# Calculation of daily, monthly, and yearly sums and averages

When a time series of radiation measurements is available, it is common practice to calculate the average of the radiation or the sum over a period. These are a priori simple operations. Actually, they are made complex by conventions in meteorology or by the possible absence of measurements in the series of measurements. Time averages must take these conventions and gaps in measurements into account. This is the subject of this section. Asking how to do it naturally leads to discussing the completion of an incomplete series of measurements.

The absence of measurements, called gaps, can be due to various factors, such as a temporary inability of the instrument, a communication fault, or a rejection of the measurement because it does not pass a quality check. A series of measurements having no gap is said to be complete. Otherwise, it is said to be incomplete. This situation very often prevails in the case of measurements made by instruments on the ground.

## Ideal case of a complete time series of measurements

Mathematically, an average is the sum of the elements divided by the number of elements. Suppose a time series of measurements, for example, hourly irradiations measured with a sampling step of 1 h, for 24h, from 01:00 to 24:00, knowing that the time stamp corresponds to the end of the duration of integration. The hourly irradiation is noted H/wur Suppose the 24 hourly values pass the quality control satisfactorily and are declared valid, in which case the time series is complete with 24 hourly values. The daily irradiation H(taY is the sum of these 24 values. The daily average of the hourly irradiation, noted Hc/ajiy_mciin jwur, is equal to this sum divided by 24. The daily average of irra- diance Е(/<п, is equal to the daily irradiation H()ay divided by 86,400s. For a given day j,

In the following, I assume that all measurements in the series are valid; i.e., the series is complete.

The monthly irradiation Hnumli,(m) is the integral of the radiation over a given month m. It is equal to the sum of the daily irradiations Hjay(j) for that month:

The monthly average of the daily irradiation H,m>mhiy_mean_m) is the average of all daily irradiations for that month. It can also be designated by the equivalent term: average daily irradiation for the month. If N(m) is the number of days in the month m, Hmonthiy_mean_day(m) is given by:

The monthly mean of irradiance Emonth(m) is given by:

In studies on climate change, it is common not to take into account the measurements taken on 29 February in leap years in the calculation of monthly quantities in order to better compare sums and averages of February between them.

The annual irradiation Hyear(m) is the integral of the radiation over a given year y. It is equal to the sum of the daily irradiations Hjay(j) for that year:

The annual average of the daily irradiation Haniwaj_meanciay(y) is the average of all daily irradiations for that year. It can also be designated by the equivalent term: average daily irradiation for the year. If N(y) is the number of days in the year y, _day{y) is given by:

The annual mean of irradiance Eyear(y) is given by:

In studies on climate change, as the measurements taken on 29 February in leap years are not taken into account, the number of days is the year is always 365. Thus, annual sums and averages can be compared between years.

In meteorology, the World Meteorological Organization recommends proceeding slightly differently when it comes to calculating annual sums and averages. The annual irradiation is the sum of the 12 monthly irradiations for that year and not the sum of the 365 (or 366) daily irradiations:

The annual average of the daily irradiation is the average of the 12 monthly averages of daily irradiation and not the average of 365 daily irradiations:

In doing so, all months have an equal weight in the sum or average regardless of its number of days. Thus, the results will differ depending on how you proceed.

## Calculation of climate normals in meteorology

When several years of measurements are available, multi-year averages can be calculated on a monthly or annual basis. Meteorologists have defined the notion of climatological normals or climate normals. For solar radiation, climate normals are annual or monthly averages of daily irradiations averaged over several years. Climate normals have two major purposes: They form a reference against which measured or estimates radiation can be assessed, and they are widely used as an indicator of the solar radiation likely to be experienced at a given location. A technical document of the World Meteorological Organization[1] details how to calculate climate normals for various weather variables and how to use them.

Before going any further, allow me to introduce some notations on the representation of the time intervals according to ISO 8601 standard on dates and times. A time interval, or a period, is defined by a start time and an end time. For dates, ISO 8601 representation of a period is YYYY-MM-DDIYYYY-MM-DD. For example, 2011-01- 01/2020-12-31 means a time interval starting on 1 January 2011 and ending on 31 December 2020, included. A duration is represented by the letter P followed by date and time: PYYYY-MM-DDThh:mm:ss. In this example, the duration is lOyears and can be represented by P0010-00-00 or in short P0010 or P10Y. The period in the previous example may be represented by the start date followed by the duration, such as 2011- 01-01/P10Y, or by the duration followed by the end date, such as P10Y/2020-12-31.

The World Meteorological Organization actually defines several normals that differ only in the number of years. The period averages are averages computed over any period of at least lOyears starting on 1 January of a year ending with the digit 1. An example is the period ranging from 2011-01-01 up to 2020-12-31 (2011-01-01/2020-12-31) of duration lOyears (P10Y). Another example is 2001-01-01/2013-12-31 of duration 13 years (PI3Y).

Climate normals are period averages computed over a period comprising at least three consecutive 10-year periods, e.g., 30, 40, or 50 years. An example is the period 2001-01-01/2030-12-31 of duration 30 years (P30Y); another one is 1961-01-01/2020-12-31 of duration 60years (P60Y).

Climatological standard normals are averages computed over specific periods of 30 consecutive years (P30Y). These periods overlap and are updated every decade. Examples are 1961-01-01/1990-12-31, 1971-01-01/2000-12-31, 1981-01-01/2010-12-31, 1991-01- 01/2020-12-31, and 2001-01-01/2030-12-31. Thirty years was the number of years having available data when the recommendation on calculation of normal was first made, and this duration P30Y has been kept over time to allow comparisons. Although it is not a formal name given by the World Meteorological Organization, the climatological standard normals for the period 1961-01-01/1990-12-31 are also called climatological standard reference normals. At the time of writing, the current climatological standard normals are for the period 1981-01-01/2010-12-31.

Normals are calculated from monthly values and not individual daily values. Monthly values may be sums (monthly irradiation) or averages (monthly mean of daily irradiation). A monthly normal for the month m for a period P is the average of all the monthly irradiations or monthly means for this month in this period. For example, the July normal in the current climatological standard normals is the average of the 30 monthly irradiations for July or monthly means in the period 1981-01-01/2010-12-31.

Annual normals are calculated by summing or averaging monthly normals and not from individual annual quantities.

• [1] World Meteorological Organization WMO No. 1203, Guidelines on the Calculation of Climate Normals, 2017, 29 p, Geneva, Switzerland.