DIVAnd.jl documentation

dbinfo = diva3d(xi,x,value,len,epsilon2,filename,varname)

Create a 3D analysis (or a series of 3D analysis) with DIVAnd using the observations value (vector) at the locations x (tuple of vectors) onto the regular grid defined by the vectors xi using the scaled observational error variance epsilon2 and the correlation length len. The result will be saved in the netCDF file filename under the variable varname.

Inputs

• xi: tuple with n elements. Every element represents a coordinate of the final grid on which the observations are interpolated

• x: tuple with n elements. Every element represents a coordinate of the observations

• value: value of the observations

• len: tuple with n elements. Every element represents the correlation length. If fitcorrlen is false (default), the correlation length should be expressed in meters. If fitcorrlen is true, then len can be the empty tuple () or a tuple containing 3 arrays of normalized correlation lengths which will be multiplied by the horizontal and vertical correlation lengths.

• epsilon2: error variance of the observations (normalized by the error variance of the background field).

epsilon2 can be a scalar (all observations have the same error variance and their errors are decorrelated), a vector (all observations can have a different error variance and their errors are decorrelated) or a matrix (all observations can have a different error variance and their errors can be correlated). If epsilon2 is a scalar, it is thus the inverse of the signal-to-noise ratio.

• filename: The output netCDF filename.

• varname: The name of the variable (used in the netCDF file).

Optional input arguments:

• bathname: path to the netCDF bathymetry (default ../../DIVAnd-example-data/Global/Bathymetry/gebco30sec16.nc relative to this source file)
• bathisglobal: true (default) is the bathymetry is a global data set
• plotres: Call-back routine for plotting ((timeindex,sel,fit,erri) -> nothing)
• timeorigin: Time origin (default DateTime(1900,1,1,0,0,0))
• moddim: modulo for cyclic dimension (vector with n elements). Zero is used for non-cyclic dimensions. Halo points should not be included for cyclic dimensions. For example if the first dimension is cyclic, then the grid point corresponding to mask[1,j] should be between mask[end,1] (left neighbor) and mask[2,j] (right neighbor). The default is [0,0,0],
• zlevel: :surface (default) for surface analysis and :floor for analysis from the bottom floor.
• ncvarattrib: dictionary of netCDF variable attributes.
• ncglobalattrib: dictionary of netCDF global attributes.
• transform: Anamorphosis transformation function (default: Anam.notransform()).
• fitcorrlen: true if the correlation length is determined from the observation (default false). Note that the parameter len is interpreted differently when fitcorrlen is set to true.
• fithorzcorrlen: true if the horizontal correlation length is determined from the observation (default: the value of fitcorrlen) Note that the parameter len is interpreted differently when fithorzcorrlen is set to true.
• fitvertcorrlen: true if the vertical correlation length is determined from the observation (default: the value of fitcorrlen) Note that the parameter len is interpreted differently when fitvertcorrlen is set to true.
• fithorz_param: dictionary with additional optional parameters for fithorzlen, for example: Dict(:smoothz => 200., :searchz => 50.).
• fitvert_param: dictionary with additional optional parameters for fitvertlen.
• distfun: function to compute the distance (default (xi,xj) -> DIVAnd.distance(xi[2],xi[1],xj[2],xj[1])).
• mask: if different from nothing, then this mask overrides land-sea mask based on the bathymetry

(default nothing).

• background: if different from nothing, then this parameter allows one

to load the background from a call-back function (default nothing). The call-back functions has the parameters (x,n,trans_value,trans) where x represent the position of the observations, n the time index, trans_value, the observations (possibly transformed) and trans the transformation function. The output of this function is the gridded background field and the observations minus the background field.

• background_epsilon2_factor: multiplication for epsilon2 when computing a vertical profile as a background estimate (default: computed internally based on the amount of data). This parameter is not used when the parameter background or background_lenz is provided.
• background_lenz: vertical correlation for background computation (default 20 m). This parameter is not used when the parameter background is provided.
• background_len: deprecated option replaced by background_lenz.
• filterbackground: number of iterations to filter the background profile (default 0, no filtering)
• memtofit: keyword controlling how to cut the domain depending on the memory remaining available for inversion. It is not the total memory (default 3). Use a large value (e.g. 100) to force the usage for the more efficient direct solver if you are not limited by the amount of RAM memory.
• minfield: if the analysed field is below minfield, its value is replace by minfield (default -Inf, i.e. no substitution is done).
• maxfield: if the analysed field is above maxfield, its value is replace by maxfield (default +Inf, i.e. no substitution is done).
• saveindex: controls if just a subset of the analysis should be saved to the netCDF file. Per default, saveindex is (:,:,:) (corresponding to longitude, latitude and depth indices) meaning that everything is saved. If however, for example the first layer should not be saved then saveindex should be (:,:,2:length(depthr)) where depthr is the 3rd element of xi.
• niter_e: Number of iterations to estimate the optimal scale factor of epsilon2 using Desroziers et al. 2005 (doi: 10.1256/qj.05.108). The default is 1 (i.e. no optimization is done).
• coeff_derivative2 (vector of 3 floats): for every dimension where this value is non-zero, an additional term is added to the cost function penalizing the second derivative. A typical value of this parameter is [0.,0.,1e-8].
• surfextend: create an additional layer on top of the surface layer so that the effective background error variance is more similar to the deep ocean. false is the default value.

Any additional keywoard arguments understood by DIVAndgo/DIVAndrun can also be used here (e.g. velocity constrain)

The output is a dictionary with the followings keys:

• :residuals: the difference between the observations and the analysis (interpolated linearly to the

location of the observations). The residual is NaN if the observations are not within the domain as defined by the mask and the coordinates of the observations x.

• :qcvalues: quality control scores (if activated)

DIVAndrun(mask,pmn,xi,x,f,len,epsilon2; <keyword arguments>)

Perform an n-dimensional variational analysis of the observations f located at the coordinates x. The array fi represent the interpolated field at the grid defined by the coordinates xi and the scales factors pmn.

Input:

• mask: binary mask delimiting the domain. true is inside and false outside.

For oceanographic application, this is the land-sea mask where sea is true and land is false.

• pmn: scale factor of the grid. pmn is a tuple with n elements. Every element represents the scale factor of the corresponding dimension. Its inverse is the local resolution of the grid in a particular dimension. For example, in two dimensions, pmn is a tuple (pm,pn) where pm is the inverse of the local resolution in first dimension and pn is the the inverse of the local resolution in second dimension.

• xi: tuple with n elements. Every element represents a coordinate of the final grid on which the observations are interpolated.

• x: tuple with n elements. Every element represents a coordinate of the observations.

• f: value of the observations minus the background estimate (vector of m elements where m is the number of observations). See also note.

• len: tuple with n elements. Every element represents the correlation length for a given dimension.

• epsilon2: error variance of the observations (normalized by the error variance of the background field). epsilon2 can be a scalar (all observations have the same error variance and their errors are decorrelated), a vector (all observations can have a different error variance and their errors are decorrelated) or a matrix (all observations can have a different error variance and their errors can be correlated). If epsilon2 is a scalar, it is thus the inverse of the signal-to-noise ratio.

Optional input arguments specified as keyword arguments

• velocity: velocity of the advection constraint. It is a tuple of n arrays and each array represents a single velocity component. The individual array should have the same size as the final grid. The first (second,..) element of the velocity is the velocity compomenent along the first (second,...) dimension. The default is no-advection constraint.

• alpha: alpha is vector of coefficients multiplying various terms in the cost function. The first element multiplies the norm. The other i-th element of alpha multiplies the (i+1)-th derivative. Per default, the highest derivative is m = ceil(1+neff/2) where neff is the effective dimension of the problem (the number of dimensions with a nonzero correlation length) and ceil is the ceiling function (rounding up).

   The values of alpha is the (m+1)th row of the Pascal triangle:
      m=0         1
      m=1       1   1
      m=1     1   2   1     (n=1,2)
      m=2   1   3   3   1   (n=3,4)
      ...

• constraints: a structure with user specified constraints (see DIVAnd_addc).

• moddim: modulo for cyclic dimension (vector with n elements). Zero is used for non-cyclic dimensions. One should not include a boundary zone (sometimes called a ghost zone or halo) for cyclic dimensions. For example if the first dimension is cyclic, then the grid point corresponding to mask[1,j] should be between mask[end,1] (left neighbor) and mask[2,j] (right neighbor).

• fracindex: fractional indices (n-by-m array). If this array is specified, then x and xi are not used.

• inversion: direct solver (:chol for Cholesky factorization) or an interative solver (:pcg for preconditioned conjugate gradient [1]) can be used.

• compPC: function that returns a preconditioner for the primal formulation if inversion is set to 'pcg'. The function has the following arguments:

       fun = compPC(iB,H,R)

  where iB is the inverse background error covariance, H the observation operator and R the error covariance of the observation. The function compPC returns the preconditioner fun(x,fx) computing fx = M \ x (the inverse of M times x) where M is a positive defined symmetric matrix [1]. Effectively, the system E⁻¹ A (E⁻¹)ᵀ (E x) = E⁻¹ b is solved for (E x) where E Eᵀ = M. Ideally, M should this be similar to A, so that E⁻¹ A (E⁻¹)ᵀ is close to the identity matrix.

