Local Least Squares Analysis of Auroral Currents
Abstract
Multispacecraft probing of geospace allows the study of physical structures on spatial scales dictated by orbital and instrumental parameters. This chapter highlights multipoint array analysis methods for constellations of two or three spacecraft such as Swarm, and also discusses multiscale techniques for the geometrical characterisation of auroral current structures using observations of stationary or weakly timedependent current structures along the tracks of individual satellites. Linear estimators are based on a least squares approach which is local in the sense that only few measurements around a reference point are considered for the reconstruction of geometrical and physical parameters. Local least squares estimators for fieldaligned currents are compared with nonlocal counterparts and also local estimators based on finite differences. Uncertainties, implementation and other practical aspects are discussed. The techniques are illustrated using selected Swarm crossings of the auroral zone.
4.1 Introduction
Coupling processes in the auroral zone are to a large extent controlled by electric currents flowing parallel to the ambient magnetic field lines (e.g., Lysak 1990; Paschmann et al. 2002; Vogt 2002). Auroral fieldaligned currents (FACs) connect remote plasmas in geospace and are associated with global magnetospheric dynamics such as largescale convection and substorms. The type of electrodynamic coupling in the auroral current circuit depends on the spatial scale L of FACs. Quasistatic coupling of the equatorial magnetosphere and the auroral ionosphere is expected on large scales \(L \gtrsim \lambda _P \sim 100\) km (Lyons 1980; Lotko et al. 1987). Alfvénic coupling should be important on intermediate scales \(L \gtrsim \lambda _A \sim 10\) km (Vogt and Haerendel 1998). On even smaller spatial scales in the kilometre range or below, auroral phenomena are typically very dynamic and shortlived, and not correlated with longrange magnetosphere–ionosphere coupling processes. The importance of spatial scales in FACs and their association with auroral processes and coupling regimes was confirmed by Lühr et al. (2015) using data from the initial phase of ESA’s threespacecraft Swarm mission when a range of interspacecraft distances was covered, see also Chap. 6 of this volume. Since April 2014, two Swarm satellites SwA and SwC orbit sidebyside in the auroral ionosphere, while the third Swarm satellite SwB moves at a higher altitude and lines up only occasionally with the lower pair to form a threespacecraft constellation. SwA and SwC can be understood as a twosatellite array that allows the study of auroral FACs on a regular basis but restricted to the spatial scale given by their orbital separation.
To resolve electric current systems or other geospace structures by means of singlespacecraft data or multispacecraft observations, analysis methods must provide an adequate spatial resolution. The singlespacecraft approach where an individual satellite is assumed to move across a stationary or at most weakly timedependent structure may resolve spatial scales given by the product of relative speed V and sampling interval \(\varDelta t\). Satellite constellations such as Cluster and Swarm allow to adopt a multipoint array perspective with the spatial resolution given by the interspacecraft distance scale \(\varDelta r\). For geospace phenomena and missions, \(V \varDelta t\) is typically smaller than \(\varDelta r\). Singlespacecraft methods allow the study of smaller scales but only in the alongtrack direction. Multipoint array techniques provide information also about the acrosstrack variability (and possibly temporal changes) but only on larger spatial scales (e.g., Russell et al. 1983; Dunlop et al. 1988; Neubauer and Glassmeier 1990; Pinçon and Lefeuvre 1991; Chanteur and Mottez 1993; Dudok de Wit et al. 1995; Bauer et al. 2000; Balikhin et al. 2001; Dunlop et al. 2002; De Keyser et al. 2007; Vogt et al. 2008a), see also the ISSI Scientific Reports SR1 and SR8 (Paschmann and Daly 1998, 2008).
