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- Abstract
- Introduction
- MSMVA estimation of the FAC density
- Synthetic FAC structures
- Auroral region crossings by Swarm
- Discussion and summary
- Conclusions and outlook
- Data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
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ANGEO | Articles | Volume 37, issue 3

Ann. Geophys., 37, 347–373, 2019

https://doi.org/10.5194/angeo-37-347-2019

© Author(s) 2019. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/angeo-37-347-2019

© Author(s) 2019. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Dynamics and interaction of processes in the Earth and its...

**Regular paper**
24 May 2019

**Regular paper** | 24 May 2019

Multiscale estimation of the field-aligned current density

^{1}Space Plasma and Magnetometry Group,Institute of Space Science, Bucharest, Romania^{2}Department of Physics and Earth Science, Jacobs University Bremen, Bremen, Germany

^{1}Space Plasma and Magnetometry Group,Institute of Space Science, Bucharest, Romania^{2}Department of Physics and Earth Science, Jacobs University Bremen, Bremen, Germany

**Correspondence**: Costel Bunescu (costel@spacescience.ro)

**Correspondence**: Costel Bunescu (costel@spacescience.ro)

Abstract

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Field-aligned currents (FACs) in the magnetosphere–ionosphere (M–I) system exhibit a range of spatial and temporal scales that are linked to key dynamic coupling processes. To disentangle the scale dependence in magnetic field signatures of auroral FACs and to characterize their geometry and orientation, Bunescu et al. (2015) introduced the multiscale FAC analyzer framework based on minimum variance analysis (MVA) of magnetic time series segments. In the present report this approach is carried further to include in the analysis framework a FAC density scalogram, i.e., a multiscale representation of the FAC density time series. The new technique is validated and illustrated using synthetic data consisting of overlapping sheets of FACs at different scales. The method is applied to Swarm data showing both large-scale and quiet aurora as well as mesoscale FAC structures observed during more disturbed conditions. We show both planar and non-planar FAC structures as well as uniform and non-uniform FAC density structures. For both synthetic and Swarm data, the multiscale analysis is applied by two scale sampling schemes, namely the linear and logarithmic scanning of the FAC scale domain. The local FAC density is compared with the input FAC density for the synthetic data, whereas for the Swarm data we cross-check the results with well-established single- and dual-spacecraft techniques. All the multiscale information provides a new visualization tool for the complex FAC signatures that complements other FAC analysis tools.

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Bunescu, C., Vogt, J., Marghitu, O., and Blagau, A.: Multiscale estimation of the field-aligned current density, Ann. Geophys., 37, 347–373, https://doi.org/10.5194/angeo-37-347-2019, 2019.

1 Introduction

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The dynamics of the magnetosphere–ionosphere (M–I) system at auroral latitudes is essentially controlled by solar wind–magnetosphere (S–M) coupling, subject to ionospheric feedback. One result of the dynamic interaction in the global S–M–I system is the accumulation of magnetic flux in different parts of the system, e.g., the magnetotail. The energy in the large-scale components is transported and dissipated to smaller-scale components of the system, e.g., in the polar ionosphere. The transfer of energy and momentum in the system is mediated by field-aligned currents (FACs) flowing along the ambient magnetic field lines and driving the formation of ionospheric (Hall and Pedersen) currents. The entire chain of the energy flow and conversion mechanisms is governed by a multiscale behavior in both time and space. The multiscale character is observed in all the measurable quantities associated with the system, like magnetic field measurements from above (spacecraft) and below (ground) the ionosphere. While above the ionosphere one measures the magnetic perturbation of the field-aligned current (closed in the ionosphere mainly by the Pedersen current), the magnetic perturbation observed on ground is related mainly to the Hall component of the ionospheric current. The multiscale character is observed also in the measurements of optical emissions, associated in turn with a multiscale particle precipitation pattern.

The spatial and temporal scales of the auroral arcs observed optically on
ground are dependent on the characteristics of the optical instruments (e.g.,
resolution, sampling frequency, coverage, exposure). Earlier statistical
measurements of the auroral arc thickness (Maggs and Davis, 1968) were based on
narrow field of view (FoV) TV camera observations and found a median of the
scale distribution around 230 m in the range of fine- and small-scale
auroral arcs (70 m–1.5 km). Later measurements (Knudsen et al., 2001) based on
All Sky Imager (ASI) observations found a maximum of the scale distribution
around 18 km in the range of mesoscale arcs (10–100 km). The TV and ASI
observations also correspond to different temporal scales because of the
large sampling frequency difference, with maxima at about ∼25 Hz for TV
and ∼0.3 Hz for ASI. Note that arcs which are not quasi-stationary at
the exposure timescales are likely to be smeared and integrated to
larger-scale structures in the optical data. More recently, Partamies et al. (2010)
showed measurements based on intermediate FoV optics (FoV of 20^{∘} and
a spatial resolution of 100 m) with a median of the arc width distribution
around 0.5–1.5 km. Partamies et al. (2010) observations fit in between the previous
fine and mesoscale arc width distributions. While these studies concentrated
on the visible arcs, Trondsen and Cogger (1997) addressed the scale distribution of the
black aurora, found to peak around 400–500 m with an average of 615 m
(range between 200 m and 1 km). A review of the optical aurora (caused by
electrons) with spatial and temporal scales below 1 km and 1 s,
respectively, is given by Sandahl et al. (2008). Overall, the results of all these
studies together indicate a rather continuous scale spectrum (Partamies et al., 2010).

FAC structures in the auroral zone are typically organized in east–west
aligned sheets. The first statistical studies (Iijima and Potemra, 1976a, b) of
the large-scale FACs separated those into the well-known poleward Region 1
(R1) and equatorward Region 2 (R2) currents with different orientation
depending on the magnetic local time (MLT) sector. This large-scale picture
was confirmed later by other studies, e.g., Peria et al. (2013) and
McGranaghan et al. (2017). Peria et al. (2013) examined the statistical properties of
stationary sheet-like FACs (thickness within 10–1000 km and densities
larger than 0.1 µA m^{−2}) observed by FAST. The McGranaghan et al. (2017)
study, based on Swarm observations, addresses the multiscale character of
FACs by separating the FAC contributions from small scale (∼ 50 km),
mesoscale (∼ 150 km), and large scale (∼ 350 km). Modeling
efforts, e.g., He et al. (2012), characterized the FAC properties (e.g.,
thickness and intensity) as a function of the solar wind properties and
geomagnetic indices (e.g., AE index). The internal structure of large-scale
FACs, associated with, e.g., discrete auroral arcs, shows variability in all
observed characteristics (e.g., the spatial and temporal scales, orientation,
geometry) depending on MLT and substorm phase. The importance of small-scale
FACs is confirmed by Peria et al. (2013), who found that the large-scale FACs
account for about 20 % of the FAC events and for about half of the total
charge transport.

Above the ionosphere, spacecraft observations provide information about the
scale distribution and main characteristics of the FACs (mapped to the
ionosphere) through the measurements of magnetic fields (upward and downward
FACs), associated electric fields (monopolar, converging or diverging
bipolar), and particle fluxes (upgoing and downgoing). A scale distribution
with a maximum between 4 and 5 km was obtained by Johansson et al. (2007) using
Cluster measurements (3–6 *R*_{E} altitude) of intense electric fields
(>0.15 V m^{−1}). Johansson et al. (2007) found that the associated FACs and
density gradients also have typical values within the 4–5 km range.
Johansson et al. (2007) (Fig. 9) also compare the scale distribution with former
results. We notice the distribution of the diverging electric fields
(Karlsson and Marklund, 1996) observed by Freja with the peak around 4 km. A statistical
study of inverted V structures (U-shaped potential drops) observed by the
FAST satellite (Partamies et al., 2008) showed typical scale widths of 20–40 km
(maximum energies of 2–4 keV). Simultaneous measurements of narrow arc
structures (down to a few kilometers) in both particle and optical data were
shown by Stenbaek-Nielsen et al. (1998) by analyzing conjugate FAST/aircraft observations. In the small-scale range we also mention the
high-resolution measurements of fine-scale FACs observed by Freja
(Lühr et al., 1994) showing a minimum FAC scale of ∼1.7 km for a specific
event.

The scale distribution of FACs reflects a variety of M–I coupling mechanisms. At large scales we have a quasi-stationary coupling (FACs closing in the ionosphere), whereas at small and fine scales a time-dependent coupling, typically provided by Alfvén waves in different regimes (e.g., shear, kinetic, inertial). The interaction of shear Alfvén waves with the auroral acceleration region (Vogt and Haerendel, 1998; Vogt, 2002) presents a maximum absorption (conversion of Poynting flux to electron energy flux) for wavelengths that are consistent with the scale size of mesoscale auroral arcs. The arc generation through inertial Alfvén waves (Chaston et al., 2003) shows scales corresponding to fine-scale auroral arcs (1 km width) near the polar cap boundary.

Multi-spacecraft missions on low-altitude polar orbits (e.g., Swarm, ST5)
offer a high coverage of the auroral oval and enable statistical studies that
address the dynamics and stationarity of FACs, more precise FAC estimates, as
well as comparison with the currents inferred by ground magnetic field
measurements or cross-check with optical observations. Forsyth et al. (2017)
computed the stability of FACs by comparing the lower-altitude Swarm
satellites' (SwA, SwC) FAC density using a shape and an amplitude correlation
and found that ∼50 % and ∼1 %–5 % of the large- and
small-scale FACs, respectively, correlate between the two spacecraft.
Previous correlation analysis using SwA/SwC
(Lühr et al., 2015) addressed the stationarity and the planar geometry assumption
and found small- and large-scale FACs stationary on 10 and 60 s,
respectively. Comparison of Swarm FAC density with ground data was done by
Juusola et al. (2016). Statistical analysis of the magnetic field perturbation
(Δ*B*) measured by the ST-5 spacecraft (Gjerloev et al., 2011) showed Δ*B* dependence on time and scale as well as on the geomagnetic conditions and
local time. For small and mesoscale structures the statistical lifetime of
the structures varies linearly with the structure scale. The same is true for
large scales; however, in this case the lifetime increases faster with the
structure scale. The ST-5 data constrained the analysis of
Gjerloev et al. (2011) to scale sizes above 20 km, which is situated in the
mesoscale range (Knudsen et al., 2001).

Due to the known statistical alignment of the large-scale and mesoscale FACs
with MLT, single-spacecraft methods typically do not consider the orientation
of the FACs in the plane perpendicular to ** B**. The assumption of
east–west alignment was verified by Gillies et al. (2014) in a statistical study
of optical observations based on the THEMIS ASI array. The Gillies et al. (2014)
survey addressed the stable presubstorm auroral arcs to infer their
multiplicity and orientation with respect to the magnetic east–west
direction. Their results show the prevalence of multiple arc systems with
respect to single arcs. Essentially, the quiet arcs show east–west alignment
around 23:00 MLT and inclination within a few degrees toward north and south at later
and earlier times, respectively. The dependence of the tilt angle on MLT is
linear, with a variation of about 1

With a few exceptions, most of the FAC studies based on Swarm use mainly the low-resolution (1 s) data, associated with a mapped scale of ∼7.6 km, whereas the full-resolution measurements (0.02 s) correspond to ∼150 m. Small-scale FACs play an important role in different stages of the aurora, and a proper multiscale analysis of the FAC density is important. High-resolution Swarm data conjugate with THEMIS ASI measurements were used by Gillies et al. (2015) for the study of small-scale pulsating aurora patches. While their findings are related to pulsating aurora, e.g., strong downward currents at the edges of the pulsating form and typically weaker upward currents inside the patches, Gillies et al. (2015) pointed out that the single-spacecraft FAC density provides better identification of the boundaries of the auroral patches, compared to the dual-spacecraft estimate. The small tilt assumption, underlying the single-spacecraft FAC density estimate, is questionable in this case, and likewise for small-scale structures, as proved by, e.g., Miles et al. (2018).

