For the most part, the mathematics of solar radiation is pretty straightforward. However, there are a couple of situations that are less intuitive than they first appear. This is definitely true in the calculation of average hourly incident solar radiation (insolation).

## Calculating Averages

A typical average is simply the cumulative sum divided by the total number of samples taken. Thus, if we have five numbers, their average is given by adding them all up and dividing the result by five. Using this approach, the average hourly solar radiation for a given day would simply be the total cumulative incident solar radiation over the day divided by 24.

To illustrate one reason why this is not appropriate for solar radiation, imagine we calculated these averages for the summer and winter solstices in San Francisco, and then compared them. The results of this, given in Table 1, show that the cumulative insolation on a horizontal surface during the summer solstice is 8.09 kWh/m^{2} compared to only 1.37 kWh/m^{2} on the winter solstice. This would give average hourly insolation values of 337.1 W/m^{2} and 56.9 W/m^{2} respectively.

21st Jun | 21st Dec | |
---|---|---|

HOUR | (W/m^{2}) |
(W/m^{2}) |

00:00 | 0 | 0 |

01:00 | 0 | 0 |

02:00 | 0 | 0 |

03:00 | 0 | 0 |

04:00 | 0 | 0 |

05:00 | 0 | 0 |

06:00 | 0 | 41 |

07:00 | 0 | 208 |

08:00 | 16 | 434 |

09:00 | 66 | 611 |

10:00 | 96 | 761 |

11:00 | 189 | 915 |

12:00 | 132 | 978 |

13:00 | 118 | 969 |

14:00 | 280 | 952 |

15:00 | 271 | 826 |

16:00 | 192 | 646 |

17:00 | 0 | 478 |

18:00 | 0 | 270 |

19:00 | 0 | 0 |

20:00 | 0 | 0 |

21:00 | 0 | 0 |

22:00 | 0 | 0 |

23:00 | 0 | 0 |

TOTALS | 1360 | 8088 |

At this point, it is entirely understandable for an observer to believe that the differences between these two values are mainly due to a lower solar altitude and a greater likelihood of cloudiness in winter. After all, the entire reason for doing such a comparison is most likely to develop a better understanding of the effects of these two climatic characteristics.

## Variation in Daylight Hours

In fact, a major contributor to their difference is seasonal variation in daylight hours. In summer, the Sun is above the horizon in San Francisco from 6am through to 6pm, a total of 12 full daylight hours. In winter it is above the horizon from 8am until 4pm, a total of only 8 full hours. Thus, the winter hourly average is only 16.9% of the summer average.

However, if we count only the hours when the Sun was actually above the horizon in each case, we get average hourly insolation values of 622.2 W/m^{2} in summer and 151.8 W/m^{2} in winter. Now the winter hourly average is 24.4% of the summer average.

The issue at this point is which of these two approaches is more meaningful. To answer this, consider a further situation in which a restricted time range is applied over which the insolation is calculated. If a time range of 2am to 10pm was selected instead of midnight to midnight the next day, the mathematical average of the two would be quite different. In the case of the example summer solstice in San Francisco, 404.5 W/m^{2} instead of 337.1 W/m^{2}.

However, this arbitrary difference doesn't really make sense. As both time ranges include all of the available daylight hours, the average hourly incident solar radiation should be a characteristic of that particular day. Obviously if we restrict the time range to just the afternoon, for example, we would expect a different hourly average. However, this afternoon result shouldn't really be different if we selected a range of 12pm to 8pm instead of 12pm to midnight.

## Averaging Hourly Solar Radiation

Thus, when averaging hourly incident solar radiation, the convention is to use what is effectively the average insolation you would expect per hour of solar exposure. This means averaging only over those hours when the sum of direct and diffuse solar radiation is greater than zero. Thus, if a 5 day period is chosen , and assuming 11 hours of daylight, the average should be calculated by dividing the total cumulative value by those 55 hours that the surface was actually exposed to solar radiation, rather than the full 120 total hours which include periods of night-time.

It is important to reiterate that any exposed surface, irrespective of its tilt angle or orientation, will receive some level of diffuse solar radiation even if it is facing in the exact opposite direction to the direct Sun. Thus, rather than using calculated sunrise/sunset times or solar altitude to determine daylight hours, the simplest method is to sum only those hours when the sum of the measured direct and diffuse solar radiation values in the weather data being used is greater than zero.