NetCDF Climate and Forecast (CF) Metadata Conventions

The NetCDF library [NetCDF] is designed to read and write data that has been structured according to well-defined rules and is easily ported across various computer platforms. The netCDF interface enables but does not require the creation of self-describing datasets. The purpose of the CF conventions is to require conforming datasets to contain sufficient metadata that they are self-describing in the sense that each variable in the file has an associated description of what it represents, including physical units if appropriate, and that each value can be located in space (relative to earth-based coordinates) and time.

An important benefit of a convention is that it enables software tools to display data and perform operations on specified subsets of the data with minimal user intervention. It is possible to provide the metadata describing how a field is located in time and space in many different ways that a human would immediately recognize as equivalent. The purpose in restricting how the metadata is represented is to make it practical to write software that allows a machine to parse that metadata and to automatically associate each data value with its location in time and space. It is equally important that the metadata be easy for human users to write and to understand.

This standard is intended for use with climate and forecast data, for atmosphere, surface and ocean, and was designed with model-generated data particularly in mind. We recognise that there are limits to what a standard can practically cover; we restrict ourselves to issues that we believe to be of common and frequent concern in the design of climate and forecast metadata. Our main purpose therefore, is to propose a clear, adequate and flexible definition of the metadata needed for climate and forecast data. Although this is specifically a netCDF standard, we feel that most of the ideas are of wider application. The metadata objects could be contained in file formats other than netCDF. Conversion of the metadata between files of different formats will be facilitated if conventions for all formats are based on similar ideas.

This convention is designed to be backward compatible with the COARDS conventions [COARDS], by which we mean that a conforming COARDS dataset also conforms to the CF standard. Thus new applications that implement the CF conventions will be able to process COARDS datasets.

We have also striven to maximize conformance to the COARDS standard, that is, wherever the COARDS metadata conventions provide an adequate description we require their use. Extensions to COARDS are implemented in a manner such that the content that doesn’t depend on the extensions is still accessible to applications that adhere to the COARDS standard.

The following principles are followed in the design of these conventions:

The terms in this document that refer to components of a netCDF file are defined in the NetCDF User’s Guide (NUG) [NUG] NUG. Some of those definitions are repeated below for convenience.

No variable or dimension names are standardized by this convention. Instead we follow the lead of the NUG and standardize only the names of attributes and some of the values taken by those attributes. Variable or dimension names can either be a single variable name or a path to a variable. The overview provided in this section will be followed with more complete descriptions in following sections. Appendix A, Attributes contains a summary of all the attributes used in this convention.

Files using this version of the CF Conventions must set the NUG defined attribute Conventions to contain the string value "CF-1.11" to identify datasets that conform to these conventions.

The general description of a file’s contents should be contained in the following attributes: title, history, institution, source, comment and references (Section 2.6.2, "Description of file contents"). For backwards compatibility with COARDS none of these attributes is required, but their use is recommended to provide human readable documentation of the file contents.

Each variable in a netCDF file has an associated description which is provided by the attributes units, long_name, and standard_name. The units, and long_name attributes are defined in the NUG and the standard_name attribute is defined in this document.

The units attribute is required for all variables that represent dimensional quantities (except for boundary variables defined in Section 7.1, "Cell Boundaries". The values of the units attributes are character strings that are recognized by UNIDATA’s UDUNITS package [UDUNITS], (with exceptions allowed as discussed in Section 3.1, "Units").

The long_name and standard_name attributes are used to describe the content of each variable. For backwards compatibility with COARDS neither is required, but use of at least one of them is strongly recommended. The use of standard names will facilitate the exchange of climate and forecast data by providing unambiguous identification of variables most commonly analyzed.

Four types of coordinates receive special treatment by these conventions: latitude, longitude, vertical, and time. Every variable must have associated metadata that allows identification of each such coordinate that is relevant. Two independent parts of the convention allow this to be done. There are conventions that identify the variables that contain the coordinate data, and there are conventions that identify the type of coordinate represented by that data.

There are two methods used to identify variables that contain coordinate data. The first is to use the NUG-defined "coordinate variables." The use of coordinate variables is required for all dimensions that correspond to one dimensional space or time coordinates. In cases where coordinate variables are not applicable, the variables containing coordinate data are identified by the coordinates attribute.

Once the variables containing coordinate data are identified, further conventions are required to determine the type of coordinate represented by each of these variables. Latitude, longitude, and time coordinates are identified solely by the value of their units attribute. Vertical coordinates with units of pressure may also be identified by the units attribute. Other vertical coordinates must use the attribute positive which determines whether the direction of increasing coordinate value is up or down. Because identification of a coordinate type by its units involves the use of an external package [UDUNITS], we provide the optional attribute axis for a direct identification of coordinates that correspond to latitude, longitude, vertical, or time axes.

Latitude, longitude, and time are defined by internationally recognized standards, and hence, identifying the coordinates of these types is sufficient to locate data values uniquely with respect to time and a point on the earth’s surface. On the other hand identifying the vertical coordinate is not necessarily sufficient to locate a data value vertically with respect to the earth’s surface. In particular a model may output data on the dimensionless vertical coordinate used in its mathematical formulation. To achieve the goal of being able to spatially locate all data values, this convention includes the definitions of common dimensionless vertical coordinates in Appendix D, Parametric Vertical Coordinates. These definitions provide a mapping between the dimensionless coordinate values and dimensional values that can be uniquely located with respect to a point on the earth’s surface. The definitions are associated with a coordinate variable via the standard_name and formula_terms attributes. For backwards compatibility with COARDS use of these attributes is not required, but is strongly recommended.

It is often the case that data values are not representative of single points in time and/or space, but rather of intervals or multidimensional cells. This convention defines a bounds attribute to specify the extent of intervals or cells. When data that is representative of cells can be described by simple statistical methods, those methods can be indicated using the cell_methods attribute. An important application of this attribute is to describe climatological and diurnal statistics.

Methods for reducing the total volume of data include both packing and compression. Packing reduces the data volume by reducing the precision of the stored numbers. It is implemented using the attributes add_offset and scale_factor which are defined in the NUG. Compression on the other hand loses no precision, but reduces the volume by not storing missing data. The attribute compress is defined for this purpose.

These conventions generalize and extend the COARDS conventions [COARDS]. A major design goal has been to maintain backward compatibility with COARDS. Hence applications written to process datasets that conform to these conventions will also be able to process COARDS conforming datasets. We have also striven to maximize conformance to the COARDS standard so that datasets that only require the metadata that was available under COARDS will still be able to be processed by COARDS conforming applications. But because of the extensions that provide new metadata content, and the relaxation of some COARDS requirements, datasets that conform to these conventions will not necessarily be recognized by applications that adhere to the COARDS conventions. The features of these conventions that allow writing netCDF files that are not COARDS conforming are summarized below.

COARDS standardizes the description of grids composed of independent latitude, longitude, vertical, and time axes. In addition to standardizing the metadata required to identify each of these axis types COARDS restricts the axis (equivalently dimension) ordering to be longitude, latitude, vertical, and time (with longitude being the most rapidly varying dimension). Because of I/O performance considerations it may not be possible for models to output their data in conformance with the COARDS requirement. The CF convention places no rigid restrictions on the order of dimensions, however we encourage data producers to make the extra effort to stay within the COARDS standard order. The use of non-COARDS axis ordering will render files inaccessible to some applications and limit interoperability. Often a buffering operation can be used to miminize performance penalties when axis ordering in model code does not match the axis ordering of a COARDS file.

COARDS addresses the issue of identifying dimensionless vertical coordinates, but does not provide any mechanism for mapping the dimensionless values to dimensional ones that can be located with respect to the earth’s surface. For backwards compatibility we continue to allow (but do not require) the units attribute of dimensionless vertical coordinates to take the values "level", "layer", or "sigma_level." But we recommend that the standard_name and formula_terms attributes be used to identify the appropriate definition of the dimensionless vertical coordinate (see Section 4.3.2, "Dimensionless Vertical Coordinate").

The CF conventions define attributes which enable the description of data properties that are outside the scope of the COARDS conventions. These new attributes do not violate the COARDS conventions, but applications that only recognize COARDS conforming datasets will not have the capabilities that the new attributes are meant to enable. Briefly the new attributes allow:

These conventions implicitly incorporate parts of the UGRID conventions for storing unstructured (or flexible mesh) data in netCDF files using mesh topologies [UGRID]. Only version 1.0 of the UGRID conventions is allowed. The UGRID conventions description is referenced from, rather than rewritten into, this document and the canonical description of how to store mesh topologies is only to be found at [UGRID]. A summary indicating how UGRID relates to other parts of the CF conventions, and which features of UGRID are excluded from CF, can be found in Section 5.9, "Mesh Topology Variables". To reduce the chance of ambiguities arising from their accidental re-use, all of the UGRID standardized attributes are specified in Appendix K, Mesh Topology Attributes and Appendix A, Attributes.

The UGRID conventions have their own conformance document, which should be used in conjunction with the CF conformance document when checking the validity of datasets.

NetCDF files should have the file name extension ".nc".

Data variables must be one of the following data types: string, char, byte, unsigned byte, short, unsigned short, int, unsigned int, int64, unsigned int64, float or real, and double (which are all the netCDF external data types supported by netCDF-4). The string type is only available in files using the netCDF version 4 (netCDF-4) format. The char and string types are not intended for numeric data. One byte numeric data should be stored using the byte or unsigned byte data types. It is possible to treat the byte and short types as unsigned by using the NUG convention of indicating the unsigned range using the valid_min, valid_max, or valid_range attributes. In many situations, any integer type may be used. When the phrase "integer type" is used in this document, it should be understood to mean byte, unsigned byte, short, unsigned short, int, unsigned int, int64, or unsigned int64.

