Volume 125, Issue 1 e2019JA027514
Technical Reports: Methods
Open Access

Identifying Radiation Belt Electron Source and Loss Processes by Assimilating Spacecraft Data in a Three-Dimensional Diffusion Model

S. Cervantes

Corresponding Author

S. Cervantes

Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany

Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany

Correspondence to: S. Cervantes,

[email protected]

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Y. Y. Shprits

Y. Y. Shprits

Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany

Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany

Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA, USA

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N. A. Aseev

N. A. Aseev

Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany

Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany

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A. Y. Drozdov

A. Y. Drozdov

Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA, USA

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A. Castillo

A. Castillo

Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany

Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany

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C. Stolle

C. Stolle

Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany

Faculty of Science, University of Potsdam, Potsdam, Germany

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First published: 28 December 2019
Citations: 14


Data assimilation aims to blend incomplete and inaccurate data with physics-based dynamical models. In the Earth's radiation belts, it is used to reconstruct electron phase space density, and it has become an increasingly important tool in validating our current understanding of radiation belt dynamics, identifying new physical processes, and predicting the near-Earth hazardous radiation environment. In this study, we perform reanalysis of the sparse measurements from four spacecraft using the three-dimensional Versatile Electron Radiation Belt diffusion model and a split-operator Kalman filter over a 6-month period from 1 October 2012 to 1 April 2013. In comparison to previous works, our 3-D model accounts for more physical processes, namely, mixed pitch angle-energy diffusion, scattering by Electromagnetic Ion Cyclotron waves, and magnetopause shadowing. We describe how data assimilation, by means of the innovation vector, can be used to account for missing physics in the model. We use this method to identify the radial distances from the Earth and the geomagnetic conditions where our model is inconsistent with the measured phase space density for different values of the invariants urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0001 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0002. As a result, the Kalman filter adjusts the predictions in order to match the observations, and we interpret this as evidence of where and when additional source or loss processes are active. The current work demonstrates that 3-D data assimilation provides a comprehensive picture of the radiation belt electrons and is a crucial step toward performing reanalysis using measurements from ongoing and future missions.

Key Points

  • We perform radiation belt reanalysis with a Kalman filter, a 3-D diffusion model, and spacecraft data
  • Several physical mechanisms are added to the model and their effect in reanalysis is explored
  • We use the innovation vector to pinpoint when and where missing physics in the model become apparent

1 Introduction

The Earth's radiation belts consist of protons and electrons trapped by the Earth's magnetic field. The inner electron belt is usually located below 2 Earth radii ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0003) and is relatively stable. In contrast, the outer belt may extend from approximately 3 to 7  urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0004, is very dynamic, can vary by several orders of magnitude over a few hours (Rothwell & McIlwain, 1960; Craven, 1966), and is correlated with geomagnetic activity. In addition, the outer belt consists of high-energy particles (from urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0005100 keV to several tens of MeV), which pose a significant hazard to satellites in space (Baker et al., 1998, 2018; Baker, 2002; Green et al., 2017).

One of the space weather impacts to satellites is known as “deep-dielectric charging.” Electrons with energies of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0006100 keV and up to multiple MeV can readily penetrate spacecraft shielding, bury themselves in dielectric materials deep within spacecraft subsystems, such as circuit boards or cable insulators, and accumulate on ungrounded metal. As charge builds up, this can lead to electrical breakdown in the vicinity of sensitive electronics (Fennell et al., 2001; Lohmeyer et al., 2015). Another kind of space weather impact is called “surface charging.” Lower-energy electrons, from 10 to 100 keV, cannot penetrate the shielding but can accumulate on satellite surfaces. Charge buildup may lead to high voltages, damaging electrostatic discharges, and electromagnetic interference that can disorient the satellite. Mazur et al. (2011) presented the high-voltage increase in the LICA instrument on board the SAMPEX satellite as an example of a surface charging anomaly.

It is evident from the above description that a wide variety of impacts on spacecraft systems can be caused by the radiation belts. Therefore, knowledge of the outer belt dynamics is of particular importance since it spatially overlaps with many communication and scientific spacecraft orbits. Understanding the mechanisms responsible for the acceleration and loss of electrons is essential for predicting the response of the radiation belts to geomagnetic disturbances.

However, the analysis of spacecraft data poses a number of challenges. Satellite measurements are often limited to a restricted range of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0007 shells, pitch angles, and energies, which complicates the data analysis geared toward reproducing the global state of the radiation belts. Additionally, the manual analysis of observations is a challenging task, while an automated analysis is complicated by the fact that measurements at different urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0008 shells are taken at different points along the spacecraft orbit and therefore at different times. Moreover, fluxes of energetic electrons in the outer belt change on timescales shorter than a typical satellite orbital period; thus, it is impossible to observe the instantaneous radial profiles of fluxes.

