Application of linear minimum variance estimation to the multi-model ensemble of atmospheric radioactive Cs-137 with observations

Great efforts have been made to simulate atmospheric pollutants, but their spatial and temporal distributions are still highly uncertain. Observations can measure their concentrations with high accuracy but cannot estimate their spatial distributions due to the sporadic locations of sites. Here, we propose an ensemble method by applying a linear minimum variance estimation (LMVE) between multi-model ensemble (MME) simulations and measurements to derive a more realistic distribution of atmospheric pollutants. The LMVE is a classical and basic version of data assimilation, although the estimation itself is still useful for obtaining the best estimates by combining simulations and observations without a large amount of computer resources, even for high-resolution models. In this study, we adopt the proposed methodology for atmospheric radioactive caesium (Cs-137) in atmospheric particles emitted from the Fukushima Daiichi Nuclear Power Station (FDNPS) accident in March 2011. The uniqueness of this approach includes (1) the availability of observed Cs-137 concentrations near the surface at approximately 100 sites, thus providing dense coverage over eastern Japan; (2) the simplicity of identifying the emission source of Cs-137 due to the point source of FDNPS; (3) the novelty of MME with the high-resolution model (3 km horizontal grid) over complex terrain in eastern Japan; and (4) the strong need to better estimate the Cs-137 distribution due to its inhalation exposure among residents in Japan. The ensemble size is six, including two atmospheric transport models: the Weather Research and Forecasting – Community Multi-scale Air Quality (WRF-CMAQ) model and non-hydrostatic icosahedral atmospheric model (NICAM). The results showed that the MME that estimated Cs-137 concentrations using all available sites had the lowest geometric mean bias (GMB) against the observations (GMB =1.53), the lowest uncertainties based on the root mean square error (RMSE) against the observations (RMSE =9.12 Bq m−3), the highest Pearson correlation coefficient (PCC) with the observations (PCC =0.59) and the highest fraction of data within a factor of 2 (FAC2) with the observations (FAC2 =54 %) compared to the single-model members, which provided higher biases (GMB =1.83–4.29, except for 1.20 obtained from one member), higher uncertainties (RMSE =19.2–51.2 Bq m−3), lower correlation coefficients (PCC =0.29–0.45) and lower precision (FAC2 =10 %–29 %). At the model grid, excluding the measurements, the MME-estimated Cs-137 concentration was estimated by a spatial interpolation of the variance used in the LMVE equation using the inverse distance weights between the nearest two sites. To test this assumption, the available measurements were divided into two categories, i.e. learning and validation data; thus, the assumption for the spatial interpolation was found to guarantee a moderate PCC value (> 0.4) within an approximate distance of at least 70 km. Extra sensitivity tests for several parameters, i.e. the site number and the weighting coefficients in the spatial interpolation, the time window in the LMVE and the ensemble size, were performed. In conclusion, the important assumptions were the time window and the ensemble size; i.e. a shorter time window (the minimum in this study was 1 h, which is the observation interval) and a larger ensemble size (the maximum in this study was six, but five is also acceptable if the members are effectively selected) generated better results.

Application of linear minimum variance estimation to the multi-model ensemble of atmospheric radioactive Cs-137 with observations

Great efforts have been made to simulate atmospheric pollutants, but their spatial and temporal distributions are still highly uncertain. Observations can measure their concentrations with high accuracy but cannot estimate their spatial distributions due to the sporadic locations of sites. Here, we propose an ensemble method by applying a linear minimum variance estimation (LMVE) between multi-model ensemble (MME) simulations and measurements to derive a more realistic distribution of atmospheric pollutants. The LMVE is a classical and basic version of data assimilation, although the estimation itself is still useful for obtaining the best estimates by combining simulations and observations without a large amount of computer resources, even for high-resolution models. In this study, we adopt the proposed methodology for atmospheric radioactive caesium (Cs-137) in atmospheric particles emitted from the Fukushima Daiichi Nuclear Power Station (FDNPS) accident in March 2011. The uniqueness of this approach includes (1) the availability of observed Cs-137 concentrations near the surface at approximately 100 sites, providing dense coverage over eastern Japan; (2) the simplicity of identifying the emission source of Cs-137 due to the point source of FDNPS; (3) the novelty of MME with the high-resolution model (3-km horizontal grid) over complex terrain in eastern Japan; and (4) the strong need to better estimate the Cs-137 distribution due to its inhalation exposure among residents in Japan. The ensemble size is six, including two atmospheric transport models (the Weather Research and Forecasting-Community Multi-scale Air Quality (WRF-CMAQ) model and non-hydrostatic icosahedral atmospheric model (NICAM)). The results showed that the MME-that estimated Cs-137 concentrations using all available sites had the lowest geometric mean bias (GMB) against the observations (GMB = 1.53), the lowest uncertainties based on the root-mean-square error (RMSE) against the observations (RMSE = 9.12 Bq m−3), the highest Pearson correlation coefficient (PCC) with the observations (PCC = 0.59) and the highest fraction of data within a factor of 2 (FAC2) with the observations (FAC2 = 54 %) compared to the single-model members, which provided higher biases (GMB = 1.20–4.29), higher uncertainties (RMSE = 19.2–51.2 Bq m−3), lower correlation coefficients (PCC = 0.29–0.45) and lower precision (FAC2 = 10–29 %). At the model grid, excluding the measurements, the MME-estimated Cs-137 concentration was estimated by a spatial interpolation of the variance used in the LMVE equation using the inverse distance weights between the nearest two sites. To test this assumption, the available measurements were divided into two categories, i.e., learning and validation data; thus, the assumption for the spatial interpolation was found to guarantee a moderate PCC value (> 0.4) within an approximate distance of 50 km. Extra sensitivity tests for several parameters, i.e., the site number and the weighting coefficients in the spatial interpolation, the time window in the LMVE and the ensemble size, were performed. The most important assumption was that the ensemble size generated remarkably better results than the single-member model as it increased. Therefore, the proposed ensemble method, with a maximum ensemble size (six in this study), can be applicable for the best estimation of the Cs-137 distribution.

Unbiased Minimum Variance Estimation for Discrete-Time Systems with Measurement Delay and Unknown Measurement Disturbance

This paper addresses the state estimation problem for stochastic systems with unknown measurement disturbances whose any prior information is unknown and measurement delay resulting from the inherent limited bandwidth. For such complex systems, the Kalman-like one-step predictor independent of unknown measurement disturbances is designed based on the linear unbiased minimum variance criterion and the reorganized innovation analysis approach. One simulation example shows the effectiveness of the proposed algorithms.