• fi0: starting field for iterative primal algorithm (same size as mask).

• f0: starting field for iterative dual algorithm (same size as the observations f).

• operatortype: Val{:sparse} for using sparse matrices (default) or Val{:MatFun} or using functions to define the constrains.

• scale_len: true (default) if the correlation length-scale should be scaled such that the analytical kernel reaches 0.6019072301972346 (besselk(1.,1.)) at the same distance than in 2D. The kernel behaves thus similar to the default kernel in two dimensions (alpha = [1,2,1]).

• alphabc: numerical value defining how the last grid points are stretched outward. If alphabc is 1, the default value mimics an infinite domain. To have previous behaviour of finite domain use alphabc equal to 0.

• btrunc: if provided defines where to truncate the calculation of the covariance matrix B. Only values up and including alpha[btrunc] will be calculated. If the iterative solution is calculated, the missing terms will be calculated on the fly during the conjugate gradient calculations. Default value is none and full covariance calculation.

Output:

• fi: the analysed field
• s: a structure with an array s.P representing the analysed error covariance

Note:

If zero is not a valid first guess for your variable (as it is the case for e.g. ocean temperature), you have to subtract the first guess from the observations before calling DIVAnd and then add the first guess back in.

Example:

see DIVAndsimpleexample.jl

References

[1] https://en.wikipedia.org/w/index.php?title=Conjugategradientmethod&oldid=761287292#Thepreconditionedconjugategradientmethod

fi, erri, residuals, qcvalues, scalefactore = DIVAndgo(mask,pmn,xi,x,f,len,epsilon2,errormethod; ...);

Input:

• Same arguments as DIVAndrun with in addition
• errormethod : you have the choice between :cpme (clever poorman's method, default method if parameter not provided), :none or :exact (only available if windowed analysis are done with DIVAndrun)
• MEMTOFIT=: keyword controlling how to cut the domain depending on the memory remaining available for inversion (not total memory)
• RTIMESONESCALES= : if you provide a tuple of length scales, data are weighted differently depending on the numbers of neighbours they have. See weight_RtimesOne for details
• QCMETHOD= : if you provide a qc method parameter, quality flags are calculated. See DIVAnd_cv for details
• solver (default :auto:). :direct for the direct solver or :auto for automatic choice between the direct solver or the iterative solver.
• overlapfactor : describes how many times the length scale is used for the overlapping. default is 3.3. use lower values ONLY for very good data coverage.

Output:

• fi: the analysed field
• erri: relative error field on the same grid as fi. () if errormethod is fixed to :none
• residuals: array of residuals at data points. For points not on the grid or on land: NaN
• qcvalues: if QCMETHOD= is provided, the output array contains the quality flags otherwise qcvalues is (). For points on land or not on the grid: 0
• scalefactore: Desroziers et al. 2005 (doi: 10.1256/qj.05.108) scale factor for epsilon2

Perform an n-dimensional variational analysis of the observations f located at the coordinates x. The array fi represent the interpolated field at the grid defined by the coordinates xi and the scales factors pmn.

IMPORTANT: DIVAndgo is very similar to DIVAndrun and is only interesting to use if DIVAndrun cannot fit into memory or if you want to parallelize. (In the latter case do not forget to define the number of workers; see addprocs for example)

fma,faanom = DIVAnd_averaged_bg(mask,pmn,xi,x,f,len,epsilon2,toaverage;moddim=[])

Input:

As for DIVAndrun, including all dimensions before averaging

additional argument:

• toaverage: Array of ndims of boolean telling if in the corresponding direction averaging must be done

Presently NO optional arguments from DIVAndrun supported except moddim

Output:

• fma: Analysis where in the directions where toaverage is true, the same value is found
• faanom: Data anomalies when the analysis is subtracted from the input field.

ncglobalattrib,ncvarattrib = SDNMetadata(metadata,fi)

Based on the information in the dictionary metadata and the analysed 4D field fi produce a list of NetCDF global and variable attributes for DIVAnd_save2.

save(filename,xyi,fi,varname;
                      ncvarattrib = Dict(), ncglobalattrib = Dict(), ...)

Save the result of the analysis in a netCDF file.

Input arguments

• filename: the name of the NetCDF file
• xyi: tuple with n vectors. Every element in this tuple represents a coordinate of the final grid on which the observations are interpolated
• fi: the analysed field
• varname: the name of the NetCDF variable

Optional arguments:

• ncglobalattrib: a dictionary with the global attributes
• ncvarattrib: a dictionary with the variable attributes
• relerr: relative error

value,lon,lat,depth,time,obsid = loadbigfile(filename)

Load data from the text file filename and returns vectors with the value, longitude, latitude, depth and time (as DateTime). A list string identifiers is also returned.

checkobs(x,v,ids)
checkobs(io::IO,x,v,ids)

Print some basic information about the coordinates x (tuple of vector) and values v (vector) having the identifier ids (vector of strings) to check erroneous data. It prints wheter NaNs or Infs are found and the minimum and maximum value.

If the argument io is provided, the information is input/output stream io.

ff = smoothfilter(x,f,scale)

Smooth the function f defined on x by solving the diffusion equation

∂ₜ ϕ = ν ∂²ₓ ϕ

scale is the spatial scales of the removed length-scales. It is defined as 2Tν where T is the integration time.

It uses the Greens functions for 1D diffusion: 1/sqrt(4 π ν t) * exp(-x^2 / (4νt))

trans,invtrans = loglin(t; epsilon = 0.)

Provide the following transform log(x + epsilon) (for x < t) and its inverse. Beyond the threshold t (x ≥ t), the function is extended linearly in a continous way.

trans,invtrans are scalar functions such that for any x (x > epsilon), x == invtrans(trans(x)).

For any array X, we have: X == invtrans.(trans.(X)).

trans,invtrans = logit(; min = 0., max = 1.)

Provide the logit transform and its inverse. Per default the logit transform maps values within the interval from 0 and 1. This can be changed with the min and max parameters. Note that trans(min) = -∞ and trans(max) = +∞. The use safety-margin might be necessary.

DIVAnd.divadoxml(filepath,varname,project,cdilist,xmlfilename;
                 ignore_errors = false,
                 WMSlayername = [],
                 previewindex = 1,
                 additionalcontacts = [],
                 additionalvars = Dict{String,Any}())

Generate the XML metadata file xmlfilename from the NetCDF file filepath (or list of files) with the NetCDF variable varname. Project is either "SeaDataNet", "EMODNET-chemistry" or "SeaDataCloud". cdilist is the file from https://emodnet-chemistry.maris.nl/download/export.zip.

The XML file contains a list of the data the originators. divadoxml will abort with an error if some combinations of EDMO code, local CDI ID are not present in the cdilist. Such errors can be ignored if ignore_errors is set to true.

Information can be overridden with the dictionary additionalvars. The keys should corresponds to the template tags found the in template directory. Template tags are the strings inside {{ and }}.

NetCDF_URL should be suppplied since it's a URL of a ZIP file which is usually not from OceanBrowser.

If filepath is a vector of file names, the argument WMSlayername can be provided to give additional information to distinguish between the NetCDF files. The elements of the vector of string will be appended to the description of the WMS layer.

The resulting XML file includes the file names (provided by filepath). Do not change the file names after running this function, otherwise the XML will still contain a reference to the old file names. If you must change the file names please do so before running this script.

If the data is present in a subfolder (e.g. "Winter") later on the OceanBrowser webserver, the filepath should also contain this subfolder (e.g. "Winter/somefile.nc"). The local directories should mirror the directory structure on OceanBrowser. Relative paths should be used, and if the Julia code isn't right above the NetCDF files, use cd("<path>") before each setting the files parameter which use paths relative to this path.

additionalcontacts is a list of dictionaries with additional condact information to be added in the XML file. Elements are typically create by the function DIVAnd.getedmoinfo.

Example

download("https://emodnet-chemistry.maris.nl/download/export.zip","export.zip")
files = [
         "Winter (January-March) - 6-year running averages/Water_body_chlorophyll-a.4Danl.nc",
         "Spring (April-June) - 6-year running averages/Water_body_chlorophyll-a.4Danl.nc",
         "Summer (July-September) - 6-year running averages/Water_body_chlorophyll-a.4Danl.nc",
         "Autumn (October-December) - 6-year running averages/Water_body_chlorophyll-a.4Danl.nc"
         ];

additionalcontacts = [
    DIVAnd.getedmoinfo(1977,"originator"), # US NODC for World Ocean Database
    DIVAnd.getedmoinfo(4630,"originator"), # CORIOLIS for CORA
]

DIVAnd.divadoxml(files,"Water_body_chlorophyll-a","EMODNET-chemistry","export.zip","test.xml";
    ignore_errors = true,
    additionalvars = Dict("abstract" => "Here goes the abstract"),
    additionalcontacts = additionalcontacts,
    WMSlayername = ["winter","spring","summer","autumn"]
)

contact = DIVAnd.getedmoinfo(edmo_code,role)

Returns a dictionary with the contact information from the EDMO registry based on the prodivided emdo_code. role is the Sextant contact information role, i.e. either "originator" or "author".

field = DIVAnd.random(mask,pmn,len,Nens)

Create Nens random fields with the correlation length len in a domain with the mask mask and the metric pmn.