Similar challenges to those mentioned above were faced by the atmospheric sciences in the 1970s (Kalnay, 2003). A powerful method, inherited from the engineering of navigation systems and referred to as “data assimilation,” was successfully applied to accomplish better weather predictions. This term denotes a process in which observations are merged together with a dynamical numerical model in order to determine the state of the atmosphere as accurately as possible (Talagrand, 1997). Data assimilation allows us to fill in the spatial and temporal gaps left by sparse measurements and to combine them with a physics-based model according to their underlying error structure, and the result is typically referred to as “reanalysis.” Some of the most popular data assimilation methods are the standard Kalman filter (KF) (Kalman, 1960), the extended Kalman filter (Jazwinski, 1970), and the ensemble Kalman filter (Evensen, 1994).

Naehr and Toffoletto (2005) were among the first to show the potential of data assimilation to significantly improve the forecasting capabilities of radiation belt models. Kondrashov et al. (2007) used the extended Kalman filter and Combined Release and Radiation Effects Satellite (CRRES) observations to estimate the electron phase space density (PSD) and to infer unknown parameters in a model. Shprits et al. (2007) and Koller et al. (2007) performed data assimilation with a simple radial diffusion model and demonstrated how it can be used to identify and adjust for missing physics in the model.

Furthermore, Ni, Shprits, Nagai, et al. (2009) used four empirical external magnetic field models and found that combined reanalyses are relatively insensitive to the choice of magnetic field model. The results also showed that the errors of PSD obtained by assimilating multiple satellite measurements at different locations can be smaller than the errors of individual satellite reconstructions. Daae et al. (2011) tested the sensitivity of the reanalysis of radiation belt PSD to the assumed outer boundary conditions and loss model and demonstrated that the KF performs remarkably well when sufficient data is available at all considered urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0009 shells for the assimilation. More recently, Shprits et al. (2012) performed a long-term multispacecraft reanalysis and found a good correlation between the location of the peak of the PSD and the plasmapause location, and investigated the link between PSD dropouts and solar wind dynamic pressure increases.

Several recent studies have employed 3-D diffusion models accounting for radial, pitch angle, and energy diffusion (Bourdarie & Maget, 2012; Kellerman et al., 2014). Such 3-D models potentially account for more physical processes and use the knowledge of the dynamics of pitch angle distributions and energy spectra. Shprits, Kellerman, et al. (2013) suggested an operator splitting method that allowed to use the KF for 3-D diffusion codes and verified it on 2-month CRRES data.

The purpose of this study is to incorporate the following processes into the 3-D Versatile Electron Radiation Belt Code (VERB-3D) data assimilation scheme, absent in previous studies: mixed pitch angle-energy diffusion, scattering by electromagnetic ion cyclotron (EMIC) waves, and magnetopause shadowing. Their relevance to the radiation belt dynamics has been evaluated in earlier works, based either on observations or model simulations (Albert & Young, 2005; Albert et al., 2009; Drozdov et al., 2015, 2017; Shprits et al., 2006, 2008, 2016, 2017; Shprits, Subbotin, et al., 2013; Subbotin et al., 2010; Turner et al., 2012; Turner, Angelopoulos, Li, et al., 2014; Turner, Angelopoulos, Morley, et al., 2014; Usanova et al., 2014; Xiang et al., 2017; Xiao et al., 2010; Yu et al., 2013). However, their effect on the reanalysis has not yet been investigated and objectively quantified.

In the current work, we perform a combination of the VERB-3D code with data from the Van Allen Probes (formerly known as the Radiation Belt Storm Probes) and the Geostationary Operational Environmental Satellites (GOES) by means of a split-operator KF (Shprits, Kellerman, et al., 2013) from 1 October 2012 to 1 April 2013. We perform multiple reanalyses by systematically adding, one by one, the above-mentioned processes. The innovation vector, a measure on how the observations and the model predictions differ, is inspected for each reanalysis and for various values of the adiabatic invariants urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0010 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0011. We interpret this measure as an evidence of the effect of these mechanisms in the dynamics of the radiation belt electrons. Furthermore, the radial distance from the Earth and the geomagnetic conditions under which each physical process operates are also identified by means of the innovation vector.

The outline of this paper is as follows: Our radiation belt model and the spacecraft databases used in this study are presented in sections 2 and 3, respectively, followed by a description of the standard KF as the overarching algorithm blending data and model predictions in section 4. Section 5 shows the 6-month reanalysis results of electron PSD, and section 6 presents how the Kalman innovation adds source and loss terms to the radiation belt model. The results are discussed and the main conclusions of the study are summarized in section 7. The definitions of adiabatic invariants are given in Appendix Appendix A.

