Time mean over area fractions which vary with time
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Description (last modified by taylor13)
Following a discussion on the mailing list, I'd like to propose adding a new example to the CF Convention document to illustrate the use of cell_methods to specify different mean quantities when using a mask which is time varying (e.g. sea_ice). The qualifier where has been introduced into the cell_methods to specify masked spatial operations, e.g. area: mean where sea_ice to represent a spatial mean over sea ice. The current convention does not explicitly comment on whether the where construct can be used with other dimensions. For the CMIP6 data request there is a requirement to specify the temporal mean of quantities averaged over sea ice, and the spatial extent of the sea ice is generally varying in time.
The proposal is to make it clear that use of where for non-spatial dimensions is allowed by adding examples in section 7. It is also necessary to provide these examples to clarify the subtle differences implied by different formulations of the cell_methods statement.
1. Replace text of section 7.3.3 (first 3 paragraphs)
By default, the statistical method indicated by cell_methods is assumed to have been evaluated over the entire horizontal area of the cell. Sometimes, however, it is useful to limit consideration to only a portion of a cell (e.g. a mean over the sea-ice area). The portion concerned is constant in time in some cases, but it could be time-varying. Grid cell “portions” that can be considered are only those permitted to be associated with the standard_name of area_type. There are two options for indicating when a quantity represents a portion of a cell.
The first method can be used for the common case that the cell_method applies to a single area-type. In this case, the cell_methods attribute may include a string of the form name: method where type. Here name could, for example, be area and type may be any of the strings permitted for a variable with a standard_name of area_type. As an example, if the method is area: mean where sea_ice, then the data would represent a mean over only the sea ice portion of the grid cell. When this first option is adopted, none of the variables in the netCDF file should be given a name identical to the string that names the area_type. This restriction is imposed so that it will be clear that the metadata should not be interpreted following the second option (described in the next paragraph), which takes precedence.
The second method for indicating that a statistic applies to only a portion of a cell is more general because it can reference multiple area-types. This may be needed when a variable has a dimension that ranges across various area types. In this case, the cell_methods entry is of the form name: method where typevar. Here typevar is a string-valued auxiliary coordinate variable or string-valued scalar coordinate variable (see Section 6.1, "Labels") with a standard_name of area_type. The variable typevar contains the name(s) of the selected portion(s) of the grid cell to which the method is applied. This method provides a convenient way to store output from land surface models, for example, since they deal with many area types within each surface grid cell (e.g., vegetation, bare_ground, snow, etc.).
2. Caption of example 7.6
If the method is mean, various ways of calculating the mean can be distinguished in the cell_methods attribute with a string of the form mean where type1 [over type2]. Here, type1 can be any of the possibilities allowed for typevar or type (as specified in the two paragraphs preceding above Example). The same options apply to type2, except it is not allowed to be the name of an auxiliary coordinate variable with a dimension greater than one (ignoring the dimension accommodating the maximum string length).
A cell_methods attribute with a string of the form area: mean where type1 over type2 indicates the mean is calculated by integrating over the type1 portion of the cell and dividing by the area of the type2 portion. When over type2 is omitted, it is assumed to be the same as type1.
3. Addtional text for masks which vary over additional dimensions (e.g. time) .. form proposed by Karl
When the “where” construct is used, and when “area” is not the only “dimension” to which it applies, the interpretation more generally is that a “weighted” mean is reported. Specifically, the quantity of interest is integrated over the additional dimension(s) with weights proportional to the fraction of “type1” area_type that exists, and then this is divided by the integral over the same dimension(s) of the fraction of “type2” area_type that exists. [Note that certain variables might be undefined if the fraction of the area_type considered is 0; for example the temperature of sea ice is not defined if there is no sea ice. In this case, a time-mean value can still be computed for cells containing some sea ice during at least a portion of the averaging interval because no matter what the value assumed for temperature when sea ice is missing, those values are given zero weight in computing the time-mean.]
Note that "`all_area_types” is one of the valid strings permitted for a variable with the standard_name area_type, so a cell_methods string of the form “area: mean over type1 where all_area_types” indicates the mean is calculated by integrating over the type1 portion of the grid cell and dividing by the entire area of the grid cell.
The following three examples illustrate cases when one might want to use “where” or “where … over” in defining the cell_methods:
- Suppose that in a grid cell the fractional sea ice varies over time, but there is interest in the time-mean surface temperature of the sea ice. The time-samples, each representing a spatially-averaged sea ice temperature can be summed and then divided by the number of samples to obtain an unweighted mean where sea ice exists. This would be indicated with:
cell_methods = “area: mean where sea_ice time: mean”
- Suppose there is interest in recording the mean fractional area covered by sea ice and the mean sea ice thickness in such a way that their product would equal the time-mean volume of sea ice in each grid cell. In this case the sea ice area would be reported as an unweighted time-mean, while the mean sea ice thickness would be calculated with time samples weighted by the fractional area of sea ice. Thus, for sea ice thickness:
cell_methods = “area: time: mean where sea_ice”
- Suppose the time-mean contributions to total heat flux from different portions of a grid cell (e.g., ice-free and ice-covered) are of interest, and there are reasons to report these in such a way that the total heat flux is the sum of the individual contributions. Then the cell_methods attribute would be defined:
cell_methods=”area: mean where sea_ice over all_area_types time: mean”
In some cases a variable referencing a specific area_type will actually be defined even in the absence of that area_type (i.e., over the entire grid cell). Consider the surface_snow_thickness, which could sensibly be considered to be 0 in the absence of snow. In this case one might in some instances want to report “area: time: mean where snow” (giving a measure of the typical snow depth when snow exists) and in other instances “area: time: mean where snow over all_area_types” (which in this case would be identical to “area: time: mean”) or “area: time: mean where snow over land”.
4. Clarification at start of section 7.3.3 (not needed if above is accepted)
Add a clarification after this sentence in the first paragraph of 7.3.3 "Sometimes, however, it is useful to limit consideration to only a portion of a cell (e.g. a mean over the sea-ice area)", to introduce the idea of time-varying area fractions:
The portion concerned is constant in time in many cases, but it could be time-varying.
5. New example for time-varying area fractions
The following new example and explanatry text should be added in section 7.3.3:
Example 7.8: Time mean over area fractions which vary with time
float simple_mean(lat,lon): simple_mean:cell_methods: area: mean where sea_ice time: mean float weighted_mean(lat,lon): weighted_mean:cell_methods: area: time: mean where sea_ice float partial_mean(lat,lon): partial_mean:cell_methods: area: mean where sea_ice over sea time: mean
When the area fraction is varying with time, there are several different ways in which a time mean can be formulated. Three of these are illustrated in this example. Suppose, for instance, we are averaging over three time steps and the data at one grid point is -10, -6, -2 with area fractions .75, .50, .25. The values of the simple_mean, weighted_mean and partial mean are, respectively, (-10 -6 -2)/3 = -6, (-10*.75 - 6*.5 -2*.25)/(.75+.5+.25) = -7.33 , and (-10*.75 - 6*.5 -2*.25)/3 = -3.667. The partial mean provides the contribution to the mean over the entire grid cell from a specified area type. The simple mean is weighting each time period equally, while the weighted mean provides equal weighting to each unit area of sea_ice.
In example 7.8, time could be replaced by any other coordinate over which an average is taken, such as an ensemble index.