Implement Inflation and Localisation for enkf DA

[BenjMy]

Improvement of spatial DA considering localisation

• Short and efficient explanation: https://epic.awi.de/id/eprint/31369/1/Offenbach.pdf
• https://www.sciencedirect.com/science/article/pii/S0309170820306138#fig0002
• Goof ref (thesis): https://theses.hal.science/tel-03117151/document
• Nice explication from https://doi.org/10.1038/s41598-020-75710-5
• Covariance localisation: https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2019.00003/full " To enhance EnKF performance during assimilation ensemble inflation and localization are applied. It has been shown by literature85,86 that ensemble-based data assimilation methods are sensitive to the size of ensemble. Generally, a larger number of ensemble members can better span the state-observation space and lead to better results but at the expense of strongly increased computation needs. To address this, ensemble inflation and localization methods are usually used to tackle filter divergent or inaccurate estimation87 for a small ensemble size and to avoid filter inbreeding. Ensemble inflation increases ensemble deviation from the ensemble-mean by applying a small coefficient ([1.1−1.3] for the parameter and state updates) to ensemble members88. Localization using the Local Analysis (LA) scheme is also applied. It performs by spatially limiting the assimilation process within a certain distance from a grid point10,89. The suggested values (3∘ ) by Khaki et al.10 are used as localization radii to achieve the best outcomes using a trial and error. "

Examples of applications in prep:

• [ ] Noble dataset (Peruzzo et al.)
• [ ] Re della Pietra dataset
• [ ] Bousval dataset
• [ ] Synthetic dataset Weill

How?

from pyCATHY.DA.cathy_DA import DA
from pyCATHY.DA import perturbate, normalise, localisation
from pyCATHY.DA.cathy_DA import perturbate_parm, dictObs_2pd

# We create a DA object
simu = DA(dirName=path2prj, prj_name=prj_name_DA)

# We read all the observations (spatially distributed)
read_observations()
# Here is the new function to add: need to localize before creation of the data covariance matrice
localisation.localisation()
make_data_cov()

# Parameters pertubation
perturbate.perturbate()

# run DA simulation
simu.run_DA_sequential()

[BenjMy]

Localisation with LenKf

LenKf: local ensemble Kalman filter

Could be done simply using zones definition (from the zone file) or by layers. Each zone will be treated individually (unique analysis and parameter update).

Improvements needed:

• parameters perturbation per layers and zone. Add a flag to indicate it;
• update the soil parameters per layer and per zone;