2 VERB Code

2.1 Model Description

The VERB-3D code models the evolution of electron PSD by solving the modified 3-D Fokker-Planck diffusion equation that incorporates radial diffusion, energy diffusion, pitch angle scattering, and mixed diffusion into the drift- and bounce-averaged particle PSD (Schulz & Lanzerotti, 1974). Diffusion is produced by resonant wave-particle interactions with various waves in the magnetosphere. Radial diffusion is caused by ultralow frequency waves, while pitch angle, energy, and mixed diffusion are caused by whistler mode and EMIC waves.

The 3-D time-dependent Fokker-Planck equation for the PSD evolution of relativistic electrons can be written in terms of the urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0012 shell, equatorial pitch angle urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0013, and relativistic momentum urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0014, following Shprits et al. (2009) and Subbotin and Shprits (2009):
where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0016 is the electron PSD, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0017 is time, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0018 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0019 are the first and second adiabatic invariants (see Appendix Appendix A), respectively, and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0020 is inversely related to the third adiabatic invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0021 and is constant along the particle's drift path. urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0022, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0023, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0024, and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0025 are the bounce-averaged radial, energy, pitch angle, and mixed pitch angle-energy diffusion coefficients, respectively. The lifetime parameter urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0026 accounts for losses of particles inside the loss cone due to collisions with atmospheric neutrals, assumed to be infinite outside the loss cone and equal to a quarter of the electron bounce time inside the loss cone. urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0027 is a function related to the bounce frequency (Schulz and Lanzerotti, 1974).
The parameterization of the radial diffusion coefficients due to magnetic fluctuations is adopted from Brautigam and Albert (2000). The bounce-averaged diffusion coefficients for hiss and dayside and nightside chorus waves are computed using the Full Diffusion Code (Shprits & Ni, 2009). The parameters for hiss are taken from Orlova et al. (2014), while for dayside and nightside chorus the parameterization of Orlova and Shprits (2014) is used. The spectral properties from Meredith et al. (2014) are used to calculate diffusion coefficients for helium band EMIC waves. In accordance with Drozdov et al. (2017), EMIC waves are included in the simulation when the solar wind dynamic pressure is greater than or equal to 3 nPa. The location of the plasmapause is calculated following Carpenter and Anderson (1992) as
where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0029 is the maximum urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0030 value in the preceding 24 hr.
The 3-D VERB code includes the Last Closed Drift Shell (LCDS) as a function of time and invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0031. The LCDS is introduced to include physics associated with magnetopause shadowing. In this study, the Tsyganenko and Sitnov (2007) model is used to determine the LCDS. Since the magnetopause shadowing effect is dependent on the drift period, we use an energy-dependent loss mechanism, as suggested by Drozdov et al. (2015). Losses due to magnetopause shadowing are simulated with an exponential decay of the electron PSD outside the LCDS location, as follows:
Here, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0033 is the electron drift period calculated as (Walt, 2005):
where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0035 for electrons, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0036, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0037, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0038 km, and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0039 is the distance from the center of the Earth to the equatorial crossing point of a magnetic field line. As urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0040 increases, the drift period decreases. In addition, as the electron energy increases, the drift period decreases.

2.2 Model Grid and Boundary Conditions

The computational grid has urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0041 points along radial, energy, and pitch angle dimension, respectively. Radial grid points are uniformly distributed, while energy and pitch angle grid points are logarithmically distributed. The urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0042 grid extends from 1 to 6.6  urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0043. The energy grid is defined by a minimum of 0.01 MeV and a maximum of 10 MeV at the outer radial boundary urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0044. The pitch angle grid is set from 0.3° to 89.7°.

For the initial PSD we use the steady state solution of the radial diffusion equation. The solution of equation 1 requires six boundary conditions, two for each variable in the equation. In order to simulate the loss of electrons to the atmosphere, a lower radial boundary condition ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0045) of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0046 is used. The PSD required for the upper radial boundary condition ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0047) is time-dependent and is obtained from the GOES observations. The lower pitch angle boundary condition is set to 0 to simulate precipitation loss of electrons into the loss cone in a weak diffusion regime. For the upper pitch angle boundary condition, a zero gradient is chosen to account for the flat pitch angle distribution observed at 90° (Horne et al., 2003). The upper energy boundary at 10 MeV is set to 0, while for the lower energy boundary, the PSD is set constant in time to represent a balance of convective source and loss processes.

3 Satellite Observations

In this study, in situ observations from four spacecraft are used in combination with the VERB-3D code: Van Allen Probes A and B, and GOES satellites 13 and 15. The observations cover a 6-month period from 1 October 2012 to 1 April 2013. In order to assimilate the data, they are converted from flux to PSD in phase space coordinates ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0048). To do so, magnetic field information is required. The in situ magnetic field measurements are used to calculate urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0049, while to calculate urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0050 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0051 the Tsyganenko and Sitnov (2007) model is employed. Similar to Kellerman et al. (2014), the intercalibration of the observations from different satellites is performed using a PSD matching algorithm.