Strings in variables may be represented one of two ways - as atomic strings or as character arrays. An n-dimensional array of strings may be implemented as a variable of type string with n dimensions, or as a variable of type char with n+1 dimensions where the last (most rapidly varying) dimension is large enough to contain the longest string in the variable. For example, a character array variable of strings containing the names of the months would be dimensioned (12,9) in order to accommodate "September", the month with the longest name. The other strings, such as "May", should be padded with trailing NULL or space characters so that every array element is filled. If the atomic string option is chosen, each element of the variable can be assigned a string with a different length. The CDL example below shows one variable of each type.

The examples in this document that use string-valued variables alternate between these two forms.

Variable, dimension, attribute and group names should begin with a letter and be composed of letters, digits, and underscores. By the word letters we mean the standard ASCII letters uppercase A to Z and lowercase a to z. By the word digits we mean the standard ASCII digits 0 to 9, and similarly underscores means the standard ASCII underscore _. Note that this is in conformance with the COARDS conventions, but is more restrictive than the netCDF interface which allows almost all Unicode characters encoded as multibyte UTF-8 characters (NUG Appendix B). The netCDF interface also allows leading underscores in names, but the NUG states that this is reserved for system use.

Case is significant in netCDF names, but it is recommended that names should not be distinguished purely by case, i.e., if case is disregarded, no two names should be the same. It is also recommended that names should be obviously meaningful, if possible, as this renders the file more effectively self-describing.

This convention does not standardize any variable or dimension names. Attribute names and their contents, where standardized, are given in English in this document and should appear in English in conforming netCDF files for the sake of portability. Languages other than English are permitted for variables, dimensions, and non-standardized attributes. The content of some standardized attributes are string values that are not standardized, and thus are not required to be in English. For example, a description of what a variable represents may be given in a non-English language using the long_name attribute (see Section 3.2, "Long Name") whose contents are not standardized, but a description given by the standard_name attribute (see Section 3.3, "Standard Name") must be taken from the standard name table which is in English.

A variable may have any number of dimensions, including zero, and the dimensions must all have different names. COARDS strongly recommends limiting the number of dimensions to four, but we wish to allow greater flexibility. The dimensions of the variable define the axes of the quantity it contains. Dimensions other than those of space and time may be included. Several examples can be found in this document. Under certain circumstances, one may need more than one dimension in a particular quantity. For instance, a variable containing a two-dimensional probability density function might correlate the temperature at two different vertical levels, and hence would have temperature on both axes.

If any or all of the dimensions of a variable have the interpretations of "date or time" (T), "height or depth" (Z), "latitude" (Y), or "longitude" (X) then we recommend, but do not require (see Section 1.5, "Relationship to the COARDS Conventions"), those dimensions to appear in the relative order T, then Z, then Y, then X in the CDL definition corresponding to the file. All other dimensions should, whenever possible, be placed to the left of the spatiotemporal dimensions.

Dimensions may be of any size, including unity. When a single value of some coordinate applies to all the values in a variable, the recommended means of attaching this information to the variable is by use of a dimension of size unity with a one-element coordinate variable. It is also acceptable to use a scalar coordinate variable which eliminates the need for an associated size one dimension in the data variable. The advantage of using either a coordinate variable or an auxiliary coordinate variable is that all its attributes can be used to describe the single-valued quantity, including boundaries. For example, a variable containing data for temperature at 1.5 m above the ground has a single-valued coordinate supplying a height of 1.5 m, and a time-mean quantity has a single-valued time coordinate with an associated boundary variable to record the start and end of the averaging period.

This convention does not standardize variable names.

NetCDF variables that contain coordinate data are referred to as coordinate variables, auxiliary coordinate variables, scalar coordinate variables, or multidimensional coordinate variables.

This standard describes many attributes (some mandatory, others optional), but a file may also contain non-standard attributes. Such attributes do not represent a violation of this standard. Application programs should ignore attributes that they do not recognise or which are irrelevant for their purposes. Conventional attribute names should be used wherever applicable. Non-standard names should be as meaningful as possible. Before introducing an attribute, consideration should be given to whether the information would be better represented as a variable. In general, if a proposed attribute requires ancillary data to describe it, is multidimensional, requires any of the defined netCDF dimensions to index its values, or requires a significant amount of storage, a variable should be used instead. When this standard defines string attributes that may take various prescribed values, the possible values are generally given in lower case. However, applications programs should not be sensitive to case in these attributes. Several string attributes are defined by this standard to contain "blank-separated lists". Consecutive words in such a list are separated by one or more adjacent spaces. The list may begin and end with any number of spaces. See Appendix A, Attributes for a list of attributes described by this standard.

Groups provide a powerful mechanism to structure data hierarchically. This convention does not standardize group names. It may be of benefit to name groups in such a way that human readers can interpret them. However, files that conform to this standard shall not require software to interpret or decode information from group names. References to out-of-group variable and dimensions shall be found by applying the scoping rules outlined below.

The units attribute is required for all variables that represent dimensional quantities (except for boundary variables defined in Section 7.1, "Cell Boundaries" and climatology variables defined in Section 7.4, "Climatological Statistics"). The units attribute is permitted but not required for dimensionless quantities (see Section 3.1.1, "Dimensionless units").

The value of the units attribute is a string that can be recognized by the UDUNITS package [UDUNITS], with the exceptions that are given in Section 3.1.1, "Dimensionless units" and Section 3.1.3, "Scale factors and offsets". Note that case is significant in the units strings. Note also that CF depends on UDUNITS only for the definition of legal units strings. CF does not assume or require that the UDUNITS software will be used for units conversion. In most units conversions, the sole operation on the data is multiplication by a scale factor. Special treatment is required in converting the units of variables that involve temperature (Section 3.1.2, "Temperature units") and the units of time coordinate variables (Section 4.4, "Time Coordinate").

The COARDS convention prohibits the unit degrees altogether, but this unit is not forbidden by the CF convention because it may in fact be appropriate for a variable containing, say, solar zenith angle. The unit degrees is also allowed on coordinate variables such as the latitude and longitude coordinates of a transformed grid. In this case the coordinate values are not true latitudes and longitudes, which must always be identified using the more specific forms of degrees as described in Section 4.1, "Latitude Coordinate" and Section 4.2, "Longitude Coordinate".

The long_name attribute is defined by the NUG to contain a long descriptive name which may, for example, be used for labeling plots. For backwards compatibility with COARDS this attribute is optional. But it is highly recommended that either this or the standard_name attribute defined in the next section be provided to make the file self-describing. If a variable has no long_name attribute then an application may use, as a default, the standard_name if it exists, or the variable name itself.

A fundamental requirement for exchange of scientific data is the ability to describe precisely the physical quantities being represented. To some extent this is the role of the long_name attribute as defined in the NUG. However, usage of long_name is completely ad-hoc. For many applications it is desirable to have a more definitive description of the quantity, which allows users of data from different sources (some of which might be models and others observational) to determine whether quantities are in fact comparable. For this reason each variable may optionally be given a "standard name", whose meaning is defined by this convention. There may be several variables in a dataset with any given standard name, and these may be distinguished by other metadata, such as coordinates (Chapter 4, Coordinate Types) and cell_methods (Section 7.3, "Cell Methods").

A standard name is associated with a variable via the attribute standard_name which takes a string value comprised of a standard name optionally followed by one or more blanks and a standard name modifier (a string value from Appendix C, Standard Name Modifiers).

The set of permissible standard names is contained in the standard name table. The table entry for each standard name contains the following:

When appropriate, the table entry also contains the corresponding GRIB parameter code(s) (from ECMWF and NCEP) and AMIP identifiers.

The standard name table is located at https://cfconventions.org/Data/cf-standard-names/current/src/cf-standard-name-table.xml, written in compliance with the XML format, as described in Appendix B, Standard Name Table Format. Knowledge of the XML format is only necessary for application writers who plan to directly access the table. A formatted text version of the table is provided at https://cfconventions.org/Data/cf-standard-names/current/build/cf-standard-name-table.html, and this table may be consulted in order to find the standard name that should be assigned to a variable. Some standard names (e.g. region and area_type) are used to indicate quantities which are permitted to take only certain standard values. This is indicated in the definition of the quantity in the standard name table, accompanied by a list or a link to a list of the permitted values.

Standard names by themselves are not always sufficient to describe a quantity. For example, a variable may contain data to which spatial or temporal operations have been applied. Or the data may represent an uncertainty in the measurement of a quantity. These quantity attributes are expressed as modifiers of the standard name. Modifications due to common statistical operations are expressed via the cell_methods attribute (see Section 7.3, "Cell Methods" and Appendix E, Cell Methods). Other types of quantity modifiers are expressed using the optional modifier part of the standard_name attribute. The permissible values of these modifiers are given in Appendix C, Standard Name Modifiers.

When one data variable provides metadata about the individual values of another data variable it may be desirable to express this association by providing a link between the variables. For example, instrument data may have associated measures of uncertainty. The attribute ancillary_variables is used to express these types of relationships. It is a string attribute whose value is a blank separated list of variable names. The nature of the relationship between variables associated via ancillary_variables must be determined by other attributes. The variables listed by the ancillary_variables attribute will often have the standard name of the variable which points to them including a modifier (Appendix C, Standard Name Modifiers) to indicate the relationship. The dimensions of an ancillary variable must be the same as or a subset of the dimensions of the variable to which it is related, but their order is not restricted, and with one exception: If an ancillary variable of a data variable that has been compressed by gathering (Section 8.2, "Lossless Compression by Gathering") does not span the compressed dimension, then its dimensions may be any subset of the data variable’s uncompressed dimensions, i.e. any of the dimensions of the data variable except the compressed dimension, and any of the dimensions listed by the compress attribute of the compressed coordinate variable.

