# Mengbin Zhu, Academy of Ocean Science and Engineering, NUDT

## The Approximate Modelling of Model Error Covariance Matrix Using Implicit Equal-Weights Particle Filter

##### Coauthors

Weimin Zhang, Xiaoqun Cao, Peter Jan van Leeuwen, Javier Amezuqua, Boheng Duan#### Abstract:

Implicit equal-weights particle filter (IEWPF) is a particle filter that has equal weights for particles and could beat the "curse of dimensionality" which has plagued the particle filter community for decades in non-linear high-dimensional systems. IEWPF shares the same idea with the earlier equivalent-weights particle filter (EWPF) which is making the particle weights equal using proposal density that depends on all the particles from the previous time step. IEWPF has shown quite a good consistent property in high-dimensional linear and non-linear Lorenz 96 systems under the metric of rank histograms and the ratio of RMSE and spread.

But there is also a long-lasting problem that troubles IEWPF and also other data assimilation methods which is the modelling of model error covariance matrix which we use $Q$ to denote here. In IEWPF theory, the $Q$ matrix is the key of constructing the gain matrix and the random term matrix. We introduce a method that constructing $Q$ matrix statistically using the residual term of IEWPF and it shows that a good approximate simulation with the real defined $Q$ matrix in high-dimensional Lorenz 96 system. The method introduced by us has a machine-learning process that could approximate $Q$ matrix from the first-given start matrix which is an appropriate ratio of background error covariance matrix here in our method. Under our system of self-learning construction of $Q$ matrix, the IEWPF also shows a very good rank histograms and the ratio of RMSE and spread. This self-learning $Q$ construction method solves the problem of constructing $Q$ matrix in real high-dimensional non-linear geophysical numerical models.