advectionDiffusion

Propagation of a Gauss-linear model and data assimilation by means of the analytical Kalman filter (KF) as well as by means of the ensemble-based Ensemble transform Kalman filter (ETKF) without and with localisation

Acknowledgement

The example and the original code is taken from the work of Gunhild Berget (NTNU).

Content

The stochastic advection diffusion equations serves as an example to compare the posteriors of the ETKF and the LETKF, respectively, with the analytical posteriors of the KF.

Requirements

We recommend the use of a conda environment with Python 3.8 (tested) and the following packages

• jupyter
• numpy
• matplotlib
• scipy
• statsmodels

Introduction to code structure

Utility functions are collected in the .py-files, but plots are generated in the .ipynb-notebooks.

Forward model and truth generation

The truth is handled as an ensemble of size 1, the corresponding experimental set-up and the observed values are stored to file in /experiment_files/.... The timestamp serves as a key for each set-up or observation, respectively. See Truth.ipynb.

Filtering

Given a set-up (by its timestamp), data assimilation can be seen in action in KFiltering.ipynb, ETKFiltering.ipynb, and SLETKFiltering.ipynb for the respective methods.

Comparison

The FilteringComparison.ipynb opposes the results of the different data assimilation methods for a single experiment by different criteria. The same critieria are evaulated for multiple repetitions in run_FilteringComparison.py with changing parameters if specified.

Reproducing resutls

The plots shown in the paper "Comparison of Ensemble-Based Data Assimilation Methods for Oceanographic Applications with Sparse Observations" are generated from this repository.

• Figure 3.1: Truth.ipynb (Note that no seed is selected, such that the truth will look different every single realisation) and the values are written to file with a certain time stamp in the /experiment_files/...-folder.
• Figure 3.2: KFiltering.ipynb (The truth is refered to by a time stamp.)
• Figure 3.3, 3.4, 3.7: FilteringComparison.ipynb (The truth is refered to by a time stamp and results are again stochastic.)
• Figure 3.5: FilteringCoverage.ipynb
• Figure 3.6: Spectrum.ipynb
• Figure 3.8, 3.9: Generate outputs with run_FilteringComparisonLocalisation.py and postprocess with Localisation_PostProcessing.ipynb
• Figure 3.10: Generate outputs with run_FilteringComparison.py

NB: The plots in the notebooks are only for reference and the final figures in the manuscript are generated with tikz such that they differ in style.