Alternatively, ancillary_variables may be used as status flags indicating the operational status of an instrument producing the data or as quality flags indicating the results of a quality control test, or some other quantitative quality assessment, performed against the measurements contained in the source variable. In these cases, the flag variable will include a standard name that differs from that of the source variable and indicates the specific type of flag the variable represents.

The standard names table includes many names intended to be used in this situation, both general names meant to be used to flexibly represent any type of status or quality assessment, as well as names for specific quality control tests commonly applied to geophysical phenomena timeseries data. Several examples are listed below:

The following example illustrates the use of three of these flags to represent two independent quality control tests and an aggregate flag that combines the results of the two tests.

The attributes flag_values, flag_masks and flag_meanings are intended to make variables that contain flag values self describing. Status codes and Boolean (binary) condition flags may be expressed with different combinations of flag_values and flag_masks attribute definitions.

The flag_values and flag_meanings attributes describe a status flag consisting of mutually exclusive coded values. The flag_values attribute is the same type as the variable to which it is attached, and contains a list of the possible flag values. The flag_meanings attribute is a string whose value is a blank separated list of descriptive words or phrases, one for each flag value. Each word or phrase should consist of characters from the alphanumeric set and the following five: '_', '-', '.', '+', '@'. If multi-word phrases are used to describe the flag values, then the words within a phrase should be connected with underscores. The following example illustrates the use of flag values to express a speed quality with an enumerated status code.

The flag_masks and flag_meanings attributes describe a number of independent Boolean conditions using bit field notation by setting unique bits in each flag_masks value. The flag_masks attribute is the same type as the variable to which it is attached, and contains a list of values matching unique bit fields. The flag_meanings attribute is defined as above, one for each flag_masks value. A flagged condition is identified by performing a bitwise AND of the variable value and each flag_masks value; a non-zero result indicates a true condition. Thus, any or all of the flagged conditions may be true, depending on the variable bit settings. The following example illustrates the use of flag_masks to express six sensor status conditions.

A variable with standard name of region, area_type or any other standard name which requires string-valued values from a defined list may use flags together with flag_values and flag_meanings attributes to record the translation to the string values. The following example illustrates this using integer flag values for a variable with standard name region and flag_values selected from the standardized region names (see section 6.1.1).

The flag_masks, flag_values and flag_meanings attributes, used together, describe a blend of independent Boolean conditions and enumerated status codes. The flag_masks and flag_values attributes are both the same type as the variable to which they are attached. A flagged condition is identified by a bitwise AND of the variable value and each flag_masks value; a result that matches the flag_values value indicates a true condition. Repeated flag_masks define a bit field mask that identifies a number of status conditions with different flag_values. The flag_meanings attribute is defined as above, one for each flag_masks bit field and flag_values definition. Each flag_values and flag_masks value must coincide with a flag_meanings value. The following example illustrates the use of flag_masks and flag_values to express two sensor status conditions and one enumerated status code.

In this case, mutually exclusive values are blended with Boolean values to maximize use of the available bits in a flag value. The table below represents the four binary digits (bits) expressed by the sensor_status_qc variable in the previous example.

Bit 0 and Bit 1 are Boolean values indicating a low battery condition and a hardware fault, respectively. The next two bits (Bit 2 and Bit 3) express an enumeration indicating abnormal sensor operating modes. Thus, if Bit 0 is set, the battery is low and if Bit 1 is set, there is a hardware fault - independent of the current sensor operating mode.

The remaining bits (Bit 2 and Bit 3) are decoded as follows:

The "12b" flag mask is repeated in the sensor_status_qc flag_masks definition to explicitly declare the recommended bit field masks to repeatedly AND with the variable value while searching for matching enumerated values. An application determines if any of the conditions declared in the flag_meanings list are true by simply iterating through each of the flag_masks and AND’ing them with the variable. When a result is equal to the corresponding flag_values element, that condition is true. The repeated flag_masks enable a simple mechanism for clients to detect all possible conditions.

Variables representing latitude must always explicitly include the units attribute; there is no default value. The recommended value of the units attribute is the string degrees_north. Also accepted are degree_north, degree_N, degrees_N, degreeN, and degreesN.

Application writers should note that the UDUNITS package does not recognize the directionality implied by the "north" part of the unit specification. It only recognizes its size, i.e., 1 degree is defined to be pi/180 radians. Hence, determination that a coordinate is a latitude type should be done via a string match between the given unit and one of the acceptable forms of degrees_north.

Optionally, the latitude type may be indicated additionally by providing the standard_name attribute with the value latitude, and/or the axis attribute with the value Y.

Coordinates of latitude with respect to a rotated pole should be given units of degrees, not degrees_north or equivalents, because applications which use the units to identify axes would have no means of distinguishing such an axis from real latitude, and might draw incorrect coastlines, for instance.

Variables representing longitude must always explicitly include the units attribute; there is no default value. The recommended value of the units attribute is the string degrees_east. Also accepted are degree_east, degree_E, degrees_E, degreeE, and degreesE.

Application writers should note that the UDUNITS package has limited recognition of the directionality implied by the "east" part of the unit specification. It defines degrees_east to be pi/180 radians, and hence equivalent to degrees_north. We recommend the determination that a coordinate is a longitude type should be done via a string match between the given unit and one of the acceptable forms of degrees_east.

Optionally, the longitude type may be indicated additionally by providing the standard_name attribute with the value longitude, and/or the axis attribute with the value X.

Coordinates of longitude with respect to a rotated pole should be given units of degrees, not degrees_east or equivalents, because applications which use the units to identify axes would have no means of distinguishing such an axis from real longitude, and might draw incorrect coastlines, for instance.

Variables representing dimensional height or depth axes must always explicitly include the units attribute; there is no default value.

The direction of positive (i.e., the direction in which the coordinate values are increasing), whether up or down, cannot in all cases be inferred from the units. The direction of positive is useful for applications displaying the data. For this reason the attribute positive as defined in the COARDS standard is required if the vertical axis units are not a valid unit of pressure (as determined by the UDUNITS package [UDUNITS]) — otherwise its inclusion is optional. The positive attribute may have the value up or down (case insensitive). This attribute may be applied to either coordinate variables or auxiliary coordinate variables that contain vertical coordinate data.

For example, if an oceanographic netCDF file encodes the depth of the surface as 0 and the depth of 1000 meters as 1000 then the axis would use attributes as follows:

If, on the other hand, the depth of 1000 meters were represented as -1000 then the value of the positive attribute would have been up. If the units attribute value is a valid pressure unit the default value of the positive attribute is down.

A vertical coordinate will be identifiable by:

Optionally, the vertical type may be indicated additionally by providing the standard_name attribute with an appropriate value, and/or the axis attribute with the value Z. If both positive and standard_name are provided, it is recommended that they should be consistent. For instance, if a depth of 1000 metres is represented by -1000 and positive is up, it would be inconsistent to give the standard_name as depth, whose definition (vertical distance below the surface) implies positive down. If an application detects such an inconsistency, the user should be warned, and the positive attribute should be used to determine the sign convention.

Recommendations: The positive attribute should be consistent with the sign convention implied by the definition of the standard_name, if both are provided.

Variables representing reference time must always explicitly include the units attribute; there is no default value. The units attribute takes a string value that follows the formatting requirements of the [UDUNITS] package. These requirements can best be described by an example with explanatory comments:

The time unit specification seconds since 1992-10-8 15:15:42.5 -6:00 indicates seconds since October 8th, 1992 at 3 hours, 15 minutes and 42.5 seconds in the afternoon in the time zone which is six hours to the west of Coordinated Universal Time (i.e. Mountain Daylight Time). The time zone specification can also be written without a colon using one or two digits (indicating hours) or three or four digits (indicating hours and minutes).

The acceptable units for time are given by the UDUNITS package. The most commonly used of these strings (and their abbreviations) includes day (d), hour (hr, h), minute (min) and second (sec, s). Plural forms are also acceptable.

UDUNITS permits a number of alternatives to the word since in the units of time coordinates. All the alternatives have exactly the same meaning in UDUNITS. For compatibility with other software, CF strongly recommends that since should be used.

The reference date/time string (appearing after the identifier since) is required. It may include date alone, or date and time, or date, time and time zone. If the time zone is omitted the default is UTC, and if both time and time zone are omitted the default is 00:00:00 UTC.

UDUNITS defines a year to be exactly 365.242198781 days (the interval between 2 successive passages of the sun through vernal equinox). It is not a calendar year. UDUNITS defines a month to be exactly year/12, which is not a calendar month. The CF standard follows UDUNITS in the definition of units, but we recommend that year and month should not be used, because of the potential for mistakes and confusion.

A reference time coordinate is identifiable from its units string alone.

Optionally, the time coordinate may be indicated additionally by providing the standard_name attribute with an appropriate value, and/or the axis attribute with the value T.

The spatiotemporal coordinates described in sections 4.1-4.4 are continuous variables, and other geophysical quantities may likewise serve as continuous coordinate variables, for instance density, temperature or radiation wavelength. By contrast, for some purposes there is a need for an axis of a data variable which indicates either an ordered list or an unordered collection, and does not correspond to any continuous coordinate variable. Consequently such an axis may be called “discrete”. A discrete axis has a dimension but might not have a coordinate variable. Instead, there might be one or more auxiliary coordinate variables with this dimension (see preamble to section 5). Following sections define various applications of discrete axes, for instance section 6.1.1 “Geographical regions”, section 7.3.3 “Statistics applying to portions of cells”, section 9.3 “Representation of collections of features in data variables”.