3.1 Van Allen Probes

The Van Allen Probes (Mauk et al., 2012; Stratton et al., 2012) were launched on 30 August 2012 to study the dynamical evolution of the radiation belts. Both spacecraft are equipped with identical suites of instruments designed for monitoring radiation belt particles and wave environment. The Radiation Belt Storm Probes-Energetic particle, Composition, and Thermal plasma suite measures particles with energies ranging from hot to ultrarelativistic (Spence et al., 2013), and it includes the Magnetic Electron Ion Spectrometer (Blake et al., 2013), and Relativistic Electron Proton Telescope (Baker et al., 2012) instruments.

3.2 GOES

The multimission GOES (Onsager et al., 1996; Singer et al., 1996) program is aimed at monitoring the near-Earth space, including operational meteorology and space weather. Data from the GOES 13 and 15 units, launched on 24 May 2006 and 4 March 2010, respectively, are used in this study. The GOES Space Environment Monitor subsystem contains multiple instruments including the Energetic Particle Sensor/High Energy Proton and Alpha Detector instrument that measures the flux of protons, alpha particles, and electrons over an extensive range of energies. In particular, we employ data from the MAGnetospheric Electron Detector (Hanser, 2011) and Energetic Proton, Electron, and Alpha Detector (Hanser, 2011; Onsager et al., 1996) instruments.

4 Data Assimilation Using the Kalman Filter

Data assimilation is an algorithm that allows for an optimal combination of model results and sparse data from various sources, contaminated by noise and systematic errors (Kalman, 1960). The purpose of data assimilation is to find the most likely estimate of the unknown true state of a dynamic system using the information provided by the physical model and the available observations, considering both their uncertainties and limitations. One popular technique for data assimilation is the Kalman filter (Kalman, 1960), an optimal recursive algorithm, which has found many applications such as the navigational system on the Apollo mission and Global Positioning System devices, operational weather forecasting, and ocean modeling (Kalnay, 2003; Lahoz et al., 2010; Sorenson, 1985). The methodology of the KF is outlined below.

4.1 Kalman Filter Methodology

For a given dynamic system described by a set of partial differential equations, the numerical algorithm can be presented in the following discrete form:
where the state vector urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0053 is composed of all model variables, which for our radiation belt model is the PSD on the numerical grid locations. urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0054 is a matrix of the numerical model, in our case, the discretized Fokker-Plank equation, and it advances the state vector urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0055 in discrete time intervals urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0056. Superscripts urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0057 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0058 refer to forecast and analysis, respectively, and the subscript urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0059 shows the time step. The urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0060 is the best estimate of the state vector at time urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0061, relying on the model and the available observations. The evolution of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0062, where the superscript urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0063 refers to true, is assumed to be given by
where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0065 is the model error, represented by a spatially correlated ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0066) white noise ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0067). The symbol E represents the expectation operator over time, and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0068 is the model covariance matrix.
The observations urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0069, where the superscript urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0070 refers to observed, are also assumed to be contaminated by errors:
where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0072 is the observational error, represented by a spatially correlated ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0073) white noise ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0074). urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0075 is the observational error covariance matrix. The observation matrix urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0076 maps the true space onto the observed space, and it accounts for the fact that usually the dimension of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0077 is less than the dimension of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0078 (i.e., only certain variables are observed).
During the update times, the forecast state vector is corrected by the data-driven innovation vector to yield the analysis state vector:
where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0080 is the innovation vector. urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0081 is the Kalman gain matrix calculated at each time step from a time-evolving forecast-error covariance matrix given by
On the analysis step the error covariance matrix is also updated:
When no observations are available at time urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0084, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0085, only the forecast step is performed, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0086 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0087. The Kalman gain matrix urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0088 in equation 7 is computed by minimizing the analysis error covariance urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0089, and it represents the optimal weights given to the observations when updating the model state vector:

The innovation vector urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0091 measures how much the new and additional information provided by the data modifies the model forecast to produce an optimal estimate of the state vector. The sign and the value of the innovation vector are determined by how much the observed and modeled values differ from each other, as well as by the estimated forecast and observational errors. If a model were perfect, it would predict exactly the incoming observations, and the innovation would be 0. From equation 11 we see that as the observational error covariance matrix urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0092 approaches 0, the Kalman gain urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0093 weights the innovation more heavily. On the contrary, as the forecast error covariance matrix urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0094 approaches 0, the gain urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0095 weights the innovation less heavily. Ghil and Malanotte-Rizzoli (1991) present a detailed description of the KF algorithm.

