

The present article was pooled from 14 world-wide locations to obtain a regression model to complete the above missing link. It was presently shown that the averaged–data based regressions are distinctly different from previously available hour-by-hour regressions.

The unique nature of solar radiation regressions

Historically speaking, a large number of research teams from around the world have produces k-kt regressions that were based on an hour-by-hour, daily, monthly or annual data. Examples that may be cited here, are Liu and Jordan (daily, and monthly-averaged daily) (Liu & Jordan ), Erbs et al. (hourly, daily and monthly-averaged daily) (Erbs et al. ), Hawas and Muneer (hour-by-hour, daily, monthly- and annual-averaged daily) (Hawas & Muneer ; Muneer & Hawas ; Muneer et al. ) and Stanhill (monthly- and annual-averaged daily) (Stanhill ).

Presently, Figs. 2 and 3 show the unique nature of hour-by-hour (Fig. 2: k − kt plot) and monthly-averaged hourly regressions (Fig. 3: ). An important point to note is that while Fig. 2 shows a convex profile, Fig. 3 demonstrates a concave behaviour. The latter two figures are based on data from common UK locations.

Fig. 2 

Hourly diffuse ratio versus clearness index for UK locations

Fig. 3 

Monthly-averaged hourly plot for UK locations (arranged in an increasing order of latitude: Camborne 50.21° N, Aldergrove 54.65° N)

While there are established models for data of Fig. 2 there are no regressions available in literature for averaged-hourly data such as those shown in Fig. 3. The object of this article is to present the latter type of regressions.

Presently developed monthly-averaged hourly regressions

Fourteen worldwide locations were chosen for this study, details of which are shown in Table 2.

Table 2

The 14 worldwide locations that were presently investigated

CountryLocationLatitudeLongitudePeriod of observation
Kingdom of BahrainBahrain26.0350.612000–2002
State of KuwaitKuwait29.2247.981996–2000
United KingdomCamborne50.215.301981–1995

Data consisted of hourly global and diffuse irradiation values for several years for each location, covering most of the range of latitude for the country. The location names have been arranged in an increasing order of latitude.

Monthly-averaged hourly values were calculated for the global and diffuse radiation considering the data period for each location. For each of them, the monthly-averaged hourly diffuse ratio and the corresponding clearness index were calculated from sunrise to sunset. The following conditions were used in each case to remove erroneously recorded data.


The monthly-averaged clearness index was then regressed against the monthly-averaged diffuse ratio for each location. Figure 4 shows one such scatter plot for Chennai and Lisbon. Furthermore, for each increment at bandwidth of clearness index of 0.05 width, the corresponding values of averaged diffuse ratio shown in Fig. 5 were obtained, shown here for pooled data from two Indian locations.

Fig. 4 

Monthly-averaged hourly diffuse ratio (y-axis) versus clearness index (x-axis) fot two locations. One Indian and other a South European

Fig. 5 

Averaged values of diffuse ratio for the locations between latitude 13-20° North

Figures 6 and 7 respectively show the regressions for locations in a narrower range of latitudes (20-42° N) and worldwide sites with a more diverse range of latitudes (13-58°N).

Fig. 6 

Averaged values of diffuse ratio for the locations between latitude 20-42° North

Fig. 7 

Averaged values of diffuse ratio for the all locations

Note that Fig. 6 shows the potential for a single regression model. Figure 7 on the other hand indicates the existence of different sub-models and these shall now be explored further.

Figures 8, 9 and 10 respectively present regressions models that were obtained by pooling data from locations with a latitude range of 13-20° N, 20-42° N and 50-58° N.

Fig. 8 

Averaged values of diffuse ratio for the locations between latitude 13-20° North (Chennai and Pune)

Fig. 9 

Averaged values of diffuse ratio for the locations between latitude 20-42° North (Bahrain, Kuwait, Almeria, Faro, Lisbon, Madrid and Girona)

Fig. 10 

Averaged values of diffuse ratio for the locations between latitude 50-58° North (Camborne, Bracknell, Aberporth, Finningley and Stornoway)

Table 3 presents regressions equations and coefficient of determination (R2) for each location.

Table 3

The regressions equations and coefficient of determination (R2) for each location

CountryLocationRegression equationsR2
IndiaChennaiy = 0.5124x2-0.9809x + 0.87330.83
Puney = 0.4083x2-0.873x + 0.8530.92
Kingdom of BahrainBahrainy = 1.4455x2-2.113x + 1.12620.98
State of KuwaitKuwaity = 0.7088x2-1.3237x + 0.82990.96
SpainAlmeriay = 1.9414x2-2.9329x + 1.36370.98
PortugalFaroy = 0.9184x2-2.2173x + 1.36540.99
Lisbony = 0.0721x2-1.3001x + 1.12460.99
SpainMadridy = 0.9087x2-2.0465x + 1.18080.99
Gironay = 0.1781x2-1.0867x + 0.8870.98
United KingdomCamborney = 0.8188x2-1.1127x + 0.93650.96
Bracknelly = 1.4394x2-1.5414x + 0.98780.97
Aberporthy = 0.9797x2-1.3032x + 0.94030.95
Finningleyy = 0.451x2-0.876x + 0.92670.99
Stornowayy = 7441x2-1.1382x + 1.01470.98

Three points are worthy of note from Figs. 8, 9 and 10:

  1. A strong correlation is observed between and with the respective coefficient of determination of 0.87, 0.80 and 0.83 (corresponding values of coefficient of correlation are 0.93, 0.89 and 0.91),
  2. In each case the shape of the regressed curve is concave, contrary to the convex profile for hour-by-hour regressions reported by research teams from around the world, and
  3. it is not possible to produce a single regressed curve for worldwide locations.

The latter point is reinforced via Fig. 11.

Fig. 11 

Monthly-averaged hourly regressions for three ranges of latitudes


Monthly-averaged daily global irradiation data are now easily available from NASA website for any terrestrial location. Using established models such as those presented by Liu and Jordan (Liu & Jordan ), Collares-Pereira and Rabl (Collares-Pereira & Rabl ), Mani and Rangarajan (Mani & Rangarajan ), Muneer and Saluja (Muneer & Saluja ) and Lloyd (Lloyd ) it is then possible to decompose the daily–to averaged-hourly global irradiation. The missing link so far has been hourly averaged diffuse irradiation. The authors report a regression model to complete the above missing link for 14 world locations and show that the averaged–data regressions are distinctly different from previously available hour-by-hour regressions.