When each of a variable’s spatiotemporal dimensions is a latitude, longitude, vertical, or time dimension, then each axis is identified by a coordinate variable.

xwind(n,k,j,i) is associated with the coordinate values lon(i), lat(j), pres(k), and time(n).

The latitude and longitude coordinates of a horizontal grid that was not defined as a Cartesian product of latitude and longitude axes, can sometimes be represented using two-dimensional coordinate variables. These variables are identified as coordinates by use of the coordinates attribute.

T(k,j,i) is associated with the coordinate values lon(j,i), lat(j,i), and lev(k). The vertical coordinate is represented by the coordinate variable lev(lev) and the latitude and longitude coordinates are represented by the auxiliary coordinate variables lat(yc,xc) and lon(yc,xc) which are identified by the coordinates attribute.

Note that coordinate variables are also defined for the xc and yc dimensions. This faciliates processing of this data by generic applications that don’t recognize the multidimensional latitude and longitude coordinates.

A "reduced" longitude-latitude grid is one in which the points are arranged along constant latitude lines with the number of points on a latitude line decreasing toward the poles. Storing this type of gridded data in two-dimensional arrays wastes space, and results in the presence of missing values in the 2D coordinate variables. We recommend that this type of gridded data be stored using the compression scheme described in Section 8.2, "Lossless Compression by Gathering". Compression by gathering preserves structure by storing a set of indices that allows an application to easily scatter the compressed data back to two-dimensional arrays. The compressed latitude and longitude auxiliary coordinate variables are identified by the coordinates attribute.

PS(n) is associated with the coordinate values lon(n), lat(n). Compressed grid index (n) would be assigned to 2D index (j,i) (C index conventions) where

Notice that even if an application does not recognize the compress attribute, the grids stored in this format can still be handled, by an application that recognizes the coordinates attribute.

This section has been superseded by the treatment of time series as a type of discrete sampling geometry in Chapter 9.

This section has been superseded by the treatment of time series as a type of discrete sampling geometry in Chapter 9.

A grid mapping variable may be referenced by a data variable in order to explicitly declare the coordinate reference system (CRS) used for the horizontal spatial coordinate values. For example, if the horizontal spatial coordinates are latitude and longitude, the grid mapping variable can be used to declare the figure of the earth (WGS84 ellipsoid, sphere, etc.) they are based on. If the horizontal spatial coordinates are easting and northing in a map projection, the grid mapping variable declares the map projection CRS used and provides the information needed to calculate latitude and longitude from easting and northing.

When the horizontal spatial coordinate variables are not longitude and latitude, it is required that further information is provided to geo-locate the horizontal position. A grid mapping variable provides this information.

If no grid mapping variable is provided and the coordinate variables for a horizontal grid are not longitude and latitude, then it is required that the latitude and longitude coordinates are supplied via the coordinates attribute. Such coordinates may be provided in addition to the provision of a grid mapping variable, but that is not required.

A grid mapping variable provides the description of the mapping via a collection of attached attributes. It is of arbitrary type since it contains no data. Its purpose is to act as a container for the attributes that define the mapping. The one attribute that all grid mapping variables must have is grid_mapping_name, which takes a string value that contains the mapping’s name. The other attributes that define a specific mapping depend on the value of grid_mapping_name. The valid values of grid_mapping_name along with the attributes that provide specific map parameter values are described in Appendix F, Grid Mappings.

The grid mapping variables are associated with the data and coordinate variables by the grid_mapping attribute. This attribute is attached to data variables so that variables with different mappings may be present in a single file. The attribute takes a string value with two possible formats. In the first format, it is a single word, which names a grid mapping variable. In the second format, it is a blank-separated list of words "<gridMappingVariable>: <coordinatesVariable> [<coordinatesVariable> …​] [<gridMappingVariable>: <coordinatesVariable>…​]", which identifies one or more grid mapping variables, and with each grid mapping associates one or more coordinatesVariables, i.e. coordinate variables or auxiliary coordinate variables.

Where an extended "<gridMappingVariable>: <coordinatesVariable> [<coordinatesVariable>]" entity is defined, then the order of the <coordinatesVariable> references within the definition provides an explicit order for these coordinate value variables, which is used if they are to be combined into individual coordinate tuples.

This order is only significant if crs_wkt is also specified within the referenced grid mapping variable. Explicit 'axis order' is important when the grid_mapping_variable contains an attribute crs_wkt as it is mandated by the OGC CRS-WKT standard that coordinate tuples with correct axis order are provided as part of the reference to a Coordinate Reference System.

Using the simple form, where the grid_mapping attribute is only the name of a grid mapping variable, 2D latitude and longitude coordinates for a projected coordinate reference system use the same geographic coordinate reference system (ellipsoid and prime meridian) as the projection is projected from.

The grid_mapping variable may identify datums (such as the reference ellipsoid, the geoid or the prime meridian) for horizontal or vertical coordinates. Therefore a grid mapping variable may be needed when the coordinate variables for a horizontal grid are longitude and latitude. The grid_mapping_name of latitude_longitude should be used in this case.

The expanded form of the grid_mapping attribute is required if one wants to store coordinate information for more than one coordinate reference system. In this case each coordinate or auxiliary coordinate is defined explicitly with respect to no more than one grid_mapping variable. This syntax may be used to explicitly link coordinates and grid mapping variables where only one coordinate reference system is used. In this case, all coordinates and auxiliary coordinates of the data variable not named in the grid_mapping attribute are unrelated to any grid mapping variable. All coordinate names listed in the grid_mapping attribute must be coordinate variables or auxiliary coordinates of the data variable.

In order to make use of a grid mapping to directly calculate latitude and longitude values it is necessary to associate the coordinate variables with the independent variables of the mapping. This is done by assigning a standard_name to the coordinate variable. The appropriate values of the standard_name depend on the grid mapping and are given in Appendix F, Grid Mappings.

A CF compliant application can determine that rlon and rlat are longitude and latitude values in the rotated grid by recognizing the standard names grid_longitude and grid_latitude. Note that the units of the rotated longitude and latitude axes are given as degrees. This should prevent a COARDS compliant application from mistaking the variables rlon and rlat to be actual longitude and latitude coordinates. The entries for these names in the standard name table indicate the appropriate sign conventions for the units of degrees.

An application can determine that x and y are the projection coordinates by recognizing the standard names projection_x_coordinate and projection_y_coordinate. The grid mapping variable Lambert_Conformal contains the mapping parameters as attributes, and is associated with the Temperature variable via its grid_mapping attribute.

When a variable has an associated coordinate which is single-valued, that coordinate may be represented as a scalar variable (i.e. a data variable which has no netCDF dimensions). Since there is no associated dimension these scalar coordinate variables should be attached to a data variable via the coordinates attribute.

The use of scalar coordinate variables is a convenience feature which avoids adding size one dimensions to variables. A numeric scalar coordinate variable has the same information content and can be used in the same contexts as a size one numeric coordinate variable. Similarly, a string-valued scalar coordinate variable has the same meaning and purposes as a size one string-valued auxiliary coordinate variable (Section 6.1, "Labels"). Note however that use of this feature with a latitude, longitude, vertical, or time coordinate will inhibit COARDS conforming applications from recognizing them.

Once a name is used for a scalar coordinate variable it can not be used for a 1D coordinate variable. For this reason we strongly recommend against using a name for a scalar coordinate variable that matches the name of any dimension in the file.

If a data variable has two or more scalar coordinate variables, they are regarded as though they were all independent coordinate variables with dimensions of size one. If two or more single-valued coordinates are not independent, but have related values (this might be the case, for instance, for time and forecast period, or vertical coordinate and model level number, Section 6.2, "Alternative Coordinates"), they should be stored as coordinate or auxiliary coordinate variables of the same size one dimension, not as scalar coordinate variables.

In this example both the analysis time and the single pressure level are represented using scalar coordinate variables. The analysis time is identified by the standard name "forecast_reference_time" while the valid time of the forecast is identified by the standard name "time".

A domain describes data locations and cell properties. It defines cells that span a collection of dimensions with cell coordinates, cell measures, and coordinate reference systems.

A data variable defines its domain via its own attributes, but a domain variable provides the description of a domain in the absence of any data values. The variable should be a scalar (i.e. it has no dimensions) of arbitrary type, and the value of its single element is immaterial. It acts as a container for the attributes that define the domain. The purpose of a domain variable is to provide domain information to applications that have no need of data values at the domain’s locations, thus removing any ambiguity when retrieving a domain from a dataset. Ancillary variables and cell methods are not part of the domain, because they are only defined in relation to data values.

The domain variable supports the same attributes as are allowed on a data variable for describing a domain, with exactly the same meanings and syntaxes, as described in Appendix A, Attributes. If an attribute is needed by a particular data variable to describe its domain, then that attribute would also be needed by the equivalent domain variable.

The dimensions of the domain must be stored with the dimensions attribute, and the presence of a dimensions attribute will identify the variable as a domain variable. Therefore the dimensions attribute must not be present on any variables that are to be interpreted as data variables. It is necessary to list these dimensions, rather than inferring them from the contents of the other attributes, as it can not be guaranteed that the referenced variables span all of the required dimensions (as could be the case for a discrete axis, for instance). The value of the dimensions attribute is a blank separated list of the dimension names. There is no restriction on the order in which the dimensions appear in the dimensions attribute string. If a domain has no named dimensions then the value of the dimensions attribute must be an empty string, as could be the case if the dimensions of the domain are all defined implicitly by scalar coordinate variables.

