chap_OBS.tex

\documentclass[../main/NEMO_manual]{subfiles}

\begin{document}

\chapter{Observation and Model Comparison (OBS)}
\label{chap:OBS}

%\subsubsection*{Changes record}
%\begin{tabular}{l||l|m{0.65\linewidth}}
%    Release   & Author        & Modifications \\
%    {\em 4.0} & {\em D. J. Lea} & {\em \NEMO\ 4.0 updates}  \\
%    {\em 3.6} & {\em M. Martin, A. Ryan} & {\em Add averaging operator, standalone obs oper} \\
%    {\em 3.4} & {\em D. J. Lea, M. Martin, ...} & {\em Initial version}  \\
%    {\em --\texttt{"}--} & {\em ... K. Mogensen, A. Vidard, A. Weaver} & {\em ---\texttt{"}---}  \\
%\end{tabular}

\chaptertoc

\paragraph{Changes record} ~\\

{\footnotesize
  \begin{tabularx}{\textwidth}{l||X|X}
    Release & Author(s) & Modifications \\
    \hline
    {\em   5.0} & {\em D. Ford} & {\em Update to accommodate interface changes} \\
    {\em   4.0} & {\em ...} & {\em ...} \\
    {\em   3.6} & {\em ...} & {\em ...} \\
    {\em   3.4} & {\em ...} & {\em ...} \\
    {\em <=3.4} & {\em ...} & {\em ...}
  \end{tabularx}
}

\clearpage

The observation and model comparison code, the observation operator (OBS), reads in observation files
(temperature profiles, salinity profiles, velocity profiles, sea surface temperature, sea surface salinity, sea level anomaly, sea surface velocity, sea ice concentration, sea ice thickness)
and calculates an interpolated model equivalent value at the observation location and nearest model time step.
The resulting data are saved in a ``feedback'' file (or files).
The code was originally developed for use with the NEMOVAR data assimilation code,
but can be used for validation or verification of the model or with any other data assimilation system.
Its use can be extended to other variables, for instance biogeochemistry, with small changes to the code.

The OBS code is called from \mdl{nemogcm} for model initialisation and to calculate the model equivalent values for observations on the 0th time step.
The code is then called again after each time step from \mdl{step}.
The code is only activated if the \nam{obs}{obs} namelist logical \np{ln_diaobs}{ln\_diaobs} is set to true.

For all data types a 2D horizontal interpolator or averager is needed to
interpolate/average the model fields to the observation location.
For {\em in situ} profiles, a 1D vertical interpolator is needed in addition to
provide model fields at the observation depths.
This now works in a generalised vertical coordinate system.

Some profile observation types (\eg\ tropical moored buoys) are made available as daily averaged quantities.
The observation operator code can be set-up to calculate the equivalent daily average model temperature fields using
the \np{nn_profdavtypes}{nn\_profdavtypes} namelist array.
Some SST observations are equivalent to a night-time average value and
the observation operator code can calculate equivalent night-time average model fields by
setting the namelist value \np{ln_night}{ln\_night} to true.
Otherwise (by default) the model value from the nearest time step to the observation time is used.

The code is controlled by the namelist \nam{obs}{obs} and multiple instances of the namelist \nam{obs_dta}{obs\_dta}, one for each observation group.
See the following sections for more details on setting up the namelists.

In \autoref{sec:OBS_example} a test example of the observation operator code is introduced, including
where to obtain data and how to setup the namelists.
In \autoref{sec:OBS_advanced} some more technical details of the available options are introduced, and we
also show more complete namelists.
In \autoref{sec:OBS_headers} example ``feedback'' file NetCDF headers are shown.
In \autoref{sec:OBS_newvars} a description of how to add an observation operator for a new variable is given.
In \autoref{sec:OBS_theory} some of the theoretical aspects of the observation operator are described including
interpolation methods and running on multiple processors.
In \autoref{sec:OBS_sao} the standalone observation operator code is described.
In \autoref{sec:OBS_obsutils} we describe some utilities to help work with the files produced by the OBS code.

%% =================================================================================================
\section{Running the observation operator code example}
\label{sec:OBS_example}

In this section an example of running the observation operator code is described using
profile observation data which can be freely downloaded.
It shows how to adapt an existing run and build of \NEMO\ to run the observation operator. Note also the observation operator and the assimilation increments code are run in the ORCA2\_ICE\_OBS SETTE test.

\begin{enumerate}
\item Compile \NEMO.

\item Download some EN4 data from \href{http://www.metoffice.gov.uk/hadobs}{www.metoffice.gov.uk/hadobs}.
  Choose observations which are valid for the period of your test run because
  the observation operator compares the model and observations for a matching date and time.

\item Compile the OBSTOOLS code in the \path{tools} directory using:
\begin{cmds}
./maketools -n OBSTOOLS -m [ARCH]
\end{cmds}

replacing \texttt{[ARCH]} with the build architecture file for your machine. Note the tools are checked out from a separate location of the repository (under \path{/utils/tools}).

\item Convert the EN4 data into feedback format:
\begin{cmds}
enact2fb.exe profiles_01.nc EN.4.1.1.f.profiles.g10.YYYYMM.nc
\end{cmds}

\item Include the following in the \NEMO\ namelist to run the observation operator on this data:
\end{enumerate}

Top-level options are defined through the \nam{obs}{obs} namelist variables.
\begin{itemize}
\item \np{ln_diaobs}{ln\_diaobs} turns on the observation operator code.
\item \np{nn_obsgroups}{nn\_obsgroups} specifies the number of observation groups and therefore number of \nam{obs_dta}{obs\_dta} namelists required; in this example it should be 1, as the temperature and salinity data are grouped together in the same input file(s).
\item The model grid points for a particular observation latitude and longitude are found using
the grid searching part of the code.
This can be expensive, particularly for large numbers of observations,
setting \np{ln_grid_search_lookup}{ln\_grid\_search\_lookup} allows the use of a lookup table which
is saved into an \np{cn_gridsearchfile}{cn\_gridsearchfile} file (or files).
This will need to be generated the first time if it does not exist in the run directory.
However, once produced it will significantly speed up future grid searches.
\np{rn_gridsearchres}{rn\_gridsearchres} specifies the resolution (in degrees) of the grid search.
\item Setting \np{ln_grid_global}{ln\_grid\_global} means that the code distributes the observations evenly between processors.
Alternatively each processor will work with observations located within the model subdomain
(see \autoref{subsec:OBS_parallel}).
\item \np{rn_dobsini}{rn\_dobsini} and \np{rn_dobsend}{rn\_dobsend} can be used to restrict the period the observation operator is run for.
\end{itemize}

A \nam{obs_dta}{obs\_dta} namelist is then needed for each observation group.
Typically, an individual variable (\eg\ sea surface temperature) would be its own group, but if variables are collocated in the input files (\eg\ temperature and salinity profiles, or meridional and zonal velocities) they would be considered as a single group.
\begin{itemize}
\item \np{cn_groupname}{cn\_groupname} is used to name the output ``feedback'' file(s).
\item \np{ln_enabled}{ln\_enabled} must be .true. for the observation operator to be run on this group, setting it to .false. allows it to be turned off without needing to remove the instance of \nam{obs_dta}{obs\_dta}.
\item Setting \np{ln_prof}{ln\_prof} specifies the observations consist of vertical profiles, as in this example.
\np{ln_surf}{ln\_surf} indicates surface data.
One and only one of \np{ln_prof}{ln\_prof} and \np{ln_surf}{ln\_surf} must be .true..
\item \np{cn_obsfiles}{cn\_obsfiles} specifies the input filename or array of filenames.
\item \np{cn_obstypes}{cn\_obstypes} specifies the observation variable(s).
In this example it should be set to 'POTM','PSAL' for profile temperature and profile salinity.
A full list of \np{cn_obstypes}{cn\_obstypes} options is given in \autoref{sec:OBS_headers}.
\end{itemize}
Other \nam{obs_dta}{obs\_dta} options are available to control specific behaviour, as detailed in \autoref{sec:OBS_advanced}.