To study the multiscale nature of auroral FACs in sufficient rigor and detail, the arsenal of space physics analysis tools ought to be amended with proper multiscale versions of classical methods. The multiscale FAC analyzer (Bunescu et al., 2015), denoted MSMVA, extends minimum variance analysis (MVA) (Sonnerup and Cahill, 1967; Sonnerup and Scheible, 1998) by providing continuous and multiscale information on the planarity and orientation of the FACs. MSMVA allows us to identify the location and characteristic scale of the planar FACs. MSMVA was used (Bunescu et al., 2017) to correlate conjugate observations of FACs by FAST and Cluster spacecraft.

This paper extends the MSMVA framework (Bunescu et al., 2015) with the addition of a FAC density scalogram, i.e., a multiscale representation of the FAC density that takes into account the orientation derived from MSMVA. The extended MSMVA framework provides a consistent visualization tool, useful for the analysis of complex FAC systems in terms of their scales. Two different scale sampling schemes are considered and tested using synthetic data and Swarm measurements. The local FAC density around the characteristic scale of the FACs, as identified by MSMVA, is compared with single-spacecraft and dual-spacecraft FAC density estimates (Ritter et al., 2013; Ritter and Lühr, 2006).

The article is organized as follows. Section 2 reviews the MSMVA and describes the multiscale current density. In Sect. 3 the method is applied to the magnetic signatures of synthetic currents showing both large- and superposed smaller-scale structures. Section 4 shows applications to Swarm events with both quiet and more dynamic, smaller-scale FAC features. A discussion is presented in Sect. 5 and the paper is concluded in Sect. 6.

2 MSMVA estimation of the FAC density

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Statistical studies of FACs are typically carried out in global geocentric
coordinate systems such as GEO. Individual crossings are often studied in
mean-field aligned (MFA) systems which are local, centered at the spacecraft,
and with the third (*z*) axis pointing along the background magnetic field
** B**. Then the

In this paper we distinguish between general MFA frames (coordinates
$x,y,z$) and reference systems of FAC sheets with coordinates
$\mathit{\xi},\mathit{\eta},\mathit{\zeta}$. Here *ξ* is along the sheet normal, *η* is
tangential to the sheet, and *ζ* points along the ambient magnetic
field. The magnetic field perturbation Δ** B** (oriented
along

FAC density estimators can be based on single-spacecraft or multi-spacecraft data (Ritter et al., 2013; Vogt et al., 2013). Here we adopt the single-spacecraft approach to construct a FAC density scalogram, i.e., a multiscale representation of FAC density. Single-spacecraft FAC estimators are based on Ampére's law, $\mathit{j}={\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}\times \mathit{B}$, with the field-aligned component given by

$$\begin{array}{}\text{(1)}& {j}_{\parallel}={j}_{z}={\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{1}}\left({\partial}_{x}{B}_{y}-{\partial}_{y}{B}_{x}\right).\end{array}$$

For a sufficiently elongated FAC sheet, in the sheet reference system, Eq. (1) reduces to

$$\begin{array}{}\text{(2)}& {j}_{\parallel}\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}{j}_{\mathit{\zeta}}\phantom{\rule{0.125em}{0ex}}\simeq \phantom{\rule{0.125em}{0ex}}{\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{1}}{\partial}_{\mathit{\xi}}{B}_{\mathit{\eta}}.\end{array}$$

The typical method used to describe the orientation of the FACs is the MVA
(Sonnerup and Scheible, 1998) applied to the magnetic field measurements. MVA is based on
the assumption of planarity and stationarity. MVA analysis for FACs can be
performed on all components of ** B** (3-D MVA) or, in a simplified case,
on the perpendicular perturbation,

The analysis performed in this paper is done in the MFA coordinates and takes
into account only the variance in *B*_{⟂}. By using this simplified
approach we get a lower variance in the data (not including
*B*_{∥}) and thus expect better results with respect to the 3-D
case. The 2-D approach is particularly useful for the case of small-scale
FACs in order to avoid ambiguous cases where *e*_{min} is
associated with a perpendicular direction rather than with the
*B*_{∥} direction. Moreover, we note that at the low-altitude
Swarm orbit *B*_{z} (or *B*_{ζ}) can be affected by large-scale remote
current systems in the ionosphere, e.g., the electrojet current. A
statistical study emphasizing the global characteristics of the Hall current
derived from Swarm observations was performed by Huang et al. (2017).

In the idealized case of an infinite planar current sheet oriented along the
east–west direction (east–west aligned auroral arcs), the FAC density is
approximated by discretizing Eq. (1) and by using the spacecraft
velocity, *v*^{sc}, to compute the spatial gradient along the
normal to the FAC structure:

$$\begin{array}{}\text{(3)}& {j}_{\parallel}=({\mathit{\mu}}_{\mathrm{0}}{v}_{\u27c2}^{\mathrm{sc}}{)}^{-\mathrm{1}}\mathrm{\Delta}{B}_{y}/\mathrm{\Delta}t.\end{array}$$

For the quasi-static FAC approximation and in the case of spacecraft crossing along the normal to the arc, Eq. (3) gives correct results. In reality, due to the orbital configuration and FAC dynamics, the crossings are not normal to the arc and the FACs show deviations from the quasi-static approximation. Equation (3) was used to obtain estimates of the FAC density for many single-spacecraft missions like Freja (e.g., Luhr et al., 1996) and FAST (Elphic et al., 1998), or more recently for single-spacecraft FAC estimates from Swarm (Ritter and Lühr, 2006).

For an east–west aligned FAC sheet, the observed sign of the slope in the
*B*_{y} time series (with the *y* axis pointing towards east) depends not only
on the FAC direction, but also on the direction of the spacecraft velocity
** V** and on the hemisphere. The sign of

$$\begin{array}{}\text{(4)}& {\mathit{\mu}}_{\mathrm{0}}\mathit{j}\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{\widehat{\mathit{n}}\times \dot{\mathit{B}}}{{V}_{n}}}\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{\widehat{\mathit{n}}\times \dot{\mathit{B}}}{\widehat{\mathit{n}}\cdot \mathit{V}}}.\phantom{\rule{0.25em}{0ex}}\end{array}$$

For an ideal (infinitely extended) sheet of FACs, we obtain

$$\begin{array}{}\text{(5)}& {\mathit{\mu}}_{\mathrm{0}}{j}_{\Vert}\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}{\mathit{\mu}}_{\mathrm{0}}\mathit{j}\cdot {\widehat{\mathit{B}}}_{\mathrm{0}}\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{|\dot{\mathit{B}}\times {\widehat{\mathit{B}}}_{\mathrm{0}}{|}^{\mathrm{2}}}{(\dot{\mathit{B}}\times {\widehat{\mathit{B}}}_{\mathrm{0}})\cdot \mathit{V}}},\phantom{\rule{0.25em}{0ex}}\end{array}$$

since $\dot{\mathit{B}}\times {\widehat{\mathit{B}}}_{\mathrm{0}}$ is aligned with $\widehat{\mathit{n}}$. Hence the FAC is positive/negative if the two vectors
$\dot{\mathit{B}}\times {\widehat{\mathit{B}}}_{\mathrm{0}}$ and ** V** form an angle
smaller/larger than 180

When multi-spacecraft information is available, one can relax part of the assumptions involved in the single-spacecraft methods to compute the FAC density. For the case of the Swarm mission, the multi-point configuration is constructed by using the low orbit SwA and SwC spacecraft. By shifting the along-track positions one can build virtual quads which make an appropriate configuration for the computation of the FAC density. Based on their computation principle, we distinguish two classes of dual-spacecraft methods. Finite differencing (FD) methods (Ritter et al., 2013; Ritter and Lühr, 2006) evaluate a discrete version of the boundary integral ${j}_{\parallel}=({\mathit{\mu}}_{\mathrm{0}}A{)}^{-\mathrm{1}}\oint \mathit{B}\cdot \mathrm{d}\mathit{s}$. Linear least squares (LS) estimators (Vogt et al., 2009, 2013) are constructed by projecting the dual-satellite measurements onto a local linear magnetic field model.

While both FD and LS methods have obvious advantages over the
single-satellite methods, they are limited with respect to the scale
resolution. The along-track separation can be varied in order to obtain
squared quads configurations, whereas the cross-track is limited by the orbit
separation. Thus, the cross-track separation defines the lower limit of the
FAC scales in the cross-track direction, whereas the limit in the along-track
direction is determined by the along-track separation, provided that the FAC
structure is quasi-stationary. The typical cross-track separation between SwA
and SwC above the auroral oval is decreasing towards poles from ∼80 to
∼50 km around latitudes of ∼60 to ∼70^{∘}, respectively.
The along-track separation of about 10 s corresponds to some 70 km.

In order to characterize the small-scale FACs, one has to rely on
single-spacecraft methods. Bunescu et al. (2015) introduced the multiscale FAC
analyzer (MSMVA) to study the FAC signatures. The MSMVA technique extends the
MVA analysis by providing continuous and multiscale information on the
planarity and orientation of the observed FACs. The continuous character over
the time domain is achieved by computing the MVA parameters (eigenvalues and
eigenvectors) over a sliding window (width *w*). The multiscale character
is achieved by repeating the procedure for an array of window widths, *w*_{k},
within a given range (resolution d*w*). The eigenvalues
(*λ*_{η}, *λ*_{ξ}), eigenvectors (*e*_{η},
*e*_{ξ}), eigenvalues ratio, $R={\mathit{\lambda}}_{\mathit{\eta}}/{\mathit{\lambda}}_{\mathit{\xi}}$, and the
orientation,
*θ*≡ $({\mathit{e}}_{\mathit{\xi}},\widehat{\mathit{x}})$, are thus 2-D quantities dependent on time and scale. Bunescu et al. (2015)
showed that the derivative of *λ*_{η} with respect to the length of
the analysis window, ∂_{w}*λ*_{η}, provides the location
(center) and scale (thickness) of the planar FAC structures. We note that the
amplitude of ∂_{w}*λ*_{η} depends on the scanning parameter *w*
which represents the along track scale. In order to obtain the amplitude
corrected derivative we use the orientation information, ${\partial}_{\mathit{\xi}}{\mathit{\lambda}}_{\mathit{\eta}}={\partial}_{w}{\mathit{\lambda}}_{\mathit{\eta}}/\mathrm{cos}\left(\mathit{\theta}\right)$. Here after in this work
we only use the amplitude corrected derivative ∂_{ξ}*λ*_{η}.

The method was checked on simple synthetic FACs (infinite and finite structures) of both uniform and nonuniform FAC density and showed good performance in identifying FAC scales. The method was applied to Cluster data showing both large-scale quiet arcs and locally planar and dynamic FAC structures (Bunescu et al., 2015), as well as for the analysis of conjugate Cluster/FAST observations (Bunescu et al., 2017).

The multiscale information provided by MSMVA can be used to compute other quantities, like the FAC density. MSMVA provides the scale-dependent orientation that can be used to compute the FAC density in the FAC's own reference system. Combined with the MSMVA results this provides a consistent tool to analyze the FAC signatures. One can compute the FAC density at each scale by discretizing Eq. (2):

$$\begin{array}{}\text{(6)}& {j}_{\parallel}={\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\Delta}{B}_{\mathit{\eta}}/\mathrm{\Delta}\mathit{\xi},\end{array}$$

where *B*_{η} is computed as the projection of ** B** along the
tangential direction, ${B}_{\mathit{\eta}}=\mathit{B}\cdot {\mathit{e}}_{\mathit{\eta}}$, whereas Δ

The amplitude of *B*_{η} at each scale *w*=Δ*t* is estimated
by fitting *B*_{η} using a simple linear regression analysis. Thus,
$\mathrm{\Delta}{B}_{\mathit{\eta}}={B}_{\mathit{\eta}}\left({t}_{b}\right)-{B}_{\mathit{\eta}}\left({t}_{a}\right)$, where *t*_{a} and *t*_{b} are
the limits of the analyzing window, *w*, at the respective position
(center *t*_{cen} of [*t*_{a}, *t*_{b}] interval). When the
analyzing scale is centered on a certain FAC structure and has the
width equal to the FAC thickness, Δ*B*_{η} approximates well
the entire perturbation across the structure. When the analysis window
is centered between two balanced FACs of similar amplitude, *j*_{0}, and
thickness, *w*_{0}, the two FACs cancel each other and provide no
contribution to the current at that position and scale, Δ*B*_{η}=0. In the case of unbalanced FAC structures, the FAC density
depends on their respective amplitudes and thicknesses.