See DIVAnd.DIVAndrun for more information about these parameters.

d = distance(lat1,lon1,lat2,lon2)

Compute the great-circle distance between the points (lat1,lon1) and (lat2,lon2). The units of all input and output parameters are degrees.

d = distance([lon1,lat1],[lon2,lat2])

The same as distance(lat1,lon1,lat2,lon2) but there the arguments are vectors and the order is longitude then latitude.

The units of all input and output parameters are degrees.

f = interp(xi,fi,x)

Interpolate field fi (n-dimensional array) defined at xi (tuble of n-dimensional arrays or vectors) onto grid x (tuble of n-dimensional arrays). The grid in xi must be align with the axis (e.g. produced by DIVAnd.ndgrid).

fun = backgroundfile(fname,varname)

Return a function fun which is used in DIVAnd to make anomalies out of observations based relative to the field defined in the NetCDF variable varname in the NetCDF file fname. It is assumed that the NetCDF variables has the variable lon, lat and depth. And that the NetCDF variable is defined on the same grid as the analysis.

fun = backgroundfile(fname,varname,TS)

Return a function fun which is used in DIVAnd to make anomalies out of observations based relative to the field defined in the NetCDF variable varname in the NetCDF file fname. It is assumed that the NetCDF variables has the variable lon, lat and depth. And that the NetCDF variable is defined on the same grid as the analysis and was generated according to the provided time selector TS (TimeSelectorYearListMonthList or TimeSelectorRunningAverage).

dupl = checkduplicates(x,value,delta,deltavalue)

Based on the coordinates x (a tuple of longitudes lons, latitudes lats, depths (zs) and times (times vector of DateTime)), search for points which are in the same spatio-temporal bounding box of length delta. delta is a vector with 4 elements corresponding to longitude, latitude, depth and time (in days). dupl a vector of vectors containing the indices of the duplicates.

dupl = checkduplicates(x1,value1,x2,v2,value2,delta,deltavalue)

Report duplicates of observations in data set (x2,v2) which are also in data set (x1,v1). x1 and x2 are tuples of vectors with the coordinates, v1 and v2 are the corresponding values.

Computes a heatmap based on locations of observations using kernel density estimation (probability density field whose integral over the domain is one)

dens,Ltuple,LCV,LSCV = DIVAnd_heatmap(mask,pmn,xi,x,inflation,Labs;Ladaptiveiterations=0,myheatmapmethod="DataKernel", optimizeheat=true,nmax=1000,otherargs...)

Input:

• mask: mask as usual

• pmn : tuple of metrics as usual

• xi: tuple of coordinates of the grid for the heatmap

• x : tuple of coordinates of observations

• inflation: array generally of ones. For some applications an observation can carry a different weight which is then encoded in the array

• Labs : the length scales for DIVAnd. Here their meaning is the spread (bandwidth) of the observations for the Kernel calculation

•          if zero is provided, the routine applies an empirical estimate, returned in the Ltuple output.

• Ladaptiveiterations: adaptive scaling where the length scales are adapted on the data density already estimated. You can iterate. Default "0"

• optimizeheat : boolean which can turn on or off an algorithmic optimisation. Results should be identical. Default is to optimize

• myheatmapmethod: can be "Automatic", "GridKernel" or "DataKernel" (Results should be very similar except near boundaries)

• nmax: maximum number of data points. If actual data size is larger, approximatively nmax superobservations are calculated and a warning issued.

• otherargs...: all other optional arguments DIVAndrun can take (advection etc)

Output:

• dens: data density field (integral is one)
• Ltuple : The bandwidthth used (either the input value or the calculated ones)
• LCV : Likelihood Cross Validation estimator value (the higher the better) leave one out approach
• LSCV : Least Square Cross Validation estimator (the lower the better) leave one out approach

xi,yi,bath = DIVAnd.load_bath(bath_name,isglobal,xi,yi)

Load the bathymetry from the netCDF file bathname. The parameter isglobal is true if the NetCDF file covers the whole globe and thus the last longitude point can be considered to be right next to the first longitude point. xi and yi are vectors containing the longitude and latitude grid onto which the bathymetry should be interpolated.

bx,by,b = DIVAnd.extract_bath(bath_name,isglobal,xi,yi)

Extract the bathymetry from the NetCDF file bathname. The parameter isglobal is true if the NetCDF file covers the whole globe and thus the last longitude point can be considered to be right next to the first longitude point. xi and yi are vectors defining the bounding box of the data. No interpolation is performed.

Convention: b is positive in the water and negative in the air.

The NetCDF file is expected to have the one dimensional variables lon and lat with the longitude (degrees East) and latitude (degrees North) and the two dimentional array bat with the digital terrain model (negative in water and positive above water). The order of the dimension should follow be: longitude and then latitude in Column-major ordering (or latitude and then longitude if the tool ncdump is used, which is based on Row-major ordering).

Example of the output of ncdump -h:

netcdf gebco_30sec_8 {
dimensions:
     lat = 2702 ;
     lon = 5400 ;
variables:
     double lat(lat) ;
          lat:long_name = "Latitude" ;
          lat:standard_name = "latitude" ;
          lat:units = "degrees_north" ;
     double lon(lon) ;
          lon:long_name = "Longitude" ;
          lon:standard_name = "longitude" ;
          lon:units = "degrees_east" ;
     float bat(lat, lon) ;
          bat:long_name = "elevation above sea level" ;
          bat:standard_name = "height" ;
          bat:units = "meters" ;

// global attributes:
          :title = "GEBCO" ;
}

xi,yi,mask = load_mask(bath_name,isglobal,xi,yi,level::Number)

Generate a land-sea mask based on the topography from the NetCDF file bathname. The parameter isglobal is true if the NetCDF file covers the whole globe and thus the last longitude point can be considered to be right next to the first longitude point. xi and yi are vectors containing the longitude and latitude grid onto which the bathymetry should be interpolated.

Convention: in the water, level is positive and in the air level is negative.

pm,pn = DIVAnd_metric(lon,lat)

Compute metric scale factors pm and pn based on the arrays longitude lon and latitude lat. The variables pm and pn represent the inverse of the local resolution in meters using the mean Earth radius.

mask,(pm,pn),(xi,yi) = domain(bathname,bathisglobal,lonr,latr)

Generate a 2D geospatial domain based on the topography from the netCDF file bathname.

mask,(pm,pn,po),(xi,yi,zi) = domain(bathname,bathisglobal,lonr,latr,depthr)

Generate a 3D geospatial domain based on the topography from the netCDF file bathname. If zlevel is :surface, then depthr is zero for the sea surface and positive in water (positive is down). If zlevel is :floor, then depthr is zero for the sea floor and positive in water (positive is up)

mask,(pm,pn,po,pp),(xi,yi,zi,ti) = domain(bathname,bathisglobal,lonr,latr,depthr,timer)

Generate a geospatial domain based on the topography from the netCDF file bathname.

mask,pmn,xyi = DIVAnd_rectdom(coord1,coord2,...)

Create a "rectangular" domain in n dimensions with the coordinates coord1coord2... assuming a Catersian metric. This functions returns the mask mask, the coordinates (xi,yi,...) and the metric (pm,pn...).

For example:

julia> mask,(pm,pn),(xi,yi) = DIVAnd_rectdom(range(0,stop=1,length=50),linspace(0,stop=1,length=50))

mask,pmn,xyi = DIVAnd_squaredom(n,coord)

Create a "square" domain in n dimensions with the coordinates coord assuming a Cartesian metric. This functions returns the mask mask, the coordinates (xi,yi,...) and the metrics (pm,pn...).

Example

mask,(pm,pn),(xi,yi) = DIVAnd_squaredom(2,range(0,stop=1,length=50))

TS = TimeSelectorYW(years,yearwindow,monthlists)

The structure TS handles the time aggregation based on years and monthlists. It is similar to TimeSelectorYearListMonthList except that the elements of yearlists are centred around years and span yearwindow years. yearlists is in fact constructed by adding and subtracting yearwindow/2 to every element of years.

TS = TimeSelectorYearListMonthList(yearlists,monthlists)

The structure TS handles the time aggregation based on yearlists and monthlists. yearlists is a vector of ranges (containing start and end years), for example [1980:1989,1990:1999,2000:2009].

monthlists is a vector of two-element vector (containing start and end months), for example [1:3,4:6,7:9,10:12].

The upper bound of a yearlist and monthlist element is considered inclusive. The range of years of 2000:2009 consideres all years upto and including the year 2009.

If a month range spans beyond December, then all Months must be specified, e.g. example [2:4,5:6,7:9,[10,11,12,1]] or [2:4,5:6,7:9,[10:12;1]]. However using [2:4,5:6,7:9,10:1] (bug!) will result in an empty month range.

Example

# seasonal climatology using all data from 1900 to 2017
# for winter (December-February), spring, summer, autumn

TS = DIVAnd.TimeSelectorYearListMonthList([1900:2017],[[12,1,2],[3,4,5],[6,7,8],[9,10,11]])

DIVAnd.saveobs(filename,xy,ids;
               type_save = Float32,
               timeorigin = DateTime(1900,1,1,0,0,0),
               used = trues(size(ids)),
               )

Save the location and time of the observation in the netCDF file filename and their identifier ids. xy is a tuple with the vectors longitude, latitude, depth and time (as a vector of DateTime).

