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    Progress in Earth and Planetary Science

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    Atmospheric and hydrospheric sciences


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

    Kenta Sueki, Seiya Nishizawa, Tsuyoshi Yamaura, Hirofumi Tomita

    Parameter estimation, Data assimilation, Ensemble Kalman filter, Atmospheric model, Cloud microphysics, Deep convective system

    Left panel: Time series of the parameter estimation where the initial parameter value is larger than the true value (red line) and it is smaller than the true value (blue line). Color shade represents the standard deviation of 13 trials for each setting; the smaller the standard deviation, the higher the estimation precision. As for the convergence speed, the shorter the time required for the estimated value to converge around the true value, the faster the convergence speed. Middle panel: Dependence of the estimation precision on the ensemble spread (ES) of the estimated parameter. Right panel: Dependence of the convergence time on the parameter ES. Description in the bottom box: Formulation of the estimation precision and convergence time using the autoregressive parameter φ and the amplitude of random perturbation ε when the estimation time series is approximated as the AR(1) model.

    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.