xdispersion

1. Introduction

The package is designed for analyzing relative dispersion behaviors of Lagrangian particle pairs (two-particle) in a statistical fashion. It is based on the data structure of xarray and hence dask, so that one can leverage the power of multi-dimensional labelled arrays and out-of-core calculations.

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2. How to install

Requirements xdispersion is developed under the environment with xarray (=version 0.15.0), dask (=version 2.11.0), numpy (=version 1.15.4), scipy (=version 1.13.1), tqdm (=version 4.66.5), xhistogram (=version 0.3.2), mpmath (=version 1.2.0). Older versions of these packages are not well tested.

Install via pip

pip install xdispersion # not yet

Install via conda

conda install -c conda-forge xdispersion # not yet

Install from github

git clone https://github.com/miniufo/xdispersion.git
cd xdispersion
python setup.py install

3. How to do this nicely and efficiently

3.1 Problems of representing the data

For Lagrangian data represented in multi-dimensional xarray.DataArray, one of the major problems is that the lengths of Lagrangian trajectories are not the same (see real drifter data in the figure). So it is impossible to use a traj dimension to represent different trajectories.

The second problem is how to represent the dataset of particle pairs. For example, if there are 4 particles P1, P2, P3, and P4, through combination one would get 6 pairs (see pair data in the figure). Pair data are still of different lengths. More importantly, each trajectory is duplicated three times so that the overall storage would be greatly increased.

3.2 Solutions to the problems

The first problem is solved by introducing the ragged array from CloudDrift project based on Awkward Array: All the trajectories are connected head to tail, forming an obs dimension. This is memory and storage efficient, but requires extra efforts to retrieve the data of each trajectory.

The second problem is solved here by grouping the basic information of particle pairs pairs (e.g., pair length, initial separations, initial time, initial locations..., see drifter pairs in memory in the figure below) into a xr.Dataset, instead of duplicating all the data. Therefore, this design is memory efficient. In addition, this allows one to filter pairs and select those satisfying specific criterions using xarray's functionality:

pairs = get_all_pairs(drifters)

# select original pairs whose initial separation r0 is within a given range
condition = np.logical_and(pairs.r0>rmin, pairs.r0<rmax)
origin_pairs = pairs.where(condition, drop=True).astype(pairs.dtypes)

Also, one can perform statistics (like histogram of r0) using pair information pairs or plot the statistics (like initial locations), without loading all the data into memory.

3.3 Calculations of dispersion measures

To calculate dispersion measures, like relative dispersion $r^2(t)=<|\mathbf{x_i}-\mathbf{x_j}|^2>$, one need to load the full data into memory. This means that we need to specify a maximum time of analysis (maxtlen): pairs will be truncated if they are longer than maxtlen, and will be padded with nans if they are shorter than maxtlen. This is shown schematically below. Notice that the time is not the actual date/time, so we call it relative time rtime.

This is not memory efficient, as already mentioned. One can sum up all the pairs online without allocating a memory of pair×rtime size. But we choose to keep the ['pair', 'rtime'] dimensions in the present design because:

• one can check intermediate results before taking average over pair dimension;
• one can get the number of observations by counting the non-nans values, which generally vary with time;
• one can perform an estimate of the errorbar through bootstrapping along the pair dimension;
• one can use chunks along pair dimension if there are many pairs, when dask is used.

Following this design, we have a function get_variable(pairs, vname) to load a variable (positions or velocities) from pairs into memory:

lon = get_variable(pairs, 'lon')
lat = get_variable(pairs, 'lat')

The returned variable is of size ['pair', 'particle', 'rtime'], and relative dispersion is then calculated as:

lon1 = lon.isel(particle=0) # 1st particle
lon2 = lon.isel(particle=1) # 2nd particle
lat1 = lat.isel(particle=0) # 1st particle
lat2 = lat.isel(particle=1) # 2nd particle

# r2(t) is only a function of rtime
r2 = (great_circle(lon1, lon2, lat1, lat2) ** 2.0).mean('pair')

which is clean and clear, as it follows almost to its mathematical expression.

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3. Examples

Notebooks are given here demonstrating how to use the package.