Optional arguments:

• type_save: the type to save the data (default Float32). However, the time is always saved as Float64.
• timeorigin: time origin for the time units attribute (default is

1900-01-01 00:00:00)

• used: allows one to subset the data to save only used variables in the netCDF file

DIVAnd.saveobs(filename,varname,value,xy,ids;
               type_save = Float32,
               timeorigin = DateTime(1900,1,1,0,0,0),
               used = trues(size(ids)),
               chunksize = 10_000,
               )

Save value and the location and time of the observation in the NetCDF file filename and their identifier ids. xy is a tuple with the vectors longitude, latitude, depth and time (as a vector of DateTime). The values will be saved in the variable called varname.

Optional arguments:

• type_save: the type to save the data (default Float32). However, the time is always saved as Float64.
• timeorigin: time origin for the time units attribute (default is

1900-01-01 00:00:00)

• used: allows to subset the data to save only used variables in the netCDF file

obsvalue,obslon,obslat,obsdepth,obstime,obsid = loadobs(T,filename,varname)

Load the variable varname from the netCDF file filename. Coordinates (the netCDF variables "obslon", "obslat", "obsdepth"), time ("obstime") and identifiers ("obsids") will also be loaded. Numeric output arguments will have the type T.

obsvalue,obslon,obslat,obsdepth,obstime,obsids = load(T,
  fname,param; qualityflags = [GOOD_VALUE, PROBABLY_GOOD_VALUE])

data,lon,lat,z,time,ids = SDN.load(T,fnames,param; qualityflags = ...)

Load all data in the vector of file names fnames corresponding to the parameter param as the data type T. Only the data with the quality flags SDN.good_data and SDN.probably_good_data are loaded per default. The output parameters correspond to the data, longitude, latitude, depth, time (as DateTime) and an identifier (as String).

data = loadvar(ds,param;
               fillvalue::T = NaN,
               qualityflags = [GOOD_VALUE, PROBABLY_GOOD_VALUE],
               qfname = param * QC_SUFFIX,
               )

Load the netCDF variable param from the NCDataset ds. Data points not having the provide quality flags will be masked by fillvalue. qfname is the netCDF variable name for the quality flags.

obsvalue,obslon,obslat,obsdepth,obstime,obsids = NCODV.load(T,fname,long_name;
     qv_flags = ["good_value","probably_good_value"])

Load all profiles in the file fname corresponding to netCDF variable with the long_name attribute equal to the parameter long_name. qv_flags is a list of strings with the quality flags to be kept. obsids is a vector of strings with the EDMO code and local CDI id concatenated by a hyphen.

data = loaddata(sheet,profile,locname,fillvalue; fillmode = :repeat)

Load a single column referred by the local name locname in the profile profile from the ODV spreadsheet sheet. Empty values are either replaced by fillvalue (if fillmode is :fill) or the previous value if repeated (if fillmode is :repeat)

dt = parsejd(t)

Convert a Chronological Julian Day Number to a DateTime object. The reference value is taken from Chronological Julian Date

From the SDN standard: "A real number representing the Chronological Julian Date, which is defined as the time elapsed in days from 00:00 on January 1 st 4713 BC. ... "

The time origin is not noon (12:00) on Monday, January 1, 4713 BC as for the Julia Date Number.

v = myparse(T,s)

Parse the string s as a type T. Unlike Julia's parse function an error message contains the string s (which could not be parsed) for debugging.

bestfactorl,bestfactore, cvval,cvvalues, x2Ddata,y2Ddata,cvinter,xi2D,yi2D = DIVAnd_cv(mask,pmn,xi,x,f,len,epsilon2,nl,ne,method;...);

Performs a cross validation to estimate the analysis parameters (correlation length and signal-to-noise ratio).

Input

Same as for DIVAndrun with three more parameters nl,ne and method

• mask: binary mask delimiting the domain. true is inside and false outside.

For oceanographic application, this is the land-sea mask.

• pmn: scale factor of the grid. pmn is a tuple with n elements. Every element represents the scale factor of the corresponding dimension. Its inverse is the local resolution of the grid in a particular dimension.

• xi: tuple with n elements. Every element represents a coordinate of the final grid on which the observations are interpolated

• x: tuple with n elements. Every element represents a coordinate of the observations

• f: value of the observations minus the background estimate (m-by-1 array). (see note)

• len: correlation length

• epsilon2: error variance of the observations (normalized by the error variance of the background field). epsilon2 can be a scalar (all observations have the same error variance and their errors are decorrelated), a vector (all observations can have a difference error variance and their errors are decorrelated) or a matrix (all observations can have a difference error variance and their errors can be correlated). If epsilon2 is a scalar, it is thus the inverse of the signal-to-noise ratio.

• nl: number of testing points around the current value of L. 1 means one additional point on both sides of the current L. 0 is allowed and means the parameter is not optimised.

• ne: number of testing points around the current value of epsilon2. 0 is allowed as for nl

• method: cross validation estimator method 1: full CV 2: sampled CV 3: GCV 0: automatic choice between the three possible ones, default value

• Optional input arguments specified via keyword arguments are the same as for DIVAnd

Output:

• bestfactorl: best estimate of the multiplication factor to apply to len

• bestfactore: best estimate of the multiplication factor to apply to epsilon2

• cvvales : the cross validation values calculated

• factors : the tested multiplication factors

• cvinter : interpolated cv values for final optimisation

• X2Data, Y2Data : coordinates of sampled cross validation in L,epsilon2 space . Normally only used for debugging or plotting

• Xi2D, Yi2D : coordinates of interpolated estimator . Normally only used for debugging or plotting

The output bestfactorl and bestfactore represent multiplication factors which should be applied to L and epsilon2.

The len and epsilon2 provided should be close the real one as the tests will be performed around.

distx,covar,corr,varx,count = empiriccovar(x,v,distbin,mincount;
                          maxpoints = 10000,
                          distfun = (xi,xj) -> sqrt(sum(abs2,xi-xj)))

Compute the covariance, correlation and variance of a cloud of data points with the value v (a vector) and the location x (a tuple of vectors) grouped by distance. Random pairs are choosen and grouped by their distance (computed by distfun) in bins defined by distbin. The function try to fill at least mincount of data points in each bin but always stop after considering maxpoints pairs.

lenxy,dbinfo = DIVAnd.fithorzlen(x,value,z)

Determines the horizontal correlation length lenxy based on the measurments value at the location x (tuple of 3 vectors corresponding to longitude, latitude and depth) at the depth levels defined in z.

Optional arguments:

• smoothz (default 100): spatial filter for the correlation scale
• searchz (default 50): vertical search distance (can also be a function of the depth)
• maxnsamp (default 5000): maximum number of samples
• limitlen (default false): limit correlation length by mean distance between observations
• limitfun (default no function): a function with with the two arguments (depth and

estimated correlation length) which returns an adjusted correlation length. For example to force the correlation length to be between 300 km and 50 km one would use the following: limitfun = (z,len) -> max(min(len,300_000),10_000)). If provided limitfun is used before and after the smoothing.

• epsilon2 (default is a vector of the same size as value with all elements equal to 1): the relative error variance of the observations. Less reliable observation would have a larger corresponding value.
• distfun: function computing the distance between the points xi and xj.

Per default it represent the Euclidian distance.

lenz,dbinfo = DIVAnd.fitvertlen(x,value,z,...)

See also DIVAnd.fithorzlen

RL = lengraddepth(pmn,h, L;
                  h2 = h,
                  hmin = 0.001
                  )

Create the relative correlation length-scale field RL based on the bathymetry h and the metric pmn (tuple of arrays). Effectively the correlation-length scale is close to zero if the relative bathymetry gradients (|∇h|/h) are smaller than the length-scale L (in consistent units as pmn).

R_L = 1 / (1 + L |∇h| / max(h2,hmin))

Per default h2 is equal to h. The depth h must be positive. hmin must have the same units as h (usually meters).

theta = DIVAnd_cvestimator(s,residual)

Computes the cross validation estimator $(d-\hat{d})^T \mathbf R^{-1} (d-\hat{d}) / ( \mathbf 1^T \mathbf R^{-1} \mathbf 1)$ where the $\hat{d}$ is the analysis not using a data point.

 weights = weight_RtimesOne(x,len)

Compute the weight of the observations at location x to reduce the influence of locally clustered data. x is a tuple with n elements: every element represents a coordinate of the observations. len is a tuple of arrays representing the correlation length. len[i] is the correlation length in the i-th dimension.