The standard formulation of the KF assumes that the model and observational error covariance matrices are known. This rarely happens in practice and simple approximations are usually made. In the current study, the initial error covariance matrices are set equal to 0. The model error covariance matrix urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0096 is a diagonal matrix with elements calculated as urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0097, where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0098 is the error and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0099 is the PSD at the time of the assimilation. A similar form is assigned to the observational error covariance matrix urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0100. We use VERB-3D with a 1-hr time step, and the assimilation is performed at the same cadence using Van Allen Probes A and B and GOES 13 and 15 observations.

5 Combined Reanalysis of Electron PSD

In this section, we present radial profiles of PSD for a 6-month interval starting on 1 October 2012, on the basis of the methodology described in section 4. We focus on the evolution of PSD for two sets of three pairs of the adiabatic invariants urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0101 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0102, yielding six pairs in total. For the first set, the invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0103 is equal to urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0104, with the following three different values of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0105: 50, 300, and 600MeV G urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0106. For the second set, the invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0107 equals urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0108, with the following three different values of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0109: 200, 1300, and 2650 MeV G urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0110. Their related equatorial pitch angle urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0111 and electron kinetic energy urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0112 dependencies on the urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0113 shell for a dipole magnetic field are plotted in Figure 1, using the definitions given in Appendix Appendix A. The pitch angle of the electrons changes only by a few degrees, whereas electrons can lose (gain) a large amount of kinetic energy by radially diffusing outward (inward) by a few urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0114 shells.

Details are in the caption following the image
Dependence of equatorial pitch angle urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0115 (a and c) and electron kinetic energy urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0116 (b and d) on urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0117 shell in a dipolar magnetic field, for the six pairs of ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0118) investigated in the present study.

At the heart of the outer radiation belt, namely, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0119, for the chosen values of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0120 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0121, the equatorial pitch angles are approximately 22° and 52°, respectively. At urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0122 for the pairs ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0123urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0124) and ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0125urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0126), electron energies are urn:x-wiley:jgra:media:jgra55451:jgra55451-math-01270.7 MeV. For the pairs ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0128urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0129) and ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0130urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0131), urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0132 is urn:x-wiley:jgra:media:jgra55451:jgra55451-math-01332.2 MeV, while for the pairs ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0134urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0135) and ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0136urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0137), electron energies are urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0138 MeV, at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0139. Such a selection of pairs of adiabatic invariants allows us to compare three different populations of electrons at low and high equatorial pitch angles, with approximately the same energy variation across the outer radiation belt.

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Evolution of electron PSD as a function of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0140 and time from 1 October 2012 to 1 April 2013: (a) Van Allen Probes and GOES data, and (b) assimilated radial profile of PSD for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0141 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0142; (c and d) same as (a) and (b) but for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0143 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0144; (e and f) same as (a) and (b) but for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0145 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0146; (g) evolution of solar wind dynamic pressure, and (h) geomagnetic activity urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0147 index. The assimilative results of the combined reanalysis of electron PSD in this figure account for 3-D diffusion, mixed pitch angle-energy diffusion, scattering by EMIC waves, and magnetopause shadowing.
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Same as Figure 2, but for the pairs of invariants urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0148 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0149 (a and b); urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0150 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0151 (c and d); and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0152 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0153 (e and f).

In total, we perform four data assimilation runs. The first one only accounts for radial diffusion due to ultralow frequency waves, and for pitch angle and energy diffusion due to chorus and hiss waves (hereinafter, 3-D diffusion), negelecting mixed diffusion, EMIC wave scattering, and losses due to magnetopause shadowing. In the other three runs, we systematically add one process at a time with respect to the previous run. For the second reanalysis we incorporate mixed pitch angle-energy diffusion, for the third one, scattering by EMIC waves, and for the last one, magnetopause shadowing.

The combined reanalyses based on the VERB-3D code and four satellite measurements, and accounting for 3-D diffusion, mixed pitch angle-energy diffusion, EMIC wave scattering, and magnetopause shadowing, are shown in Figures 2 and 3. From top to bottom, we present the 1-hr-averaged electron PSD versus urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0154 distribution for the four satellites and the assimilated radial profile of PSD at the above-mentioned pairs of ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0155). The last two panels show the solar wind dynamic pressure and the geomagnetic index urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0156, respectively, obtained from the online OMNIWEB database with 1-hr resolution. As illustrated by the urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0157 index, there are several geomagnetic storms occurring during the six-month period of our study, the strongest ones reaching urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0158 on 8 October 2012 and 17 March 2013. The Van Allen Probes measurements provide the electron PSD below urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0159, and the GOES measurements above urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0160, establishing a good radial coverage of observed PSD. Although the data clearly show a number of electron dropouts and enhancements, the distribution is still sparse in space and time. After blending the observations with the VERB-3D code, the gaps are filled, as illustrated in panels (b), (d), and (f).