The ensemble of the resulting estimates ${j}_{\parallel}={j}_{\parallel}({t}_{\mathrm{cen}},w)={j}_{\parallel}({t}_{\mathrm{cen}},\mathrm{\Delta}t)$ yields a
multiscale representation of FAC density in (*t*_{cen},*w*) space.
We refer to this graphical representation as the FAC density
scalogram, in analogy to the terminology used for wavelet transforms
(Torrence and Compo, 1998).

The multiscale information can be separated into invariant information, which
depends only on quantities in the local (*ξ*, *η*) frame, and
non-invariant information, which depends also on variables in the (*x*, *y*)
frame. All multiscale information depends on *w*, which is the scale length
along the spacecraft track ((*x*, *y*) frame) and thus a non-invariant
variable. In order to obtain the dependencies on the perpendicular scale (FAC
thickness), one has to correct the scale array *w*_{k} by projection
along the $\widehat{\mathit{\xi}}$ direction, *w*_{k}cos (*θ*_{k}).
Regarding the amplitude of the MSMVA quantities, we notice that invariant
information is given by *R*_{λ} and *θ*, but non-invariant
information by uncorrected *j*_{∥} and ∂_{w}*λ*_{η}.
Corrections to the scale are applied for the individual profiles (dependence
at a certain time or position; see Sects. 3 and
4) and not to the scalograms of MSMVA quantities. As long as
both synthetic and observed FACs are essentially east–west aligned
(Bunescu et al., 2015), the method cannot be properly tested and validated for
inclined structures. In Sect. 3 we perform tests on inclined
synthetic FACs, whereas in Sect. 4 we also apply the method to
inclined FAC observations by Swarm.

We use two different FAC scanning procedures (scaling schemes) for the
discretization of the FAC scale domain. The scheme implemented by
Bunescu et al. (2015) implies a linear sampling of both scale and time domain,
i.e., linearly varying width for scale space and sliding for the time space.
At a given time the discretization of the scale domain is similar to the
nested MVA analysis (Sonnerup and Scheible, 1998) used to study the stationarity of the
MVA parameters. The minimum scale, *w*_{min}, is given by three points
(one point on each side of the central point). Iteratively, the scale
increases by adding an equal number of points (depending on d*w* resolution)
to the sides of the previous scale, yielding thus an array of odd numbers
*w*_{k}=3, 5, 7, … for the highest-resolution scanning. For the Swarm
high-resolution magnetic field data (Sect. 4) we look in the
range between *w*_{min}=0.1 s and *w*_{max}=5 min, which for
an ionospheric mapping factor of 1.1 corresponds roughly to an ionospheric
scale of about ∼760 m and ∼2000 km, respectively. This scheme has
the advantage that one can scan all the FAC scales present in the data and
provide the high resolution needed in the FAC scale/position identification
(Bunescu et al., 2015). As discussed in Sects. 3 and 4,
this high-resolution linear scanning introduces a large degree of correlation
in the results. Indeed, for an infinite planar sheet of width *w*_{0}, this is
sampled many times for all scales *w*_{k}≤*w*_{0}. When searching for FAC
scale/location this proved to be fine, since ∂_{w}*λ*_{η}
maximized at *w*_{0} for essentially east–west aligned FACs.

The second FAC scanning scheme uses successive intervals that do not overlap
at a certain scale; the length of the intervals is varied logarithmically to
provide information at different scales. This scheme is similar to the one
used in Haar wavelet decomposition. All scales (interval widths) spanning
*w*_{k}=2^{k} data points, where *k*=2, *N* (*N* the highest power of 2 that fits
into the data interval) are considered. When dealing with large scales one
can use zero padding of the data interval. Practically, in an ideal auroral
oval configuration with balanced R1/R2 FACs, the largest scale samples the entire
oval, and in the second-largest scale the interval is split into two and
addresses separately the R1/R2 regions. The segmentation of the data interval
repeats down to the smallest-scale *w*_{min}. For the case of Swarm
events (Sect. 4) we take *w*_{min}=0.04 s (two points)
and *w*_{max}=21.8 min corresponding to a total number of 16
decomposition levels. One sensitive point of this scheme is the centering of
the data interval because in reality we do not have an ideal oval; e.g., one
can have a tangential crossing through the oval. One can manually center the
analysis interval on the border between the R1 and R2 regions. The main
advantage of this logarithmic scheme is that it is much faster than the
linear scheme and provides a more intuitive understanding of the multiscale
FAC density. In each computation cell of width Δ*ξ*^{(k)} we have the
current density ${j}_{\parallel}^{\left(k\right)}={\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\Delta}{B}_{\mathit{\eta}}^{\left(k\right)}/\mathrm{\Delta}{\mathit{\xi}}^{\left(k\right)}$ and the integrated current
${J}_{\parallel}^{\left(k\right)}={\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\Delta}{B}_{\mathit{\eta}}^{\left(k\right)}$. The FAC density
${j}_{\parallel}^{\left(k\right)}$ reflects the slope of *B*_{η}, whereas
${J}_{\parallel}^{\left(k\right)}$ reflects the jump of *B*_{η} over the respective
scale, *w*_{k}. Both ${j}_{\parallel}^{\left(k\right)}$ and ${J}_{\parallel}^{\left(k\right)}$ offer
complementary useful information. In the following we concentrate on
${j}_{\parallel}^{\left(k\right)}$, similar to the linear sampling scheme.

As it is constructed, the multiscale FAC density provides estimates of the
average FAC across scales, as well as an indication of the dominant scales,
given by peaks in ∂_{ξ}*λ*_{η}. Both scale sampling schemes
rely on non-orthogonal basis functions because the aim is to precisely infer
the scale and location of the FAC as well as the respective current density.
As a consequence, one cannot simply integrate over scales to obtain a global
FAC density estimate that can be compared with the single- and
dual-spacecraft FAC estimates – which provide convoluted information about
the FAC scales larger than the discretization interval (single-spacecraft) or
the virtual quad scale (dual-spacecraft). As compared to the orthogonal
decompositions, e.g., orthogonal wavelet decomposition, where the signal is
recovered easily by integration over scales, in our case such an integration
would require a proper weighting scheme of the multiscale information. This
development is considered for a future study.

3 Synthetic FAC structures

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In this section we apply the multiscale FAC density technique to synthetic
structures consisting of superposed FAC activity. We define complex FAC
structures by superposing FACs of different scales (thickness), amplitudes
(FAC intensity), and directions of the current flow (upward and downward).
Additionally, we consider the orientation of the FAC structures in the plane
perpendicular to ** B**. The total FAC density in the (

$$\begin{array}{}\text{(7)}& {\displaystyle}{j}_{\parallel}\left(\mathit{\xi}\right)=\sum _{k}{s}^{\left(k\right)}{j}_{\parallel}^{\left(k\right)}\left(\mathit{\xi},{\mathit{\sigma}}_{\u27c2}^{\left(k\right)}\right)\end{array}$$

where ${j}_{\parallel}^{\left(k\right)}$ denotes the elementary current associated with a
single FAC element; *s*^{(k)} is the sign of the FAC element, ± for the
upward/downward FACs. For the case of uniform FAC density structures
${j}_{\parallel}^{\left(k\right)}=$ const; ${j}_{\parallel}^{\left(k\right)}$ is parameterized below
by thickness, position, intensity, and orientation.

In the following, we define ${j}_{\parallel}^{\left(k\right)}$ elements according to a
nonuniform FAC density depending on *ξ* by a Gaussian function in the
(*ξ*,*η*) frame.

$$\begin{array}{}\text{(8)}& {\displaystyle}{j}_{\parallel}^{\left(k\right)}(\mathit{\xi},{J}_{\mathrm{0}}^{\left(k\right)},{\mathit{\sigma}}_{\u27c2}^{\left(k\right)})={\displaystyle \frac{{J}_{\mathrm{0}}^{\left(k\right)}}{{\mathit{\sigma}}_{\u27c2}^{\left(k\right)}\sqrt{\mathrm{2}\mathit{\pi}}}}{e}^{-{\left(\mathit{\xi}\right)}^{\mathrm{2}}/\left(\mathrm{2}{\left({\mathit{\sigma}}_{\u27c2}^{\left(k\right)}\right)}^{\mathrm{2}}\right)}\end{array}$$

The parameter *J*_{0} indicates the integrated sheet current (integral across
the arc per unit of east–west length) of a FAC element; *σ*_{⟂} is
the standard deviation and controls the perpendicular scale of the FAC
element. The Gaussian profile is consistent with the FAC structures observed
in the auroral region. Studies on the FAC scales (Johansson et al., 2007; Karlsson and Marklund, 1996)
estimated the FAC density profile by a Gaussian function, and the scale is
approximated by the full-width-at-half-maximum (fwhm) estimate,
fwhm $=\mathrm{2}\sqrt{\mathrm{2}\mathrm{ln}\left(\mathrm{2}\right)}{\mathit{\sigma}}_{\u27c2}\approx \mathrm{2.35}{\mathit{\sigma}}_{\u27c2}$. The fwhm
estimate is typically used also when estimating the auroral thickness from
optical emissions intensity (Partamies et al., 2010). In Sects. 3.1
and 3.2 we also compare fwhm FAC thickness with
${w}_{\mathrm{1}\mathit{\sigma}}=\mathrm{2}{\mathit{\sigma}}_{\u27c2}$.

Equations (7) and (8) do not include the
orientation since the FACs are defined in the (*ξ*, *η*) frame. By using
the coordinate transformation (rotation and translation) to (*x*, *y*)
defined as, $\mathit{\xi}=(x-{x}_{\mathrm{0}})\mathrm{cos}\left(\mathit{\theta}\right)$, we introduce ${x}_{\mathrm{0}}^{\left(k\right)}$ and
*θ*^{(k)} parameters which control the location and orientation of the
FAC elements. Note that the relevant angle *θ*^{(k)} is made by the
satellite trajectory with the direction normal to the current sheet. For
simplicity, we consider here that the satellite trajectory coincides with the
*x* axis (pointing north), therefore the angle *θ*^{(k)} is provided
directly by MVA (otherwise, one should subtract the angle made by the
satellite trajectory with the *x* axis).

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{j}_{\parallel}^{\left(k\right)}(x,{J}_{\mathrm{0}}^{\left(k\right)},{\mathit{\sigma}}_{\u27c2}^{\left(k\right)},{x}_{\mathrm{0}}^{\left(k\right)},{\mathit{\theta}}^{\left(k\right)})\\ \text{(9)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}={\displaystyle \frac{{J}_{\mathrm{0}}^{\left(k\right)}}{{\mathit{\sigma}}_{\u27c2}^{\left(k\right)}\sqrt{\mathrm{2}\mathit{\pi}}}}{e}^{-{\left((x-{x}_{\mathrm{0}}^{\left(k\right)})\mathrm{cos}\left({\mathit{\theta}}^{\left(k\right)}\right)\right)}^{\mathrm{2}}/\left(\mathrm{2}({\mathit{\sigma}}_{\u27c2}^{\left(k\right)}{)}^{\mathrm{2}}\right)}\end{array}$$

where the FAC density of each FAC element depends on a set of four parameters
(*x*_{0}, *J*_{0}, *σ*_{⟂}, *θ*).