Rtimesx!(coord,LS,x,Rx)

Gaussian type R matirx in ndim dimensions applied to vector x of length ndata. The Gaussian scale differs in each direction k : LS[k] Coordinates of point i are coord[i,1],coord[i,2],...,coord[i,ndim] To avoid an ndata² complexity a grid is set up first so as to allow only the calculation of covariances when distances are smaller than 3*LS

Adapted from DIVA3D/src/Fortran/Util/Rtimesx_weighting.f90

urn"SDN:x:y:z'

Resolve a SeaDataNet URN (Uniform Resource Name) using https://www.seadatanet.org/urnurl/

collection = Vocab.CFVocab()
collection = Vocab.CFVocab(url = url)

Create a Dict-like object represeting the NetCDF CF Standard Name vocabulary. If the url is not provided then current CF Standard Name list http://cfconventions.org/Data/cf-standard-names/current/src/cf-standard-name-table.xml is used. Individual standard names are retirved by indexing which return an object of the type CFEntry:

collection = Vocab.CFVocab()
entry = collection["sea_water_temperature"]

bool = haskey(collection::CFVocab,stdname)

Return true if stdname is part of the NetCDF CF Standard Name vocabulary collection.

collection = SDNCollection(name)

Open the SeaDataNet collection with the name name at the URL http://www.seadatanet.org/urnurl/collection/ The collection can be indexed with brackets using the identifier.

using DIVAnd
collection = Vocab.SDNCollection("P01")
concept = collection["PSALPR01"]
@show Vocab.prefLabel(concept)

s = Vocab.prefLabel(c::Vocab.Concept)

Return the preferred label of a concept c

s = Vocab.prefLabel(urn::AbstractString)

Return the preferred label of a concept usings it URN (Uniform Resource Name)

s = Vocab.altLabel(c::Vocab.Concept)

Return the alternative label of a concept c

s = Vocab.altLabel(urn::AbstractString)

Return the alternative label of a concept usings it URN (Uniform Resource Name)

s = Vocab.notation(c::Vocab.Concept)

Return the identifier of a concept c

s = Vocab.notation(urn::AbstractString)

Return the identifier of a concept usings it URN (Uniform Resource Name)

s = Vocab.definition(c::Vocab.Concept)

Return the definition of a concept c

s = Vocab.definition(urn::AbstractString)

Return the definition of a concept usings it URN (Uniform Resource Name)

entry = Vocab.resolve(urn)

Resolve a SeaDataNet URN (Uniform Resource Name) and returns the corresponding EDMO entry or Vocabulary concept. For example:

concept = Vocab.resolve("SDN:P021:current:TEMP")

concepts = Vocab.find(c::Concept,name,collection)

Return a list of related concepts in the collection collection. name can be the string "related", "narrower", "broader".

str = description(entry::CFEntry)
str = canonical_units(entry::CFEntry)

Return the description or the canonical units of the `entry`.

str = description(entry::CFEntry)
str = canonical_units(entry::CFEntry)

Return the description or the canonical units of the `entry`.

collection,tag,key = Vocab.splitURL(url)

Split a concept URL into collection, tag and key. url must finishe with a slash.

sv = statevector_init((mask1, mask2, ...))

Initialize structure for packing and unpacking multiple variables given their corresponding land-sea mask.

Input: mask1, mask2,...: land-sea mask for variable 1,2,... Sea grid points correspond to one and land grid points to zero. Every mask can have a different shape.

Output: sv: structure to be used with statevectorpack and statevectorunpack.

Note: see also statevectorpack, statevectorunpack

Author: Alexander Barth, 2009,2017 <a.barth@ulg.ac.be> License: GPL 2 or later

x = pack(sv,(var1, var2, ...))

Pack the different variables var1, var2, ... into the vector x where sv is a statevector under the control of a mask. Only sea grid points are retained.

Input: sv: structure generated by statevector_init. var1, var2,...: variables to pack (with the same shape as the corresponding masks).

Output: x: vector of the packed elements. The size of this vector is the number of elements of all masks equal to 1.

Notes: If var1, var2, ... have an additional trailing dimension, then this dimension is assumed to represent the different ensemble members. In this case x is a matrix and its last dimension is the number of ensemble members.

var1, var2, ... = unpack(sv,x)
var1, var2, ... = unpack(sv,x,fillvalue)

Unpack the vector x into the different variables var1, var2, ... where sv is a statevector.

Input: sv: structure generated by statevector_init. x: vector of the packed elements. The size of this vector is the number of elements equal to 1 in all masks.

Optional input parameter: fillvalue: The value to fill in var1, var2,... where the masks correspond to a land grid point. The default is zero.

Output: var1, var2,...: unpacked variables.

Notes: If x is a matrix, then the second dimension is assumed to represent the different ensemble members. In this case, var1, var2, ... have also an additional trailing dimension.

ind = statevector_sub2ind(sv,subscripts)

Compute from a tuple of subscripts the linear index in the packed state vector. The first element of the subscript indicates the variable index and the remaining the spatial subscripts.

subscripts = ind2sub(sv,index)

Compute from linear index in the packed state vector a tuple of subscripts. The first element of the subscript indicates the variable index and the remaining the spatial subscripts.

number of points per node it is always zero for non-leaf nodes

c = DIVAnd_constr_fluxes(s,topographyforfluxes,fluxes,epsfluxes,pmnin)

Creates integral constraints for each latitude so that a barotropic correction step leads to an additional flux prescribed.

Input: s: structure topographyforfluxes: tuple of two 2D arrays with the bottom topography used for the flux calculations DO NOT USE NaN in it. If an array is replaced by a scalar zero, the constraint is not used. for fluxes calculated with geostrophy apply g/f to h fluxes: tuple of two arrays of fluxes. The barotropic correction on elevation should be such that Sum over longitude at each latidute of Sum h δ(eta)/δx δx = - fluxes[1] Sum over latitude at each longitude of Sum h δ(eta)/δy δ y = -fluxes[2] WARNING: This has been coded to directly use geostrophy.jl output and flux directions epsfluxes: error variance on constraint. Scaling to be verified pmnin: metrics from the calling routine

Output: c: structure to be used by DIVAnd_addc with the following fields: R (a covariance matrix), H (extraction operator) and yo (specified value for the constrain).

c = DIVAnd_constr_constcoast(mask,eps2)

Constrain imposing that the gradients along the coastline defined by mask are close to zero controlled by the parameter eps2 which represents the scaled error variance on the gradients.

This constrain is useful to indirectly impose that a stream function does not have a current component perpendicular to the coastline.

p = listSDNparam(ODVData)

Return a list of SeaDataNet P01 parameters in a ODV spreadsheet ODVData.

 obsvalue,obslon,obslat,obsdepth,obstime,obsids = load(T,fnames,datanames;
    qv_flags = [DIVAnd.ODVspreadsheet.GOOD_VALUE,
                DIVAnd.ODVspreadsheet.PROBABLY_GOOD_VALUE],
    nametype = :P01,
    qvlocalname = "QV:SEADATANET")

Load all the profiles from every files listed in the array fnames corresponding to one of the parameter names datanames. If nametype is :P01 (default), the datanames are P01 vocabulary names with the SDN prefix. If nametype is :localname, then they are the ODV column header without units.

For example if the column header is Water body salinity [per mille], then datenames should be ["Water body salinity"]. The resulting vectors have the data type T (expect times and ids which are vectors of DateTime and String respectively). Only values matching the quality flag qv_flags are retained. qv_flags is a vector of Strings (based on http://vocab.nerc.ac.uk/collection/L20/current/, e.g. "1" means "good value"). One can also use the constants these constants (prefixed with DIVAnd.ODVspreadsheet.):

qvlocalname is the column name to denote quality flags. It is assumed that the quality flags follow immediately the data column.

If the ODV does not contain a semantic header (e.g. for the aggregated ODV files), then local names must be used.

julia> data,obslon,obslat,obsdepth,obstime,obsids = DIVAnd.ODVspreadsheet.load(Float64,["data_from_med_profiles_non-restricted_v2.txt"],
      ["Water body salinity"]; nametype = :localname );

In order to read ODV spreasheet containing World Ocean Database file odvfile, one can use a command like:

julia> obsval,obslon,obslat,obsdepth,obstime,obsid = ODVspreadsheet.load(Float64,[odvfile],
                           ["Temperature"]; qv_flags=["0", "1"], nametype = :localname, qvlocalname = "QV:WOD");

i.e.,

• explicitely specifying the accepted flags qv_flags
• set qvlocalname as "QV:WOD".

Note: no checks are performed to ensure the units are consistent.

 profiles,lons,lats,depths,times,ids = load(T,dir,P01names)

Load all the ODV files under the directory dir corresponding the one of the parameter names P01names. The resulting vectors have the data type T (expect times and ids which are vectors of DateTime and String, respectively).

No checks are done to ensure the units are consistent.

list = localnames(sheet,P01name)

Return a list list of all local names mapping to the specified P01name in the ODV spreadsheet sheet without the prefix "SDN:LOCAL:".

list = localnames(sheet)

Return a list list of all local names in the ODV spreadsheet sheet without the prefix "SDN:LOCAL:" in the order as they appear in the ODV file.

Define composite type that will contain:

• the metadata (dictionary),
• SDN parameter mapping (dictionary)
• the column labels (array) and
• the profiles (array of arrays).

 data,data_qv,obslon,obslat,obsdepth,obsdepth_qv,obstime,obstime_qv,EDMO,LOCAL_CDI_ID =
 loadprofile(T,sheet,iprofile,dataname; nametype = :P01)

Load a iprofile-th profile from the ODV spreadsheet sheet of the parameter dataname. If nametype is :P01 (default), the dataname is the P01 vocabulary name with the SDN prefix. If nametype is :localname, then it is the ODV column header. The resulting vectors have the data type T (expect the quality flag and obstime) .

data,data_qv = loaddataqv(sheet,profile,locname,fillvalue; fillmode = :repeat)

The same as loaddata, but now the quality flag are also loaded.

profile[i][j] is the j-th column of the i-th row of a profile. profile[i,j] is the i-th column of the j-th row of a profile.