The temporal evolution of the assimilated PSD is strongly correlated with the geomagnetic activity as shown in Figures 2 and 3. In accordance with previous reanalysis studies (Ni, Shprits, Nagai, et al., 2009; Ni, Shprits, Thorne, et al., 2009; Ni et al., 2013; Shprits et al., 2012), sudden and substantial dropouts in PSD are observed during the main phase of geomagnetic storms, depleting electron PSD even down to urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0161, as on 17 March 2013, and lasting a few hours. These decreases occur right after sudden pulses of solar wind dynamic pressure, which are in turn associated with clear and pronounced compressions of the magnetopause. For example, on 17 January 2013 the solar wind dynamic pressure increased up to 14.8 nPa, and the LCDS moved inward down to urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0162, well below the geosynchronous orbit. On the other hand, most of the PSD peaks are seen during the recovery phase of the storms. Such buildups of PSD are gradual and in general extend from 4 to 5  urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0163.

In addition, the dependence on energy of our magnetopause shadowing loss mechanism is evident when comparing panels (b), (d), and (f) of Figures 2 and 3. The loss effect is more pronounced at higher values of the invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0168 (lowermost reanalyses, panel f), where electron PSD is depleted faster than for lower values of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0169 (uppermost reanalyses, panel b).

6 Analysis of the Innovation Vector

As mentioned in section 4, the innovation vector adds or subtracts PSD from the predicted values, and it can be regarded as an indicator of the missing electron loss and source processes in the model. Shprits et al. (2007), Koller et al. (2007), and Daae et al. (2011) used it as a tool to understand the physical mechanisms responsible for the acceleration of electrons and interpreted its peaks as evidence of local acceleration, absent in the 1-D model employed in their studies.

To objectively investigate the effect of introducing several processes in our model, we calculated the hourly innovation vector for each urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0170, binned it according to the urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0171 index, and computed the average innovation vector as a function of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0172 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0173 for each of our four 6-month reanalyses and each of our six pairs of adiabatic invariants. The distribution of the number of measurements that were used for the reanalyses in terms of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0174 is shown in Figure 4. Owing to the highly skewed distribution of samples toward low values of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0175, and the lack of measurements at high urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0176 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0177, we restricted our analysis of the innovation vector to the intervals with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0178. Evaluation of the innovation including intervals of high geomagnetic activity will be the subject of future research.

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Distribution of the number of samples employed in the reanalyses of PSD binned in urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0164 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0165 for the indicated pairs of adiabatic invariants urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0166 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0167.

The average innovation vector as a function of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0179 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0180 is shown in Figures 5 and 6, for the pairs of invariants with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0181 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0182, respectively. Positive (negative) values denote an additional source (loss) term missing from the radiation belt model, and thus the KF adds (subtracts) PSD in order to compensate and to match the observations. In other words, the innovation is positive (negative) when the observations are significantly higher (lower) than the forecast, hence our model underestimates (overestimates) the electron PSD.

6.1 Average Innovation for PSD at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0183 and Different Values of the Invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0184

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Innovation vector averaged over the interval October 2012 to March 2013 as function of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0185 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0186 for electron PSD for the following pairs of invariants: urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0187 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0188 (first row), urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0189 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0190 (second row), and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0191 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0192 (third row). The first column corresponds to radial, pitch angle and energy diffusion (3-D) reanalysis, the second to reanalysis with the addition of mixed pitch angle-energy diffusion, the third to reanalysis including scattering by EMIC waves, and the last to reanalysis adding magnetopause shadowing. The solid black line indicates the estimated average location of the plasmapause, computed following Carpenter and Anderson (1992).

For the electrons with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0193 (Figure 5, first row), the average innovation using our model with 3-D diffusion shows enhanced overestimation of PSD extending from urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0194 to urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0195 for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0196, and for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0197 between urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0198 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0199. The addition of mixed pitch angle-energy diffusion significantly reduces this overestimation, and furthermore, it emphasizes a region of underestimation of PSD at large radial distances, extending from urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0200 to the outer boundary. The inclusion of EMIC waves does not change the average innovation vector, as the lower energy electrons ( urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0201 1 MeV) are in general unaffected by them (Horne & Thorne, 1998; Meredith et al., 2003). Lastly, incorporating magnetopause shadowing driven by the LCDS accentuates a region of large innovation at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0202 for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0203. This is indicative of a missing source in our radiation belt model, namely, earthward magnetospheric convection of electrons with keV energies from the tail region to the plasma sheet.