The integration of the Ampere law (Eq. 2) yields the magnetic
field associated with the FAC density (Eq. 7) given by
${B}_{\mathit{\eta}}={\mathit{\mu}}_{\mathrm{0}}\int {j}_{\parallel}d\mathit{\xi}$. Considering the superposition of FACs
(Eq. 7), this yields ${B}_{\mathit{\eta}}={\sum}_{k}{B}_{\mathit{\eta}}^{\left(k\right)}$, where
${B}_{\mathit{\eta}}^{\left(k\right)}$ is the magnetic field of the *k* FAC element derived as the
integral of the Gaussian function and expressed in terms of error function:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{B}_{\mathit{\eta}}^{\left(k\right)}={\displaystyle \frac{{\mathit{\mu}}_{\mathrm{0}}{J}_{\mathrm{0}}^{\left(k\right)}}{\mathrm{2}}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{erf}\left({\displaystyle \frac{\mathit{\xi}}{{\mathit{\sigma}}_{\u27c2}^{\left(k\right)}\sqrt{\mathrm{2}}}}\right)\\ \text{(10)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\equiv {\displaystyle \frac{{\mathit{\mu}}_{\mathrm{0}}{J}_{\mathrm{0}}^{\left(k\right)}}{\mathrm{2}}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{erf}\left({\displaystyle \frac{(x-{x}_{\mathrm{0}}^{\left(k\right)})\mathrm{cos}\left({\mathit{\theta}}^{\left(k\right)}\right)}{{\mathit{\sigma}}_{\u27c2}^{\left(k\right)}\sqrt{\mathrm{2}}}}\right),\end{array}$$

where the second and third term show the dependence in the (*ξ*, *η*)
and (*x*,*y*) frame, respectively. In order to obtain the *B*_{x} and *B*_{y}
components we rotate ${B}_{\mathit{\eta}}^{\left(k\right)}$ for each FAC element with the
*θ*^{(k)} angle
($({\mathit{\lambda}}_{\mathit{\xi}},\widehat{\mathit{x}})\equiv $ $({\mathit{\lambda}}_{\mathit{\eta}},\widehat{\mathit{y}})$). A positive/negative angle indicates a tilt toward
south/north. The MSMVA analysis is thus applied to the following components
of ** B**:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{B}_{x}=-\sum _{k}{B}_{\mathit{\eta}}^{\left(k\right)}\mathrm{sin}\left({\mathit{\theta}}^{\left(k\right)}\right),\\ \text{(11)}& {\displaystyle}& {\displaystyle}{B}_{y}=\sum _{k}{B}_{\mathit{\eta}}^{\left(k\right)}\mathrm{cos}\left({\mathit{\theta}}^{\left(k\right)}\right).\end{array}$$

We note that for synthetic data the magnetic field perturbation is defined as
a function of the spatial coordinate, *x*, but for the Swarm data
(Sect. 4) as a time series. The computation of *j*_{∥}
for Swarm is done using Eq. (6), which includes the amplitude
correction due to the orientation. In the case of synthetic data the
amplitude is also corrected, ${j}_{\parallel}={\partial}_{x}{B}_{\mathit{\eta}}/\mathrm{cos}\left(\mathit{\theta}\right)$.

By using the above equations we construct two particular cases of synthetic structures. In the first case we consider a simple balanced FAC structure, consisting of upward and downward FAC elements of the same thickness and amplitude, but of different orientation. The second case consists of superposed FACs; smaller-scale FACs of different orientations are embedded in larger FACs. We show how the multiscale FAC estimate can be used to visualize the FACs. The simple case of a pair of FACs resembles the large-scale R1/R2 system as well as the basic cell of a multiple arc system (Gillies et al., 2014; Wu et al., 2017). In the second case, the embedded smaller-scale superposition can be associated with the analysis of the auroral oval with embedded smaller-scale FACs, e.g., multiple arc systems, or pulsating auroras.

In the following we consider the current system consisting of the
downward/upward (labeled FD/FU) current regions. The value of the thickness
parameter, *σ*_{⟂}, for both FAC structures is 50 km. Typical
values of Δ*B* for the auroral region are in the range of a few
100 nT. Each 100 nT in the measured Δ*B* corresponds to an integrated
sheet current *J*_{0}∼0.1 A m^{−1}. For this synthetic case we consider
${J}_{\mathrm{0}}=\pm \mathrm{0.63}$ A m^{−1} for the downward/upward current. The current
elements are located at ${x}_{\mathrm{0}}^{\left(\mathrm{1}\right)}=\mathrm{600}$ and ${x}_{\mathrm{0}}^{\left(\mathrm{2}\right)}=\mathrm{800}$ km. We
introduce a variation of the orientation from
${\mathit{\theta}}^{\left(\mathrm{1}\right)}=\mathrm{0}{}^{\circ}$ at FD to ${\mathit{\theta}}^{\left(\mathrm{2}\right)}=\mathrm{40}{}^{\circ}$ at FU.
According to observations (Gillies et al., 2014) the value of
${\mathit{\theta}}^{\left(\mathrm{2}\right)}=\mathrm{40}{}^{\circ}$ is a rather extreme case for a stable auroral arc.

Figure 1 shows the results of both linear- and
logarithmic-scale sampling for this simple FAC structure. Panel (a) shows the
input current density, *j*_{∥}, of FD (magenta), FU (blue), and the
total current (black). Panel (b) shows the *B*_{x} (blue) and *B*_{y} (green)
components of the obtained magnetic field (Eq. 11). This
magnetic field contains a superposed normal distributed noise signal with
zero mean and sigma of 3 nT. The maximum FAC density at the center of the
two structures is ∼5 µA m^{−2}. The results of linear MSMVA
scanning of the FAC system are shown in panels (c), (d), (e), and (f) by the
planarity *R*_{λ}, the derivative ∂_{ξ}*λ*_{η}, the
orientation *θ*, and the linear multiscale FAC density, respectively.
The width array used in the linear MSMVA is between 1 and ∼400 km with
a step of ∼0.6 km. We note the smooth variation of all quantities
specific to this sampling scheme. On each spectrum we indicate the position
and scale or the input FACs by the black circles (diameter equal to
*σ*_{⟂}). ∂_{ξ}*λ*_{η} correctly identifies the scale
of FD around fwhm =117 km. For FU we get a larger estimate because of the
dependence of ∂_{ξ}*λ*_{η} on the non-invariant *w* variable
(length along the track). The sections at the FAC centers shown below are
represented as a function of the corrected scale, obtained by projection of
the scale array on $\widehat{\mathit{\xi}}$ using the orientation
(Sect. 2.3). *θ* scalogram (panel e) correctly
identifies the orientation, ${\mathit{\theta}}^{\left(\mathrm{1}\right)}=\mathrm{0}{}^{\circ}$ and
${\mathit{\theta}}^{\left(\mathrm{2}\right)}=\mathrm{40}{}^{\circ}$. We note that *R*_{λ} shows a signature with a
rather flat maximum extending to large scales, with the local maxima for
FD/FU regions not coincident with ∂_{ξ}*λ*_{η} maxima. This
behavior is influenced by the smoothness of Δ** B** for each FAC and
by the constant Δ

Panels (g)–(l) show the results of logarithmic FAC scale scanning. For this
case the analysis is centered in the middle of the FAC structure, indicated
by the vertical black line. The sampled scale array covers 13 logarithmic
levels from *w*_{min}=0.2 to *w*_{max}=820 km. The logarithmic
scheme shows a more discrete character due to the non-overlapping sampling
intervals at each scale. Qualitatively, we observe a good agreement with the
linear scheme for the orientation (panel k) and FAC density (panel l). The
multiscale FAC density (panel l) shows at the largest scale a close to zero
current because the two structures have similar amplitudes and compensate
each other. At around 100 km we observe the separation of the two branches
of the current centered at 600 and 800 km. The distinction between the two
regions is very clear down to smaller scales of a few kilometers. Higher FAC
intensity is observed around the centers of the FACs for scales smaller than
about ∼50 km.

Quantitative estimates are obtained trough vertical cuts into the MSMVA
scalograms shown in panels (m)–(p) of Fig. 1. The black/red
line shows the profiles in the center of FD/FU structures, whereas the
solid/dashed lines indicate the results for the linear/logarithmic sampling
scheme. The vertical dashed lines show the scales *w*_{1σ}=100 km and
fwhm =117 km. As discussed in Sect. (2.3), for all
multiscale parameters we correct the scale variable (multiplication of the
scale array by *c**o**s*(*θ*)) to get the dependence on the perpendicular
scale. For both FACs ∂_{ξ}*λ*_{η} (n) shows that the scale is
more consistent with fwhm estimate. We notice that for this simple FAC system
both the linear and logarithmic sampling scheme provide consistent results,
the scale is precisely identified in both cases. The orientation of the two
FACs (o) at fwhm scale is consistent with the input parameters, 0^{∘}
and 40^{∘} for FD and FU, respectively. We note that the scale
corrected ∂_{ξ}*λ*_{η} does not depend on the FAC's
orientation. The similarity of ∂_{ξ}*λ*_{η} amplitudes for FD
and FU indicates a good amplitude correction for FU structure. *R*_{λ}
profile (m) does not have a maximum at the same scale as
∂_{ξ}*λ*_{η}. This shift is dependent on the noise level since
*R*_{λ} contains also dependence on *λ*_{ξ}. The local FAC density
(p) at FD and FU locations provides also quantitative indication about the
FAC scale. Around the FAC scale we observe a slight change of the slope of
*j*_{∥} for the linear scheme and also a decrease for the logarithmic
scheme. At a given FAC center *j*_{∥} shows a rather constant plateau
and starts to decrease when the scanning reaches its characteristic scale.
This behavior is more evident for uniform FAC density structures (see
Sect. 4.2). The FAC density for FD and FU FACs shows values
of about ±4.5 and $\sim \pm \mathrm{3.5}$ µA m^{−2} for the linear and
logarithmic sampling, respectively, i.e., 10 % and 30 % smaller than
the input FAC density (5 µA m^{−2}).

We start again with a large-scale current system similar to the previous
synthetic case. Two FAC elements FD and FU with ${\mathit{\sigma}}_{\u27c2}^{l}=\mathrm{50}$ km and
${J}_{\mathrm{0}}^{l}=\pm \mathrm{0.63}$ A m^{−1} are placed at *x*_{0}=700 and *x*_{0}=900 km. The
orientation of FD and FU structures is ${\mathit{\theta}}_{l}^{\left(\mathrm{1}\right)}=\mathrm{0}$ and
${\mathit{\theta}}_{l}^{\left(\mathrm{2}\right)}=\mathrm{40}{}^{\circ}$. A number of three small-scale FACs are
superposed on each large-scale FAC structure. We consider equal scales of the
embedded FACs given by ${\mathit{\sigma}}_{\u27c2}^{s}=\mathrm{5}$ km. The small-scale FACs
embedded in FD have ${J}_{\mathrm{0}}^{\left(k\right)}$ parameters defined as ${J}_{\mathrm{0}}^{l}/\mathrm{6}$, ${J}_{\mathrm{0}}^{l}/\mathrm{3}$,
and ${J}_{\mathrm{0}}^{l}/\mathrm{6}$, alternatively positive and negative. Similarly the small-scale
FACs superposed onto FU also have a central more intense FAC of amplitude
${J}_{\mathrm{0}}^{l}/\mathrm{4}$ and two side FACs of intensities ${J}_{\mathrm{0}}^{l}/\mathrm{8}$. For simplicity, we
consider all small-scale FACs to have ${\mathit{\theta}}_{s}^{\left(k\right)}=\mathrm{0}$^{∘}. The
small-scale FACs introduce alternatively positive and negative amplitude
changes in the current density of the large-scale FAC system.