The dimensions listed by the dimensions attribute constrain the dimensions that may be spanned by variables referenced from any of the other attributes, in the same way that the array dimensions perform that role for a data variable. For instance, all variables named by the cell_measures attribute (Section 7.2, "Cell Measures") of a domain variable must span a subset of zero or more of the dimensions given by the dimensions attribute.

It is optional for coordinate variables to be listed by a domain variable’s coordinates attribute. Any coordinate variable that shares its name with a dimension given by the dimensions attribute will be considered as part of the domain definition.

It is recommended that a domain variable has a long_name attribute to describe its contents.

It is recommended that a domain variable does not have any of the attributes marked in Appendix A, Attributes as applicable to data variables except those which are also marked as applicable to domain variables.

Multiple domain variables may exist in a file with, or without, data variables. Note that the data variable attributes describing its domain can not be replaced by a reference to a domain variable.

A mesh topology variable defines the geospatial topology of cells arranged in two or three dimensions in real space but indexed by a single dimension. It explicitly describes the topological relationships between cells, i.e. spatial relationships which do not depend on the cell locations, via a mesh of connected nodes. A mesh topology variable may provide the topology for one or more domains, defined at the nodes, edges, or faces of the mesh. See the Domain topology construct and Cell connectivity construct descriptions in the CF data model for more details, including on how the mesh relates to the cells of the domain.

The canonical definitions of mesh topology variables and location index set variables are given externally by the UGRID conventions [UGRID], but their standardized attributes, many of which are optional, are listed in Appendix K, Mesh Topology Attributes and Appendix A, Attributes. Some features of the UGRID conventions [UGRID] are not currently recognized by the CF conventions: mesh topology volume cells (that are used to describe fully three-dimensional unstructured mesh topologies); and the "boundary node connectivity" variable (that specifies an index variable identifying the nodes that define where boundary condtions have been provided).

A data or domain variable may use one of a mesh topology variable’s domains by referencing the mesh topology variable with the mesh attribute; along with the identity of required domain provided by the location attribute (see example A two-dimensional UGRID mesh topology variable).

The variables containing the coordinate values for cells indexed by the mesh topology are defined by the mesh topology variable but are equivalent to one-dimensional auxiliary coordinate variables, and so may also be provided by the data or domain variable’s coordinates attribute. Note that the mesh topology variable allows cell bounds to be provided without any cell coordinate values, via its node_coordinates attribute.

A location index set variable defines a subset of locations of a mesh topology variable, e.g. only special locations like weirs and gates. It is provided as a space saving device to prevent the need to redefine parts of an existing mesh topology variable, and as such is logically equivalent to a mesh topology variable. A data or domain variable references a location index set variable via its location_index_set attribute.

Character strings can be used to provide a name or label for each element of an axis. This is particularly useful for discrete axes (section 4.5). For instance, if a data variable contains time series of observational data from a number of observing stations, it may be convenient to provide the names of the stations as labels for the elements of the station dimension (Section H.2, "Time Series Data"). There are several other uses for labels in CF. For instance, Northward heat transport in Atlantic Ocean shows the use of labels to indicate geographic regions.

Character strings labelling the elements of an axis are regarded as string-valued auxiliary coordinate variables. The coordinates attribute of the data variable names the variable that contains the string array. An application processing the variables listed in the coordinates attribute can recognize a string-valued auxiliary coordinate variable because it has a type of char or string. If the variable has a type of char, the inner dimension (last dimension in CDL terms) is the maximum length of each string, and the other dimensions are axis dimensions. If an auxiliary coordinate variable has a type of string and has no dimensions, or has a type of char and has only one dimension (the maximum length of the string), it is a string-valued scalar coordinate variable (see Section 5.7, "Scalar Coordinate Variables"). As such, it has the same information content and can be used in the same contexts as a string-valued auxiliary coordinate variable of a size one dimension. This is a convenience feature.

In some situations a dimension may have alternative sets of coordinates values. Since there can only be one coordinate variable for the dimension (the variable with the same name as the dimension), any alternative sets of values have to be stored in auxiliary coordinate variables. For such alternative coordinate variables, there are no mandatory attributes, but they may have any of the attributes allowed for coordinate variables.

To represent cells we add the attribute bounds to the appropriate coordinate variable(s). The value of bounds is the name of the variable that contains the vertices of the cell boundaries. We refer to this type of variable as a "boundary variable." A boundary variable will have one more dimension than its associated coordinate or auxiliary coordinate variable. The additional dimension should be the most rapidly varying one, and its size is the maximum number of cell vertices.

A boundary variable inherits the values of some attributes from its parent coordinate variable. If a coordinate variable has any of the attributes marked "BI" (for "inherit") in the "Use" column of Appendix A, Attributes, they are assumed to apply to its bounds variable as well. It is recommended that BI attributes not be included on a boundary variable. If a BI attribute is included, it must also be present in the parent variable, and it must exactly match the parent attribute’s data type and value. A boundary variable can only have inheritable attributes if they are also present on its parent coordinate variable. A bounds variable may have any of the attributes marked "BO" for ("own") in the "Use" column of Appendix A, Attributes. These attributes take precedence over any corresponding attributes of the parent variable. In these cases, the parent variable’s attribute does not apply to the bounds variable, regardless of whether the latter has its own attribute.

If a parametric coordinate variable with a formula_terms attribute (section 4.3.2) also has a bounds attribute, its boundary variable must have a formula_terms attribute too. In this case the same terms would appear in both (as specified in Appendix D), since the transformation from the parametric coordinate values to physical space is realized through the same formula. For any term that depends on the vertical dimension, however, the variable names appearing in the formula terms would differ from those found in the formula_terms attribute of the coordinate variable itself because the boundary variables for formula terms are two-dimensional while the formula terms themselves are one-dimensional.

Whenever a formula_terms attribute is attached to a boundary variable, the formula terms may additionally be identified using a second method: variables appearing in the vertical coordinates' formula_terms may be declared to be coordinate, scalar coordinate or auxiliary coordinate variables, and those coordinates may have bounds attributes that identify their boundary variables. In that case, the bounds attribute of a formula terms variable must be consistent with the formula_terms attribute of the boundary variable. Software digesting legacy datasets (constructed prior to version 1.7 of this standard) may have to rely in some cases on the first method of identifying the formula term variables and in other cases, on the second. Starting from version 1.7, however, the first method will be sufficient.

Note that the boundary variable for a set of N contiguous intervals is an array of shape (N,2). Although in this case there will be a duplication of the boundary coordinates between adjacent intervals, this representation has the advantage that it is general enough to handle, without modification, non-contiguous intervals, as well as intervals on an axis using the unlimited dimension.

Applications that process cell boundary data often times need to determine whether or not adjacent cells share an edge. In order to facilitate this type of processing the following restrictions are placed on the data in boundary variables.

For rectangular grids, two-dimensional cells can be expressed as Cartesian products of one-dimensional cells of the type in the preceding example. However for non-rectangular grids a "rectangular" cell will in general require specifying all four vertices for each cell.

For some calculations, information is needed about the size, shape or location of the cells that cannot be deduced from the coordinates and bounds without special knowledge that a generic application cannot be expected to have. For instance, in computing the mean of several cell values, it is often appropriate to "weight" the values by area. When computing an area-mean each grid cell value is multiplied by the grid-cell area before summing, and then the sum is divided by the sum of the grid-cell areas. Area weights may also be needed to map data from one grid to another in such a way as to preserve the area mean of the field. The preservation of area-mean values while regridding may be essential, for example, when calculating surface heat fluxes in an atmospheric model with a grid that differs from the ocean model grid to which it is coupled.

In many cases the areas can be calculated from the cell bounds, but there are exceptions. Consider, for example, a spherical geodesic grid composed of contiguous, roughly hexagonal cells. The vertices of the cells can be stored in the variable identified by the bounds attribute, but the cell perimeter is not uniquely defined by its vertices (because the vertices could, for example, be connected by straight lines, or, on a sphere, by lines following a great circle, or, in general, in some other way). Thus, given the cell vertices alone, it is generally impossible to calculate the area of a grid cell. This is why it may be necessary to store the grid-cell areas in addition to the cell vertices.

In other cases, the grid cell-volume might be needed and might not be easily calculated from the coordinate information. In ocean models, for example, it is not uncommon to find "partial" grid cells at the bottom of the ocean. In this case, rather than (or in addition to) indicating grid cell area, it may be necessary to indicate volume.

To indicate extra information about the spatial properties of a variable’s grid cells, a cell_measures attribute may be defined for a variable. This is a string attribute comprising a list of blank-separated pairs of words of the form "measure: name". For the moment, "area" and "volume" are the only defined measures, but others may be supported in future. The "name" is the name of the variable containing the measure values, which we refer to as a "measure variable". The dimensions of a measure variable must be the same as or a subset of the dimensions of the variable to which it is related, but their order is not restricted, and with one exception: If a cell measure variable of a data variable that has been compressed by gathering (Section 8.2, "Lossless Compression by Gathering") does not span the compressed dimension, then its dimensions may be any subset of the data variable’s uncompressed dimensions, i.e. any of the dimensions of the data variable except the compressed dimension, and any of the dimensions listed by the compress attribute of the compressed coordinate variable. In the case of area, for example, the field itself might be a function of longitude, latitude, and time, but the variable containing the area values would only include longitude and latitude dimensions (and the dimension order could be reversed, although this is not recommended). The variable must have a units attribute and may have other attributes such as a standard_name.

For rectangular longitude-latitude grids, the area of grid cells can be calculated from the bounds: the area of a cell is proportional to the product of the difference in the longitude bounds of the cell and the difference between the sine of each latitude bound of the cell. In this case supplying grid-cell areas via the cell_measures attribute is unnecessary because it may be assumed that applications can perform this calculation, using their own value for the radius of the Earth.