The category of analysis methods addressed in this report may be termed local least squares (LS) techniques. We are concerned mainly with multipoint array estimation of auroral FACs using vector magnetometer data from satellite missions such as Swarm, but also with multiscale geometrical characterisation of current structures in the data of individual spacecraft. In local LS analysis, the least squares principle is applied to a confined region of interest, typically comprising only a few measurement points, equivalent to general (nonlocal) least squares modelling with localised basis functions but requiring less computational effort. Compared to local estimators based on finite differencing, local LS estimators turn out to be more robust with regard to nonregular (skewed or stretched) satellite constellations. To cover the multiscale nature of auroral processes, in particular also intermediate and possibly even smaller scales, a multiscale analysis technique developed by Bunescu et al. (2015, 2017) is included here as a localised version of the popular singlespacecraft minimum variance analysis or MVA that has its roots also in the least squares approach (Sonnerup and Scheible 1998).
Multispacecraft array techniques based on the local LS principle are presented in Sect. 4.2 (methodology) and Sect. 4.3 (implementation and applications). The multiscale variant of MVA is discussed in Sect. 4.4 before we conclude in Sect. 4.5 with a summary.
4.2 Methodology of Multispacecraft Array Techniques
Multispacecraft estimation of spatial gradients, electric currents, or boundary parameters is based on a set of satellite positions and corresponding observations that are interpreted as array measurements. In the simplest and most straightforward case, each satellite of a constellation contributes one position vector to the array, and all measurements are taken at the same time. This perspective was adopted, e.g., by Dunlop et al. (1988), Chanteur (1998), Harvey (1998), as well as Vogt and Paschmann (1998) in the preparation phase of ESA’s fourspacecraft mission Cluster to develop analysis methods for electric currents and spatial gradients. The corresponding threespacecraft case of the LS approach was addressed by Vogt et al. (2009). The resulting estimators are instantaneous, thus perfectly localised in time, and also local in space at the lower resolution limit given by the interspacecraft separation scale that for Cluster ranged between 100 km and 10,000 km.
For the Swarm mission with only two satellites SwA and SwC close enough to be considered a multipoint array on a regular basis, the instantaneous approach was relaxed to include additional measurements shifted in time, thus generating a virtual planar fourpoint satellite array (Ritter and Lühr 2006; Ritter et al. 2013; Shen et al. 2012a; Vogt et al. 2013). Virtual alongtrack separations approximately equal to the distance between SwA and SwC of somewhat less than 100 km in the auroral zone are obtained using time shifts of about 10 s, considerably smaller than the variation time scales of FACs associated with quasistatic coupling. Localisation in space is characterised by the effective interspacecraft separation in the virtual quad, typically of the order of 100 km. In the context of this report, we refer to this class of methods as local analysis techniques.
When spatial distributions of electrodynamic variables for large parts of or even the entire auroral region are to be reconstructed from satellite crossings (e.g., He et al. 2012; Amm et al. 2015), possibly in combination with other groundbased data, the methodology is usually termed modelling rather than analysis, and should be referred to as a regional method (spatial extent of the order of 1000 km), applicable to structures that do not vary on time scales less than the satellite crossing time of the order of a few minutes. The most popular modelling approach is based on the least squares principle that we choose as a starting point for our discussion.
4.2.1 General Linear Least Squares
Regional least squares modelling of auroral fieldaligned currents was carried out by He et al. (2012, 2014) who condensed ten years of CHAMP (Reigber et al. 2002) magnetic field measurements into the empirical FAC model MFACE using a set of dataadaptive basis functions called empirical orthogonal functions (EOFs). In the first modelling step, the set of EOFs was constructed from the data in a coordinate frame centred on the dynamic auroral oval to capture its magnetic local timedependent expansion and contraction during magnetospheric activity. The EOFs then serve as basis functions in an expansion of the form \(j_\Vert (\varDelta \beta  \mathbf {p}) = \sum _\nu a_\nu (\mathbf {p}) f_\nu (\varDelta \beta )\) where the parameter vector \(\mathbf {p}\) is formed by a set of predictor variables (magnetic local time, seasonal and solar wind parameters, AE), and \(\varDelta \beta = \beta  \beta _\text {ACC}\) is magnetic latitude in auroral oval coordinates, i.e. relative to the latitude of the auroral current centre \(\beta _\text {ACC}\). In a second step, the functional dependences \(\beta _\text {ACC} = \beta _\text {ACC} (\mathbf {p})\) and \(a_\nu = a_\nu (\mathbf {p})\) were determined also through least squares regression. The geophysical parameters that drive the model (such as the IMF, solar wind parameters and geomagnetic indices) are obtained from NASA’s OMNIWeb service. Two sample outputs are shown in Fig. 4.1. The times chosen for producing the diagrams correspond to selected Swarm auroral crossings that are used also for demonstrating multipoint FAC estimators in Sect. 4.3.2 and multiscale MVA analysis of FACs in Sect. 4.4.1.