Figure 2 shows the overall contribution of the two scales
to a rather complex FAC density profile shown by the black line in panel (a)
and the corresponding magnetic field perturbation in panel (b). The FAC
elements are indicated in panel (a) with blue/magenta for the
positive/negative FAC densities at both scales. The attenuation
(compensation)/intensification (addition) of the local FAC density from the
two FAC systems is reflected in slower/steeper gradients of Δ** B**.
We note that the superposition of scales (Eq. 11) affects
the orientation and the scale information for both large- and small-scale
FACs. Thus, in general we do not expect to find the exact input angles and
scales. The superposed normal distributed noise signal has

Panels (c)–(f) and (i)–(l) show the MSMVA decomposition into the linear and
logarithmic schemes, respectively. We notice the same characteristics of the
two schemes, namely smooth and coarse results in the linear and logarithmic
scannings, respectively. Panel (c) shows a high decrease in the planarity
level for FU as compared with the previous case (Sect. 3.1).
We note regions of high *R*_{λ} at both large- and small-scale FAC
systems. The relative combination of angles and amplitudes of ** B** from
the two scales leads to three signatures of high

The local FAC density scalogram in both scanning schemes (f and l) provides a
consistent view of the input FAC density, with well-delimited FAC elements of
both the large- and small-scale FAC systems. In panels (m)–(p) we show
vertical cuts through the scalograms at the centers of
attenuation/intensification of the FD/FU FAC density by the superposition of
the two scales, indicated by vertical dashed lines in panels (a)–(l). The
profiles show a more complex situation with respect to the previous synthetic
case. The input scales of the two FAC systems are indicated by the vertical
black (large scales) and blue (small scales) lines at *w*_{1σ} and fwhm.
We observe a good correlation of *R*_{λ} and ∂_{ξ}*λ*_{η}
maxima for the small-scale FACs. ∂_{ξ}*λ*_{η} shows
well-defined peaks for the small-scale FACs consistent with the input scales,
but for the large-scale FACs rather broad maxima, also around the expected
scales. The orientations are roughly consistent with the input setup,
${\mathit{\theta}}_{l}^{\left(\mathrm{1}\right)}=\mathrm{0}$^{∘} for FD and
${\mathit{\theta}}_{l}^{\left(\mathrm{2}\right)}=\sim $37–42^{∘} for FU. At small scales we also have
consistency, ${\mathit{\theta}}_{s}^{\left(\mathrm{1}\right)}=\mathrm{0}$ and ${\mathit{\theta}}_{s}^{\left(\mathrm{2}\right)}=\sim \mathrm{5}$^{∘}
for the small-scale FACs inside FD and FU, respectively. The local FAC
density for FD/FU is
4 µA m^{−2} ∕ −4.5 µA m^{−2}, in good
agreement with the input of ±5 µA m^{−2}. For the
small-scale FACs centered on FD/FU we have
−10 µA m^{−2} ∕ −15 µA m^{−2}, which is
roughly consistent with the input FAC density of
$\sim -\mathrm{16}$ µA m^{−2} ∕ −12 µA m^{−2}. We get
higher/lower deviations for the small-scale FACs centered in FD/FU, in
agreement with their weaker/stronger signatures in
∂_{ξ}*λ*_{η}.

In the case of superposed FACs the signatures of both large- and small-scale FACs are qualitatively reflected by the MSMVA information. The results also show some limitations of the method. One cannot expect to find a perfect decomposition of the FAC system, because of (a) the use of piece-wise linear functions of a certain length (scale) with a corresponding FAC density profile given by a step function, which is not fully suitable for the smooth Gaussian functions; and (b) the results are actually dependent on the relative parameters (e.g., intensities, orientations, scales, locations) of the superposed FAC elements.

The combined use of *R*_{λ}, ∂_{ξ}*λ*_{η}, *θ*, and
*j*_{∥} scalograms allows the identification of the geometry, scales,
orientations, and estimates of the local FAC densities present at the
respective scales. The linear approach shows a high precision in the
identification of both FAC scale (d) and local FAC density (f). The
logarithmic scheme lacks resolution in the FAC scale identification and
subsequently gives a poor estimate of the local current. However, this scheme
provides quick results that capture qualitatively similar features. More
advanced data processing can include, e.g., filtering
∂_{ξ}*λ*_{η} by the planarity *R*_{λ}, to remove non-planar
FAC structures, and applying a similar mask to current density.

A more systematic study of superposed FAC sheets is required, e.g., by varying the relative parameters of a FAC system consisting of broad and narrow FAC sheets. In this context, we note that a better approach might be to iteratively identify the FACs based on their intensity and to apply MSMVA to the successive residuals obtained by separating the identified FAC signatures (fitting the data at each iteration by model FAC functions, e.g., planar FACs, as indicated by the MSMVA parameters). However, the problem might not be uniquely determined, and before engaging in such a development, we rather apply the present procedure to several real events, three of which are detailed in the next section.

4 Auroral region crossings by Swarm

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The FAC density scalogram introduced in Sect. 2 and the other components of the multiscale FAC analyzer framework are now applied to three auroral crossings of the Swarm satellites, namely, a stable linear east–west aligned current sheet, an auroral pattern with sharp changes in inclination, and small-scale auroral structures embedded in a large-scale current.

The Swarm mission (Friis-Christensen et al., 2008; Olsen et al., 2013) consists of three spacecraft equipped with identical instruments and placed on polar orbits. The primary objective of the Swarm mission is to study the Earth's magnetic field, e.g., mapping, modeling, or separation of the different sources of the measured field. The satellites are equipped with both a vector field magnetometer (VFM) and an absolute scalar magnetometer (ASM) (Hulot et al., 2015) which provide high-accuracy and high-resolution magnetic field measurements. ASM data are used mainly for the calibration of VFM.

In this work we mainly use the VFM measurements to study the FACs. Because we
address the multiscale aspect of the FAC signatures and in order to have good
statistics also at smaller scales, we use the highest-resolution data
provided by VFM, namely the 50 Hz data (0.02 s sampling). The resolution of
the data is directly related to the scale of the structures that can be
resolved by MSMVA. For a minimum scale of five points in the MSMVA analysis,
we obtain an along-track scale mapped to the ionosphere of about 700 m
(spacecraft velocity of 7.6 km s^{−1} and linear mapping factor of
∼1.09).

One major point of the Swarm constellation is its orbital configuration. Two
spacecraft, SwA and SwC, are flying side by side at 460 km altitude with a
cross-track separation (longitudinal separation) of 1.4^{∘} which
amounts to about 50–80 km above the auroral oval. The measurements provided
by these satellites are combined in the two-satellite methods to estimate the
FAC density (Ritter and Lühr, 2006; Ritter et al., 2013). The other spacecraft, SwB, is flying at
higher altitude and periodically forms a close three-satellite configuration
with the lower pair. When this is the case, it is possible to compute the FAC
density by using also a three-spacecraft method (Vogt et al., 2009). In the
following, for each event we cross-check the local FAC density provided by
MSMVA with the single- and dual-spacecraft estimates.

The single- and dual-spacecraft FAC estimates provided by ESA (part of the Swarm L2 products available at ftp://swarm-diss.eo.esa.int/, last acces: June 2018) are based on the FD approach and available with 1 s resolution. The single-spacecraft FAC density corresponds to a resolution of the mapped ionospheric scale of ∼7 km. The computation of the FD dual-spacecraft FAC estimate is done with a filtered magnetic field perturbation. The filtering is used to remove the FACs with scales smaller than ∼20 s, corresponding to along-track scales smaller than ∼150 km (Lühr et al., 2016). Thus, we expect a good agreement between the single- and dual-spacecraft FAC density estimates for scales larger than 150 km.

The second type of data used in this study is provided by the THEMIS ASI ground network. THEMIS ASI network (Donovan et al., 2006; Mende et al., 2009) was installed to complement spacecraft observations, in particular by the THEMIS mission, related to substorms and, more generally, to auroral phenomena. With a number of 22 stations, the network covers a large region of northern Canada, Alaska, and Greenland. The THEMIS ASI locations were chosen based on an earlier statistical study (Frey et al., 2004) of the auroral substorm onsets inferred from IMAGE spacecraft. Each ASI provides frames of 256×256 pixels at a time resolution of 3 s (exposure time 1 s). All ASI are based on fish-eye lenses that provide wide angle optical observations. Due to the fish-eye lenses the pixels at the center cover a smaller sky surface element as compared to the pixels located towards the edges. Thus, the best resolution is at the center, of about 1 km. The events included in this study make use of optical data from Sanikiluaq (SNKQ), Rankin Inlet (RANK), and Fort Smith (FSMI).

One basic operation is the mapping of the spacecraft orbit into the image plane, done by using the THEMIS TDAS software (http://themis.ssl.berkeley.edu/, last access: June 2018) where the field line tracing is implemented by different versions of the Tsyganenko magnetic field model. In this paper we use the Tsyganenko T04 model (Tsyganenko and Sitnov, 2005) with the solar wind parameters provided by OMNI (http://omniweb.gsfc.nasa.gov, last access: June 2018) and the DST index from WDG at Kyoto (http://wdc.kugi.kyoto-u.ac.jp/, last access: June 2018). The footprints of Swarm are projected onto the optical frames provided by the THEMIS ground stations.

The measured magnetic field is transformed to the MFA reference system. The
magnetic field perturbation, Δ** B**, is obtained by subtracting a
model magnetic field from the measured data. The internal magnetic field
parameterization is taken from CHAOS-6 (Olsen et al., 2014; Finlay et al., 2016), whereas the
lithospheric (e.g., crust and uppermost mantle) and external magnetospheric
(e.g., ring current) contributions are taken from the Pomme 10
(Maus et al., 2006, 2010) model. The results obtained for various events in
different geomagnetic conditions showed good consistency when using this
setup. Ideally, after the subtraction of the magnetic field model we should
remain with the perturbation caused by the large-scale R1/R2 currents, the
embedded mesoscale and small-scale FACs, as well as the influence of the
ionospheric current systems. Another option is to separate the embedded
small-scale FACs from the large-scale FACs (R1/R2) by filtering the data.
Bunescu et al. (2015) computed a model magnetic field proxy from the measured field
using an average over a sliding window (with tapering at the ends). This
procedure excludes roughly the scales larger than a certain percent of the
sliding window width (depending on the tapering extent). The disadvantage of
this approach is that it can introduce additional low-amplitude fluctuations.
Thus, in the following we analyze Δ

On 17 February 2015 the Swarm spacecraft crossed the auroral oval toward north over the FoV of the SNKQ station. The event is observed around 03:25 UT at ∼1 h after an intermediate substorm intensification/onset following ∼6 h of quasi-steady magnetospheric convection. The AE index is ∼200 nT, and DST $\sim -\mathrm{26}$ nT.

Figure 3 shows the ionospheric footprints of the spacecraft (mapped at 110 km altitude) superposed on the SNKQ optical observations. The optical frames are mapped to geographic coordinates and show rather stable and east–west elongated arc structures. We distinguish two large-scale upward FACs located northward and, respectively, southward of the station. Between these two upward FACs we observe a mesoscale upward FAC with an east–west extent covering the westward FoV of SNKQ. Swarm crosses along the westward edge of the ASI over all three visible arcs. While the thick northward and narrow mesoscale structures are highly planar, the thick southward structure looks curled around the spacecraft tracks. Because Swarm crosses near its center, the magnetic field perturbation for this structure looks similar to that of a planar FAC.

Figure 4 shows the SNKQ keogram, Swarm Δ** B**,
FAC density estimates (L2 products), and the hodogram representation of
Δ

Panel (d) shows different FAC density estimates. The green and red line shows
the L2 single-spacecraft FAC density obtained using the unfiltered magnetic
field data from SwA and SwC, respectively. The L2 single-spacecraft FAC
estimate (Ritter et al., 2013) with 1 s resolution (∼7.5 km ionospheric
scale) is computed with the assumption that the main magnetic perturbation is
in the east–west *B*_{y} component. The dual-spacecraft FAC density that
combines the information provided by SwA and SwC using the FD method of
Ritter et al. (2013) is indicated by the black line. This estimate is computed
over the filtered data that remove scales smaller than 150 km. The
two-spacecraft method shows an average of the FAC density over the quad area
and does not capture small-scale FACs. Both single- and dual-spacecraft FAC
estimates are used as a qualitative reference for our multiscale FAC density
technique.

