Climatological statistics may be derived from corresponding portions of the annual cycle in a set of years, e.g., the average January temperatures in the climatology of 1961-1990, where the values are derived by averaging the 30 Januarys from the separate years. Portions of the climatological cycle are specified by references to dates within the calendar year. However, a calendar year is not a well-defined unit of time, because it differs between leap years and other years, and among calendars. Nonetheless for practical purposes we wish to compare statistics for months or seasons from different calendars, and to make climatologies from a mixture of leap years and other years. Hence we provide special conventions for indicating dates within the climatological year. Climatological statistics may also be derived from corresponding portions of a range of days, for instance the average temperature for each hour of the average day in April 1997. In addition the two concepts may be used at once, for instance to indicate not April 1997, but the average April of the five years 1995-1999.

Climatological variables have a climatological time axis. Like an ordinary time axis, a climatological time axis may have a dimension of unity (for example, a variable containing the January average temperatures for 1961-1990), but often it will have several elements (for example, a climatological time axis with a dimension of 12 for the climatological average temperatures in each month for 1961-1990, a dimension of 3 for the January mean temperatures for the three decades 1961-1970, 1971-1980, 1981-1990, or a dimension of 24 for the hours of an average day). Intervals of climatological time are conceptually different from ordinary time intervals; a given interval of climatological time represents a set of subintervals which are not necessarily contiguous. To indicate this difference, a climatological time coordinate variable does not have a `bounds` attribute. Instead, it has a `climatology` attribute, which names a variable with dimensions (n,2), n being the dimension of the climatological time axis. Using the units and calendar of the time coordinate variable, element (i,0) of the climatology variable specifies the beginning of the first subinterval and element (i,1) the end of the last subinterval used to evaluate the climatological statistics with index i in the time dimension. The time coordinates should be values that are representative of the climatological time intervals, such that an application which does not recognise climatological time will nonetheless be able to make a reasonable interpretation.

The COARDS standard offers limited support for climatological time. For compatibility with COARDS, time coordinates should also be recognised as climatological if they have a `units` attribute of time-units relative to midnight on 1 January in year 0 i.e. ```since 0-1-1``` in udunits syntax , and provided they refer to the real-world calendar. We do not recommend this convention because (a) it does not provide any information about the intervals used to compute the climatology, and (b) there is no standard for how dates since year 1 will be encoded with units having a reference time in year 0, since this year does not exist; consequently there may be inconsistencies among software packages in the interpretation of the time coordinates. Year 0 may be a valid year in non-real-world calendars, and therefore cannot be used to signal climatological time in such cases.

A climatological axis may use different statistical methods to represent variation among years, within years and within days. For example, the average January temperature in a climatology is obtained by averaging both within years and over years. This is different from the average January-maximum temperature and the maximum January-average temperature. For the former, we first calculate the maximum temperature in each January, then average these maxima; for the latter, we first calculate the average temperature in each January, then find the largest one. As usual, the statistical operations are recorded in the `cell_methods` attribute, which may have two or three entries for the climatological time dimension.

Valid values of the `cell_methods` attribute must be in one of the forms from the following list. The intervals over which various statistical methods are applied are determined by decomposing the date and time specifications of the climatological time bounds of a cell, as recorded in the variable named by the `climatology` attribute. (The date and time specifications must be calculated from the time coordinates expressed in units of "time interval since reference date and time".) In the descriptions that follow we use the abbreviations y, m, d, H, M, and S for year, month, day, hour, minute, and second respectively. The suffix 0 indicates the earlier bound and 1 the latter.

time: method1 `within years`   time: method2 `over years`

method1 is applied to the time intervals (mdHMS0-mdHMS1) within individual years and method2 is applied over the range of years (y0-y1).

time: method1 `within days`   time: method2 `over days`

method1 is applied to the time intervals (HMS0-HMS1) within individual days and method2 is applied over the days in the interval (ymd0-ymd1).

time: method1 `within days`   time: method2 `over days`   time: method3 ```over years```

method1 is applied to the time intervals (HMS0-HMS1) within individual days and method2 is applied over the days in the interval (md0-md1), and method3 is applied over the range of years (y0-y1).

The methods which can be specified are those listed in Appendix E, Cell Methods and each entry in the `cell_methods` attribute may also, as usual, contain non-standardised information in parentheses after the method. For instance, a mean over ENSO years might be indicated by "```time: mean over years (ENSO years)```".

When considering intervals within years, if the earlier climatological time bound is later in the year than the later climatological time bound, it implies that the time intervals for the individual years run from each year across January 1 into the next year e.g. DJF intervals run from December 1 0:00 to March 1 0:00. Analogous situations arise for daily intervals running across midnight from one day to the next.

When considering intervals within days, if the earlier time of day is equal to the later time of day, then the method is applied to a full 24 hour day.

We have tried to make the examples in this section easier to understand by translating all time coordinate values to date and time formats. This is not currently valid CDL syntax.