A number of utilities are now provided to plot the feedback files, convert and recombine the files.
These are explained in more detail in \autoref{sec:OBS_obsutils}.
Utilities to convert other input data formats into the feedback format are also described in
\autoref{sec:OBS_obsutils}.

%% =================================================================================================
\section{Technical details and full namelist options}
\label{sec:OBS_advanced}

Here we show more complete information about the available options in the \nam{obs_dta}{obs\_dta} namelists.
Each item in \nam{obs}{obs} has already been described in \autoref{sec:OBS_example}.
The first six entries in \nam{obs_dta}{obs\_dta}, which are the ones most likely to be modified by users,
have also been described in \autoref{sec:OBS_example}.

\begin{listing}
  \nlst{namobs}
  \caption{\forcode{&namobs}}
  \label{lst:namobs}
\end{listing}

\begin{listing}
  \nlst{namobs_dta}
  \caption{\forcode{&namobs_dta}}
  \label{lst:namobs_dta}
\end{listing}

The remaining options in \nam{obs_dta}{obs\_dta} offer more specialist control, with some only
relevant to specific observation types.

The following are relevant to all observation types:
\begin{itemize}
\item Setting \np{ln_nea}{ln\_nea} rejects observations within a grid cell that neighbours land.
\item Setting \np{ln_bound_reject}{ln\_bound\_reject} rejects observations within a grid cell that neighbours a boundary point.
\item Setting \np{ln_ignmis}{ln\_ignmis} merely issues a warning if a file listed in \np{cn_obsfiles}{cn\_obsfiles} is missing,
otherwise execution of the code ceases with an error.
\item \np{nn_2dint}{nn\_2dint} sets the type of horizontal interpolation method, see \autoref{sec:OBS_theory} for details.
\end{itemize}

The following are only relevant to profile observations:
\begin{itemize}
\item \np{nn_1dint}{nn\_1dint} sets the type of vertical interpolation method, see \autoref{sec:OBS_theory} for details.
\item \np{nn_profdavtypes}{nn\_profdavtypes} is an array of profile observation types corresponding to entries in the
STATION\_TYPE variable in the feedback file (see \autoref{sec:OBS_headers}) which should be treated as
daily averages rather than at a specific time.
Set to -1 if no daily averaged types.
\item Setting \np{ln_all_at_all}{ln\_all\_at\_all} means that if there is more than one variable in
\np{cn_obstypes}{cn\_obstypes}, then a model counterpart is calculated for all variables at any location
for which there is an observation of any of the variables.
\end{itemize}

The following are only relevant to surface observations:
\begin{itemize}
\item Setting \np{ln_fp_indegs}{ln\_fp\_indegs} means that the footprint of the observations (see options below)
is in units of degrees rather than metres.
\item \np{rn_avglamscl}{rn\_avglamscl} and \np{rn_avgphiscl}{rn\_avgphiscl} specify the east/west and north/south
footprint of the observations respectively.
This is only relevant if \np{nn_2dint}{nn\_2dint} is set to 5 or 6.
See \autoref{sec:OBS_theory} for details.
\item Setting \np{ln_night}{ln\_night} calculates a night-time average model counterpart rather than one at
the nearest model time step.
\item Setting \np{ln_time_mean_bkg}{ln\_time\_mean\_bkg} calculates a model counterpart averaged over
\np{rn_time_mean_period}{rn\_time\_mean\_period} (in hours) rather than one at the nearest model time step.
\item Setting \np{ln_obsbias}{ln\_obsbias} allows a NetCDF file containing an estimate of the observation bias
on the model grid to be read in, and this bias to be removed from the observation values.
\np{cn_obsbiasfiles}{cn\_obsbiasfiles} is a list of observation bias file names.
\np{cn_type_to_biascorrect}{cn\_type\_to\_biascorrect} is the variable in \np{cn_obstypes}{cn\_obstypes} to bias correct.
\np{cn_obsbiasfile_varname}{cn\_obsbiasfile\_varname} is the NetCDF variable to read from \np{cn_obsbiasfiles}{cn\_obsbiasfiles}.
\end{itemize}

The following are only relevant to sea level anomaly observations ('SLA' is in \np{cn_obstypes}{cn\_obstypes}),
which require use of a mean dynamic topography (MDT) (see \autoref{sec:OBS_headers}) which may need bias correcting:
\begin{itemize}
\item Setting \np{ln_altbias}{ln\_altbias} means that the MDT used is bias corrected using a bias file on the
model grid in the NetCDF file \np{cn_altbiasfile}{cn\_altbiasfile}.
\item \np{nn_msshc}{nn\_msshc} sets the mean sea surface height (MSSH) correction to apply to the MDT.
0 means no correction is applied.
1 means a correction based on the mean difference between the MDT and model SSH is applied.
2 means a correction of \np{rn_mdtcorr}{rn\_mdtcorr} is applied.
Either correction is only applied at latitudes between \np{rn_mdtcutoff}{rn\_mdtcutoff} degrees north and
\np{rn_mdtcutoff}{rn\_mdtcutoff} degrees south.
\end{itemize}

The following is only relevant to temperature and salinity observations ('POTM', 'PSAL', 'SST', or 'SSS' in
\np{cn_obstypes}{cn\_obstypes}):
\begin{itemize}
\item Setting \np{ln_output_clim}{ln\_output\_clim} means that as well as a model counterpart, an equivalent
climatological counterpart is output.
The climatology used is the same as used for tracer damping (see \autoref{sec:TRA_dmp}).
Therefore, this option also requires \np{ln_tradmp}{ln\_tradmp} to be set in \nam{tra_dmp}{tra\_dmp}.
\end{itemize}

The following is only relevant to sea ice freeboard observations ('FBD' is in \np{cn_obstypes}{cn\_obstypes}):
\begin{itemize}
\item \np{rn_radar_snow_penetr}{rn\_radar\_snow\_penetr} sets the snow depth penetration factor for the
radar ice freeboard conversion.
\end{itemize}

%% =================================================================================================
\section{Example feedback type observation file headers}
\label{sec:OBS_headers}

The observation operator code uses the feedback observation file format for all data types.
All the observation files must be in NetCDF format.
Some example headers (produced using \mbox{\textit{ncdump~-h}}) for temperature and salinity profile data,
sea level anomaly and sea surface temperature are in the following subsections.