Figure 4f shows the hodogram representation, *B*_{y} as a
function of *B*_{x}, for SwA. The hodogram is represented with the time
interval running from blue to red (rainbow color scale). On this trace we
indicate the FAC segments with the same color used in panels (a)–(e) to mark
the beginning of the respective time interval. We observe different regions
of the hodogram that consist of linear segments which indicate FAC structures
of constant orientation (linear polarization of Δ** B**). The U1,
U2, U3, and U4 FACs are indicated by the black, blue, green, and red lines,
respectively. The MSMVA is used to find and characterize such segments of
linear polarization of Δ

The left part of Fig. 5 shows the results of the linear
MSMVA for SwA. The planarity, shown by the *R*_{λ} scalogram (Fig. 5b),
indicates regions of high planarity for several large-/small-scale FACs,
e.g., U1–3 and D1–2. The scalogram of ∂_{ξ}*λ*_{η} (Fig. 5c)
shows the location and thickness of FAC structures, whereas their orientation
(Fig. 5d) confirms the optically observed alignment of the normal with the
northerly direction, $\mathit{\theta}\approx \mathrm{0}{}^{\circ}$. Some typical threshold
values of *R*_{λ} associated with planar structures are about 10–30
(for 3-D MVA). Because we use the 2-D MVA (*B*_{⟂} perturbation),
*R*_{λ} shows larger values, consistent with a reduced variance. We
note that the investigation of the relationship between the longitudinal
extension of FACs and the *R*_{λ} ratio can actually be done by using
correlation analysis of the two longitudinally separated Swarm spacecraft.
One expects that *R*_{λ} will be able to provide a more quantitative
indication of the FAC east–west length. This topic is considered for a
future study.

Panel (e) shows the newly introduced linear multiscale FAC density (Sect. 2). We can easily see the different regions of upward and downward currents at different scales, e.g., large-scale R1/R2 FACs at scales larger than 100 s, better visible in the logarithmic sampling, and smaller-scale FACs (U1-3, D1-2) at lower values, better visible in the linear sampling. The negative/positive large-scale trend is associated with upward/downward FACs, consistent with the statistical FAC model (Iijima and Potemra, 1976b) around 22:00 MLT. An alternative identification of the large-scale FACs is done by Wu et al. (2017) directly with single-spacecraft FAC density by computing the ratio of the upward and downward currents to the total current.

These representations provide a new visualization of the FAC currents dependent on scale. The linear scanning of FACs uses a large number of scales sampled at high resolution. As already mentioned, one limitation in the integrated FAC estimate for this approach is that it does not rely on an orthogonal basis and thus the integration over scales does not provide a global FAC density similar to the single- and dual-spacecraft FD methods. In order to partially improve the analysis towards an orthogonal basis we computed the same parameters also for the logarithmic scanning procedure (Sect. 2). Panels (f)–(j) show MSMVA quantities for the logarithmic scanning. In this case, the scale range extends to higher values (∼1000 s = 16.6 min) and from about 200 s (1381 km mapped to ionosphere) up one can see a close to zero net current. While the resolution is not suitable to obtain precise information on the scale dependence of these quantities, the results are in good qualitative agreement with the linear scanning.

Figure 6 shows a more quantitative comparison of the MSMVA
quantities, including FAC density given by the two scanning schemes. We show
the scale dependence of *R*_{λ}, ∂_{ξ}*λ*_{η}, *θ*,
and *j*_{∥} at the center of the FACs as identified by
∂_{ξ}*λ*_{η} and indicated by the solid lines in
Fig. 5. The selected times are *t*_{U1}= 03:24:43
and *t*_{D1}= 03:25:00, associated with U1 and D1, respectively. All
quantities are represented as a function of the corrected scale, similarly to
the synthetic data (Sect. 3) and neglecting the small
inclination of the Swarm trajectory with respect to the *x* axis (direction
pointing north). One can see that all quantities have local maxima around the
same scale, indicated by the vertical dashed lines at 22 and 10 s for U1 and
D1, respectively. These scales correspond to about 153 and 70 km in the
ionosphere. *R*_{λ} shows a high planarity at these two scales, with
values larger than 100 (threshold indicated by the horizontal blue line) for
both FACs in the linear sampling. The logarithmic sampling shows smaller
values, with a smoothing of the linear profile and values below the threshold
for D1. For the logarithmic sampling ∂_{ξ}*λ*_{η} shows a
similar scale, 16 s (110 km at ionosphere), for both U1 and D1 FACs. The
orientation is consistent for both linear and logarithmic sampling,
*θ*=10^{∘} for U1 and ∼2^{∘} for D1. In the case of
rather uniform FAC density (U1 and D1) we observe that the maxima of
∂_{ξ}*λ*_{η} are almost aligned with local maxima in
*R*_{λ}, which is consistent with the intuitive expectation that the
planarity of a sheet-like FAC structure maximizes around the scale
(thickness) of the sheet.

The FAC density at U1 and D1 is around −2.7 and
4.5 µA m^{−2}, respectively, for the linear sampling. In the
case of logarithmic sampling, *j*_{∥} (dashed lines in panel d) shows
roughly similar results where the respective scales are properly sampled. We
have agreement for U1 ($\sim -\mathrm{2.5}$ µA m^{−2}) and a close to zero
FAC density for D1. The zero estimate of the current for D1 in the
logarithmic scanning is caused by imperfect centering at that scale with
respect to the linear scanning. Most likely, it is evaluated between U1 and
D1, where we have a compensation of the currents from the two FACs. The
profile of *j*_{∥} for D1 corresponds to the same scale, but it is
evaluated at a different point with respect to *t*_{D1}. For the
logarithmic scheme, precise comparison with the linear scheme can be obtained
at the centers of the sampled intervals (Fig. 5j).

Due to the high planarity and relatively large thickness, U1 and D1
structures satisfy the assumptions of the single- and dual-spacecraft
methods. The FAC density in the single-spacecraft approximation (panel d in
Fig. 4) at *t*_{U1} and *t*_{D1} shows
values of −2.37 and 4.02 µA m^{−2}, respectively, whereas the
dual-spacecraft FAC estimate indicates values of −3.21 and
2.58 µA m^{−2}. These values indicate deviations of the local
FAC density (linear) with respect to single-spacecraft FAC
($\mathrm{100}\cdot ({j}_{\parallel}^{\mathrm{MSMVA}}-{j}_{\parallel}^{\mathrm{sc}})/{j}_{\parallel}^{\mathrm{sc}}$)
of about 14 % and 12 % for U1 and D1, respectively. The same
estimates with respect to the dual-spacecraft FAC density are −15 % and
−74 %. The main characteristics of U1 and D1 FACs, including the
percentage differences between the FAC density estimates (multiscale,
single-, and dual-spacecraft) are summarized in Table 1. The
deviation of the local FAC with respect to the dual-spacecraft FAC density is
consistent with the scale information, low/high deviation for
large/small-scale FACs. While U1 scale (153 km) is close to the resolution
limit (150 km) of the dual-spacecraft method, the scale of D1 is below this
limit. Considering the uncertainties in the scale definition and estimate of
the FAC density, we consider that the differences between the local FAC
density and the single-spacecraft estimate (<15 %) indicate a good
agreement.

Through the continuous and multiscale MSMVA analysis we identify the discrete
FAC elements associated with the measured magnetic field perturbation. The
sections in the MSMVA scalograms quantify how much current one has at the
respective FAC structure. The results show the difficulty of dealing at the
same time with a meaningful local FAC density estimate at a given scale and
the need for orthogonality in the MSMVA basis functions. While FAC density is
correctly inferred locally, one cannot compute a global FAC density estimate
by integration over scales due to the lack of orthogonality of the basis
functions. The sections shown in Fig. 6 were selected around
the local maxima of ∂_{ξ}*λ*_{η}. The sharp maxima of
∂_{ξ}*λ*_{η} for U1 and D1 agree with structures of constant
current densities (Bunescu et al., 2015), also expected from the Δ** B**
profile. The gray shaded area in Fig. 6 shows the range of
scales for which

A comparison of the regular single-satellite FAC density with the
MVA-corrected FAC density product, albeit without scale dependence, is also
included in Gillies et al. (2015) for nine events of pulsating aurora.
Gillies et al. (2015) found consistent results between the two estimates at the
edges of the patches associated with
${R}_{\mathit{\lambda}}={\mathit{\lambda}}_{\mathrm{int}}/{\mathit{\lambda}}_{\mathrm{min}}>\mathrm{10}$ for which the
infinite FAC sheet approximation was considered valid, whereas within the
patch the criterion *R*_{λ}>10 was fulfilled for only five out of
nine events.

The multiscale FAC density benefits from the orientation computed at each scale. For the case of east–west aligned FACs, this may have less influence, even though one cannot exclude the possibility that some FAC elements, in a certain range of scales, are not east–west aligned. The more so, one can expect differences for events of inclined FACs. Typically, the quiet aurora during the growth phase has the normal direction aligned with the northerly direction. By using the multiscale approach one can check whether this is true also for the embedded small-scale FACs. During the onset, expansion, or early recovery phase the aurora is typically dominated by 2-D forms, possibly including locally planar small-scale FACs. By using the multiscale estimates, one can better quantify the FACs with respect to their orientation as a function of scale. This might help to quantify whether the embedded FACs are forced to have the same orientation as the large-scale FACs and, further on, possible relationships between the respective mechanisms. The FAC density scalogram combined with the other information of MSMVA provides a more intuitive and visual representation that can help to search the data for particular information.

This event was observed by Swarm and RANK station of the THEMIS ASI network
on 15 January 2015 around 07:39 UT. The event was observed after a long
quiet period, during the growth phase of a substorm with maximum ∼1 h
later and, possibly, during/after pseudo-breakup activity. The AE index is
∼70 nT and DST between −5 and −8 nT. The optical frames under the
spacecraft track (07:39:27–07:39:54) are shown in Fig. 7.
The optical frames from the southward pass of Swarm over RANK were not
included since the structures are not clearly visible. Δ** B** shown
below indicates locally planar FACs also in this region. Overall, the optical
data show a larger-scale structure inclined with respect to the east–west
direction (the angle between the normal to the FAC and north is about
−20

Figure 8 shows Swarm measurements of Δ** B**,
FAC density estimates, and Δ

Figure 9 shows the results of the multiscale analysis for
SwA. The left/right plots show the comparison of linear/logarithmic scanning
schemes. *R*_{λ} (Fig. 9b) shows high values for some of the mesoscale
FAC structures in the southern part of the RANK location, not well visible
optically. Higher values are also associated with the crossing of the FAC
system in the center of the FoV. By comparing *R*_{λ} values with the
previous event we observe a decrease in planarity level by half, consistent
with the sub-structuring of aurora, finite east–west aligned FACs. We also
observe the alternation of high- and low-planarity regions, well correlated
with regions of upward and downward currents, respectively, in the mesoscale
range. High planarity at small-scale FACs is embedded also in the downward
current regions. The scalogram of ∂_{ξ}*λ*_{η} (Fig. 9c) shows
high intensity for the U4 and D4 regions. The scale associated with the U4
and D4 regions is around 10 s (70 km). The orientation (Fig. 9d and i) at
these scales is $\sim -\mathrm{20}$ and ∼0^{∘}, respectively, qualitatively
consistent with the optical data. The *j*_{∥} scalograms (Fig. 9e and
j) show well the embedded regions of upward and downward directed currents.
One can zoom into this display to get information at smaller scales, e.g.,
the region adjacent to the equatorward part of the track.

Similar to the previous event, in Fig. 10 we show sections
into MSMVA scalograms to infer quantitative estimates of the scales and
current densities for a few selected FAC elements. The times of the sections
are 07:38:41 (blue), 07:39:09 (green), and 07:39:33 (red). These times,
indicated by the solid lines in Fig. 9, are all located
in upward current regions, U2, U3, and U4, intervals. As before, for all
profiles we show the dependence on the corrected scale (taking into account
the inclination). *R*_{λ} shows values larger than 100 for all selected
upward FACs. The maxima of ∂_{ξ}*λ*_{η} at larger scales
correspond to remote FAC elements, e.g., U4 and D4 (see
Fig. 9). We note the slight shift between *R*_{λ}
local maxima and the maxima of ∂_{ξ}*λ*_{η} and
*j*_{∥}. We have good agreement between the linear and logarithmic
sampling for the identification of the scale for U2 and U4, whereas U3 is not
properly sampled by the logarithmic scheme. We note scales of ∼12–14 s
(84–98 km ionospheric scale) for the three FACs. The scale dependence at
these sections shows again clearly that a masking procedure based on
*R*_{λ} would be effective in removing the features associated with
remote FACs crossed earlier or later. The orientation (panel c) shows an
inclination of about −18^{∘} for U2 (blue), 4.5^{∘} for U3
(green) and −25^{∘} for U4 (red), with roughly similar values in the
two sampling schemes and consistent with the optical data.

