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

Abstract

In this study we present a first attempt to identify errors of the 1-D radial diffusion model for relativistic electron phase space density (PSD). In practice, the model error and characteristics of satellite observations are poorly known, which may cause failure of a Kalman filter algorithm. Correct specification of model errors statistics is necessary for the development of the next generation of radiation belt specification models providing the effective PSD reconstruction and hence the prediction and mitigation of space weather effects in the hazardous space environment. The proposed approach to the identification of errors statistics is based on estimating the unknown bias and the covariance matrix of model errors from the sparse CRRES observations over a period of 441 days, from 28 July 1990 to 11 October 1991. With our technique we demonstrate that model errors are biased. Neglecting the bias when applying a data assimilation algorithm to radiation belt electrons can cause significant errors of the PSD estimate during data gaps. Both the identified bias and the covariance matrix of model errors increase with increase of L shell. Sensitivity of the PSD reconstruction to model errors statistics and advances of the improved physical-based model based on the model errors identification are illustrated by a number of representative examples of the PSD reanalysis. Identification of satellite observations characteristics, and filtration and smoothing algorithms are discussed in the companion paper.

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

“…In this section, we develop the identification technique of the measurement noise covariance matrix R to incorporate it into a Kalman filter for data assimilation of CRRES observations in the radiation belts. The temporal evolution of the PSD is obtained by solving the 1‐D radial diffusion equation for L shell from 3 to 5 presented in the companion paper [ Podladchikova et al , , equation ] and is given by the following state equation Here X j is an 11‐dimensional state vector of the PSD at various L shell bins L 1 =3, L 2 =3.2,…, L 10 =4.8, L 11 =5 with a 10 min time step j . The transition state matrix Φ j + 1, j relates a current state vector X j to a one‐step ahead‐predicted state vector X j + 1 .…”

“…The proposed approach to the identification of the measurement noise covariance matrix R is based on the same principles as the identification of the bias, q , and the covariance matrix, Q , of model noise w presented in the companion paper [ Podladchikova et al , ]. The general principle of the proposed identification technique includes the following requirements: Construction of residuals characterizing the mismatch between an observation and an auxiliary estimate of state vector (the pseudomeasurement vector) that does not statistically depend on this observation.…”

“…Figure shows the identification results of coefficients of proportionality (solid line), α 1 , α 1 ,…, α 11 , obtained as square roots of the components of vector Y given by equation . We should note that vector C in equation contains two terms, and , depending on the model noise bias q and the model noise covariance matrix Q that were identified in the companion paper [ Podladchikova et al , ]. As shown in Figure , the observation error is set to 40% of the PSD observations at L shell L = 3.…”

“…A popular algorithm of data assimilation of electron radiation belts observations is a Kalman filter [ Kalman , ]. Over a period of less than 10 years, there has been a steady increase in Kalman filter applications for radiation belts, reviewed in the companion paper [ Podladchikova et al , ]. However, a major problem of the Kalman filter application is related to the specification of uncertainties in radiation belt models.…”

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

“…Recent analysis by Podladchikova et al . [] show that improper model errors and error biases have a strong effect on the PSD estimate if observations are sparse in time. Observations assimilated for this analysis are available at every time step, reducing the effect of the unbiased error assumptions.…”