Ionospheric Multi-Spacecraft Analysis Tools pp 219-232 | Cite as

# Recent Progress on Inverse and Data Assimilation Procedure for High-Latitude Ionospheric Electrodynamics

## Abstract

Polar ionospheric electrodynamics plays an important role in the Sun–Earth connection chain, acting as one of the major driving forces of the upper atmosphere and providing us with a means to probe physical processes in the distant magnetosphere. Accurate specification of the constantly changing conditions of high-latitude ionospheric electrodynamics has long been of paramount interest to the geospace science community. The Assimilative Mapping of Ionospheric Electrodynamics procedure, developed with an emphasis on inverting ground-based magnetometer observations for historical reasons, has long been used in the geospace science community as a way to obtain complete maps of high-latitude ionospheric electrodynamics by overcoming the limitations of a given geospace monitoring system. This Chapter presents recent technical progress on inverse and data assimilation procedures motivated primarily by availability of regular monitoring of high-latitude electrodynamics by space-borne instruments. The method overview describes how electrodynamic state variables are represented with polar-cap spherical harmonics and how coefficients are estimated from the point of view of the Bayesian inferential framework. Some examples of the recent applications to analysis of SuperDARN plasma drift, Iridium, and DMSP magnetic fields, as well as DMSP auroral particle precipitation data are included to demonstrate the method.

## 10.1 Introduction

The most dynamic electromagnetic energy and momentum exchange processes between the upper atmosphere and the magnetosphere take place in the polar ionosphere. Physical processes producing aurora involve ionization and excitation of atmospheric constituents due to energetic charged particles precipitating into the upper atmosphere from the magnetosphere along the geomagnetic field lines, which in turn modulates the ionosphere’s ability to conduct electric currents. Polar ionospheric electrodynamics plays an important role in the Sun–Earth connection chain, acting as one of the major driving forces of the upper atmosphere and providing us with a means to probe physical processes in the distant magnetosphere. Accurate specification of the constantly changing conditions of high-latitude ionospheric electrodynamics has long been of paramount interest to the geospace science community.

Global monitoring of high-latitude geospace has dramatically improved thanks to a recent expansion of ground-based and space-based observing capability. International consortiums of ground-based instrumentation such as the Super Dual Auroral Radar Network (SuperDARN) (e.g., Greenwald et al. 1995), International Real-Time Magnetic Observatory Network (e.g., Love 2013) and SuperMAG (e.g., Gjerloev 2009) have made a large volume of quality-controlled, standardized data accessible to the public. Acquisition, processing, and distribution of engineering-grade magnetometer data from the Iridium satellite constellation for scientific purposes by the Active Magnetosphere and Polar Electrodynamics Response Experiment (AMPERE) program (Anderson et al. 2000) have been instrumental in making continuous, global monitoring of geomagnetic-field-aligned currents (FAC) possible. Defense Meteorological Satellite Program (DMSP) space environment instruments have long been providing valuable measurements of precipitating electron and ion particles, magnetic fields, and ultraviolet spectrographic images (e.g., Rich 1984; Hardy et al. 1984; Paxton et al. 2002). And the Swarm multi-satellite mission (Friis-Christensen et al. 2006) provides high precision measurements of magnetic fields that complement theses existing geospace observing systems.

Data assimilation techniques such as the Assimilative Mapping of Ionospheric Electrodynamics (AMIE) procedure of Richmond and Kamide (1988) have long been used in the geospace science community as a way to obtain complete maps of high-latitude ionospheric electrodynamics by overcoming the limitations of a given geospace monitoring system. The procedure combines a number of different types of space-based and ground-based observations with an empirical model of ionospheric electrodynamics to infer distributions of ionospheric electric fields and currents, FAC, associated geomagnetic perturbation fields at both ground and low-Earth-orbit altitudes, Hall and Pedersen conductance, and Joule heating. AMIE maps have yielded a number of important insights into the coupling of the magnetosphere, ionosphere, and thermosphere that takes place at high latitudes. Lu (2017) provides a comprehensive overview of AMIE applications.

This paper presents an overview of the recent technical developments of the inverse and data assimilation procedure for high-latitude electrodynamics. Some of these developments are a consequence of a reformulation of the best linear unbiased estimation problem presented in Richmond and Kamide (1988) as a Bayesian estimation problem (Matsuo et al. 2005). Under the assumption that electrodynamic variables are Gaussian distributed, these two estimation problems are equivalent. A Bayesian perspective has helped to clarify the role of the prior model (background) error covariance as a key component in the modeling of Gaussian processes, and thus guided modeling and estimation of prior covariance functions from a large volume of SuperDARN data (Cousins et al. 2013a), DMSP particle precipitation data (McGranaghan et al. 2015, 2016), and Iridium magnetic perturbation data (Cousins et al. 2015b; Shi et al. 2019). Even though ionospheric conductivity serves as a critical linkage in electromagnetic energy and momentum exchange processes, direct monitoring of this conductivity is almost nonexistent. Another notable development led by McGranaghan et al. (2016) is an assimilative mapping of the conductance using the auroral ionization derived from DMSP electron energy flux spectra with help of the GLobal airglOW (GLOW) model (Solomon et al. 1988) without the assumption of Maxwellian distribution. Since the AMIE has been developed with an emphasis on inverting ground-based magnetometer observations for historical reasons (Kamide et al. 1981; Richmond and Kamide 1988), it is not tailored to analyses of space-based magnetometer data from DMSP, Iridium, and Swarm. In order to solve the optimization problem in terms of electrostatic potential, the space-based magnetometer data first need to be converted to electrostatic potential through the application of Ohm’s law and current continuity. To minimize the impact of conductance on the inversion of space-based magnetometer data for FAC, the optimization problem is now being solved in terms of both magnetic potential and electrostatic potential (Matsuo et al. 2015; Cousins et al. 2015a).