The VARIABLES in the feedback files must match those specified in \np{cn_obstypes}{cn\_obstypes} in \nam{obs_dta}{obs\_dta}.
The following variable names can be used:

\begin{itemize}
\item POTM: temperature profiles
\item PSAL: salinity profiles
\item UVEL: zonal velocities (profile or surface)
\item VVEL: meridional velocities (profile or surface)
\item SST: sea surface temperature
\item SLA: sea level anomaly
\item SSS: sea surface salinity
\item ICECONC: sea ice concentration
\item SIT: sea ice thickness
\item FBD: sea ice freeboard
\end{itemize}

Other variables, such as biogeochemical variables from PISCES, are not currently available.
However, to define a new variable a user need only add an extra option to the relevant SELECT CASE statement in the routine dia\_obs, that matches a new observation type with the desired model variable (see \autoref{sec:OBS_newvars}).

%% =================================================================================================
\subsection{Temperature and salinity profile feedback file}

\begin{clines}
netcdf profiles_01 {
dimensions:
     N_OBS = 603 ;
     N_LEVELS = 150 ;
     N_VARS = 2 ;
     N_QCF = 2 ;
     N_ENTRIES = 1 ;
     N_EXTRA = 1 ;
     STRINGNAM = 8 ;
     STRINGGRID = 1 ;
     STRINGWMO = 8 ;
     STRINGTYP = 4 ;
     STRINGJULD = 14 ;
variables:
     char VARIABLES(N_VARS, STRINGNAM) ;
          VARIABLES:long_name = "List of variables in feedback files" ;
     char ENTRIES(N_ENTRIES, STRINGNAM) ;
          ENTRIES:long_name = "List of additional entries for each variable in feedback files" ;
     char EXTRA(N_EXTRA, STRINGNAM) ;
          EXTRA:long_name = "List of extra variables" ;
     char STATION_IDENTIFIER(N_OBS, STRINGWMO) ;
          STATION_IDENTIFIER:long_name = "Station identifier" ;
     char STATION_TYPE(N_OBS, STRINGTYP) ;
          STATION_TYPE:long_name = "Code instrument type" ;
     double LONGITUDE(N_OBS) ;
          LONGITUDE:long_name = "Longitude" ;
          LONGITUDE:units = "degrees_east" ;
          LONGITUDE:_Fillvalue = 99999.f ;
     double LATITUDE(N_OBS) ;
          LATITUDE:long_name = "Latitude" ;
          LATITUDE:units = "degrees_north" ;
          LATITUDE:_Fillvalue = 99999.f ;
     double DEPTH(N_OBS, N_LEVELS) ;
          DEPTH:long_name = "Depth" ;
          DEPTH:units = "metre" ;
          DEPTH:_Fillvalue = 99999.f ;
     int DEPTH_QC(N_OBS, N_LEVELS) ;
          DEPTH_QC:long_name = "Quality on depth" ;
          DEPTH_QC:Conventions = "q where q =[0,9]" ;
          DEPTH_QC:_Fillvalue = 0 ;
     int DEPTH_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
          DEPTH_QC_FLAGS:long_name = "Quality flags on depth" ;
          DEPTH_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
     double JULD(N_OBS) ;
          JULD:long_name = "Julian day" ;
          JULD:units = "days since JULD_REFERENCE" ;
          JULD:Conventions = "relative julian days with decimal part (as parts of day)" ;
          JULD:_Fillvalue = 99999.f ;
     char JULD_REFERENCE(STRINGJULD) ;
          JULD_REFERENCE:long_name = "Date of reference for julian days" ;
          JULD_REFERENCE:Conventions = "YYYYMMDDHHMMSS" ;
     int OBSERVATION_QC(N_OBS) ;
          OBSERVATION_QC:long_name = "Quality on observation" ;
          OBSERVATION_QC:Conventions = "q where q =[0,9]" ;
          OBSERVATION_QC:_Fillvalue = 0 ;
     int OBSERVATION_QC_FLAGS(N_OBS, N_QCF) ;
          OBSERVATION_QC_FLAGS:long_name = "Quality flags on observation" ;
          OBSERVATION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          OBSERVATION_QC_FLAGS:_Fillvalue = 0 ;
     int POSITION_QC(N_OBS) ;
          POSITION_QC:long_name = "Quality on position (latitude and longitude)" ;
          POSITION_QC:Conventions = "q where q =[0,9]" ;
          POSITION_QC:_Fillvalue = 0 ;
     int POSITION_QC_FLAGS(N_OBS, N_QCF) ;
          POSITION_QC_FLAGS:long_name = "Quality flags on position" ;
          POSITION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          POSITION_QC_FLAGS:_Fillvalue = 0 ;
     int JULD_QC(N_OBS) ;
          JULD_QC:long_name = "Quality on date and time" ;
          JULD_QC:Conventions = "q where q =[0,9]" ;
          JULD_QC:_Fillvalue = 0 ;
     int JULD_QC_FLAGS(N_OBS, N_QCF) ;
          JULD_QC_FLAGS:long_name = "Quality flags on date and time" ;
          JULD_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          JULD_QC_FLAGS:_Fillvalue = 0 ;
     int ORIGINAL_FILE_INDEX(N_OBS) ;
          ORIGINAL_FILE_INDEX:long_name = "Index in original data file" ;
          ORIGINAL_FILE_INDEX:_Fillvalue = -99999 ;
     float POTM_OBS(N_OBS, N_LEVELS) ;
          POTM_OBS:long_name = "Potential temperature" ;
          POTM_OBS:units = "Degrees Celsius" ;
          POTM_OBS:_Fillvalue = 99999.f ;
     float POTM_Hx(N_OBS, N_LEVELS) ;
          POTM_Hx:long_name = "Model interpolated potential temperature" ;
          POTM_Hx:units = "Degrees Celsius" ;
          POTM_Hx:_Fillvalue = 99999.f ;
     int POTM_QC(N_OBS) ;
          POTM_QC:long_name = "Quality on potential temperature" ;
          POTM_QC:Conventions = "q where q =[0,9]" ;
          POTM_QC:_Fillvalue = 0 ;
     int POTM_QC_FLAGS(N_OBS, N_QCF) ;
          POTM_QC_FLAGS:long_name = "Quality flags on potential temperature" ;
          POTM_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          POTM_QC_FLAGS:_Fillvalue = 0 ;
     int POTM_LEVEL_QC(N_OBS, N_LEVELS) ;
          POTM_LEVEL_QC:long_name = "Quality for each level on potential temperature" ;
          POTM_LEVEL_QC:Conventions = "q where q =[0,9]" ;
          POTM_LEVEL_QC:_Fillvalue = 0 ;
     int POTM_LEVEL_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
          POTM_LEVEL_QC_FLAGS:long_name = "Quality flags for each level on potential temperature" ;
          POTM_LEVEL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          POTM_LEVEL_QC_FLAGS:_Fillvalue = 0 ;
     int POTM_IOBSI(N_OBS) ;
          POTM_IOBSI:long_name = "ORCA grid search I coordinate" ;
     int POTM_IOBSJ(N_OBS) ;
          POTM_IOBSJ:long_name = "ORCA grid search J coordinate" ;
     int POTM_IOBSK(N_OBS, N_LEVELS) ;
          POTM_IOBSK:long_name = "ORCA grid search K coordinate" ;
     char POTM_GRID(STRINGGRID) ;
          POTM_GRID:long_name = "ORCA grid search grid (T,U,V)" ;
     float PSAL_OBS(N_OBS, N_LEVELS) ;
          PSAL_OBS:long_name = "Practical salinity" ;
          PSAL_OBS:units = "PSU" ;
          PSAL_OBS:_Fillvalue = 99999.f ;
     float PSAL_Hx(N_OBS, N_LEVELS) ;
          PSAL_Hx:long_name = "Model interpolated practical salinity" ;
          PSAL_Hx:units = "PSU" ;
          PSAL_Hx:_Fillvalue = 99999.f ;
     int PSAL_QC(N_OBS) ;
          PSAL_QC:long_name = "Quality on practical salinity" ;
          PSAL_QC:Conventions = "q where q =[0,9]" ;
          PSAL_QC:_Fillvalue = 0 ;
     int PSAL_QC_FLAGS(N_OBS, N_QCF) ;
          PSAL_QC_FLAGS:long_name = "Quality flags on practical salinity" ;
          PSAL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          PSAL_QC_FLAGS:_Fillvalue = 0 ;
     int PSAL_LEVEL_QC(N_OBS, N_LEVELS) ;
          PSAL_LEVEL_QC:long_name = "Quality for each level on practical salinity" ;
          PSAL_LEVEL_QC:Conventions = "q where q =[0,9]" ;
          PSAL_LEVEL_QC:_Fillvalue = 0 ;
     int PSAL_LEVEL_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
          PSAL_LEVEL_QC_FLAGS:long_name = "Quality flags for each level on practical salinity" ;
          PSAL_LEVEL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          PSAL_LEVEL_QC_FLAGS:_Fillvalue = 0 ;
     int PSAL_IOBSI(N_OBS) ;
          PSAL_IOBSI:long_name = "ORCA grid search I coordinate" ;
     int PSAL_IOBSJ(N_OBS) ;
          PSAL_IOBSJ:long_name = "ORCA grid search J coordinate" ;
     int PSAL_IOBSK(N_OBS, N_LEVELS) ;
          PSAL_IOBSK:long_name = "ORCA grid search K coordinate" ;
     char PSAL_GRID(STRINGGRID) ;
          PSAL_GRID:long_name = "ORCA grid search grid (T,U,V)" ;
     float TEMP(N_OBS, N_LEVELS) ;
          TEMP:long_name = "Insitu temperature" ;
          TEMP:units = "Degrees Celsius" ;
          TEMP:_Fillvalue = 99999.f ;