SDNparse!(col,fillmode,fillvalue,data)

Parse the list of String col into the corresponding data type of the vector data. Empty values are either replaced by fillvalue (if fillmode is :fill) or the previous value if repeated (if fillmode is :repeat)

cn = colnumber(sheet,localname)

Return the column number cn of the first column with the local name localname (without the prefix "SDN:LOCAL:") in the ODV spreadsheet sheet.

n = nprofiles(ODVData)

Return the number of profiles in a ODV Spreadsheet ODVData loaded by readODVspreadsheet.

H,out = sparse_interp(mask,I)

Create interpolation matrix from mask and fractional indexes I.

Input: mask: 0 invalid and 1 valid points (n-dimensional array) I: fractional indexes (2-dim array n by mi, where mi is the number of points to interpolate) Ouput: H: sparse matrix with interpolation coefficients out: true if value outside of grid outbbox: 1 if outise bouding box onland: 1 if point touches land (where mask == 0)

sparse_interp(x,mask,xi) Interpolate from x onto xi

diffx = sparse_diff(sz1,m,cyclic)

Sparse operator for differentiation along dimension m for "collapsed" matrix of the size sz1. cyclic is true if domain is cyclic along dimension m. false is the default value

T = matfun_trim(sz1,m)

Create an operator which trim first and last row (or column) in The field is a "collapsed" matrix of the size sz1. m is the dimension to trim.

S = matfun_stagger(sz1,m,cyclic)

Create an operator for staggering a field in dimension m. The field is a "collapsed" matrix of the size sz1.

Input: sz1: size of rhs m: dimension to stagger cyclic: true if domain is cyclic along dimension m. False is the default value

Operator for differentiation.

diffx = matfun_diff(sz1,m,cyclic)

Operator for differentiation along dimension m for "collapsed" matrix of the size sz1.

Input: sz1: size of rhs m: dimension to differentiate cyclic: true if domain is cyclic along dimension m. False is the default value

Operator shifting a field in a given dimension.

function S = matfun_shift(sz1,m,cyclic)

Operator shifting a field in the dimension m. The field is a "collapsed" matrix of the size sz1.

Input: sz1: size of rhs m: dimension to shift cyclic: true if domain is cyclic along dimension m. False is the default value

quadtree (of the higher-dimensional equivalent) T the type of the coordinates TA the type of the attributes N number of dimensions

recursive split

sucess = add!(qt,x,attrib,max_cap = 10) Add point x with the attribute attrib to the quadtree qt. sucess is true if x is within the bounds of the quadtree node qt (otherwise false and the point has not been added)

attribs = within(qt,min,max)

Search all the points within a bounding box defined by the vectors min and max.

Test if the n-th bit in a is set. The least significant bit is n = 1.

inside(x0,x1,y)

Returns true of the point y is inside the rectange defined by x0 and x1.

             x1
  +----------+
  |          |
  |   +      |
  |   y      |
  +----------+
 x0

Test if the rectanges defined by x0,x1 and y0,y1 intersects/overlap

             x1
  +----------+
  |          |
  |   +----------+ y1
  |   |      |   |
  +----------+   |
 x0   |          |
      |          |
      +----------+
     y0

split a single node

x,cgsuccess,niter = conjugategradient(fun!,b)

Solve a linear system with the preconditioned conjugated-gradient method: A x = b where A is a symmetric positive defined matrix and b is a vector. Equivalently the solution x minimizes the cost function J(x) = ½ xᵀ A x - bᵀ x.

The function fun!(x,fx) computes fx which is equal to A*x. For example:

function fun!(x,fx)
    fx[:] = A*x
end

Note that the following code will NOT work, because a new array fx would be created and it would not be passed back to the caller.

function fun!(x,fx)
    fx = A*x # bug!
end

The function fun! works in-place to reduce the amount of memory allocations.

Optional input arguments

• x0: starting vector for the interations
• tol: tolerance on |Ax-b| / |b|
• maxit: maximum of interations
• pc!: the preconditioner. The functions pc(x,fx) computes fx = M⁻¹ x (the inverse of M times x) where M is a symmetric positive defined matrix. Effectively, the system E⁻¹ A (E⁻¹)ᵀ (E x) = E⁻¹ b is solved for (E x) where E Eᵀ = M. Ideally, M should this be similar to A, so that E⁻¹ A (E⁻¹)ᵀ is close to the identity matrix. The function pc! should be implemented in a similar way than fun! (see above).

Output

• x: the solution
• cgsuccess: true if the interation converged (otherwise false)
• niter: the number of iterations

pc_none!(x,fx)

Dummy call-back function when no preconditioner is used. fx will be equal to x.

xAy, yATx = checksym(n,fun!)

Check if the the function fun! represents a symmetric matrix when applied on random vectors of size n.

Create the laplacian operator.

Lap = DIVAnd_laplacian(mask,pmn,nu,iscyclic)

Form a Laplacian using finite differences assumes that gradient is zero at "coastline"

Input: mask: binary mask delimiting the domain. 1 is inside and 0 outside. For oceanographic application, this is the land-sea mask. pmn: scale factor of the grid. nu: diffusion coefficient of the Laplacian field of the size mask or cell arrays of fields

Output: Lap: sparce matrix represeting a Laplacian

Dx1,Dx2,...,Dxn = DIVAnd.DIVAnd_gradient(operatortype,mask,pmn,iscyclic)

Form the gradient using finite differences in all n-dimensions. mask is a binary mask delimiting the domain. 1 is inside and 0 outside. For oceanographic application, this is the land-sea mask. pmn is a tuple of arrays with the scale factor of the grid. The output Dx1,Dx2,...,Dxn are sparse matrices represeting a gradient along different dimensions.

Dx1,Dx2,...,Dxn = DIVAnd.DIVAnd_gradient(operatortype,mask,pmn,iscyclic)

Form the gradient using finite differences in all n-dimensions. mask is a binary mask delimiting the domain. 1 is inside and 0 outside. For oceanographic application, this is the land-sea mask. pmn is a tuple of arrays with the scale factor of the grid. The output Dx1,Dx2,...,Dxn are sparse matrices represeting a gradient along different dimensions.

R = DIVAnd_obscovar(epsilon2,m)

Create a matrix representing the observation error covariance R of size m x m.

If epsilon2 is a scalar, then R = epsilon2 * I If epsilon2 is a vector, then R = diag(epsilon2) If epsilon2 is a matrix, then R = epsilon2

factor = DIVAnd_adaptedeps2(s,fi);

Input:

• s: structure returned by DIVAndrun
• fi: analysis returned by DIVAndrun

Output:

• factor : multiplicative factor to apply to epsilon2

Using Deroziers adaptive approach provides a multiplicative factor for the current epsilon2 value so that factor*epsilon2 is a better estimate of the R matrix. If you cannot use DIVAndrun but use DIVAndgo, the latter provides automatically this pamater as result.

DIVAnd_adaptedeps2(yo, residual, diagR, ignoreobs)

Using Deroziers adaptive approach provides a multiplicative factor for the current epsilon2 value so that factor*epsilon2 is a better estimate of the R matrix.

yo the observations (minus the background), residual the obserations minus the analysis, diagR, the diagonal of the rel. obs. error covariance matrix and ignoreobs is true if an observation is out of the grid or should be ignored for other reasons.

For unscaled R and assuming that the background is zero, Deroziers showed that:

mean((yᵒ - Hxᵃ) ⋅ yᵒ) = ϵ² mean(yᵒ ⋅ yᵒ) = σ² + ϵ²

mean(yᵒ ⋅ yᵒ) / mean((yᵒ - Hxᵃ) ⋅ yᵒ) = σ²/ϵ² + 1 λ = σ²/ϵ² = 1 - mean(yᵒ ⋅ yᵒ) / mean((yᵒ - Hxᵃ) ⋅ yᵒ)

ϵ² / σ² = 1 / λ

diagonalterms = DIVAnd_diagHKobs(s)

Computes the diagonal terms of the so called hat-matrix HK, using the already solved analysis and it structure s. Warning: might take some time

This version only uses the real data (not those related to additional constraints)

dataresidual = DIVAnd_residual(s,fi)

Computes the generalized residual yo - H xa using the analysis on the grid fi and the solution structure s.

s = DIVAnd_addc(s,c)

Add a constraint c to the cost function defined by s. The structure s is typically created by DIVAnd_background and the contrain c has the following fields: R (a covariance matrix), H (extraction operator) and yo (specified value for the constrain). The added contrain Jc(x) is quadratic and has the following structure.