A variable referenced by cell_measures is not required to be present in the file containing the data variable. If the cell_measures variable is located in another file (an "external file"), rather than in the file where it is referenced, it must be listed in the external_variables attribute of the referencing file (Section 2.6.3).

To describe the characteristic of a field that is represented by cell values, we define the cell_methods attribute of the variable. This is a string attribute comprising a list of blank-separated words of the form "name: method". Each "name: method" pair indicates that for an axis identified by name, the cell values representing the field have been determined or derived by the specified method. For example, if data values have been generated by computing time means, then this could be indicated with cell_methods="t: mean", assuming here that the name of the time dimension variable is "t".

In the specification of this attribute, name can be a dimension of the variable, a scalar coordinate variable, a valid standard name, or the word "area". (See Section 7.3.4, "Cell methods when there are no coordinates" concerning the use of standard names in cell_methods.) The values of method should be selected from the list in Appendix E, Cell Methods, which includes point, sum, mean, among others. Case is not significant in the method name. Some methods (e.g., variance) imply a change of units of the variable, as is indicated in Appendix E, Cell Methods.

It must be remembered that the method applies only to the axis designated in cell_methods by name, and different methods may apply to other axes. If, for instance, a precipitation value in a longitude-latitude cell is given the method maximum for these axes, it means that it is the maximum within these spatial cells, and does not imply that it is also the maximum in time. Furthermore, it should be noted that if any method other than "point" is specified for a given axis, then bounds should also be provided for that axis (except for the relatively rare exceptions described in Section 7.3.4, "Cell methods when there are no coordinates").

The default interpretation for variables that do not have the cell_methods attribute specified depends on whether the quantity is extensive (which depends on the size of the cell) or intensive (which does not). Suppose, for example, the quantities "accumulated precipitation" and "precipitation rate" each have a time axis. A variable representing accumulated precipitation is extensive in time because it depends on the length of the time interval over which it is accumulated. For correct interpretation, it therefore requires a time interval to be completely specified via a boundary variable (i.e., via a bounds attribute for the time axis). In this case the default interpretation is that the cell method is a sum over the specified time interval. This can be (optionally) indicated explicitly by setting the cell method to sum. A precipitation rate on the other hand is intensive in time and could equally well represent either an instantaneous value or a mean value over the time interval specified by the cell. In this case the default interpretation for the quantity would be "instantaneous" (which, optionally, can be indicated explicitly by setting the cell method to point). More often, however, cell values for intensive quantities are means, and this should be indicated explicitly by setting the cell method to mean and specifying the cell bounds.

Because the default interpretation for an intensive quantity differs from that of an extensive quantity and because this distinction may not be understood by some users of the data, it is recommended that every data variable include for each of its dimensions and each of its scalar coordinate variables the cell_methods information of interest (unless this information would not be meaningful). It is especially recommended that cell_methods be explicitly specified for each spatio-temporal dimension and each spatio-temporal scalar coordinate variable.

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.

For compatibility with the COARDS standard, a climatological time coordinate in the default standard and julian calendars may be indicated by setting the date/time reference string in the time coordinate’s units attribute to midnight on 1 January in year 0 (i.e., since 0-1-1). This convention is deprecated because it does not provide any information about the intervals used to compute the climatology, and there may be inconsistencies among software packages in the interpretation of the time coordinates with a reference time of year 0. Use of year 0 for this purpose is impossible in all other calendars, because year 0 is a valid year.

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.

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.

For many geospatial applications, data values are associated with a geometry, which is a spatial representation of a real-world feature, for instance a time-series of areal average precipitation over a watershed. Polygonal cells with an arbitrary number of vertices can be described using Section 7.1, "Cell Boundaries", but in that case every cell must have the same number of vertices and must be a single polygon ring. In contrast, each geometry may have a different number of nodes, the geometries may be lines (as alternatives to points and polygons), and they may be multipart, i.e., include several disjoint parts. While line and point geometries don’t describe an interval along a dimension as the traditional cell bounds described above do, they do describe the extent of a geometry or real-world feature so are included in this section. The approach described here specifies how to encode such geometries following the pattern in 9.3.3 Contiguous ragged array representation and attach them to variables in a way that is consistent with the cell bounds approach.

All geometries are made up of one or more nodes. The geometry type specifies the set of topological assumptions to be applied to relate the nodes (see Table 7.1). For example, multipoint and line geometries are nearly the same except nodes are interpreted as being connected for lines. Lines and polygons are also nearly the same except that the first and last nodes are assumed to be connected for polygons. Note that CF does not require the first and last node to be identical but allows them to be coincident if desired. Polygons that have holes, such as waterbodies in a land unit, are encoded as a collection of polygon ring parts, each identified as exterior or interior polygons. Multipart geometries, such as multiple lines representing the same river or multiple islands representing the same jurisdiction, are encoded as collections of unconnected points, lines, or polygons that are logically grouped into a single geometry.

Any data variable can be given a geometry attribute that indicates the geometry for the quantity held in the variable. One of the dimensions of the data variable must be the number of geometries to which the data applies. As shown in Example 7.15, if the data variable has a discrete sampling geometry, the number of geometries is the length of the instance dimension (Section 9.2).

A geometry container variable acts as a container for attributes that describe a set of geometries. The geometry attribute of the data variable contains the name of a geometry container variable. The geometry container variable must hold geometry_type and node_coordinates attributes. The grid_mapping and coordinates attributes can be carried by the geometry container variable provided they are also carried by the data variables associated with the container.

The geometry_type attribute indicates the type of geometry present. Its allowable values are: point, line, polygon. Multipart geometries are allowed for all three geometry types. For example, polygon geometries could include single part geometries like the State of Colorado and multipart geometries like the State of Hawaii.

The node_coordinates attribute contains the blank-separated names of the variables that contain geometry node coordinates (one variable for each spatial dimension). The geometry node coordinate variables must each have an axis attribute whose allowable values are X, Y, and Z.

If a coordinates attribute is carried by the geometry container variable or its parent data variable, then those coordinate variables that have a meaningful correspondence with node coordinates are indicated as such by a nodes attribute that names the corresponding node coordinates, but only if the grid_mapping associated with the geometry node variables is the same as that of the coordinate variables. If a different grid mapping is used, then the provided coordinates must not have the nodes attribute.

Whether linked to normal CF space-time coordinates with a nodes attribute or not, inclusion of such coordinates is recommended to maintain backward compatibility with software that has not implemented geometry capabilities.

The geometry node coordinate variables must all have the same single dimension, which is the total number of nodes in all the geometries. The nodes must be stored consecutively for each geometry and in the order of the geometries, and within each multipart geometry the nodes must be stored consecutively for each part and in the order of the parts. Polygon exterior rings must be stored before any interior rings they may contain. Nodes for polygon exterior rings must be ordered using the right-hand rule, e.g., anticlockwise in the lon-lat plane as viewed from above. Polygon interior rings must be in clockwise order. They are put in opposite orders to facilitate calculation of area and consistency with the typical implementation pattern.

When more than one geometry instance is present, the geometry container variable must have a node_count attribute that contains the name of a variable indicating the count of nodes per geometry. The node count is the total number of nodes in all the parts. The exception is when all geometries are single part point geometries, in which case a node count is not needed since each geometry contains a single node. However in that case, the dimension of the node coordinate variables must be one of the dimensions of the data variable (because it serves also as the instance dimension for geometries).

For multipart lines, multipart polygons, and polygons with holes, the geometry container variable must have a part_node_count attribute that indicates a variable of the count of nodes per geometry part. Note that because multipoint geometries always have a single node per part, the part_node_count is not required for point geometry types. The single dimension of the part node count variable must equal the total number of parts in all the geometries.

For polygon geometries with holes, the geometry container variable must have an interior_ring attribute that contains the name of a variable that indicates if the polygon parts are interior rings (i.e., holes) or not. This interior ring variable must contain the value 0 to indicate an exterior ring polygon and 1 to indicate an interior ring polygon. The single dimension of the interior ring variable must be the same dimension as that of the part node count variable. The geometry types included in this convention are listed in Table 7.1.

Table 7.1. Dimensionality, description, and additional required attributes for geometry_types.

At the current time the netCDF interface does not provide for packing data. However a simple packing may be achieved through the use of the optional NUG defined attributes scale_factor and add_offset. After the data values of a variable have been read, they are to be multiplied by the scale_factor, and have add_offset added to them. If both attributes are present, the data are scaled before the offset is added. When scaled data are written, the application should first subtract the offset and then divide by the scale factor. The units of a variable should be representative of the unpacked data.

This standard is more restrictive than the NUG with respect to the use of the scale_factor and add_offset attributes; ambiguities and precision problems related to data type conversions are resolved by these restrictions.

When packed data is written, the scale_factor and add_offset attributes must be of the same type as the unpacked data, which must be either float or double. Data of type float must be packed into one of these types: byte, unsigned byte, short, unsigned short. Data of type double must be packed into one of these types: byte, unsigned byte, short, unsigned short, int, unsigned int.

When packed data is read, it should be unpacked to the type of the scale_factor and add_offset attributes, which must have the same type if both are present. For guidance only, we suggest that packed data which does not conform to the rules of this section regarding the types of the data variable and attributes should be unpacked to double type, in order to minimise the risk of loss of precision.

When data to be packed contains missing values the attributes that indicate missing values (_FillValue, valid_min, valid_max, valid_range) must be of the same data type as the packed data. See Section 2.5.1, "Missing data, valid and actual range of data" for a discussion of how applications should treat variables that have attributes indicating both missing values and transformations defined by a scale and/or offset.