4.2.2 Local LS Estimators of Spatial Gradients
The position vectors in an array of S spacecraft are denoted as \(\mathbf {r}_\sigma , \sigma = 1, \ldots , S\), and the difference vectors are \(\mathbf {r}_{\sigma \tau } = \mathbf {r}_\tau  \mathbf {r}_\sigma \). The average position or mesocentre \(\mathbf {r}_*\; = \; (1/S) \, \sum _\sigma \mathbf {r}_\sigma \) may be chosen to coincide with the origin. In such a mesocentric coordinate system, the position tensor \({\mathsf {R}} = \sum _\sigma (\mathbf {r}_\sigma  \mathbf {r}_*) (\mathbf {r}_\sigma  \mathbf {r}_*)^\mathsf {T}\) simplifies to \({\mathsf {R}} = \sum _\sigma \mathbf {r} \, \mathbf {r}^\mathsf {T}\). Note that the volumetric tensor introduced by Harvey (1998) differs by a factor of 1/S from the position tensor defined here.
Invertible position tensor
Singular position tensor, planar spacecraft array
The singular position tensor case is most relevant for the Swarm mission: here the gradient vector cannot be resolved fully from the measurements, and additional information has to be taken into account to reconstruct its outofplane component. Constraints may in principle be incorporated in the least squares framework using Lagrange multipliers. The approach chosen by Vogt et al. (2009) is based on geometrical considerations, and offers the possibility to choose between three types of constraints: (1) gradient parallel to a given direction \(\hat{\mathbf {e}}\), (2) gradient perpendicular to a given direction \(\hat{\mathbf {e}}\), (3) the physical structure is stationary in a reference frame moving with a known velocity relative to the spacecraft array. Of particular importance for studies of fieldaligned currents is a fourth constraint that combines the forcefree condition \(\mathbf {B} \varvec{\times }(\nabla \varvec{\times }\mathbf {B}) = \mathbf {0}\) with \(\nabla \varvec{\cdot }\mathbf {B} = 0\) to estimate the full magnetic gradient matrix from spacecraft measurements in one plane (Shen et al. 2012a; Vogt et al. 2013).
4.2.3 Local LS Estimators of Electric Currents
4.2.4 Related Local Estimators of Gradients and Currents
In preparation of the Cluster mission, gradient analysis methods were derived from several different principles such as discretised boundary integration (Dunlop et al. 1988), spatial interpolation (Chanteur 1998), and least squares estimation (Harvey 1998). For the Swarm mission, curl estimators based on finite differences (FD) and also on discretised boundary integrals (BI) were developed by Ritter and Lühr (2006), see also Shen et al. (2012a) and Ritter et al. (2013). Although the underlying principles differ, the resulting estimators may still turn out to be identical. Chanteur and Harvey (1998) demonstrated that in the regular (tetrahedral) fourspacecraft case, spatial interpolation yields the same analysis scheme as unconstrained least squares. Considering virtual fourpoint almost planar configurations relevant for the Swarm mission, (Vogt et al. 2013) compared different estimators for the normal component of the curl (corresponding to the radial current density) and found that the FD and BI estimators are algebraically identical.
4.2.5 Errors and Limitations
Spatial gradient estimates produced from multipoint measurements are affected by several errors and limitations: (a) measurement errors, (b) positional errors, (c) imperfections of the assumed linear model. In the planar spacecraft array case when additional information has to be considered to reconstruct the normal gradient, and (d) uncertainties in the imposed geometrical or physical conditions give rise to additional errors.
