estimating model error covariances using particle filters

​ Zhu M, Van Leeuwen P J, Zhang W. Estimating Model Error Covariances Using Particle Filters[J]. Quarterly Journal of the Royal Meteorological Society, 2017.

For high-dimensional systems the covariance is never calculated explicitly.

Several methods to estimate state covariance have been developed.

Todling R. 2015a. A complementary note to lag-1 smoother approach to system-error estimation: The intrinsic limitations of residual diagnostics. Q. J. R. Meteorol. Soc. 141: 2917–2922. Todling R. 2015b. A lag-1 smoother approach to system-error estimation: Sequential method. Q. J. R. Meteorol. Soc. 141: 1502–1513.

Todling gives an alternative method to diagnoise model error, which requires two overlapping DA systems, one sequential filter and one fixed lag-l smoother.

Three solutions have been presented in the literature:

1. localized particle filter ( A localized particle filter for high-dimensional nonlinear, A non-parametric ensemble transform method for Bayesian inference)
2. explore the proposal density freedom ( Implicit equal-weights particle filter)
3. combine particle filter with ensemble kalman filter

A problem with localized particle filter is that use localization is that the localization radius has to be taken very small to avod this weight collapse.

Furthermore, issues with localization as mentioned above for ensemble Kalman filters play a role here too.

Estimating the model error covariance

The model equations are denoted as

[x^n = f(x^{n-1}) + beta^n]

Then:

[y^n - H(f_i )= y^n - H()]