The average innovation vector of the 3-D reanalysis of the electrons with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0204 (Figure 5, second row) exhibits a region of large overestimation of PSD at all urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0205 for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0206. Such missing loss is then added when mixed pitch angle-energy diffusion is incorporated into our scheme, in particular at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0207. Furthermore, scattering by EMIC waves effectively brings loss into our model, as the average innovation vector decreases for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0208 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0209, which for this population of electrons corresponds to energies greater than 1.9 MeV. Note that the effect of EMIC waves is only observed close to the plasmapause and is negligible below 1.9 MeV. The innovation, when magnetopause shadowing and outward diffusion induced by it are included, acts as a loss process, especially for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0210 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0211.

Finally, for the electrons with first invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0212 (Figure 5, third row), the PSD is largely overestimated when relying only on our 3-D model, hence the KF substracts PSD, particularly in the region extending from urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0213 to urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0214 at all urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0215. The addition of both mixed pitch angle-energy diffusion and EMIC wave scattering acts as a loss process, mostly for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0216 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0217, where the energy of electrons is larger than 2.2 MeV. Moreover, taking into account magnetopause shadowing reduces to some extent the overestimation of electron PSD at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0218 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0219.

6.2 Average Innovation for PSD at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0220 and Different Values of the Invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0221

Details are in the caption following the image
Same as Figure 6, for the pairs of invariants urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0222 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0223 (first row), urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0224 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0225 (second row), and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0226 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0227 (third row).

For the 3-D reanalysis of the electrons with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0228 (Figure 6, first row), the average innovation vector shows how our model accounting only for 3-D diffusion significantly overestimates PSD. As a result, the KF subtracts PSD between urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0229 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0230 for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0231, and from urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0232 to urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0233 for quieter geomagnetic conditions. Such a missing loss process is added when mixed pitch angle-energy diffusion is incorporated into our framework, particularly for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0234. Due to the low energy of this population of electrons, the inclusion of EMIC wave scattering does not modify the average innovation vector. Moreover, considering magnetopause shadowing shows that a missing source process (magnetospheric convection) is operating between urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0235 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0236, similarly to the case with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0237 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0238.

The average innovation of the electrons with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0239 (Figure 6, second row) shows that for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0240 and all geomagnetic conditions, PSD is overestimated by the 3-D model. However, introducing mixed pitch angle-energy diffusion does not significantly affect this population. The addition of scattering by EMIC waves corrects for the PSD overestimation in the region between urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0241 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0242 for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0243, where the energy of the electrons is greater than 2.3 MeV. Considering magnetopause shadowing brings further losses into our model, in particular for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0244 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0245.

Lastly, the innovation of the 3-D reanalysis of the electrons with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0246 (Figure 6, third row) shows, on average, a region of overestimation at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0247 for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0248, and for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0249 at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0250. For this population, mixed pitch angle-energy diffusion does not introduce losses as was the case for the electrons with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0251. In contrast, EMIC wave scattering effectively decreases PSD for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0252 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0253, where electrons have energies larger than 2.3 MeV, and magnetopause shadowing partially removes PSD at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0254 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0255.

7 Discussion and Conclusions

In this study, we show how data assimilation by means of a standard KF allows us to fill in the spatial and temporal gaps left by sparse in situ measurements, combine them with our radiation belt model, and as result, reconstruct electron PSD. In comparison to previous works, our model potentially accounts for more physical loss processes, and in the future, will allow us to further understand the dynamical evolution of radial profiles of PSD. Data assimilation can also be applied to identify time intervals where our model predictions are significantly lower or higher than the observations. In this regard, the innovation vector is used to pinpoint which mechanisms are absent in our model, where they operate, and which population of electrons they affect the most.

Mixed pitch angle-energy diffusion subtracts PSD from our model, and its effect is stronger at higher urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0256 (section 6.1), than at lower urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0257 (section 6.2). Similar results have been reported by others (Albert & Young, 2005; Albert et al., 2009; Subbotin et al., 2010; Xiao et al., 2010) relying on 2-D (pitch angle and energy diffusion) and 3-D simulations. The contribution of mixed pitch angle-energy diffusion at low urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0258 and high urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0259 (last two cases in section 6.2) warrants a more detailed study in the future.

The scattering effect induced by EMIC waves on electrons with energies above urn:x-wiley:jgra:media:jgra55451:jgra55451-math-02602 MeV and equatorial pitch angles less than urn:x-wiley:jgra:media:jgra55451:jgra55451-math-026140° to 60° (both second and third cases in sections 6.1 and 6.2) is consistent with previous modeling and observational studies (Drozdov et al., 2017; Shprits, Subbotin, et al., 2013; Shprits et al., 2016; Xiang et al., 2017). Visual analysis of the corresponding average innovation plots, before and after including EMIC waves for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0262 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0263, and for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0264 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0265, show that, for a fixed urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0266, the losses brought by introducing EMIC waves, are larger for higher urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0267 than for lower urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0268. Similar effects are observed for urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0269. Specifically, such losses are mostly evident in the region with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0270 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0271, on average. Our findings provide supporting evidence that EMIC waves certainly provide an additional loss mechanism and play an essential role in the dynamics of the ultrarelativistic electron population.