Measurement errors
Positional errors
Random uncertainties in spacecraft positions are called positional errors or geometrical errors. Isotropic and uncorrelated positional errors can be incorporated in the parameter covariances obtained from considering only measurement errors by replacing \((\delta h)^2 \rightarrow (\delta h)^2 + \nabla h^2 (\delta r)^2\) (Vogt and Paschmann 1998; Vogt et al. 2009). In the case of magnetic gradient and electric current estimates based on Swarm magnetic field data, positional errors have a much smaller impact than measurement errors because \(\nabla \mathbf {B} \delta r \ll \delta B\) and can thus be neglected. This statement remains valid if positional inaccuracies in virtual fourpoint configurations imposed by timeshift errors \(\delta t\) (in the order of ms) are taken into account, then \(\delta r \sim V \delta t\) where V is the spacecraft speed.
Model imperfections
Nonlinear variations of the observable over the spatial region covered by the spacecraft array cause deviations from the assumed linear model, and the spatial gradient is not perfectly uniform. In contrast to the statistical nature of measurement errors and positional errors which become less important for larger interspacecraft separation distances, gradient estimation errors due to deviations from linearity tend to increase with spacecraft separations (Robert et al. 1998). In the auroral zone the problem implies that current structures with variation scales (sheet widths) smaller than the array extension cannot be resolved and are effectively smeared out.
Uncertainties of imposed conditions
In the planar spacecraft array case (position tensor has rank 2) where only the inplane component can be directly estimated from the measurements, the constraint equations used to reconstruct the normal gradient may not be perfectly satisfied, producing additional errors. The quality of the normal gradient can be assessed by means of error indicators (Vogt et al. 2009, 2013). The assumption that the full gradient is aligned with a given direction \(\hat{\mathbf {e}}\) (parallel constraint) can lead to large uncertainties if \(\hat{\mathbf {e}} \varvec{\times }\hat{\mathbf {n}}\) is small. The normal gradient estimate resulting from the perpendicular constraint (full gradient perpendicular to a given direction \(\hat{\mathbf {e}}\)) should be taken with care if the error indicator \(\hat{\mathbf {e}} \varvec{\cdot }\hat{\mathbf {n}}\) is small. An error indicator for the forcefree case that should not become too small is \(\hat{\mathbf {B}}_0 \varvec{\cdot }\hat{\mathbf {n}}\). Since in the auroral zone the magnetic field forms a small angle with the radial vector, the error indicator for the virtual fourpoint configuration constructed from Swarm dualspacecraft positions is close to unity and thus wellbehaved. Larger uncertainties are expected at low latitudes.
4.3 Multispacecraft Array Techniques in Practice
This section is concerned with the implementation and applications of local LS estimators for the planar array case, i.e. threespacecraft arrays and virtual fourpoint configuration constructed from the positions of SwA and SwC. Then the canonical base vectors \(\mathbf {q}_\sigma \) are minimumnorm solutions of \({\mathsf {R}} \mathbf {q}_\sigma = \mathbf {r}_\sigma \), and Eqs. (4.17) and (4.18) produce the inplane gradient and the normal curl component directly from the measurements. The threespacecraft LS gradient estimator was tested by Vogt et al. (2009) and applied to Cluster pressure measurements in the magnetail. The dualsatellite LS FAC estimator was validated by Vogt et al. (2013) using Cluster observations of a forcefree plasma structure in the solar wind that had previously been studied and characterised in detail by means of multispacecraft timing analysis (Vogt et al. 2011). Below in Sect. 4.3.2 we present selected applications of the threespacecraft and the dualsatellite LS FAC estimator to Swarm magnetic field measurements, after discussing the implementation of local LS estimators for planar arrays in Sect. 4.3.1.
4.3.1 Implementation of Planar Multipoint Array Estimators
Local LS gradient and/or curl estimation using measurements of a planar spacecraft array involves the following steps.
Construction of canonical base vectors
Estimation of planar gradient and/or normal current components
Quality indicators of planar gradient and normal current estimates
The stability of the (constrained) matrix inversion that yields the canonical base vectors is controlled by the effective condition number \(\text {CN} ({\mathsf {R}}) = \rho _1/\rho _2\) of the position tensor \({\mathsf {R}}\).