## 10.2 Method Overview

### 10.2.1 Representation of Electrodynamic State Variables Using Scalar and Vector Polar-Cap Spherical Harmonic Basis Functions

**x**is a column vector of 244 elements and \(\mathbf{\Psi }\) is an \(n\times 244\) matrix, where

*n*is the number of grid points. Using the Nyquist sampling rate, the effective resolution is \(15^{\circ }\) longitude and \(2.5^{\circ }\) latitude. Let’s suppose that the electrostatic potential \(\Phi \) at \(\phi _m\) and \(\lambda _m\) is given by

*J*by

### 10.2.2 Bayesian State Estimation for Gaussian Processes

*j*observations that may consist of electric field, ground-based magnetic field, and/or space-based magnetic field measurements at discrete observation locations. By evaluating the polar-cap spherical harmonics and their derivatives at observation locations, \(\mathbf{y}\) can be expressed as

### 10.2.3 Nonstationary Covariance Modeling

*p*principal components are expressed by a linear combination of the polar-cap spherical harmonic basis functions of Richmond and Kamide (1988), and each component is estimated sequentially by a back-fitting technique along with orthonormalization of the regression coefficients for each component. Each EOF can be expressed as \(\varvec{\Psi } \varvec{\beta }\), where \(\varvec{\beta }\) is a \(244\times p\) matrix. Then \(\mathbf{C}_b\) is given as

*p*is set to 30. It is evident that the correlation structures are highly anisotropic with a larger correlation length scales in the zonal direction in comparison to the meridional direction, and correlations vary depending on reference point locations. These are features of strong nonstationary correlation, which will enable the data assimilation procedure to spatially distribute the impact of observations with consideration of realistic location-specific correlation structures of SuperDARN plasma drifts or electric fields.

## 10.3 Analysis of Electrostatic Potential and Electric Fields

Cousins et al. (2013b) presents an inverse and data assimilation procedure designed to specifically estimate \(\mathbf{x}_\mathrm{E}\) as defined in (10.6) and (10.7) from SuperDARN data. A comprehensive cross-validation study (Cousins et al. 2013b) wherein observations are systematically set aside for validation and compared to predictions by data assimilation outperforms the standard SuperDARN mapping procedure (Ruohoniemi and Baker 1998; Shepherd and Ruohoniemi 2000). The inverse and data assimilation procedure is found to reduce median prediction errors by up to 43% as compared to the standard SuperDARN mapping procedure. The procedure is built using the prior covariance modeled with EOFs obtained by Cousins et al. (2013a) and the prior mean specified by the empirical plasma convection model of Cousins and Shepherd (2010). Figure 10.2 compares the maps of electrostatic potentials obtained by the standard SuperDARN mapping procedure (Ruohoniemi and Baker 1998; Shepherd and Ruohoniemi 2000 to the ones by Cousins et al. (2013b) along with maps of the uncertainty associated with assimilative mapping as given by the diagonal elements of \(\mathbf{C}_a\) (10.15). The uncertainty reflects the observation distributions with higher uncertainty found in the area of the SuperDARN data gap. The comparison also highlights the role of the nonstationary covariance in the inverse and data assimilation procedure that help regularize assimilative mapping analysis.

## 10.4 Analysis of Toroidal Magnetic Potential and Field-Aligned Currents

## 10.5 Dual Optimization Approach

## 10.6 Summary

This paper demonstrates that simultaneous analysis of multiple types of space-based and ground-based global geospace observations enabled by the inverse and data assimilation procedure provides a global perspective of high-latitude ionospheric electrodynamics. The paper summarizes important technical developments that have been made in response to the expansion of high-latitude geospace observing systems. The primary areas of the methodological extension to the AMIE (Richmond and Kamide 1988) are (a) the optimization in terms of both magnetic and electrostatic potential to minimize the impact of conductance on the inversion of space-based Iridium and DMSP magnetometer data for FAC mapping (Matsuo et al. 2015; Cousins et al. 2015a; (b) the use of realistic prior error covariance estimated from a large data set of SuperDARN (Cousins et al. 2013a), DMSP (McGranaghan et al. 2015) and Iridium magnetic perturbation data (Shi et al. 2019; (c) improved assimilative conductance/conductivity mapping (McGranaghan et al. 2016).

## Notes

### Acknowledgements

This study is supported by the NSF grants PLR-1443703 and ICER-1541010. We thank the International Space Science Institute (ISSI) in Bern, Switzerland for supporting the Working Group “Multi Satellite Analysis Tools—Ionosphere” from which this chapter resulted. The Editors thank Gang Lu for her assistance in evaluating this chapter.

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