To save space in the netCDF file, it may be desirable to eliminate points from data arrays that are invariably missing. Such a compression can operate over one or more adjacent axes, and is accomplished with reference to a list of the points to be stored. The list is constructed by considering a mask array that only includes the axes to be compressed, and then mapping this array onto one dimension without reordering. The list is the set of indices in this one-dimensional mask of the required points. In the compressed array, the axes to be compressed are all replaced by a single axis, whose dimension is the number of wanted points. The wanted points appear along this dimension in the same order they appear in the uncompressed array, with the unwanted points skipped over. Compression and uncompression are executed by looping over the list.

The list is stored as the coordinate variable for the compressed axis of the data variable. Thus, the list variable and its dimension have the same name. If any auxiliary coordinate variable has all the dimensions to be compressed, adjacent and in the same order as in the data variable, and if the auxiliary coordinate variable has missing data at all the points which are to be eliminated from the data variable, then the affected dimensions can optionally be replaced by the list dimension for the auxiliary coordinate variable just as for the data variable. The list variable has a string attribute compress, containing a blank-separated list of the dimensions which were affected by the compression in the order of the CDL declaration of the uncompressed array. The presence of this attribute identifies the list variable as such. The list, the original dimensions and coordinate variables (including boundary variables), and the compressed variables with all the attributes of the uncompressed variables are written to the netCDF file. The uncompressed variables can be reconstituted exactly as they were using this information. The list variable must not have an associated boundary variable.

In A single timeseries with time-varying deviations from a nominal point spatial location, two auxiliary coordinate variables are compressed as described in this section, although their data variable is not.

For some applications the coordinates of a data variable can require considerably more storage than the data itself. Space may be saved in the netCDF file by storing a subsample of the coordinates that describe the data. The uncompressed coordinate and auxiliary coordinate variables can be reconstituted by interpolation, from the subsampled coordinate values to the domain of the data (i.e. the target domain). This process will likely result in a loss in accuracy (as opposed to precision) in the uncompressed variables, due to rounding and approximation errors in the interpolation calculations, but it is assumed that these errors will be small enough to not be of concern to users of the uncompressed dataset. The creator of the compressed dataset can control the accuracy of the reconstituted coordinates through the degree of subsampling and the choice of interpolation method, see Appendix J, Coordinate Interpolation Methods.

The subsampled coordinates are called tie points and are stored in tie point coordinate variables.

In addition to the tie point coordinate variables themselves, metadata defining the coordinate interpolation method is stored in attributes of the data variable and of the associated interpolation variable. The partitioning of metadata between the data variable and the interpolation variable has been designed to minimise redundancy and maximise the reusability of the interpolation variable within a dataset.

The metadata that define the interpolation formula and its inputs are complete, so that the results of the coordinate reconstitution process are well defined and of a predictable accuracy.

Each type of discrete sampling geometry (point, time series, profile or trajectory) is defined by the relationships among its spatiotemporal coordinates. We refer to the type of discrete sampling geometry as its featureType. The term “ feature ” refers herein to a single instance of the discrete sampling geometry (such as a single time series). The representation of such features in a CF dataset was supported previous to the introduction of this chapter using a particular convention, which is still supported (that described by section 9.3.1). This chapter describes further conventions which offer advantages of efficiency and clarity for storing a collection of features in a single file. When using these new conventions, the features contained within a collection must always be of the same type; and all the collections in a CF file must be of the same feature type. (Future versions of CF may allow mixing of multiple feature types within a file.) Table 9.1 presents the feature types covered by this chapter. Details and examples of storage of each of these feature types are provided in Appendix H, as indicated in the table.

Table 9.1. Logical structure and mandatory coordinates for discrete sampling geometry featureTypes. Other space-time coordinates may be included which are not mandatory.

In Table 9.1 the spatial coordinates x and y typically refer to longitude and latitude but other horizontal coordinates could also be used (see sections 4 and 5.6).   The spatial coordinate z refers to vertical position. The time coordinate is indicated as t. The space-time coordinates that are indicated for each feature are mandatory. However a featureType may also include other space-time coordinates which are not mandatory (notably the z coordinate, and for instance a forecast_reference_time coordinate in addition to a mandatory time coordinate). The array subscripts that are shown illustrate only the logical structure of the data. The subscripts found in actual CF files are determined by the specific type of representations (see section 9.3).

The designation of dimensions as mandatory precludes the encoding of data variables where geo-positioning cannot be described as a discrete point location. Problematic examples include:  

Future versions of CF will generalize the concepts of geolocation to encompass these cases. As of CF version 1.6 such data can be stored using the representations that are documented here by two means: 1) by utilizing the orthogonal multidimensional array representation and omitting the featureType attribute; or 2) by assigning arbitrary coordinates to the mandatory dimensions. For example a globally-averaged latitude position (90s to 90n) could be represented arbitrarily (and poorly) as a latitude position at the equator.

In Table 9.1 the dimension with subscript i identifies a particular feature within a collection of features. It is called the instance dimension. One-dimensional variables in a Discrete Geometry CF file, which have only this dimension (such as x(i), y(i) and z(i) for a timeseries), are instance variables. Instance variables provide the metadata that differentiates individual features.

The subscripts o and p distinguish the data elements that compose a single feature. For example in a collection of timeSeries features, each time series instance, i, has data values at various times, o. In a collection of profile features, the subscript, o, provides the index position along the vertical axis of each profile instance. We refer to data values in a feature as its elements, and to the dimensions of o and p as element dimensions. Each feature can have its own set of element subscripts o and p. For instance, in a collection of timeSeries features, each individual timeSeries can have its own set of times. The notation t(i,o) means there is a set of times with subscripts o for the elements of each feature i.   Feature instances within a collection need not have the same numbers of elements. If the features do all have the same number of elements, and the sequence of element coordinates is identical for all features, savings in simplicity and space are achievable by storing only one copy of these coordinates. This is the essence of the orthogonal multidimensional representation (see section 9.3.1).

If there is only a single feature to be stored in a data variable, there is no need for an instance dimension and it is permitted to omit it. The data will then be one-dimensional, which is a special (degenerate) case of the multidimensional array representation. The instance variables will be scalar coordinate variables; the data variable and other auxiliary coordinate variables will have only an element dimension and not have an instance dimension, e.g. data(o) and t(o) for a single timeSeries.

The individual features within a collection need not necessarily contain the same number of elements.   For instance observed in situ time series will commonly contain unique numbers of time points, reflecting different deployment dates of the instruments.   Other data sources, such as the output of numerical models, may commonly generate features of identical size. CF offers multiple representations to allow the storage to be optimized for the character of the data. Four types of representation are utilized in this chapter:

In the multidimensional array representations, data variables have both an instance dimension and an element dimension. The dimensions may be given in any order. If there is a need for either the instance or an element dimension to be the netCDF unlimited dimension (so that more features or more elements can be appended), then that dimension must be the outer dimension of the data variable i.e. the leading dimension in CDL.

In the ragged array representations, the instance dimension (i), which sequences the individual features within the collection, and the element dimension, which sequences the data elements of each feature (o and p), both occupy the same dimension (the sample dimension).   If the sample dimension is the netCDF unlimited dimension, new data can be appended to the file.  

In all representations, the instance dimension (which is also the sample dimension in ragged representations) may be set initially to a size that is arbitrarily larger than what is required for the features which are available at the time that the file is created.   Allocating unused array space in this way (pre-filled with missing values — see also section 9.6, Missing data), can be useful as a means to reserve space that will be available to add features at a later time.

A global attribute, featureType, is required for all Discrete Geometry representations except the orthogonal multidimensional array representation, for which it is highly recommended. The exception is allowed for backwards compatibility, as discussed in 9.3.1. A Discrete Geometry file may include arbitrary numbers of data variables, but (as of CF v1.6) all of the data variables contained in a single file must be of the single feature type indicated by the global featureType attribute, if it is present.1   The value assigned to the featureType attribute is case-insensitive;  it must be one of the string values listed in the left column of Table 9.1.

Every feature within a Discrete Geometry CF file must be unambiguously associated with an extensible collection of instance variables that identify the feature and provide other metadata as needed to describe it. Every element of every feature must be unambiguously associated with its space and time coordinates and with the feature that contains it. The coordinates attribute must be attached to every data variable to indicate the spatiotemporal coordinate variables that are needed to geo-locate the data.

Where feasible, one of the coordinate or auxiliary coordinate variables of a discrete sampling geometry should have an attribute named cf_role. This attribute has no other function in the CF convention (despite its general-sounding name), and its only permitted values are timeseries_id, profile_id, and trajectory_id. The variable carrying the cf_role attribute may have any data type. When a variable is assigned this attribute, it must provide a unique identifier for each feature instance.   CF files that contain timeSeries, profile or trajectory featureTypes, should include only a single occurrence of a cf_role attribute;  CF files that contain timeSeriesProfile or trajectoryProfile may contain two occurrences, corresponding to the two levels of structure in these feature types.

It is not uncommon for observational data to have two sets of coordinates for particular coordinate axes of a feature: a nominal point location and a more precise location that varies with the elements in the feature. For example, although an idealized vertical profile is measured at a fixed horizontal position and time, a realistic representation might include the time variations and horizontal drift that occur during the duration of the sampling. Similarly, although an idealized time series exists at a fixed lat-long position, a realistic representation of a moored ocean time series might include the “watch cycle” excursions of horizontal position that occur as a result of tidal currents.

