Sequential Markov Chain Monte Carlo for Lagrangian Data Assimilation with Applications to Unknown Data Locations

We illustrate the performance of our Algorithm (SMCMC) in the paper (https://arxiv.org/abs/2305.00484) published in QJRMS in four different cases; two linear cases and two nonlinear cases (rotating shallow-water equations):

1. A linear Gaussian model 1. Here we show both efficiency and accuracy compared to competing ensemble methodologies: EnKF, ETKF and ESTKF. In contrast to how practitioners use the latter, we use moderately high number of ensembles 500-10^4. Note that for linear Gaussian models these methods are consistent and increasing the number of ensembles improves accuracy. The goal is to compare these methods at a higher accuracy. To run this case do the following:

   • First we run the KF to generate the data and the true filter for the given parameters in example_input.yml , simply run the following in the terminal:

      cd linear_Gaussian_model1
      python3 ./run_KF.py
     

     This will create a folder named "example_dx=#_dy=#_sigx=#_sigy=#" with # corresponds to the parameters dx, dy, sig_x and sig_y from example_input.yml . It will also generate the observations (the data) data.h5 and save it in the subfolder data to be used by the other methods. It will also save the true filter (the Kalman Filter) kalman_filter.h5 in a subfolder kf.
   • Then run the rest of the methods from the same main folder as follows:

     python3 ./run_smcmc_filter.py
     python3 ./run_enkf.py
     python3 ./run_etkf.py
     python3 ./run_estkf.py
     

     These runs will generate subfolders with the corresponding filter.
   • After finishing running the rest of the methods, open the matlab file read_h5.m and edit the parameters then run the following matlab code to generate histogram plots of the absolute errors.

     matlab ./read_h5.m
     

2. A linear Gaussian model 2. Here we show both efficiency and accuracy compared to the R-localaized EnKF methodology. This is similar to the previous example. First run the Kalman Filter code to generate the data and the true filter. Then call run_smcmc_filter.py and run_enkf_loc.py. Finally, edit the matlab file read_h5.m and run to generate histogram plots.

The code for the shallow-water cases is available upon request.