// global attributes:
          :title = "NEMO observation operator output" ;
          :Convention = "NEMO unified observation operator output" ;
}
\end{clines}

%% =================================================================================================
\subsection{Sea level anomaly feedback file}

\begin{clines}
netcdf sla_01 {
dimensions:
     N_OBS = 41301 ;
     N_LEVELS = 1 ;
     N_VARS = 1 ;
     N_QCF = 2 ;
     N_ENTRIES = 1 ;
     N_EXTRA = 1 ;
     STRINGNAM = 8 ;
     STRINGGRID = 1 ;
     STRINGWMO = 8 ;
     STRINGTYP = 4 ;
     STRINGJULD = 14 ;
variables:
     char VARIABLES(N_VARS, STRINGNAM) ;
          VARIABLES:long_name = "List of variables in feedback files" ;
     char ENTRIES(N_ENTRIES, STRINGNAM) ;
          ENTRIES:long_name = "List of additional entries for each variable in feedback files" ;
     char EXTRA(N_EXTRA, STRINGNAM) ;
          EXTRA:long_name = "List of extra variables" ;
     char STATION_IDENTIFIER(N_OBS, STRINGWMO) ;
          STATION_IDENTIFIER:long_name = "Station identifier" ;
     char STATION_TYPE(N_OBS, STRINGTYP) ;
          STATION_TYPE:long_name = "Code instrument type" ;
     double LONGITUDE(N_OBS) ;
          LONGITUDE:long_name = "Longitude" ;
          LONGITUDE:units = "degrees_east" ;
          LONGITUDE:_Fillvalue = 99999.f ;
     double LATITUDE(N_OBS) ;
          LATITUDE:long_name = "Latitude" ;
          LATITUDE:units = "degrees_north" ;
          LATITUDE:_Fillvalue = 99999.f ;
     double DEPTH(N_OBS, N_LEVELS) ;
          DEPTH:long_name = "Depth" ;
          DEPTH:units = "metre" ;
          DEPTH:_Fillvalue = 99999.f ;
     int DEPTH_QC(N_OBS, N_LEVELS) ;
          DEPTH_QC:long_name = "Quality on depth" ;
          DEPTH_QC:Conventions = "q where q =[0,9]" ;
          DEPTH_QC:_Fillvalue = 0 ;
     int DEPTH_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
          DEPTH_QC_FLAGS:long_name = "Quality flags on depth" ;
          DEPTH_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
     double JULD(N_OBS) ;
          JULD:long_name = "Julian day" ;
          JULD:units = "days since JULD_REFERENCE" ;
          JULD:Conventions = "relative julian days with decimal part (as parts of day)" ;
          JULD:_Fillvalue = 99999.f ;
     char JULD_REFERENCE(STRINGJULD) ;
          JULD_REFERENCE:long_name = "Date of reference for julian days" ;
          JULD_REFERENCE:Conventions = "YYYYMMDDHHMMSS" ;
     int OBSERVATION_QC(N_OBS) ;
          OBSERVATION_QC:long_name = "Quality on observation" ;
          OBSERVATION_QC:Conventions = "q where q =[0,9]" ;
          OBSERVATION_QC:_Fillvalue = 0 ;
     int OBSERVATION_QC_FLAGS(N_OBS, N_QCF) ;
          OBSERVATION_QC_FLAGS:long_name = "Quality flags on observation" ;
          OBSERVATION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          OBSERVATION_QC_FLAGS:_Fillvalue = 0 ;
     int POSITION_QC(N_OBS) ;
          POSITION_QC:long_name = "Quality on position (latitude and longitude)" ;
          POSITION_QC:Conventions = "q where q =[0,9]" ;
          POSITION_QC:_Fillvalue = 0 ;
     int POSITION_QC_FLAGS(N_OBS, N_QCF) ;
          POSITION_QC_FLAGS:long_name = "Quality flags on position" ;
          POSITION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          POSITION_QC_FLAGS:_Fillvalue = 0 ;
     int JULD_QC(N_OBS) ;
          JULD_QC:long_name = "Quality on date and time" ;
          JULD_QC:Conventions = "q where q =[0,9]" ;
          JULD_QC:_Fillvalue = 0 ;
     int JULD_QC_FLAGS(N_OBS, N_QCF) ;
          JULD_QC_FLAGS:long_name = "Quality flags on date and time" ;
          JULD_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          JULD_QC_FLAGS:_Fillvalue = 0 ;
     int ORIGINAL_FILE_INDEX(N_OBS) ;
          ORIGINAL_FILE_INDEX:long_name = "Index in original data file" ;
          ORIGINAL_FILE_INDEX:_Fillvalue = -99999 ;
     float SLA_OBS(N_OBS, N_LEVELS) ;
          SLA_OBS:long_name = "Sea level anomaly" ;
          SLA_OBS:units = "metre" ;
          SLA_OBS:_Fillvalue = 99999.f ;
     float SLA_Hx(N_OBS, N_LEVELS) ;
          SLA_Hx:long_name = "Model interpolated sea level anomaly" ;
          SLA_Hx:units = "metre" ;
          SLA_Hx:_Fillvalue = 99999.f ;
     int SLA_QC(N_OBS) ;
          SLA_QC:long_name = "Quality on sea level anomaly" ;
          SLA_QC:Conventions = "q where q =[0,9]" ;
          SLA_QC:_Fillvalue = 0 ;
     int SLA_QC_FLAGS(N_OBS, N_QCF) ;
          SLA_QC_FLAGS:long_name = "Quality flags on sea level anomaly" ;
          SLA_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          SLA_QC_FLAGS:_Fillvalue = 0 ;
     int SLA_LEVEL_QC(N_OBS, N_LEVELS) ;
          SLA_LEVEL_QC:long_name = "Quality for each level on sea level anomaly" ;
          SLA_LEVEL_QC:Conventions = "q where q =[0,9]" ;
          SLA_LEVEL_QC:_Fillvalue = 0 ;
     int SLA_LEVEL_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
          SLA_LEVEL_QC_FLAGS:long_name = "Quality flags for each level on sea level anomaly" ;
          SLA_LEVEL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          SLA_LEVEL_QC_FLAGS:_Fillvalue = 0 ;
     int SLA_IOBSI(N_OBS) ;
          SLA_IOBSI:long_name = "ORCA grid search I coordinate" ;
     int SLA_IOBSJ(N_OBS) ;
          SLA_IOBSJ:long_name = "ORCA grid search J coordinate" ;
     int SLA_IOBSK(N_OBS, N_LEVELS) ;
          SLA_IOBSK:long_name = "ORCA grid search K coordinate" ;
     char SLA_GRID(STRINGGRID) ;
          SLA_GRID:long_name = "ORCA grid search grid (T,U,V)" ;
     float MDT(N_OBS, N_LEVELS) ;
          MDT:long_name = "Mean Dynamic Topography" ;
          MDT:units = "metre" ;
          MDT:_Fillvalue = 99999.f ;