Jc(x) = (H x - yo)ᵀ R⁻¹ (H x - yo)

errorvariance = DIVAnd_erroratdatapoints(s;restrictedlist=[])

Computes the error at the real data locations using the analysis structure s

If a restricedlist is provided erros are only calculated at the indexes where restricedlist==true

Computes an estimate of the mean value of the diagonal of HK using GCV and the already solved analysisand it structure s

Kii = DIVAnd_GCVKii(s);

 stepsize,overlapping,isdirect = DIVAnd_fittocpu(Lpmnrange,gridsize,latercsteps,moddim,MEMTOFIT;forcedirect=false,overlapfactor=3.3);

Creates a list of windows for subsequent domain decomposition

Also calculates already the subsampling steps csteps for the preconditionners

Input:

• Lpmnrange: for every dimension the minimum and maximum correlation length scaled by the local resolution (i.e. the product between L and pm (pn,...))
• gridsize: number of points in each direction (size(mask))
• latercsteps: coarsening steps used later if a lower resolution model is used for preconditioning.
• moddim: modulo for cyclic dimension (vector with n elements). Zero is used for non-cyclic dimensions.
• memtofit : parameter describing how much memory is expected to be available in Gb
• forcedirect : if true forces direct solver even if iterative solver might allow for larger tiles
• overlapfactor : describes how many times the length scale is used for the overlapping. default is 3.3. use lower values ONLY for very good data coverage.

Output:

• stepsize: spatial (and temporal) shift in grid points between subdomains for every dimension (?)
• overlapping: number of overlapping grid points for every dimension
• isdirect: true is the direct solver is activated

Form the inverse of the background error covariance matrix. s = DIVAnd_background(mask,pmn,Labs,alpha,moddim) Form the inverse of the background error covariance matrix with finite-difference operators on a curvilinear grid

Input:

• mask: binary mask delimiting the domain. 1 is inside and 0 outside. For oceanographic applications, this is the land-sea mask.
• pmn: scale factor of the grid.
• Labs: correlation length
• alpha: dimensional coefficients for norm, gradient, laplacian,... alpha is usually [1,2,1] in 2 dimensions.

Output:

• s: structure containing
  • s.iB: inverse of the background error covariance
  • s.L: spatially averaged correlation length
  • s.n: number of dimensions
  • s.coeff: scaling coefficient such that the background variance diag(inv(iB)) is one far away from the boundary.

s = DIVAnd_obs(s,xi,x,R,I)

Include the constrain from the observations. It is assumed that each coordinate depends only on one index. If this is not the case, then matrix I must be provided.

Input: s: structure created by DIVAnd_background xi: coordinates of observations (tuple of vectors) x: coordinates of grid (tuple of arrays) R: obs. error covariance matrix (normalized) I (optional): fractional indexes of location of observation within the grid

Output: s: structure to be used by DIVAnd_factorize

Note: make sure not to mix Float32 and Float64 for DIVAnd_constrain.

Computes the diagonal terms of the so called hat-matrix HK, using the already solved analysis and it structure s. Warning: might take some time

diagonalterms = DIVAnd_diagHK(s);

mu,K,len_scale = DIVAnd_kernel(n,alpha)

Return the analytical kernel and normalization factor.

Analytical (normalized) kernels K for infinite domain in dimension n and for coefficients alpha and normalization factor mu.

K(r) is the kernel function (function of the normalized distance r), len_scale is the distance at which K(len_scale) = 0.6019072301972346 (which is besselk(1,1))

dataresidual = DIVAnd_residualobs(s,fi);

Computes the residual yo - H xa only at real data points using the analysis. on the grid fi and the solution structure s.

aexerr,Bref,fa,sa = DIVAnd_aexerr(mask,pmn,xi,x,f,len,epsilon2;...);

Input: same as for DIVAndrun

• mask: binary mask delimiting the domain. true is inside and false outside. For oceanographic application, this is the land-sea mask.

• pmn: scale factor of the grid. pmn is a tuple with n elements. Every element represents the scale factor of the corresponding dimension. Its inverse is the local resolution of the grid in a particular dimension.

• xi: tuple with n elements. Every element represents a coordinate of the final grid on which the observations are interpolated

• x: tuple with n elements. Every element represents a coordinate of the observations

• f: value of the observations minus the background estimate (m-by-1 array). (see note)

• len: correlation length

• epsilon2: error variance of the observations (normalized by the error variance of the background field). epsilon2 can be a scalar (all observations have the same error variance and their errors are decorrelated), a vector (all observations can have a difference error variance and their errors are decorrelated) or a matrix (all observations can have a difference error variance and their errors can be correlated). If epsilon2 is a scalar, it is thus the inverse of the signal-to-noise ratio.

Optional input arguments specified as keyword arguments also as for DIVAnd

Output:

• aexerr: the almost exact error

• Bref: the background error for error scaling by background aexerr./Bref

• fa: the analysis (with low impact fake data): DO NOT USE UNLESS YOU KNOW WHAT YOU ARE DOING

• sa: the associated structure

Compute a variational analysis of arbitrarily located observations to calculate the almost exact error

cpme = DIVAnd_cpme(mask,pmn,xi,x,f,len,epsilon2;...);

Input: Same as for DIVAndrun

• mask: binary mask delimiting the domain. true is inside and false outside.

For oceanographic application, this is the land-sea mask.

• pmn: scale factor of the grid. pmn is a tuple with n elements. Every element represents the scale factor of the corresponding dimension. Its inverse is the local resolution of the grid in a particular dimension.

• xi: tuple with n elements. Every element represents a coordinate of the final grid on which the observations are interpolated

• x: tuple with n elements. Every element represents a coordinate of the observations

• f: value of the observations minus the background estimate (m-by-1 array). (see note)

• len: correlation length

• epsilon2: error variance of the observations (normalized by the error variance of the background field). epsilon2 can be a scalar (all observations have the same error variance and their errors are decorrelated), a vector (all observations can have a difference error variance and their errors are decorrelated) or a matrix (all observations can have a difference error variance and their errors can be correlated). If epsilon2 is a scalar, it is thus the inverse of the signal-to-noise ratio.

• keywords : undocumented for the moment how to use iterative solver with coarser grid as preconditionner. see DIVAndjog for csteps, lmask and alphapcparameters

Optional input arguments specified as keyword arguments also as for DIVAnd

Output:

• cpme: the clever poor mans error

Perform an n-dimensional variational analysis of the observations f located at the coordinates x. The array cpme represent the error field at the grid defined by the coordinates xi and the scales factors pmn. If you cannot run DIVAndrun you can use DIVAndgo with error field calculation :cpme

erri = DIVAnd_cpme_go(mask,pmn,xi,x,f,len,epsilon2; ...);

Input:

• Same arguments as DIVAndrun with in addition
• MEMTOFIT=: keyword controlling how to cut the domain depending on the memory remaining available for inversion (not total memory)
• RTIMESONESCALES= : if you provide a tuple of length scales, data are weighted differently depending on the numbers of neighbours they have. See weight_RtimesOne for details

Output:

• erri: relative error field using the clever poor man's error approach. Result on the same grid as fi.

ONLY USE THIS VERSION IF YOU CANNOT RUN DIVAndgo with :cmpe activated (or directly DIVAnd_cpme if you can run DIVAndrun)

xn,fn,indexes,Rn = DIVAnd_datainboundingbox(xi,x,f;Rmatrix=())

Input:

xi: tuple with n elements. Every element represents a coordinate of the final grid on which the observations are interpolated

• x: tuple with n elements. Every element represents a coordinate of the observations

• f: value of the observations

• Rmatrix: error variance of the observations (normalized by the error variance of the background field). epsilon2 can be a scalar (all observations have the same error variance and their errors are decorrelated), a vector (all observations can have a difference error variance and their errors are decorrelated) or a matrix (all observations can have a difference error variance and their errors can be correlated). If epsilon2 is a scalar, it is thus the inverse of the signal-to-noise ratio.

Output:

xn: tuple with n elements. Every element represents a coordinate of the observations which falls in the bounding box defined by xi fn: the corresponding data indexes: the indexes in the original array retained Rn: the new error variance

Lpmnrange = DIVAnd_Lpmnrange(pmn,len);

In each direction, searches for the minimum and maximum value of the length scale times the metric in this direction

So it basically looks at the worst and the best resolution found in the grid

Input:

• pmn: scale factor of the grid. pmn is a tuple with n elements. Every element represents the scale factor of the corresponding dimension. Its inverse is the local resolution of the grid in a particular dimension.

• len: correlation length

Output:

• Lpmnrange: Array of range tuples (minimum and maximum of L times metric)

Compute a preconditioner using the Cholesky decomposition.

[M1,M2] = DIVAnd_pc_michol(iB,H,R)

Compute preconditioner matrices M1 and M2 based on the Cholesky decomposition of iB. The matrices H and R are not used. M2 is the transpose of M1 for this preconditioner.

fun = DIVAnd_pc_none(iB,H,R)

Dummy function for requiring that no preconditioner is used in DIVAnd.

See also: diavndpcsqrtiB

Kii = DIVAnd_GCVKiiobs(s)

Computes an estimate of the mean value of the diagonal of HK using GCV and the already solved analysis and it structure s Only using real data locations.

windowlist,csteps,lmask,alphapc = DIVAnd_cutter(Lpmnrange,gridsize,moddim,MEMTOFIT);

Creates a list of windows for subsequent domain decomposition. Also calculates already the subsampling steps csteps for the preconditionners as well as the mask lmask to apply to the length scales in the preconditionner, allowing to reduce the problem size

Input:

• Lpmnrange:
• gridsize: number of points in each direction (size(mask))
• moddim:
• MEMTOFIT
• overlapfactor : describes how many times the length scale is used for the overlapping. default is 3.3. use lower values ONLY for very good data coverage.