Construction of the full gradient matrix and/or the full current vector
In order to obtain the full gradient, the component normal to the spacecraft plane has to be constructed in addition to the planar gradient estimate. This can be achieved by means of suitable constraint equations as discussed in Sect. 4.2.2, see also Vogt et al. (2009). The curl vector \(\nabla \varvec{\times }\mathbf {B}\) can then be read directly from the components of the skewsymmetric part of \(\nabla \mathbf {B}\). In the special case of the forcefree condition, the current is parallel to the ambient magnetic field \(\mathbf {B}_0\), and the fieldaligned current density \(j_\Vert \) can be computed directly from the normal current density \(j_n\) through \(j_\Vert \simeq j_n / \hat{\mathbf {n}} \varvec{\cdot }\hat{\mathbf {B}}_0\). The construction of normal gradients and planar curl components should be critically assessed using error indicators as discussed in Sect. 4.2.5.
4.3.2 Application to Swarm Auroral Crossings
Figure 4.5 shows magnetic field measurements of the three Swarm satellites in the Southern hemisphere on 24 July 2014, 02:44–02:50 UT, together with different FAC estimates and quality indicators. Clearly visible are negative parallel (downward) currents between \(\sim \)02:47:30 and \(\sim \)02:48:00 followed by positive parallel (upward) current until \(\sim \)02:48:30. The dualsatellite LS FAC estimates are very close to the Level2 FAC product J_L2_AC apart from several smallerscale deviations. They are due to the fact that the Level2 dualsatellite FAC product is based on filtered Swarm magnetic field observations whereas here the least squares estimator processed unfiltered data as input. The smallerscale deviations are much less pronounced if the dualsatellite LS estimate of the FAC profile is also computed after application of a suitable filter. The output produced by the threespacecraft LS estimator, also based on unfiltered magnetic field measurements, also shows smallerscale variability but otherwise follows the other two profiles quite well apart from an apparent time shift, caused by different mesocentres of the threespacecraft array and the fourpoint configuration. Condition numbers for the current structure crossing are moderate (below 5 for the dualsatellite estimator, and not much larger than 10 for the threespacecraft array). The angle between the ambient magnetic field and the normal direction of the threespacecraft plane assumes tolerable values far from \(90^\circ \).
Data from the second Swarm auroral crossing on 29 May 2014, 13:36–13:44 UT, are displayed in Fig. 4.6. The largest current densities are observed between \(\sim \)13:39:30 and \(\sim \)13:41:30. Again apart from smallerscale deviations due to differences in filtering of the input magnetic field measurements, the two dualsatellite FAC estimates (both based on data from SwA and SwC) are very similar. The FAC profile produced by the threespacecraft LS estimator differs significantly at around 13:41, despite reasonable values of the quality indicators. Closer inspection of the SwB magnetic field profile reveals a substructure at 13:41 that is not present in the measurements of SwA and SwC, indicating nonuniform currents on the interspacecraft separation scale that are inconsistent with the linear model assumption. Eigenvalue ratios are 11, 16, 13 and thus somewhat smaller than for the first crossing, and also the sheet orientations obtained from singlespaceraft MVA show larger differences up to about 10 degrees.
4.4 SingleSpacecraft Multiscale Analysis
Satellite measurements of the magnetic field allow the study of planar geospace structures such as current sheets or boundary layers through the eigenvalues and eigenvectors of the data covariance matrix. This type of principal axis decomposition is known as principal component analysis (PCA) or empirical orthogonal function (EOF) analysis in the statistical literature, and as minimum variance analysis (MVA) in space physics (Sonnerup and Cahill 1967). MVA can be derived using constrained least squares estimation (Sonnerup and Scheible 1998) and is usually applied to the entire geospace structure of interest. (Bunescu et al. 2015, 2017) introduced a multiscale version by applying the MVA procedure using a range of sliding windows, thus producing local estimates of key MVA parameters such as the eigenvalue ratio and the angle characterising sheet orientation. The novel multiscale version of MVA is described below in Sect. 4.4.2, followed by an application to Swarm magnetic field measurements in Sect. 4.4.3. The starting point of our discussion are the principles of MVA as summarised in Sect. 4.4.1.