// global attributes:
          :title = "NEMO observation operator output" ;
          :Convention = "NEMO unified observation operator output" ;
}
\end{clines}

To use Sea Level Anomaly (SLA) data the mean dynamic topography (MDT) must be provided in a separate file defined on
the model grid called \textit{slaReferenceLevel.nc}.
The MDT is required in order to produce the model equivalent sea level anomaly from the model sea surface height.
Below is an example header for this file (on the ORCA025 grid).

\begin{clines}
dimensions:
        x = 1442 ;
        y = 1021 ;
variables:
        float nav_lon(y, x) ;
                nav_lon:units = "degrees_east" ;
        float nav_lat(y, x) ;
                nav_lat:units = "degrees_north" ;
        float sossheig(y, x) ;
                sossheig:_FillValue = -1.e+30f ;
                sossheig:coordinates = "nav_lon nav_lat" ;
                sossheig:long_name = "Mean Dynamic Topography" ;
                sossheig:units = "metres" ;
                sossheig:grid = "orca025T" ;
\end{clines}

%% =================================================================================================
\subsection{Sea surface temperature feedback file}

\begin{clines}
netcdf sst_01 {
dimensions:
     N_OBS = 33099 ;
     N_LEVELS = 1 ;
     N_VARS = 1 ;
     N_QCF = 2 ;
     N_ENTRIES = 1 ;
     STRINGNAM = 8 ;
     STRINGGRID = 1 ;
     STRINGWMO = 8 ;
     STRINGTYP = 4 ;
     STRINGJULD = 14 ;
variables:
     char VARIABLES(N_VARS, STRINGNAM) ;
          VARIABLES:long_name = "List of variables in feedback files" ;
     char ENTRIES(N_ENTRIES, STRINGNAM) ;
          ENTRIES:long_name = "List of additional entries for each variable in feedback files" ;
     char STATION_IDENTIFIER(N_OBS, STRINGWMO) ;
          STATION_IDENTIFIER:long_name = "Station identifier" ;
     char STATION_TYPE(N_OBS, STRINGTYP) ;
          STATION_TYPE:long_name = "Code instrument type" ;
     double LONGITUDE(N_OBS) ;
          LONGITUDE:long_name = "Longitude" ;
          LONGITUDE:units = "degrees_east" ;
          LONGITUDE:_Fillvalue = 99999.f ;
     double LATITUDE(N_OBS) ;
          LATITUDE:long_name = "Latitude" ;
          LATITUDE:units = "degrees_north" ;
          LATITUDE:_Fillvalue = 99999.f ;
     double DEPTH(N_OBS, N_LEVELS) ;
          DEPTH:long_name = "Depth" ;
          DEPTH:units = "metre" ;
          DEPTH:_Fillvalue = 99999.f ;
     int DEPTH_QC(N_OBS, N_LEVELS) ;
          DEPTH_QC:long_name = "Quality on depth" ;
          DEPTH_QC:Conventions = "q where q =[0,9]" ;
          DEPTH_QC:_Fillvalue = 0 ;
     int DEPTH_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
          DEPTH_QC_FLAGS:long_name = "Quality flags on depth" ;
          DEPTH_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
     double JULD(N_OBS) ;
          JULD:long_name = "Julian day" ;
          JULD:units = "days since JULD_REFERENCE" ;
          JULD:Conventions = "relative julian days with decimal part (as parts of day)" ;
          JULD:_Fillvalue = 99999.f ;
     char JULD_REFERENCE(STRINGJULD) ;
          JULD_REFERENCE:long_name = "Date of reference for julian days" ;
          JULD_REFERENCE:Conventions = "YYYYMMDDHHMMSS" ;
     int OBSERVATION_QC(N_OBS) ;
          OBSERVATION_QC:long_name = "Quality on observation" ;
          OBSERVATION_QC:Conventions = "q where q =[0,9]" ;
          OBSERVATION_QC:_Fillvalue = 0 ;
     int OBSERVATION_QC_FLAGS(N_OBS, N_QCF) ;
          OBSERVATION_QC_FLAGS:long_name = "Quality flags on observation" ;
          OBSERVATION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          OBSERVATION_QC_FLAGS:_Fillvalue = 0 ;
     int POSITION_QC(N_OBS) ;
          POSITION_QC:long_name = "Quality on position (latitude and longitude)" ;
          POSITION_QC:Conventions = "q where q =[0,9]" ;
          POSITION_QC:_Fillvalue = 0 ;
     int POSITION_QC_FLAGS(N_OBS, N_QCF) ;
          POSITION_QC_FLAGS:long_name = "Quality flags on position" ;
          POSITION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          POSITION_QC_FLAGS:_Fillvalue = 0 ;
     int JULD_QC(N_OBS) ;
          JULD_QC:long_name = "Quality on date and time" ;
          JULD_QC:Conventions = "q where q =[0,9]" ;
          JULD_QC:_Fillvalue = 0 ;
     int JULD_QC_FLAGS(N_OBS, N_QCF) ;
          JULD_QC_FLAGS:long_name = "Quality flags on date and time" ;
          JULD_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          JULD_QC_FLAGS:_Fillvalue = 0 ;
     int ORIGINAL_FILE_INDEX(N_OBS) ;
          ORIGINAL_FILE_INDEX:long_name = "Index in original data file" ;
          ORIGINAL_FILE_INDEX:_Fillvalue = -99999 ;
     float SST_OBS(N_OBS, N_LEVELS) ;
          SST_OBS:long_name = "Sea surface temperature" ;
          SST_OBS:units = "Degree centigrade" ;
          SST_OBS:_Fillvalue = 99999.f ;
     float SST_Hx(N_OBS, N_LEVELS) ;
          SST_Hx:long_name = "Model interpolated sea surface temperature" ;
          SST_Hx:units = "Degree centigrade" ;
          SST_Hx:_Fillvalue = 99999.f ;
     int SST_QC(N_OBS) ;
          SST_QC:long_name = "Quality on sea surface temperature" ;
          SST_QC:Conventions = "q where q =[0,9]" ;
          SST_QC:_Fillvalue = 0 ;
     int SST_QC_FLAGS(N_OBS, N_QCF) ;
          SST_QC_FLAGS:long_name = "Quality flags on sea surface temperature" ;
          SST_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          SST_QC_FLAGS:_Fillvalue = 0 ;
     int SST_LEVEL_QC(N_OBS, N_LEVELS) ;
          SST_LEVEL_QC:long_name = "Quality for each level on sea surface temperature" ;
          SST_LEVEL_QC:Conventions = "q where q =[0,9]" ;
          SST_LEVEL_QC:_Fillvalue = 0 ;
     int SST_LEVEL_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
          SST_LEVEL_QC_FLAGS:long_name = "Quality flags for each level on sea surface temperature" ;
          SST_LEVEL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
          SST_LEVEL_QC_FLAGS:_Fillvalue = 0 ;
     int SST_IOBSI(N_OBS) ;
          SST_IOBSI:long_name = "ORCA grid search I coordinate" ;
     int SST_IOBSJ(N_OBS) ;
          SST_IOBSJ:long_name = "ORCA grid search J coordinate" ;
     int SST_IOBSK(N_OBS, N_LEVELS) ;
          SST_IOBSK:long_name = "ORCA grid search K coordinate" ;
     char SST_GRID(STRINGGRID) ;
          SST_GRID:long_name = "ORCA grid search grid (T,U,V)" ;

