Issue in particle filter DA method

[LIS-navari]

Bug Description

In the EnKF all the ensemble members at time step t are updated using the same approximated Kalman gain and the innovation factor which depends on the perturbed measurements and the observation model. On the other hand, in the particle filter all the particles at time step t are weighted and resampled (Plaza et al 2011).

In the current Particle Filter (PF) implementation within LIS, we treated the PF similarly to EnKF. This involved computing the increment and adding it to the state variables. However, I believe this implementation is incorrect.

The updated weights provide a discrete estimate of the posterior PDF, which can be used to determine posterior statistics, for example, those representing the central tendency (e.g., mean and median). The posterior PF median (at a given time) is determined by sorting the state variables (e.g, SWE , soil moisture) replicates (particles) and summing up the corresponding weights to get the replicate with an integrated value closest to 50% (Margulis 2015).

Correct implementation:

We should compute the posterior weights, resample the particles (ensembles) based on these weights, and initiate the simulation after the assimilation step using selected particles without updating the state variables.