The values of the FAC density at these FAC segments are about
−0.6 µA m^{−2} for U2, −0.67 µA m^{−2} for U3,
and −1.49 µA m^{−2} for U4 in the linear sampling. Roughly
similar currents are obtained in the logarithmic sampling. The FAC density
given by the single-spacecraft L2 estimate (Fig. 8) for
U2, U3, and U4 is −0.23, −0.58, and −0.66 µA m^{−2},
whereas the dual-spacecraft FAC estimate is −0.35, −0.52, and
−0.87 µA m^{−2}, respectively. For U2, U3, and U4 we have
deviations of the local multiscale FAC density of 161 %, 15 %, and
126 % with respect to the single-spacecraft L2 estimate.

The L2 single-spacecraft FAC product and the MSMVA analysis provide estimates
of the FAC density at different resolutions, are based on a slightly
different computation procedure, and address the scale aspect in a different
way. The L2 single-spacecraft FAC density is provided at 1 s resolution and
thus enables a characterization of the current at scales larger than or equal
to 1 s and typically takes into account only the east–west component of
Δ** B** (

MSMVA FAC estimates, selected based on ∂_{ξ}*λ*_{η}, correspond
to the overall (average) current at the mesoscale U2–U4 FACs. When simply
compared with the instantaneous values of the L2 single-spacecraft FAC
density, the differences are significant (e.g., U2 and U4) due to the
mismatch of the compared scales. In order to properly compare the FAC
estimates, they should correspond to similar scales. Thus, we also computed
an estimate of the current at a scale similar to the U2 and U4 thickness by
simply smoothing the L2 single-spacecraft current using a boxcar running
average of 12 s width. Figure 11 shows Δ** B**
and FAC density estimates from SwA. We notice that most of the mesoscale FACs
(including U2 and U4) have an internal structure. For both the U2 and U4
regions we have embedded perturbations visible through the slope change in
Δ

Comparison of the local multiscale FAC density with the dual-spacecraft estimate gives 71 %, 29 %, and 71 % for the three FACs. Thus, we have again a lower difference for U3 and higher for U2 and U4. The percentages for U2 and U4 are still smaller than when comparing with the single-spacecraft estimates. Part of the differences is probably related to the resolution limit of the dual-spacecraft FAC estimates, larger than our scale of ∼83 km. The comparison between the FAC density estimates is summarized in Table 2.

Both the linear and logarithmic samplings provide consistent information. We have similar results for the orientation and the local FAC density, whereas the scale identification can sometimes be missed in the logarithmic sampling (e.g., U3) due to the limitations of this scanning by non-overlapping intervals. This event indicates that care is needed when designing an automatic procedure for the analysis of FACs on a statistical basis. The two-spacecraft methods can average over different structures; moreover, some assumptions of the methods are possibly not fulfilled.

The relation between multiple arc systems and their FAC signatures was addressed recently by Wu et al. (2017) based on Swarm/THEMIS ASI observations. Wu et al. (2017) selected events with clearly identifiable stable arcs and separated the observations into two categories, unipolar (multiple arcs embedded in a single large upward FAC) and multipolar events (a collection of multiple arcs and related pairs of upward and downward FACs). Arcs associated with multipolar FAC events were found to be broader and more separated than those associated with unipolar FAC events. In this section we perform MSMVA analysis for a unipolar event investigated by Wu et al. (2017).

The event occurred on 27 September 2014 around 06:00 UT, in the evening sector (∼22 MLT) and was observed simultaneously by Swarm and FSMI ASI in Canada. The event was observed during a very active period, with multiple substorms and an average AE of ∼500 nT over the hours around the event. The AE index is ∼550 and DST = −23 nT.

The mapped optical frames and the superposed spacecraft tracks, shown in
Fig. 12, indicate the crossing towards the Equator of a
thick auroral structure (∼ 05:59:48–06:00:00 = 12 s) followed by
some small-scale less intense arcs and an intense structure around
06:00:09–06:00:12. Since the crossing is near the edge of the ASI's FoV, in
the following we do not attempt to make a one-to-one matching between the
optical observations and Δ** B** or FAC signatures.

The MLT location of the event and the optical data indicate the crossing near
the Harang discontinuity region. Following the Swarm track (north to south),
the statistical model of FACs (Iijima and Potemra, 1976b) indicates the crossing of the
large-scale downward, upward, and downward FACs. Figure 13
shows again Δ** B**, FAC density, and the Δ

The hodogram (panel e) shows the typical characteristics observed for the
previous events. The interval color is given by the color of the left
vertical dashed line, which for this case is related to Δ** B** from
SwC. The prevalence of Δ

Figure 14 shows the results of the linear and logarithmic
FAC scanning. The intensity of the scalograms for the linear scheme is also
shown in logarithmic scale to emphasize the small-scale FACs. The highly
planar FACs at small-scales are confirmed by *R*_{λ} scalogram (Fig. 14b
and g), consistent with the hodogram. The general description from the
previous events applies also here. In the following, we select and analyze in
more detail a few small-scale FACs, indicated by the solid vertical lines.
The black/green color indicates downward/upward FACs. Here we do not
distinguish between these small-scale downward FACs and just infer a range of
the parameters. Figure 15 shows the sections in the MSMVA
scalograms at the respective times. We observe again that remote FACs have a
smaller impact on *R*_{λ} as compared to ∂_{ξ}*λ*_{η}. All
selected small-scale FACs show a high degree of planarity. The dependence of
∂_{ξ}*λ*_{η} on scale indicates a range of scales between 1.8
(12.4 km) and 4 s (27.6 km) (shown by the vertical red lines) for the
selected FACs. The orientation shows values from $\sim -\mathrm{40}$ to
∼10^{∘}. The FAC density has values of about
−7.5 µA m^{−2} for the upward region (green) and between 4 and
6 µA m^{−2} for the four downward FACs. For this event we can
make just a qualitative comparison with the single-spacecraft FAC estimates
since the scales are well below the resolution of the dual-spacecraft
estimate. The single-spacecraft FAC density is −4.24 µA m^{−2}
for the upward FAC and between 3.84 and 6.08 µA m^{−2} for the
downward FACs. Thus, we have roughly similar values for downward FACs with
small inclination (<10^{∘}) and higher deviations for the highly
inclined FACs, e.g., upward FAC element. The detailed analysis (not shown)
indicates that the selected times are associated with local maxima of the
single-spacecraft FAC density and this indicates the consistency of
∂_{ξ}*λ*_{η} information at small scales.

When going to smaller scales, non-stationary effects become more important
and can be characterized by using the nested MVA analysis. This procedure is
implicitly included in the MSMVA technique since at each point we perform a
nested MVA in the linear scanning. The standard nested MVA (Sonnerup and Scheible, 1998)
is applied in 3-D and investigates the scale dependence of orientation and
projections of ** B** on the eigenvectors. In this study, we extended
MSMVA analysis by the density scalogram and showed quantitative estimates of
the local current density for FACs observed by Swarm.

5 Discussion and summary

Back to toptop
A good fraction of the FAC signatures above the auroral oval consists of rapidly varying FAC features, associated with time-dependent discrete auroras, superposed on slowly varying FAC structures (R1 and R2 currents). Using a fixed window analysis approach to study the FACs which occur at different scales has limitations. Instead, one can use varying window sizes to capture both the fast and slowly varying FACs. The long/short analysis windows are appropriate for large-/small-scale FACs.

The MSMVA technique was previously applied to auroral oval crossings by
Cluster and FAST spacecraft. The main goal of Bunescu et al. (2015) was to
introduce the technique for the scale identification capability by
∂_{w}*λ*_{η}. Bunescu et al. (2015) showed large-scale planar and
stable arcs as well as more dynamic aurora (locally planar), but did not
address in detail the superposition of scales or the inclined FACs. The
magnetic field was filtered such that the large-scale R1 and R2 FACs were
practically removed. Thus, the method was effectively showing the sequence of
crossings of mesoscale FACs, whereas the small-scale FACs were not analyzed
in detail in terms of localization and orientation.

In order to explore all scales, the analysis was previously applied using a linear-scale sampling, covered typically with high resolution. While some small-scale structures are seen, as they should be, in their scale range, they also contribute to the variance at large scales. One large scale is identified as planar at any scale smaller than its thickness. By using this method one can self-consistently derive various information on the planarity, scale, and also current density, which is particularly useful when optical data are not available in event studies.

In this study we compared the local multiscale FAC density estimates with well-established methods used routinely for the computation of the FAC density. The goal was to show that the multiscale FAC density provided results consistent with other methods, in particular Ritter et al. (2013). For the case of synthetic FACs (Sect. 3), the comparison of the input parameters with the local output of the MSMVA parameters indicated specific limitations of the method (related, e.g., to the accuracy of resolving different scales and the respective orientations), to be explored more closely by upcoming work.

The analysis presented in this work offers a new visualization tool for the
FAC density that helps to explore current structures embedded in larger-scale
FACs. The main goal of the paper is to enable the visualization of the
multiscale FAC density. Based on this framework we can easily visualize the
discrete constituents of a measured FAC signature. *R*_{λ} dependence on
scale in the center of FAC structures allows us to separate the
instantaneously crossed FACs from remote FACs. Thus we can separate the
near-field FACs from the far-field FACs. The accuracy of the identification
depends on the relative distance between the FACs and their planarity. The
complex FAC signatures can thus be deconvoluted into a discrete sequence of
FAC elements.

The extended MSMVA framework, and the FAC density scalogram in particular, can be compared with other spectral techniques offering spectral resolution together with time localization. The most prominent examples are the dynamical Fourier spectra produced by a windowed Fourier transformation (WFT) and wavelet techniques. In a so-called orthogonal or discrete wavelet transform (DWT) such as the Haar transform or the Daubechies transform (Daubechies, 1992), the signal is represented using a family of mutually uncorrelated (and hence orthogonal with respect to the canonical scalar product) basis functions. The orthogonality condition facilitates signal reconstruction but puts severe constraints on the selection of scale and time parameters that are then usually arranged in a manner similar to the logarithmic-scale sampling scheme chosen for the FAC density scalogram (Sect. 2.4). A so-called continuous wavelet transform (CWT) does not aim at a compact signal representation and hence can be based on a function family that is not constrained by orthogonality conditions. A CWT produces a (redundant) set of signal correlations with basis functions that depend on scale and time, e.g., Gaussian wave packets in the Morlet wavelet transform. In contrast to DWTs, the flexible choice of time and scale parameters in CWTs allows for a smooth representation of the time-varying scale dependence of the signal.

The FAC density scalogram of the extended MSMVA framework introduced in this
paper takes into account scale-dependent current structure information such
as sheet inclination, reflected in both perpendicular components of the
magnetic field perturbation Δ** B**, and thus yields a more
comprehensive FAC representation than straightforward wavelet transforms.
Selected elements of wavelet transform are adopted in our constructions of
FAC density scalograms, e.g., logarithmic-scale sampling, and the
construction of time series segments for multiscale MVA. At the largest scale
we sample the entire auroral oval, perform a separate sampling of R1 and R2
at the second scale, and then a progressive decrease in the analysis window
appropriate for small-scale FACs, without overlapping segments. The
logarithmic scheme is faster and consumes fewer computational resources, but
the analysis intervals cannot be expected to properly capture the location
and extent of FAC structures.