4.4.1 MVA Applied to Auroral Current Sheets
The quality of \(\hat{\mathbf {n}}\) estimates is associated with eigenvalue ratios. In the case of auroral FAC sheets, magnetic perturbations are in the plane perpendicular to the ambient magnetic field. The problem reduces to two spatial dimensions with two relevant eigenvalues \(\lambda _1 \ge \lambda _2\) and two eigenvectors \(\hat{\mathbf {e}}_1,\hat{\mathbf {e}}_2\). The eigenvalue ratio \(\lambda _1 / \lambda _2\) can be understood as a measure of planarity and should be sufficiently large. The sheet orientation is given by tangential vectors \(\hat{\mathbf {B}}_0\) (direction of the ambient magnetic field) and \(\hat{\mathbf {e}}_1\), and the normal unit vector \(\hat{\mathbf {n}} = \hat{\mathbf {e}}_2\). The orientation of auroral current sheets can be concisely characterised by the (inclination) angle formed by the sheet normal with magnetic north, or the spacecraft velocity vector (approximately geographic north for polar orbiting satellites such as CHAMP or Swarm).
4.4.2 Multiscale FieldAligned Current Analyzer

A range of window widths w with linear resolution \(\mathrm {d}w\) is defined.

At each time t of the magnetic field series, MVA is applied to an array of data segments of width w within a predefined range and centred at t, thus yielding a series of key MVA parameters \(\lambda _1 = \lambda _\text {max}\), \(\lambda _2 = \lambda _\text {min}\), \(R_\lambda = \lambda _1/\lambda _2\) (eigenvalue ratio), and an inclination angle. All parameters are functions of time t and scale w.

In addition to these MVA parameters, the derivative of the largest eigenvalue \(\lambda _1 = \lambda _\text {max}\) with respect to scale w is computed numerically to yield \(\partial _w \lambda _\text {max}\).

The continuous and multiscale MVA parameters are displayed as functions of time t and scale w in a suitable twodimensional graphical representation, either as a contour plot and/or using an appropriate colour bar. Important scales are found to show up well in colour plots of \(\partial _w \lambda _\text {max}\).
4.4.3 Application of MSMVA to Swarm Auroral Crossings
Figure 4.7 shows the MSMVA results for the two Swarm auroral crossings considered already in Sect. 4.3.2. In both cases, MSMVA was applied to magnetic field data from SwA (panels 1 and 5), here given in the meanfieldaligned (MFA) coordinate frame (note that in Figs. 4.5 and 4.6 the magnetic field was displayed in NEC coordinates). The eigenvalue ratio \(R_\lambda \) is displayed in panels 2 and 6. The current sheet inclination is shown in panels 4 and 8. The multiscale nature of the FAC sheets is visualised very clearly in the panels 3 and 7 showing \(\partial _w \lambda _\text {max}\).
4.5 Summary
The local least squares approach to the estimation of spatial derivatives from multispacecraft magnetic field measurements yields a generic framework for the analysis for auroral FACs and their errors. This report reviewed the underlying principles, estimation procedures, uncertainties, limitations, and practical aspects. The array geometry defines the position tensor with eigenvalues controlling the quality of gradient and curl estimates. Linear estimators can be uniquely qualified through their set of canonical base vectors, facilitating error analysis and comparison with alternative approaches. In planar spacecraft array configurations, reconstruction of the full gradient and curl vectors requires additional information that can be supplemented in the form of geometrical or physical constraints. The multiscale nature of auroral currents can be investigated using a multiscale version of the wellestablished minimum variance analysis. Analysis techniques were illustrated using selected auroral crossings of the Swarm satellites.
Notes
Acknowledgements
Financial support by the Deutsche Forschungsgemeinschaft in the context of the DFG Special Programme SPP 1788 DynamicEarth through grant VO 855/41 is acknowledged. The authors thank the International Space Science Institute in Bern, Switzerland, for supporting the ISSI Working Group MultiSatellite Analysis Tools, Ionosphere, from which this chapter resulted. The editors thank Chao Shen for his assistance in evaluating this chapter.
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