// global attributes:
          :title = "NEMO observation operator output" ;
          :Convention = "NEMO unified observation operator output" ;
}
\end{clines}

%% =================================================================================================
\section{Adding code for a new variable}
\label{sec:OBS_newvars}

As detailed in \autoref{sec:OBS_headers}, the OBS interface is only defined for specific physics variables.
However, the internal observation operator code is largely generic, and for observation types which can be
directly matched to model variables, is relatively straightforward to implement for additional variables.
This could include biogeochemical variables from PISCES or another coupled model.

In the routine dia\_obs in \mdl{diaobs} (\path{src/OCE/OBS/diaobs.F90}) there are two \textbf{SELECT CASE}
constructs, one for surface variables and one for profile variables.
To implement an observation operator for a new variable, an extra \textbf{CASE} need simply be added to the
relevant \textbf{SELECT CASE} construct.
To give a generic example for a profile variable:

\begin{forlines}
CASE('OBSNAME')
   zprofvar(:,:,:) = MODELVARIABLE(:,:,:,Kmm)
\end{forlines}

where MODELVARIABLE is the model variable to be matched to the observation type, and OBSNAME is the name of
the observation variable used in the feedback file and in \np{cn_obstypes}{cn\_obstypes} in the
\nam{obs_dta}{obs\_dta} namelist.
OBSNAME must have a maximum of eight characters.

Then, a corresponding namelist and feedback file(s) just need to be supplied, following the method
described in \autoref{sec:OBS_example}.

%% =================================================================================================
\section{Theoretical details}
\label{sec:OBS_theory}

%% =================================================================================================
\subsection{Horizontal interpolation and averaging methods}

For most observation types, the horizontal extent of the observation is small compared to the model grid size and so
the model equivalent of the observation is calculated by interpolating from
the four surrounding grid points to the observation location.
Some satellite observations (\eg\ microwave satellite SST data, or satellite SSS data) have a footprint which
is similar in size or larger than the model grid size (particularly when the grid size is small).
In those cases the model counterpart should be calculated by averaging the model grid points over
the same size as the footprint.
\NEMO\ therefore has the capability to specify either an interpolation or an averaging
(for surface observation types only).

The main namelist option associated with the interpolation/averaging is \np{nn_2dint}{nn\_2dint}.
This default option can be set to values from 0 to 6.
Values between 0 to 4 are associated with interpolation while values 5 or 6 are associated with averaging.
\begin{itemize}
\item \np[=0]{nn_2dint}{nn\_2dint}: Distance-weighted interpolation
\item \np[=1]{nn_2dint}{nn\_2dint}: Distance-weighted interpolation (small angle)
\item \np[=2]{nn_2dint}{nn\_2dint}: Bilinear interpolation (geographical grid)
\item \np[=3]{nn_2dint}{nn\_2dint}: Bilinear remapping interpolation (general grid)
\item \np[=4]{nn_2dint}{nn\_2dint}: Polynomial interpolation
\item \np[=5]{nn_2dint}{nn\_2dint}: Radial footprint averaging with diameter specified in the namelist as
  \texttt{rn\_avglamscl} in degrees or metres (set using \texttt{ln\_fp\_indegs})
\item \np[=6]{nn_2dint}{nn\_2dint}: Rectangular footprint averaging with E/W and N/S size specified in
  the namelist as \texttt{rn\_avglamscl} and \texttt{rn\_avgphiscl} in degrees or metres
  (set using \texttt{ln\_fp\_indegs})
\end{itemize}

Below is some more detail on the various options for interpolation and averaging available in \NEMO.

%% =================================================================================================
\subsubsection{Horizontal interpolation}

Consider an observation point ${\mathrm P}$ with longitude and latitude (${\lambda_{}}_{\mathrm P}$, $\phi_{\mathrm P}$) and
the four nearest neighbouring model grid points ${\mathrm A}$, ${\mathrm B}$, ${\mathrm C}$ and ${\mathrm D}$ with
longitude and latitude ($\lambda_{\mathrm A}$, $\phi_{\mathrm A}$),($\lambda_{\mathrm B}$, $\phi_{\mathrm B}$) etc.
All horizontal interpolation methods implemented in \NEMO\ estimate the value of a model variable $x$ at point $P$ as
a weighted linear combination of the values of the model variables at the grid points ${\mathrm A}$, ${\mathrm B}$ etc.:

\begin{align*}
  {x_{}}_{\mathrm P} =
\frac{1}{w} \left( {w_{}}_{\mathrm A} {x_{}}_{\mathrm A} +
{w_{}}_{\mathrm B} {x_{}}_{\mathrm B} +
{w_{}}_{\mathrm C} {x_{}}_{\mathrm C} +
{w_{}}_{\mathrm D} {x_{}}_{\mathrm D} \right)
\end{align*}

where ${w_{}}_{\mathrm A}$, ${w_{}}_{\mathrm B}$ etc. are the respective weights for the model field at
points ${\mathrm A}$, ${\mathrm B}$ etc., and $w = {w_{}}_{\mathrm A} + {w_{}}_{\mathrm B} + {w_{}}_{\mathrm C} + {w_{}}_{\mathrm D}$.