Example 7.8. Climatological seasons

This example shows the metadata for the average seasonal-minimum temperature for the four standard climatological seasons MAM JJA SON DJF, made from data for March 1960 to February 1991.

```dimensions:
time=4;
nv=2;
variables:
float temperature(time,lat,lon);
temperature:long_name="surface air temperature";
temperature:cell_methods="time: minimum within years time: mean over years";
temperature:units="K";
double time(time);
time:climatology="climatology_bounds";
time:units="days since 1960-1-1";
double climatology_bounds(time,nv);
data:  // time coordinates translated to date/time format
time="1960-4-16", "1960-7-16", "1960-10-16", "1961-1-16" ;
climatology_bounds="1960-3-1",  "1990-6-1",
"1960-6-1",  "1990-9-1",
"1960-9-1",  "1990-12-1",
"1960-12-1", "1991-3-1" ;
```

Example 7.9. Decadal averages for January

Average January precipitation totals are given for each of the decades 1961-1970, 1971-1980, 1981-1990.

```dimensions:
time=3;
nv=2;
variables:
float precipitation(time,lat,lon);
precipitation:long_name="precipitation amount";
precipitation:cell_methods="time: sum within years time: mean over years";
precipitation:units="kg m-2";
double time(time);
time:climatology="climatology_bounds";
time:units="days since 1901-1-1";
double climatology_bounds(time,nv);
data:  // time coordinates translated to date/time format
time="1965-1-15", "1975-1-15", "1985-1-15" ;
climatology_bounds="1961-1-1", "1970-2-1",
"1971-1-1", "1980-2-1",
"1981-1-1", "1990-2-1" ;

```

Example 7.10. Temperature for each hour of the average day

Hourly average temperatures are given for April 1997.

```dimensions:
time=24;
nv=2;
variables:
float temperature(time,lat,lon);
temperature:long_name="surface air temperature";
temperature:cell_methods="time: mean within days time: mean over days";
temperature:units="K";
double time(time);
time:climatology="climatology_bounds";
time:units="hours since 1997-4-1";
double climatology_bounds(time,nv);
data:  // time coordinates translated to date/time format
time="1997-4-1 0:30", "1997-4-1 1:30", ... "1997-4-1 23:30" ;
climatology_bounds="1997-4-1 0:00",  "1997-4-30 1:00",
"1997-4-1 1:00",  "1997-4-30 2:00",
...
"1997-4-1 23:00", "1997-5-1 0:00" ;

```

Example 7.11. Extreme statistics and spell-lengths

Number of frost days during NH winter 2007-2008, and maximum length of spells of consecutive frost days. A "frost day" is defined as one during which the minimum temperature falls below freezing point (0 degC). This is described as a climatological statistic, in which the minimum temperature is first calculated within each day, and then the number of days or spell lengths meeting the specified condition are evaluated. In this operation, the standard name is also changed; the original data are `air_temperature `.

```variables:
float n1(lat,lon);
n1:standard_name="number_of_days_with_air_temperature_below_threshold";
n1:coordinates="threshold time";
n1:cell_methods="time: minimum within days time: sum over days";
float n2(lat,lon);
n2:standard_name="spell_length_of_days_with_air_temperature_below_threshold";
n2:coordinates="threshold time";
n2:cell_methods="time: minimum within days time: maximum over days";
float threshold;
threshold:standard_name="air_temperature";
threshold:units="degC";
double time;
time:climatology="climatology_bounds";
time:units="days since 2000-6-1";
double climatology_bounds(time,nv);
data: // time coordinates translated to date/time format
time="2008-1-16 6:00";
climatology_bounds="2007-12-1 6:00", "2000-8-2 6:00";
threshold=0.;

```

Example 7.12. Temperature for each hour of the typical climatological day

This is a modified version of the previous example, "Temperature for each hour of the average day". It now applies to April from a 1961-1990 climatology.

```variables:
float temperature(time,lat,lon);
temperature:long_name="surface air temperature";
temperature:cell_methods="time: mean within days ",
"time: mean over days time: mean over years";
temperature:units="K";
double time(time);
time:climatology="climatology_bounds";
time:units="days since 1961-1-1";
double climatology_bounds(time,nv);
data:  // time coordinates translated to date/time format
time="1961-4-1 0:30", "1961-4-1 1:30", ..., "1961-4-1 23:30" ;
climatology_bounds="1961-4-1 0:00", "1990-4-30 1:00",
"1961-4-1 1:00", "1990-4-30 2:00",
...
"1961-4-1 23:00", "1990-5-1 0:00" ;
```

Example 7.13. Monthly-maximum daily precipitation totals

Maximum of daily precipitation amounts for each of the three months June, July and August 2000 are given. The first daily total applies to 6 a.m. on 1 June to 6 a.m. on 2 June, the 30th from 6 a.m. on 30 June to 6 a.m. on 1 July. The maximum of these 30 values is stored under time index 0 in the precipitation array.

```dimensions:
time=3;
nv=2;
variables:
float precipitation(time,lat,lon);
precipitation:long_name="Accumulated precipitation";
precipitation:cell_methods="time: sum within days time: maximum over days";
precipitation:units="kg";
double time(time);
time:climatology="climatology_bounds";
time:units="days since 2000-6-1";
double climatology_bounds(time,nv);
data:  // time coordinates translated to date/time format
time="2000-6-16", "2000-7-16", "2000-8-16" ;
climatology_bounds="2000-6-1 6:00:00", "2000-7-1 6:00:00",
"2000-7-1 6:00:00", "2000-8-1 6:00:00",
"2000-8-1 6:00:00", "2000-9-1 6:00:00" ;
```