Magnetopause shadowing driven by the LCDS effectively brings losses into our model, as concluded by earlier studies, which relied either on observations or simulations (Shprits et al., 2006; Turner et al., 2012; Yu et al., 2013; Turner, Angelopoulos, Morley, et al., 2014; Xiang et al., 2017). In our case, the average innovation plots show that this mechanism is more efficient in the intervals with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0272 and in the region with urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0273. For the two populations of electrons with the lowest urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0274 values (both first cases in section 6.1 and 6.2), the incorporation of this process into the data assimilative scheme clearly illustrates a region of large underestimation extending from urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0275 up to the boundary at urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0276, corresponding to the absence of magnetospheric convection in our reanalysis. Such a process is accounted for by other models, such as the VERB-4D code (Aseev et al., 2016; Shprits et al., 2015), which solves the modified Fokker-Planck equation with additional convection terms.

Previous works have shed some light on the role and importance of mixed pitch angle-energy diffusion, scattering by EMIC waves, and magnetopause shadowing in the dynamics of the radiation belt electrons, and the current study is in agreement with their findings. Nevertheless, quantitative assessments of the contribution of each process have been scarce (Yu et al., 2013). The innovation vector certainly provides a tool to objectively determine how much loss is caused by each mechanism and to estimate the relative percentage of their contribution to the total dynamics. This analysis will be included in our future studies.

Lastly, we plan to perform a long-term reanalysis (several years spanning different levels of geomagnetic activity) and reconstruct the radial profiles of radiation belt electron PSD for the entire duration of current missions like the Van Allen Probes and Arase (Miyoshi et al., 2018). Data assimilation will play a crucial role in the analysis of such measurements and will be used to quantitatively report on the contribution of different mechanisms to the dynamical evolution of electron PSD. This will ultimately allow us to achieve a better understanding of the physical processes causing acceleration, transport, and losses in the radiation belts.


The authors acknowledge use of NASA/GSFC's Space Physics Data Facility's OMNIWeb service, and OMNI data (https://omniweb.gsfc.nasa.gov/). The urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0306 index was provided by GFZ Potsdam (https://www.gfz-potsdam.de/kp-index/). The authors are grateful to the RBSP-ECT team for the provision of Van Allen Probes observations (http://rbspect.lanl.gov/). The authors thank the developers of the IRBEM library which was adapted for use in the current study (http://irbem.sourceforge.net). We thank Adam Kellerman for the development of the initial codes. This research has been funded by the Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1294 “Data Assimilation,” Project B06 “Novel methods for the 3D reconstruction of the dynamic evolution of the Van Allen belts using multiple satellite measurements,” NASA Grant 80NSSC18K0663, and the Helmholtz Association Recruiting Initiative. The reanalysis files used in this study are available online (ftp://ftp.gfz-potsdam.de/pub/home/rbm/reanalysis).

    Appendix A: Adiabatic Invariants

    The first adiabatic invariant is associated with the gyration of a particle around the field line and may be expressed as
    where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0278 is the relativistic momentum in the direction perpendicular to the direction of the magnetic field, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0279 is the electron rest mass, and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0280 is the magnitude of the local magnetic field.
    The second adiabatic invariant associated with the bounce motion between the mirror points may be expressed as
    where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0282 is the relativistic momentum in the direction parallel to the direction of the magnetic field, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0283 is the magnetic field strength at the mirror point, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0284 and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0285 are the distances along the field line from the equator to the mirror point, and urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0286 is the distance element along the field line.
    The urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0287 invariant is a combination of the first two invariants, and it does not depend on the charge or the mass of the particle. It is usually expressed as:
    The third adiabatic invariant is associated with the drift motion around the Earth and may be expressed in terms of the magnetic flux through the orbit:
    The Roederer parameter urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0290 (Roederer, 1970) is commonly used instead of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0291:
    where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0293 is the Earth's magnetic moment.
    In a dipolar magnetic field, the equatorial pitch angle of electrons urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0294 can be related to the invariant urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0295 as (Schulz & Lanzerotti, 1974):
    where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0297.
    The kinetic energy of a particle can be related to the first adiabatic invariant as
    At any urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0299 shell PSD urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0300 may be related to the differential flux as (Rossi and Olbert, 1970):
    where urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0302 is the differential flux in units of urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0303, urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0304 is in units of MeV, and PSD is in urn:x-wiley:jgra:media:jgra55451:jgra55451-math-0305.