Output:

• windowlist: vector of tuples (iw1,iw2,isol1,isol2,istore1,istore2,) where (iw1,iw2) correspond to the start and end indices in the (global) grid (isol1,isol2) correspond to the start and end indices solution to be retained in the window (not all is retained due to overlapping) and (istore1,istore2) correspond to the start and end indices of the solution relative to the global grid. They define thus where the local solution has to be stored in the combined global solution.
• csteps: Array of steps for the coarse grid preconditionner. csteps is zero for the direct solver.
• lmask: Array of multiplication factors for length scale of preconditionner
• alphapc: Norm defining coefficients for preconditionner

qcvalues = DIVAnd_qc(fi,s,method);

Perform a quality control of the observations using the interpolated field.

Input:

• fi : interpolated field from a DIVAndrun execution
• s: corresponding structure returned by DIVAnd
• method : optional argument, which describes the method to be used:

1 as for standard cross validation, 3 as for GCV, 4 with CV estimator to be used outside the routine, 5 Poor man's GCV using data instead of random vector, 0 automatic selection of method.

Output

• qcvalues: quality check values, one for each data point.

The higher the value, the more suspect a data point is. Absolute values of qcvalues might be not robust when analysis parameters are uncertain. The ranking is however quite robust.

If you cannot run DIVAndrun but use DIVAndgo (which does not provide a structure s at the output), the latter provides qcvalues if you call DIVAndgo with a keyword parameter QCMETHOD=

Solve the variational problem.

 fi = DIVAnd_solve(s)

Derive the analysis based on all contraints included in s and using the observations yo

Input: s: structure created by DIVAnd_factorize fi0: starting point for iterative primal methods f0: starting point for the iterative dual method

btrunc: the value at which the stored value of s.iB was truncated and needs to be completed on the fly using jmBix

Output: fi: analyzed field

samplesteps = DIVAnd_sampler(pmn,len);

Defines steps for sub-sampling in the discrete grid which would still allow

one to resolve the provided lengthscales

Input:

• pmn: scale factor of the grid. pmn is a tuple with n elements. Every element represents the scale factor of the corresponding dimension. Its inverse is the local resolution of the grid in a particular dimension.

• len: correlation length

Output:

• samplesteps: vector of integers with steps in subsampling [1 2 4 1] means every grid point in x direction, every fifth in y etc

Compute a variational analysis of arbitrarily located observations.

 fi,s = DIVAndjog(mask,pmn,xi,x,f,len,epsilon2,csteps,lmask; alphapc=[1,2,1], otherargs...);

Perform an n-dimensional variational analysis of the observations f located at the coordinates x. The array fi represent the interpolated field at the grid defined by the coordinates xi and the scales factors pmn.

Input:

• Same parameters as for divarun. * Two additional parameters: * csteps: array of ndims values defining the sampling steps for the preconditionner * lmask: array of ndims mutilplications factors for length scales * One additional optional parameter * alphapc: The coefficients for the norm used in the preconditionner

Output:

• fi: the analysed field
• s: structure with an array s.P representing the analysed error covariance

iB = DIVAnd_background_components(s,D,alpha; kwargs...)

Form the different components of the background error covariance matrix. Compute the components of the background error covariance matrix s.iB_ and their sum based on alpha (the adimensional coefficients for norm, gradient, laplacian,...).

If the optional arguments contains btrunc, the calculation of iB is limited to the term up and including alpha[btrunc]

meanx,stdx = stats(sumx,sumx2,N)

Computes the mean meanx and the standard deviation stdx from the sum (sumx) and the sum of squares (sumx2) from N numbers.

meanx,meany,stdx,stdy,covar,corr = stats(sumx,sumx2,sumy,sumy2,sumxy,N)

Computes the mean meanx and the standard deviation stdx from the sum (sumx) and the sum of squares (sumx2) from N numbers and similarly for the variable y. The function computes also the Pearson correlation corr and covariance covar between x and y.

ulon,ulat = statpos(lon,lat)

Return unique positions (ulon, ulat) as well their mean, standard deviation and count of the vector of observations val located at the positions lon and lat.

ulon,ulat,meanval,stdval,count = statpos(val,lon,lat)

Return unique positions (ulon, ulat) as well as their mean, standard deviation and count of the vector of observations val located at the positions lon and lat.

concatenate diagonal matrices

findfirst(c::Concept,name,collection)

Return the first related concepts in the collection collection. name can be the string "related", "narrower", "broader".

display size as a string

interp!(xi,fi,x,f)

Interpolate field fi (n-dimensional array) defined at xi (tuble of n-dimensional arrays or vectors) onto grid x (tuble of n-dimensional arrays). The interpolated field is stored in f. The grid in xi must be align with the axis (e.g. produced by DIVAnd.ndgrid).

cfilled = ufill(c,valex)

Replace values in c equal to valex by averages of surrounding points. valex should not be NaN; use ufill(c,isfinite.(c)) or ufill(c,.!isnan.(c)) instead.

ufill(c::Array{T,2},mask::AbstractArray{Bool}) where T

mask is true where c is valid.

hx,hy = cgradient(pmn,h)

fzero(f,x0,x1,eps; maxiter = Inf) find the zero of the function f between x0 and x1 assuming x0 < x1 at a precision eps.

Derive fractional indices on a separable grid.

I = localize_separable_grid(xi,mask,x)

xi is a tuple of vectors and x and tuple of n-dimensional arrays, e.g.

x1,x2 = ndgrid(2 * collect(1:5),collect(1:6)) x = (x1,x2)

Derive fractional indices where xi are the points (typical discrete observations) to localize in the separable grid x (every dimension in independent on other dimension). The output I is an n-by-m array where n number of dimensions and m number of observations. The corresponding element of I is negative if xi is outside of the grid defined by x.

work1, work2: size of mask

Symmetric matrix

SB = √(β) (1 + α L)^(nmax / 2) W^{-1}

where W is the volumne of the corresponding grid cell. The background error covariance matrix B is SB W SB

Variational analysis similar to 3D-var

Input:

x0: start vector for iteration, at output it is the last state of the iteration. Note that x0 is related to the analysis xa by xa = SB^½ * W^½ * xa

| x + W^½ * SB^½ * H' * (R \ (H * SB^½ * W^½ * x )) - W^½ SB^{½} * H' * (R \ yo) | < tol * s.sv.n / length(yo) * | W^½ SB^{½} * H' * (R \ yo) |

Kernel is the solution of the n-dimensional diffusion equation

∂c/∂t = ∇ ⋅ (D ∇ c)

n-dimensional Green’s function

G(x,x',t) = (4πDt)^(-n/2) exp( - |x -x'|² / (4Dt))

G(x,x',t) = det(D)^(-½) (4π t)^(-n/2) exp( - (x -x')ᵀ D⁻¹ (x -x') / (4t))

http://www.rpgroup.caltech.edu/~natsirt/aph162/diffusion_old.pdf

Len = len_harmonise(len,mask)

Produce a tuple of arrays of the correlation length len which can be either a scalar (homogeneous and isotropic case), a tuple of scalar (homogeneous case) or already a tuple of arrays (general case). The the later case the size of the arrays are veryfied.

neff, alpha = alpha_default(Labs,alpha)

Return a default value of alpha.

DIVAnd_save(ds,filename,xyi,fi,varname;
                  ncvarattrib = Dict(), ncglobalattrib = Dict(), ...)

Save the result of the analysis in a netCDF file .

Input arguments

• ds: the NetCDF dataset
• filename: the name of the NetCDF file
• mask: binary mask delimiting the domain. true is inside and false outside. For oceanographic application, this is the land-sea mask where sea is true and land is false.
• xyi: tuple with n elements. Every element represents a coordinate of the final grid on which the observations are interpolated
• fi: the analysed field
• varname: the name of the NetCDF variable

Optional arguments:

• ncglobalattrib: a dictionary with the global attributes
• ncvarattrib: a dictionary with the variable attributes
• relerr: relative error
• timeorigin: time origin for the time units attribute (default is 1900-01-01 00:00:00)

ncvar, ncvar_relerr, ncvar_Lx, fi, relerr, index)

White a slice of data in a NetCDF given by the index index. The variable relerr can be nothing.

encode parameters as key-value separated by : and +

db = loadoriginators(fname)

Load the CDI list from the file fname (zip with a csv file, or csv file directly).

Computes an N-dimensional volume integral

DIVAnd_integral(mask,pmn,fieldin)

Input:

• mask: mask as usual
• pmn : tuple of metrics as usual
• fieldin: field of the same dimensions as mask and which is integrated over the domain

Output:

• integratedfield: The integral

Computes a relative length based on the mask, metrics and a density field, typically measuring the observation density calculated with DIVAnd_heatmap

lambda = DIVAnd_scaleL(mask,pmn,dens)

Input:

• mask: mask as usual
• pmn : tuple of metrics as usual
• dens: field of the same dimensions as mask. Higher values of dens will result in lower values of lambda.

Output:

• lambda: field to be applied to a reference length field. Values are around 1 so some regions will have smaller L and some higher L