Precision and convergence speed of the ensemble Kalman filter-based parameter estimation: setting parameter uncertainty for reliable and efficient estimation

Determining physical process parameters in atmospheric models is critical to obtaining accurate weather and climate simulations; estimating optimal parameters is essential for reducing model error. Recently, automatic parameter estimation using the ensemble Kalman filter (EnKF) has been tested instead of conventional manual parameter tuning. To maintain uncertainty for the parameters to be estimated and avoid filter divergence in EnKF-based methods, some inflation techniques should be applied to parameter ensemble spread (ES). When ES is kept constant through the estimation using an inflation technique, the precision and convergence speed of the estimation vary depending on the ES assigned to estimated parameters. However, there is debate over how to determine an appropriate constant ES for estimated parameters in terms of precision and convergence speed. This study examined the dependence of precision and convergence speed of an estimated parameter on the ES to establish a reliable and efficient method for EnKF-based parameter estimation. This was carried out by conducting idealized experiments targeting a parameter in a cloud microphysics scheme. In the experiments, there was a threshold value for ES where any smaller values did not result in any further improvements to the estimation precision, which enabled the determination of the optimal ES in terms of precision. On the other hand, the convergence speed accelerates monotonically as ES increases. To generalize the precision and convergence speed, we approximated the time series of parameter estimation with a first-order autoregression (AR(1)) model. We demonstrated that the precision and convergence speed may be quantified by two parameters from the AR(1) model: the autoregressive parameter and the amplitude of random perturbation. As the ES increases, the autoregressive parameter decreases, while the random perturbation amplitude increases. The estimation precision was determined based on the balance between the two values. The AR(1) approximation provides quantitative guidelines to determine the optimal ES for the precision and convergence speed of the EnKF-based parameter estimation.