The model functions implicitly employed to represent the magnetic field
measurements are piece-wise linear functions of a certain length *w*,
interpreted as the scale of the underlying current structure. The
corresponding FAC density profile is a step function of the same width *w*,
and centered at the same reference time *t*_{cen}. This approach is
compatible with established FAC estimators based on finite differencing.
Actual magnetic profiles in the auroral zone are quite similar to these
underlying piece-wise linear model functions, at least closer than perfectly
smooth functions such as the ones employed for producing the synthetic data
in Sect. 3 (which are preferred there because of analytic tractability).
Hence, we assume that our FAC scalogram performs actually better on real data
than on the synthetic examples. Nonzero correlations among different
piece-wise linear model functions lead to the non-orthogonal behavior. The
overall implications, however, depend on the particular subset of model
functions associated with the chosen sampling scheme: (a) if for a given
scale *w* all available center times *t*_{cen} are used, model
functions with neighboring *t*_{cen} are strongly correlated,
resulting in a highly redundant and very non-orthogonal representation. This
scale sampling scheme we call linear. (b) If for a given scale *w* the chosen
center times *t*_{cen} are separated by the scale *w*, the model
functions are only weakly correlated, resulting in a representation that is
much less redundant and closer to orthogonality. This scale sampling scheme
we call logarithmic. The underlying logic is the same as for the Haar wavelet
transform. By comparing the results of linear versus logarithmic-scale
sampling for synthetic data, one finds that localization of center
time/location and scale is more accurate with the linear sampling scheme. In
logarithmic sampling, the center location of a current structure is heavily
constrained by the scale *w* that thus effectively constitutes the
uncertainty of the *t*_{cen} (note also the uncertainty relation in
wavelet analysis). Here our emphasis is on constraining FAC scales and center
locations using a visualization tool, not on a full reconstruction of the FAC
profile, thus we prefer to use a highly redundant set of model functions
instead of an orthogonal and thus non-redundant one. Since the synthetic data
are smooth profiles, and the scales are the widths of Gaussian profiles, we
cannot expect that the piece-wise linear model functions identify the
parameters perfectly.

Stasiewicz and Potemra (1998) made use of DWT analysis to study the multiscale properties
of magnetic field gradients and plasma density perturbations observed by
Freja. Another option to get scale information is by filtering the measured
perturbation or the FAC density (obtained within the constraints of the
methods). Using Swarm data, the study of McGranaghan et al. (2017) separates the
contributions from scales ∼50, ∼150, and ∼350 km based on
filtering (Hanning window) of the FAC density with window lengths of 8, 20,
and 48 s, respectively. As compared to these techniques that rely on
filtering – assuming the variations of the magnetic field perturbation (or
the FAC density) are approximated by certain basis functions – we here
compute the FAC density without removing the relative influence of
large-/small-scale FACs on the small/large FACs that are present in the
measured (or simulated) magnetic field perturbation. To distinguish between
the scales, we mainly rely on the ∂_{ξ}*λ*_{η} information.

Because MSMVA is based on the statistical MVA analysis, it can be affected by two types of error, namely the statistical and discretization errors. At short analysis windows, well suited for the fine-scale FACs, the MVA is affected by the increase in the statistical error (noise level). Longer analysis windows, suitable for the mesoscale and large-scale FACs, are associated with a lower statistical error in the MSMVA, because the analysis window includes a large number of measurements. However, for long analysis windows there is an increase in the discretization error. The discretization error is caused by the use of analysis windows larger than the FAC signature, in which case the FAC is not well sampled. The error analysis of MSMVA is not the subject of this paper, and it will be addressed in a future publication.

For the dual-spacecraft FD and LS, estimates can be unreliable in the case of dynamic and/or inclined FACs with embedded smaller-scale structures, as shown in Sect. (4.3). Gillies et al. (2015) also pointed out that the two-spacecraft products can be compromised in regions of diffuse aurora, typically observed near the equatorward border of the auroral oval, around the midnight sector following substorms. When applied to pulsating aurora (Gillies et al., 2015), the dual-spacecraft approach does not precisely identify the boundaries of the auroral patches associated with FAC reversals, whereas the single-spacecraft precisely identifies these boundaries. One region of interest to apply MSMVA is adjacent to the polar cap. This region is known for the high variability of the FAC geometries, typically filamentary. For such dynamic and non-planar events two-spacecraft methods are used to derive the FAC density (Lühr et al., 2016). By using MSMVA one could obtain visual information on the consistency of these results as well as on (non)planarity.

6 Conclusions and outlook

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The technique presented in this paper extends the multiscale framework of Bunescu et al. (2015) and provides a multiscale version of the single-spacecraft FAC density estimate. The main goal of this technique is to assist the studies of aurora by an improved visualization of the FAC structures.

The MSMVA scalograms can be used to visualize and characterize the spacecraft
measurements of the auroral field-aligned current structures. One can
separate the planar FACs through the *R*_{λ} scalogram and check their
orientations by the *θ* scalogram. Using the local information about the
magnetic field perturbation, the along-track FAC thickness and the
orientation we obtain the FAC density scalogram. One can intuitively
(visually) distinguish between the currents of different orientation (upward
and downward) and their dependence on scale (e.g., get information about the
FACs extent in time domain or the along-track thickness from the scale
domain). The time and scale dependence of FAC density and
∂_{ξ}*λ*_{η} scalograms are compared to obtain the local
average current at the characteristic scales of the FAC signatures.
Technically, the computation of the FAC density scalogram can be summarized
in a few steps: (a) setup of the scanning parameters (e.g., scale range,
discretization scheme, scanning steps in the time and scale domains) that
suit the observed FACs. For the logarithmic scheme one needs to define the
smallest scale, the number of scales, and the center time of the largest
scale; (b) computation of MVA over the time and scale grid defined in (a) to
obtain the 2-D data structures with information about, e.g., planarity,
orientation, eigenvalues. The local average FAC density is computed at each
iteration on the same time and scale grid used for MVA; (c) computation of
the scale derivatives of the 2-D quantities, e.g.,
∂_{ξ}*λ*_{η}; (d) quantitative information about selected FACs
is derived for related cuts of the scalogram by looking at the dependence of
the amplitude corrected quantities (e.g., ∂_{ξ}*λ*_{η},
*j*_{∥}) as a function of the corrected scale.

The application of the technique to measured Swarm data showed that the
multiscale FAC density can provide results that are consistent with the
typically used methods. The local multiscale FAC density shows good
consistency with the dual-spacecraft FAC estimate, with deviations within
15 % for larger scales (>150 km), which are well resolved by the
dual-spacecraft estimate. In the mesoscale and small-scale range, deviations
can be larger than 70 %. The comparison of the local multiscale FAC
density with the single-spacecraft estimate shows better consistency also at
small scales, since the single-spacecraft estimate has a resolution of
∼7.5 km (to be compared with the local multiscale FAC estimate
resolution of 0.7 km, when high-frequency magnetic field data are used). We
observe higher deviations in the case of inclined structures, since the
single-spacecraft estimate neglects the north–south component of
Δ** B**.

We applied the MSMVA technique for the computation of multiscale FAC density
using two scale scanning procedures, linear and logarithmic. The logarithmic
sampling scheme shows consistent information for both synthetic and observed
FACs. We observe that the orientation and the local FAC density are typically
in good agreement with the linear sampling scheme. However, the location and
scale information provided by ∂_{ξ}*λ*_{η} is affected by the
non-overlapping of the intervals in the logarithmic scheme and can provide
inconsistent information at scales that are not properly sampled, e.g., if
the scale is not centered on the FAC element.

Future work will address an error analysis of the multiscale information. We plan to use the bootstrap method to evaluate the impact of the error level of the input magnetic field perturbation on the output multiscale information. The multiscale approach offers a good setup to study the distribution of the statistical error (noise level), predominant at small scales, and of the discretization error (imperfect sampling of the FACs), predominant at large scales, where the analysis window becomes larger than the FAC signature. We expect thus to properly distinguish between different error sources of the multiscale information, e.g., multiscale FAC density, FAC localization, thickness, and orientation.

So far, the method does not properly take into account the geometry of the FAC structures. At this point one can select thresholds in the planarity and, accordingly, apply a mask to other quantities. Masked results were not included since they affect the overall structure of the displayed quantities. However, the masks are suitable for selecting a certain type of FAC, e.g., planar or non-planar. Further improvement might address a thorough study on finite structures to properly quantify the influence of the scale and how the planarity can better weight the results. In this respect we plan to extend the method to a dual-spacecraft multiscale analysis by using Swarm observations. With two satellites one can correlate the quantities, e.g., planarity, in the longitudinal direction.

Swarm provides an appropriate platform to quantify and check the planarity as derived by eigenvalue ratios, based on the similarity of the results obtained from the two longitudinally separated measurement points. This analysis is particularly useful for the fine structure of the aurora which cannot be addressed by other dual-spacecraft methods due to the spacecraft configuration limitations.

At present, the scale dependence of FAC properties can be investigated using Swarm, FAST, and Cluster high-resolution measurements. The technique can also be adapted to other more recently launched missions, like MMS, particularly for conjugate measurements, e.g., MMS/Swarm.

Data availability

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Data availability.

We acknowledge the use of the conjunction finder interface available at https://swarm-aurora.com/conjunctionFinder/ (Swarm-Aurora Conjunction Finder, last access: June 2018). We acknowledge the use of the Swarm magnetic field data as well as the L2 products provided by ESA at ftp://swarm-diss.eo.esa.int/ (Swarm Data Access, last access: June 2018). We acknowledge NASA contract NAS5-02099 and Vassillis Angelopoulos for the use of data from the THEMIS Mission, specifically, Stephen Mende and Eric Donovan for the use of the ASI data (http://themis.ssl.berkeley.edu/data/themis/thg/, THEMIS, last access: June 2018), the CSA for logistical support in fielding and data retrieval from the GBO stations, and the NSF for support of GIMNAST through grant AGS-1004736. We acknowledge the use of the THEMIS Data Analysis Software available at http://themis.ssl.berkeley.edu/ (THEMIS Science Data Analysis Software, last access: June 2018). The Tsyganenko magnetic field model involves the use of the DST and OMNI data. The DST data are provided by the World Data Center for Geomagnetism, Kyoto (http://wdc.kugi.kyoto-u.ac.jp/, World Data Center for Geomagnetism, last access: June 2018). The OMNI data were obtained from the GSFC/SPDF OMNIWeb interface at http://omniweb.gsfc.nasa.gov (The Space Physics Data Facility (SPDF), last access: June 2018).

Author contributions

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Author contributions.

CB developed the multiscale FAC analysis framework and extended it by including the multiscale FAC density scalogram. The development of this analysis includes the software development, modeling and testing on synthetic FAC structures, and the application to Swarm events, including the Swarm events selection based on optical data. CB did the analysis of results, figures, text writing and editing, as well as answered the referees during the evaluation process. In the initial stage of this project, JV contributed theoretical elements of the multiscale FAC scalogram construction method. He also suggested the inclusion of the logarithmic FAC scanning to increase the speed of the technique and to partially improve the analysis towards an orthogonal decomposition. OM contributed through discussions about the modeling of synthetic FAC structures, namely to the inclusion of the orientation. AB contributed through discussions and with software to read the Swarm data and compute the magnetic field perturbation. All the coauthors provided feedback before the initial submission and during the evaluation process.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Dynamics and interaction of processes in the Earth and its space environment: the perspective from low Earth orbiting satellites and beyond”. It is not associated with a conference.

Acknowledgements

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Acknowledgements.

Costel Bunescu acknowledges the hospitality of Jacobs University Bremen, where this study was initiated.

Financial support

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Financial support.

We acknowledge the support from the Deutsche Forschungsgemeinschaft (DFG) through grant VO 855/4-1 MuSICAL in the context of DFG Priority Programme SPP 1788 DynamicEarth, SIFACIT ESA contract 4000118383/16/I–EF, and STAR EXPRESS contract 119/2017 with the Romanian Space Agency.

Review statement

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Review statement.

This paper was edited by Rumi Nakamura and reviewed by two anonymous referees.

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