Four different possibilities are available for computing the weights.

\begin{enumerate}
\item {\bfseries Great-Circle distance-weighted interpolation.}
  The weights are computed as a function of the great-circle distance $s(P, \cdot)$ between $P$ and
  the model grid points $A$, $B$ etc.
  For example, the weight given to the field ${x_{}}_{\mathrm A}$ is specified as the product of the distances
  from ${\mathrm P}$ to the other points:

  \begin{alignat*}{2}
    {w_{}}_{\mathrm A} = s({\mathrm P}, {\mathrm B}) \, s({\mathrm P}, {\mathrm C}) \, s({\mathrm P}, {\mathrm D})
  \end{alignat*}

  where

  \begin{alignat*}{2}
    s\left({\mathrm P}, {\mathrm M} \right) & = & \hspace{0.25em} \cos^{-1} \! \left\{
               \sin {\phi_{}}_{\mathrm P} \sin {\phi_{}}_{\mathrm M}
             + \cos {\phi_{}}_{\mathrm P} \cos {\phi_{}}_{\mathrm M}
               \cos ({\lambda_{}}_{\mathrm M} - {\lambda_{}}_{\mathrm P})
                   \right\}
   \end{alignat*}

   and $M$ corresponds to $B$, $C$ or $D$.
   A more stable form of the great-circle distance formula for small distances ($x$ near 1)
   involves the arcsine function (\eg\ see p.~101 of \citet{daley.barker_bk01}:

   \begin{alignat*}{2}
     s\left( {\mathrm P}, {\mathrm M} \right) = \sin^{-1} \! \left\{ \sqrt{ 1 - x^2 } \right\}
   \end{alignat*}

   where

   \begin{alignat*}{2}
     x = {a_{}}_{\mathrm M} {a_{}}_{\mathrm P} + {b_{}}_{\mathrm M} {b_{}}_{\mathrm P} + {c_{}}_{\mathrm M} {c_{}}_{\mathrm P}
   \end{alignat*}

   and

   \begin{alignat*}{3}
   & {a_{}}_{\mathrm M} & = && \quad \sin {\phi_{}}_{\mathrm M}, \\
   & {a_{}}_{\mathrm P} & = && \quad \sin {\phi_{}}_{\mathrm P}, \\
   & {b_{}}_{\mathrm M} & = && \quad \cos {\phi_{}}_{\mathrm M} \cos {\phi_{}}_{\mathrm M}, \\
   & {b_{}}_{\mathrm P} & = && \quad \cos {\phi_{}}_{\mathrm P} \cos {\phi_{}}_{\mathrm P}, \\
   & {c_{}}_{\mathrm M} & = && \quad \cos {\phi_{}}_{\mathrm M} \sin {\phi_{}}_{\mathrm M}, \\
   & {c_{}}_{\mathrm P} & = && \quad \cos {\phi_{}}_{\mathrm P} \sin {\phi_{}}_{\mathrm P}.
   \end{alignat*}

\item {\bfseries Great-Circle distance-weighted interpolation with small angle approximation.}
  Similar to the previous interpolation but with the distance $s$ computed as
  \begin{alignat*}{2}
    s\left( {\mathrm P}, {\mathrm M} \right)
    & = & \sqrt{ \left( {\phi_{}}_{\mathrm M} - {\phi_{}}_{\mathrm P} \right)^{2}
                                      + \left( {\lambda_{}}_{\mathrm M} - {\lambda_{}}_{\mathrm P} \right)^{2}
                                      \cos^{2} {\phi_{}}_{\mathrm M} }
  \end{alignat*}
  where $M$ corresponds to $A$, $B$, $C$ or $D$.

\item {\bfseries Bilinear interpolation for a regular spaced grid.}
  The interpolation is split into two 1D interpolations in the longitude and latitude directions, respectively.

\item {\bfseries Bilinear remapping interpolation for a general grid.}
  An iterative scheme that involves first mapping a quadrilateral cell into
  a cell with coordinates (0,0), (1,0), (0,1) and (1,1).
  This method is based on the \href{https://github.com/SCRIP-Project/SCRIP}{SCRIP interpolation package}.

\end{enumerate}

%% =================================================================================================
\subsubsection{Horizontal averaging}

For each surface observation type:
\begin{itemize}
\item The standard grid-searching code is used to find the nearest model grid point to the observation location
  (see next subsection).
\item The maximum number of grid points required for that observation in each local grid domain is calculated. Some of these points may later turn out to have zero weight depending on the shape of the footprint.
\item The longitudes and latitudes of the grid points surrounding the nearest model grid box are extracted using
  existing MPI routines.
\item The weights for each grid point associated with each observation are calculated,
  either for radial or rectangular footprints.
  For grid points completely within the footprint, the weight is one;
  for grid points completely outside the footprint, the weight is zero.
  For grid points which are partly within the footprint the ratio between the area of the footprint within
  the grid box and the total area of the grid box is used as the weight.
\item The weighted average of the model grid points associated with each observation is calculated,
  and this is then given as the model counterpart of the observation.
\end{itemize}

Examples of the weights calculated for an observation with rectangular and radial footprints are shown in
\autoref{fig:OBS_avgrec} and~\autoref{fig:OBS_avgrad}.

\begin{figure}
  \centering
  \includegraphics[width=0.66\textwidth]{OBS_avg_rec}
  \caption[Observational weights with a rectangular footprint]{
    Weights associated with each model grid box (blue lines and numbers)
    for an observation at -170.5\deg{E}, 56.0\deg{N} with a rectangular footprint of 1\deg\ x 1\deg.}
  \label{fig:OBS_avgrec}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[width=0.66\textwidth]{OBS_avg_rad}
  \caption[Observational weights with a radial footprint]{
    Weights associated with each model grid box (blue lines and numbers)
    for an observation at -170.5\deg{E}, 56.0\deg{N} with a radial footprint with diameter 1\deg.}
  \label{fig:OBS_avgrad}
\end{figure}

%% =================================================================================================
\subsection{Grid search}

