Noise statistics identification for Kalman filtering of the electron radiation belt observations: 2. Filtration and smoothing

Abstract

In this study we present the further improvement of data assimilation using the 1‐D radial diffusion model for relativistic electron phase space density (PSD) and observations of CRRES satellite. The main purpose of our study is estimation of the radiation belt dynamics for the prediction and mitigation of space weather effects in the hazardous space environment. We develop further noise statistics identification technique presented in the companion paper to estimate the observation error statistics that are crucially important for optimal performance of data assimilation. Assimilation of satellite observations into first‐principles physics model of radiation belts, when both model and observation error statistics are poorly known, may cause large errors in the PSD estimation and lead to failure of a data assimilation algorithm. We identify the coefficients of proportionality characterizing the dependence of observation errors on satellite observations. The effectiveness of the proposed identification technique is illustrated by applying the Kalman filter with optimal identified and nonoptimal observation errors statistics to the sparse CRRES observations over a period of 441 days, from 28 July 1990 to 11 October 1991. Further improvement and the accuracy increase of PSD reconstruction is demonstrated by the implementation of the backward smoothing procedure applied to the forward Kalman filter estimates.

Simultaneous event‐specific estimates of transport, loss, and source rates for relativistic outer radiation belt electrons

“…Additionally, observational errors are assumed to be proportional to the observations and have values of 40% observational error for L < =3, 10% for L > =5, and linearly interpolated for 3 < L < 5, as suggested by Podladchikova et al . []. The true observational error is extremely difficult to quantify, and these errors are roughly consistent with previous studies [ Shprits et al ., , ; Koller et al ., ; Kondrashov et al ., ; Daae et al ., ; Schiller et al ., ].…”

Noise statistics identification for Kalman filtering of the electron radiation belt observations I: Model errors

“…Kalman filter output can be improved by applying the smoothing algorithm in inverse time, which may result in a significant reduction of the PSD reanalysis errors. The advantages of the PSD reanalysis by applying the smoothing procedure, as well as the identification of R , are discussed in the companion paper [ Podladchikova et al , ]. It provides further refinement in our knowledge of the state and evolution of the electron radiation belts.…”

“…It does not provide an opportunity to estimate the covariance matrix R of measurement noise η on the basis of residual observations ν i with sufficient accuracy. A specific modification of the identification technique to determine R is discussed in the companion paper [ Podladchikova et al , ].…”

“…Second, nonoptimal q and Q are incorporated into the Kalman filter algorithm. The identification of the measurement noise covariance matrix R and the sensitivity of PSD reconstruction to its choice is discussed in the companion paper [ Podladchikova et al , ].…”

“…Note that high sensitivity of the reanalysis to noise statistics is exhibited even in conditions of an empirical choice of the measurement noise covariance matrix R assuming it equals Q . The PSD reanalysis with optimal R is discussed in the companion paper [ Podladchikova et al , ].…”