DIVAnd

DIVAnd (Data-Interpolating Variational Analysis in n dimensions) performs an n-dimensional variational analysis/gridding of arbitrarily located observations. Observations will be interpolated/analyzed on a curvilinear grid in 1, 2, 3 or more dimensions. In this sense it is a generalization of the original two-dimensional DIVA version (still available here https://github.com/gher-ulg/DIVA but not further developed anymore).

The method bears some similarities and equivalences with Optimal Interpolation or Krigging in that it allows to create a smooth and continous field from a collection of observations, observations which can be affected by errors. The analysis method is however different in practise, allowing to take into account topological features, physical constraints etc in a natural way. The method was initially developped with ocean data in mind, but it can be applied to any field where localized observations have to be used to produce gridded fields which are "smooth".

Please cite this paper as follows if you use DIVAnd in a publication:

Barth, A., Beckers, J.-M., Troupin, C., Alvera-Azcárate, A., and Vandenbulcke, L.: DIVAnd-1.0: n-dimensional variational data analysis for ocean observations, Geosci. Model Dev., 7, 225-241, doi:10.5194/gmd-7-225-2014, 2014.

(click here for the BibTeX entry).

Installing

Under Linux you will also need the packages make, gcc and netcdf which you can install under Debian/Ubuntu with:

apt-get install make gcc libnetcdf-dev netcdf-bin

You need Julia (version 1.0 or 1.4) to run DIVAnd. The command line version is sufficient for DIVAnd. Inside Julia, you can download and install the package by issuing:

using Pkg
Pkg.add(PackageSpec(name="DIVAnd", rev="master"))

It is not recommended to download the source of DIVAnd.jl directly (using the green Clone or Download button above) because this by-passes Julia's package manager and you would need to install the dependencies of DIVAnd.jl manually.

Updating DIVAnd

To update DIVAnd, run the following command and restart Julia (or restart the jupyter notebook kernel):

using Pkg
Pkg.add(PackageSpec(name="DIVAnd", rev="master"))

Testing

A test script is included to verify the correct functioning of the toolbox. The script should be run in a Julia session. Make sure to be in a directory with write-access (for example your home directory). You can change the directory to your home directory with the cd(homedir()) command.

Pkg.test("DIVAnd")

All tests should pass without error.

INFO: Testing DIVAnd
Test Summary: | Pass  Total
  DIVAnd      |  427    427
INFO: DIVAnd tests passed

The test suite will download some sample data. You need to have Internet access and run the test function from a directory with write access.

Documentation

The main routine of this toolbox is called DIVAnd which performs an n-dimensional variational analysis of arbitrarily located observations. Type the following in Julia to view a list of parameters:

using DIVAnd
?DIVAndrun

Example

DIVAnd_simple_example_4D.jl is a basic example in fours dimensions. The call to DIVAndrun looks like this:

fi,s = DIVAndrun(mask,(pm,pn,po,pq),(xi,yi,zi,ti),(x,y,z,t),f,len,epsilon2);

where mask is the land-sea mask, usually obtained from the bathymetry/topography, (pm,pn,po,pq) is a n-element tuple (4 in this case) containing the scale factors of the grid, (xi,yi,zi,ti) is a n-element tuple containing the coordinates of the final grid, (x,y,z,t) is a n-element tuple containing the coordinates of the observations, f is the data anomalies (with respect to a background field), len is the correlation length and epsilon2 is the error variance of the observations.

The call returns fi, the analyzed field on the grid (xi,yi,zi,ti).

Note on which analysis function to use

DIVAndrun is the core analysis function in n dimensions. It does not know anything about the physical parameters or units you work with. Coordinates can also be very general. The only constraint is that the metrics (pm,pn,po,...) when multiplied by the corresponding length scales len lead to non-dimensional parameters. Furthermore the coordinates of the output grid (xi,yi,zi,...) need to have the same units as the observation coordinates (x,y,z,...).

DIVAndgo is only needed for very large problems when a call to DIVAndrun leads to memory or CPU time problems. This function tries to decide which solver (direct or iterative) to use and how to make an automatic domain decomposition. Not all options from DIVAndrun are available.

diva3D is a higher-level function specifically designed for climatological analysis of data on Earth, using longitude/latitude/depth/time coordinates and correlations length in meters. It makes the necessary preparation of metrics, parameter optimizations etc you normally would program yourself before calling the analysis function DIVAndrun.

Note about the background field

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.

Determining the analysis parameters

The parameter epsilon2 and parameter len are crucial for the analysis.

epsilon2 corresponds to the inverse of the signal-to-noise ratio. epsilon2 is the normalized variance of observation error (i.e. divided by the background error variance). Therefore, its value depends on how accurate and how representative the observations are. len corresponds to the correlation length and the value of len can sometimes be determined by physical arguments. Note that there should be one correlation length per dimension of the analysis.

One statistical way to determine the parameter(s) is to do a cross-validation.

choose, at random, a relatively small subset of observations (about 5%). This is the validation data set.
make the analysis without your validation data set
compare the analysis to your validation data set and compute the RMS difference
repeat steps 2 and 3 with different values of the parameters and try to minimize the RMS difference.

You can repeat all steps with a different validation data set to ensure that the optimal parameter values are robust. Tools to help you are included in (DIVAnd_cv.jl).

Note about the error fields

DIVAnd allows the calculation of the analysis error variance, scaled by the background error variance. Though it can be calculated "exactly" using the diagonal of the error covariance matrix s.P, it is too costly and approximations are provided. Two version are recommended, DIVAnd_cpme for a quick estimate and DIVAnd_aexerr for a version closer the theoretical estimate (see Beckers et al 2014 )

Advanced usage

Additional constraint

An arbitrary number of additional constraints can be included to the cost function which should have the following form:

J(x) = ∑_i (C_ix - z_i)ᵀ Q_i^-1 (C_ix - z_i)

For every constrain, a structure with the following fields is passed to DIVAnd:

yo: the vector z_i
H: the matrix C_i
R: the matrix Q_i (symmetric and positive defined)

Internally the observations are also implemented as constraint defined in this way.

Run notebooks on a server which has no graphical interface

On the server, launch the notebook with:

jupyter-notebook --no-browser --ip='0.0.0.0' --port=8888

where the path to jupyter-notebook might have to be adapted, depending on your installation. The ip and port parameters can also be modified.

Then from the local machine it is possible to connect to the server through the browser.

Thanks to Lennert and Bart (VLIZ) for this trick.

Example data

Some examples in DIVAnd.jl use a quite large data set which cannot be efficiently distributed through git. This data can be downloaded from the URL https://dox.ulg.ac.be/index.php/s/Bo01EicxnMgP9E3/download. The zip file should be decompressed and the directory DIVAnd-example-data should be placed on the same level than the directory DIVAnd.jl.

Fun

An educational web application has been developed to reconstruct a field based on point "observations". The user must choose in an optimal way the location of 10 observations such that the analysed field obtained by DIVAnd based on these observations is as close as possible to the original field.