CF Discrete Geometries provides a mechanism to encode both the nominal and the precise positions, while retaining the semantics of the idealized feature type. Only the set of coordinates which are regarded as the nominal (default or preferred) positions should be indicated by the attribute axis, which should be assigned string values to indicate the orientations of the axes (X, Y, Z, or T). See A single timeseries with time-varying deviations from a nominal point spatial location (a single timeseries with time-varying deviations from a nominal point spatial location): Auxiliary coordinate variables containing the nominal and the precise positions should be listed in the relevant coordinates attributes of data variables. In orthogonal representations the nominal positions could be  coordinate variables, which do not need to be listed in the coordinates attribute, rather than auxiliary coordinate variables.

Coordinate bounds may optionally be associated with coordinate variables and auxiliary coordinate variables using the bounds attribute, following the conventions described in section 7.1. Coordinate bounds are especially important for accurate representations of model output data using discrete geometry representations; they record the boundaries of the model grid cells.

If there is a vertical coordinate variable or auxiliary coordinate variable, it must be identified by the means specified in section 4.3.   The use of the attribute axis=Z is recommended for clarity. A standard_name attribute (see section 3.3) that identifies the vertical coordinate is recommended, e.g. "altitude", "height", etc. (See the CF Standard Name Table).

In data for discrete sampling geometries written according to the rules of this section, wherever there are unused elements in data storage, the data variable and all its auxiliary coordinate variables (spatial and time) must contain missing values. This situation may arise for the incomplete multidimensional array representation, and in any representation if the instance dimension is set to a larger size than the number of features currently stored. Data variables should (as usual) also contain missing values to indicate when there is no valid data available for the element, although the coordinates are valid.

Similarly, for indices where the instance variable identified by cf_role contains a missing value indicator, all other instance variables should also contain missing values.

where p(k) is the pressure at gridpoint (k), p0 is a reference pressure, lev(k) is the dimensionless coordinate at vertical gridpoint (k).

The format for the formula_terms attribute is

The standard_name of p0 is reference_air_pressure_for_atmosphere_vertical_coordinate. The computed_standard_name is air_pressure.

where p(n,k,j,i) is the pressure at gridpoint (n,k,j,i), ptop is the pressure at the top of the model, sigma(k) is the dimensionless coordinate at vertical gridpoint (k), and ps(n,j,i) is the surface pressure at horizontal gridpoint (j,i)and time (n).

The format for the formula_terms attribute is

The standard_name of ptop is air_pressure_at_top_of_atmosphere_model, and of ps is surface_air_pressure. The computed_standard_name is air_pressure.

The format for the formula_terms attribute is

where a is replaced by ap if appropriate.

The hybrid sigma-pressure coordinate for level k is defined as a(k)+b(k) or ap(k)/p0+b(k), as appropriate.

The standard_name of p0 is reference_air_pressure_for_atmosphere_vertical_coordinate, and of ps is surface_air_pressure. The computed_standard_name is air_pressure. No standard_name has been defined for a, b or ap.

where z(n,k,j,i) is the height above the datum (e.g. the geoid, which is approximately mean sea level) at gridpoint (k,j,i) and time (n), orog(n,j,i) is the height of the surface above the datum at (j,i) and time (n), and a(k) and b(k) are the coordinates which define hybrid height level k. a(k) has the dimensions of height and b(i) is dimensionless.

The format for the formula_terms attribute is

The standard_name of orog may be surface_altitude (i.e. above the geoid) or surface_height_above_geopotential_datum. The computed_standard_name is altitude or height_above_geopotential_datum in these cases respectively. No standard_name has been defined for b. There is no dimensionless coordinate because a, which has the standard_name of atmosphere_hybrid_height_coordinate, is the best choice for a level-dependent but geographically constant coordinate.

where z(n,k,j,i) is the height above the datum (e.g. the geoid, which is approximately mean sea level) at gridpoint (k,j,i) and time (n), ztop is the height of the top of the model above the datum, and a(k), b1(k), and b2(k) are the dimensionless coordinates which define hybrid level k. zsurf1(n,j,i) and zsurf2(n,j,i) are respectively the large-scale and small-scale components of the topography, and a, b1 and b2 are all functions of the dimensionless SLEVE coordinate. See Shaer et al [SCH02] for details.

The format for the formula_terms attribute is

The standard_name of ztop may be altitude_at_top_of_atmosphere_model (i.e. above the geoid) or height_above_geopotential_datum_at_top_of_atmosphere_model. The computed_standard_name is altitude or height_above_geopotential_datum in these cases respectively. No standard_name has been defined for b1, zsurf1, b2 or zsurf2.

where z(n,k,j,i) is height (positive upwards) relative to the datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the sea surface (positive upwards) relative to the datum at gridpoint (n,j,i), sigma(k) is the dimensionless coordinate at vertical gridpoint (k), and depth(j,i) is the distance (a positive value) from the datum to the sea floor at horizontal gridpoint (j,i).

The format for the formula_terms attribute is

The standard_names for eta and depth and the computed_standard_name must be one of the consistent sets shown in Table D.1.

where z(n,k,j,i) is height (positive upwards) relative to the datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the sea surface (positive upwards) relative to the datum at gridpoint (n,j,i), s(k) is the dimensionless coordinate at vertical gridpoint (k), and depth(j,i) is the distance (a positive value) from the datum to the sea floor at horizontal gridpoint (j,i). The constants a, b, and depth_c control the stretching. The constants a and b are dimensionless, and depth_c must have units of length.

The format for the formula_terms attribute is

The standard_names for eta and depth and the computed_standard_name must be one of the consistent sets shown in Table D.1. No standard_name has been defined for a, b or depth_c.

where z(n,k,j,i) is height, positive upwards, relative to ocean datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the ocean surface, positive upwards, relative to ocean datum at gridpoint (n,j,i), s(k) is the dimensionless coordinate at vertical gridpoint (k) with a range of -1 <= s(k) <= 0, s(0) corresponds to eta(n,j,i) whereas s(-1) corresponds to depth(j,i); C(k) is the dimensionless vertical coordinate stretching function at gridpoint (k) with a range of -1 <= C(k) <= 0, C(0) corresponds to eta(n,j,i) whereas C(-1) corresponds to depth(j,i); the constant depth_c, (positive value), is a critical depth controlling the stretching and depth(j,i) is the distance from ocean datum to sea floor (positive value) at horizontal gridpoint (j,i).

The format for the formula_terms attribute is

The standard_names for eta and depth and the computed_standard_name must be one of the consistent sets shown in Table D.1. No standard_name has been defined for C or depth_c.

where z(n,k,j,i) is height, positive upwards, relative to ocean datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the ocean surface, positive upwards, relative to ocean datum at gridpoint (n,j,i), s(k) is the dimensionless coordinate at vertical gridpoint (k) with a range of -1 <= s(k) <= 0, S(0) corresponds to eta(n,j,i) whereas s(-1) corresponds to depth(j,i); C(k) is the dimensionless vertical coordinate stretching function at gridpoint (k) with a range of -1 <= C(k) <= 0, C(0) corresponds to eta(n,j,i) whereas C(-1) corresponds to depth(j,i); the constant depth_c, (positive value), is a critical depth controlling the stretching and depth(j,i) is the distance from ocean datum to sea floor (positive value) at horizontal gridpoint (j,i).

The format for the formula_terms attribute is

The standard_names for eta and depth and the computed_standard_name must be one of the consistent sets shown in Table D.1. No standard_name has been defined for C or depth_c.

The description of this type of parametric vertical coordinate is defective in version 1.8 and earlier versions of the standard, in that it does not state what values the vertical coordinate variable should contain. Therefore, in accordance with the rules, all versions of the standard before 1.9 are deprecated for datasets that use the "ocean sigma over z" coordinate.

where z(n,k,j,i) is height, positive upwards, relative to ocean datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the ocean surface, positive upwards, relative to ocean datum at gridpoint (n,j,i), and depth(j,i) is the distance from ocean datum to sea floor (positive value) at horizontal gridpoint (j,i).

The parameter sigma(k) is defined only for the nsigma layers nearest the ocean surface, while zlev(k) is defined for the nlayer - nsigma deeper layers, where 0 <= nsigma <= nlayer and nlayer is the size of the dimension of the vertical coordinate variable. Both sigma and zlev must have this dimension. For any k, whichever of sigma(k) or zlev(k) is undefined must contain missing data, while the other must not.

The format for the formula_terms attribute is

The standard_names for eta, depth, zlev and the computed_standard_name must be one of the consistent sets shown in Table D.1. The standard_name for sigma is ocean_sigma_coordinate. No standard_name has been defined for depth_c or nsigma. The nsigma parameter is deprecated and optional in formula_terms; if supplied, it must equal the number of elements of zlev which contain missing data.

The standard_name for the vertical coordinate variable is ocean_sigma_z_coordinate. This variable should contain sigma(k)*depth_c for the layers where sigma is defined and zlev(k) for the other layers, with units of length. The layers must be arranged so that the vertical coordinate variable contains a strictly monotonic set of indicative values for the heights of the levels relative to the datum, either increasing or decreasing, and the direction must be indicated by the positive attribute, in the usual way for a vertical coordinate variable.

where z(k,j,i) is height (positive upwards) relative to the datum (e.g. mean sea level) at gridpoint (k,j,i), sigma(k) is the dimensionless coordinate at vertical gridpoint (k) for k <= k_c, and depth(j,i) is the distance (a positive value) from the datum to sea floor at horizontal gridpoint (j,i). z1, z2, a, and href are constants with units of length.

The format for the formula_terms attribute is

The standard_name for depth and the computed_standard_name must be one of the consistent sets shown in Table D.1. No standard_name has been defined for z1, z2, a, href or k_c.

This grid mapping defines the canonical 2D geographical coordinate system based upon latitude and longitude coordinates. It is included so that the figure of the Earth can be described.