For many grids used by the \NEMO\ model, such as the ORCA family, the horizontal grid coordinates $i$ and $j$ are not simple functions of latitude and longitude.
Therefore, it is not always straightforward to determine the grid points surrounding any given observational position.
Before the interpolation can be performed, a search algorithm is then required to determine the corner points of
the quadrilateral cell in which the observation is located.
This is the most difficult and time consuming part of the 2D interpolation procedure.
A robust test for determining if an observation falls within a given quadrilateral cell is as follows.
Let ${\mathrm P}({\lambda_{}}_{\mathrm P} ,{\phi_{}}_{\mathrm P} )$ denote the observation point,
and let ${\mathrm A}({\lambda_{}}_{\mathrm A} ,{\phi_{}}_{\mathrm A} )$, ${\mathrm B}({\lambda_{}}_{\mathrm B} ,{\phi_{}}_{\mathrm B} )$,
${\mathrm C}({\lambda_{}}_{\mathrm C} ,{\phi_{}}_{\mathrm C} )$ and ${\mathrm D}({\lambda_{}}_{\mathrm D} ,{\phi_{}}_{\mathrm D} )$
denote the bottom left, bottom right, top left and top right corner points of the cell, respectively.
To determine if P is inside the cell, we verify that the cross-products
\begin{align*}
  \begin{array}{lllll}
    {{\mathbf r}_{}}_{\mathrm PA} \times {{\mathbf r}_{}}_{\mathrm PC}
    & = & [({\lambda_{}}_{\mathrm A}\; -\; {\lambda_{}}_{\mathrm P} )
          ({\phi_{}}_{\mathrm C}   \; -\; {\phi_{}}_{\mathrm P} )
          - ({\lambda_{}}_{\mathrm C}\; -\; {\lambda_{}}_{\mathrm P} )
          ({\phi_{}}_{\mathrm A}   \; -\; {\phi_{}}_{\mathrm P} )] \; \widehat{\mathbf k} \\
    {{\mathbf r}_{}}_{\mathrm PB} \times {{\mathbf r}_{}}_{\mathrm PA}
    & = & [({\lambda_{}}_{\mathrm B}\; -\; {\lambda_{}}_{\mathrm P} )
          ({\phi_{}}_{\mathrm A}   \; -\; {\phi_{}}_{\mathrm P} )
          - ({\lambda_{}}_{\mathrm A}\; -\; {\lambda_{}}_{\mathrm P} )
          ({\phi_{}}_{\mathrm B}   \; -\; {\phi_{}}_{\mathrm P} )] \; \widehat{\mathbf k} \\
    {{\mathbf r}_{}}_{\mathrm PC} \times {{\mathbf r}_{}}_{\mathrm PD}
    & = & [({\lambda_{}}_{\mathrm C}\; -\; {\lambda_{}}_{\mathrm P} )
          ({\phi_{}}_{\mathrm D}   \; -\; {\phi_{}}_{\mathrm P} )
          - ({\lambda_{}}_{\mathrm D}\; -\; {\lambda_{}}_{\mathrm P} )
          ({\phi_{}}_{\mathrm C}   \; -\; {\phi_{}}_{\mathrm P} )] \; \widehat{\mathbf k} \\
    {{\mathbf r}_{}}_{\mathrm PD} \times {{\mathbf r}_{}}_{\mathrm PB}
    & = & [({\lambda_{}}_{\mathrm D}\; -\; {\lambda_{}}_{\mathrm P} )
          ({\phi_{}}_{\mathrm B}   \; -\; {\phi_{}}_{\mathrm P} )
          - ({\lambda_{}}_{\mathrm B}\; -\; {\lambda_{}}_{\mathrm P} )
          ({\phi_{}}_{\mathrm D}  \;  - \; {\phi_{}}_{\mathrm P} )] \; \widehat{\mathbf k} \\
  \end{array}
  % \label{eq:OBS_cross}
\end{align*}
point in the opposite direction to the unit normal $\widehat{\mathbf k}$
(\ie\ that the coefficients of $\widehat{\mathbf k}$ are negative),
where ${{\mathbf r}_{}}_{\mathrm PA}$, ${{\mathbf r}_{}}_{\mathrm PB}$, etc. correspond to
the vectors between points P and A, P and B, etc..
The method used is similar to the method used in the \href{https://github.com/SCRIP-Project/SCRIP}{SCRIP interpolation package}.

In order to speed up the grid search, there is the possibility to construct a lookup table for a user specified resolution.
This lookup table contains the lower and upper bounds on the $i$ and $j$ indices to
be searched for on a regular grid.
For each observation position, the closest point on the regular grid of this position is computed and
the $i$ and $j$ ranges of this point searched to determine the precise four points surrounding the observation.

%% =================================================================================================
\subsection{Parallel aspects of horizontal interpolation}
\label{subsec:OBS_parallel}

For horizontal interpolation, there is the basic problem that
the observations are unevenly distributed on the globe.
In \NEMO\ the model grid is divided into subgrids (or domains) where
each subgrid is executed on a single processing element with explicit message passing for
exchange of information along the domain boundaries when running on a massively parallel processor (MPP) system.

For observations there is no natural distribution since the observations are not equally distributed on the globe.
Two options have been made available:
1) geographical distribution;
and 2) round-robin.

%% =================================================================================================
\subsubsection{Geographical distribution of observations among processors}

\begin{figure}
  \centering
  \includegraphics[width=0.66\textwidth]{OBS_obsdist_local}
  \caption[Observations with the geographical distribution]{
    Example of the distribution of observations with
    the geographical distribution of observational data}
  \label{fig:OBS_local}
\end{figure}

This is the simplest option in which the observations are distributed according to
the domain of the grid-point parallelization.
\autoref{fig:OBS_local} shows an example of the distribution of the {\em in situ} data on processors with
a different colour for each observation on a given processor for a 4 $\times$ 2 decomposition with ORCA2.
The grid-point domain decomposition is clearly visible on the plot.

The advantage of this approach is that all information needed for horizontal interpolation is available without
any MPP communication.
This is under the assumption that we are dealing with point observations and only using a $2 \times 2$ grid-point stencil for
the interpolation (\eg\ bilinear interpolation).
For higher order interpolation schemes this is no longer valid.
A disadvantage with the above scheme is that the number of observations on each processor can be very different.
If the cost of the actual interpolation is expensive relative to the communication of data needed for interpolation,
this could lead to load imbalance.

%% =================================================================================================
\subsubsection{Round-robin distribution of observations among processors}

\begin{figure}
  \centering
  \includegraphics[width=0.66\textwidth]{OBS_obsdist_global}
  \caption[Observations with the round-robin distribution]{
    Example of the distribution of observations with
    the round-robin distribution of observational data.}
  \label{fig:OBS_global}
\end{figure}

An alternative approach is to distribute the observations equally among processors and
use message passing in order to retrieve the stencil for interpolation.
The simplest distribution of the observations is to distribute them using a round-robin scheme.
\autoref{fig:OBS_global} shows the distribution of the {\em in situ} data on processors for
the round-robin distribution of observations with a different colour for each observation on a given processor for
a 4 $\times$ 2 decomposition with ORCA2 for the same input data as in \autoref{fig:OBS_local}.
The observations are now clearly randomly distributed on the globe.
In order to be able to perform horizontal interpolation in this case,
a subroutine has been developed that retrieves any grid points in the global space.

%% =================================================================================================
\subsection{Vertical interpolation operator}

Vertical interpolation is achieved using either a cubic spline or linear interpolation.
For the cubic spline, the top and bottom boundary conditions for the second derivative of
the interpolating polynomial in the spline are set to zero.
At the bottom boundary, this is done using the land-ocean mask.

For profile observation types we do both vertical and horizontal interpolation. \NEMO\ has a generalised vertical coordinate system this means the vertical level depths can vary with location. Therefore, it is necessary first to perform vertical interpolation of the model value to the observation depths for each of the four surrounding grid points. After this the model values, at these points, at the observation depth, are horizontally interpolated to the observation location.