Vectorial Association of Linearly Oriented Residua: Application to Swarm High-latitude Magnetic Field Measurements

We introduce the methodological framework VALOR for the analysis of ﬁeld-aligned current (FAC) 6 sheets based on dual-spacecraft magnetic ﬁeld observations. VALOR generalizes existing approaches in 7 two respects. First, it takes advantage of the primary observables at both spacecraft, namely, the full 8 magnetic ﬁeld vectors, instead of derived scalar quantities like single- or dual-spacecraft FAC densities. 9 Second, rather than being restricted to linear correlation measures, VALOR allows to deﬁne a 10 custom-made statistical measure of association between the signals at both spacecraft, that is chosen to 11 be particularly sensitive to small-scale ﬂuctuations for this ﬁrst demonstration. Additionally, VALOR 12 oﬀers to consider the linear orientation of the magnetic ﬁeld residual vectors, i.e., the sheets’ 13 polarization, in order to infer their orientation and to accentuate the association measure. This paper 14 illustrates the method on the basis of an exemplary auroral oval crossing on March 1, 2019 by the lower 15 pair of ESA’s multi-spacecraft mission Swarm. Explicit comparisons to competing methods are 16 performed to give credence to the results of the case study while the general capability of VALOR is 17 demonstrated on the basis of a statistical study comprising around 9700 auroral oval crossings from 18 2014–2020. One suitable area of application is the analysis of meso- to small-scale (tens of km to below 19 one km) auroral arc systems in combination with optical data from modern all-sky imagers. 20 Conceptually, VALOR is neither bound to the auroral zone nor to Swarm and can be applied by any 21 multi-spacecraft geospace mission with suitable instrumentation and orbit conﬁguration.

In the case of multi-spacecraft missions such as ST5 or Swarm, scale-dependent patterns in FAC density 56 profiles measured at different locations can be statistically compared to infer their degree of association. at an auroral current sheet, i.e., the directional preference of the magnetic field vector, to infer the sheet 74 orientation. 75 We proceed as follows. In the Methods section, the VALOR approach is developed and illustrated on the 76 basis of an exemplary auroral oval crossing by the lower pair of Swarm spacecraft. The section closes with 77 an explanation of how the data base for the multi-event study is compiled. The Results section presents 78 a detailed evaluation of the case study followed by the statistical results. The same structure is kept 79 throughout the Discussion & Outlook section where we explicitly compare VALOR to related methods 80 and give an outlook on future work before we present our Conclusions.

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To provide context and a reference analysis technique, linear cross-correlation analysis is summarized 83 before the VALOR methodology is explained. The general VALOR framework for vectorial time series 84 is presented, and then we discuss how polarization information comes into play. The rest of the section 85 is concerned with data selection and preprocessing in preparation of our statistical study of auroral 86 crossings.

Linear correlation analysis
Linear (cross-)correlation analysis rests on the linear correlation coefficient r = r(X, Y τ ) between one 89 signal X = X(t) and a lagged version of another signal Y τ = Y (t + τ ). In the context of Swarm auroral 90 FAC density studies, Forsyth et al (2017) pointed out that r quantifies only the degree of linear dependence 91 between the two signals. Additionally, they considered the slope m = m(X, Y τ ) of the regression line 92 to study if the amplitudes of the signals are comparable, and argued that both r ≈ 1 and m ≈ 1 are 93 required for a good match of two FAC density profiles.

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As shown in Appendix A, the two quality parameters r and m can be combined into a single measure of 95 signal association, namely, the mean squared deviation (MSD) of X and Y τ : Here, • t and• both denote time averaging, and (∆X) 2 and (∆Y τ ) 2 are the variances of the two signals.

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Hence for the MSD to be small, both r and m must be close to one, and the time series meansX and 98Ȳ τ must not differ much. The third term cancels if centered time series X c = X −X and Y τ,c = Y τ −Ȳ τ 99 are considered instead (see Appendix A). With the MSD being close to zero for a good match the inverse 100 MSD as a function of τ is expected to display a clear peak at optimum lag τ 0 . This idea is picked up 101 by the VALOR methodology with respect to the construction of an association measure as a function of 102 both t and τ .

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The VALOR analysis framework 104 A straightforward generalization of the MSD from the scalar to the vectorial case gives MSD(X,  Here, X(t) and Y (t) refer to the ionospheric magnetic perturbations (b σ ) registered at SwA (σ = 1) and 110 SwC (σ = 2) as they traverse the auroral ovals. The perturbations are extracted from the vector field 111 magnetometer data (B σ , Level-1b, MAG[A/C] LR 1B) recorded at 1 Hz resolution: The ambient magnetic field (B * ) is modeled as a superposition of contributions from the core, the 113 lithospheric and the magnetospheric magnetic fields and evaluated at the spacecraft positions (r σ ) using 114 predictions from CHAOS-7 (Finlay et al 2020).

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The magnetic perturbations during one such auroral oval crossing on March 1, 2019 around 02:00 UTC 116 are shown in Figure 1a where they are given in the geographic North (N) -East (E) -Center (C) local FAC TMS 2F, orange) following the low-pass filtering (cutoff period at 20 s) of b σ prior to its calculation.

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Since the j σ estimators are more closely related to along-track differences of b σ than to b σ themselves, 128 the pattern function P is constructed as an average of temporal derivativesḃ σ (obtained using centered 129 finite differencing) which provides a parametric family of reference perturbation signatures in the centered spacecraft frame.

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Here, the time series are restricted to a window t = [−t max , . . . , t max = 150 s] selected based on a 132 probabilistic interpretation of the normalized j 12 magnitude (Fig. 1c, details in the end of the section).

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The t|τ -sampling domain is queried with respect to the optimum spacecraft lag (τ 0 ) as well as the center 136 (t 0 ) and spatial scale (L) of the FAC signature. These tasks are conveniently addressed by evaluating 137 the scalar association measure (4) Statistically robust estimates of t 0 and L, i.e., ones with limited dependency on the specific form of 141 P (t|τ 0 ), are obtained by interpreting the normalized association measure at optimum lag C(t|τ 0 ) as 142 probability density function (PDF) with quartiles Q1, Q2 (median) and Q3: Here, IQR is the interquartile range and v is the FAC velocity in the centered spacecraft frame calculated 144 under the assumption that the FAC doesn't move with respect to the ground. The spacecraft velocities 145 v σ are obtained from centered finite differences of the respective positions r σ .

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Note that the VALOR framework is not dependent on a particular choice of association measure. The In an ideal planar FAC sheet geometry, magnetic perturbation vectors are perpendicular to bothq and We expect the association measure (Eq. 4) to further accentuate when the current sheet polarization is 160 taken into account in the construction of an improved pattern function P p . To this end, P = P (t|τ ) is 161 projected onto the candidate polarization vectorp c to yield The anticipated merit of this projection is to reduce contributions from incorrect candidate polarization 163 directions to the association measure. Once the final triplet (q,n 0 ,p 0 ) has been selected from the 164 candidates based on (t 0 ,τ 0 ), the FAC's spatial scale in the direction ofn 0 can be calculated as using the updated IQR and leaving v (Eq. 8) unchanged.

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Preprocessing for multi-event analysis (c.f., Fig. 1a,b). Note that the agreement between t 0 and the position of maximum pattern amplitude sheets encountered on outbound orbits ("p twds W, out N" like sample event) as on inbound ones.

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The consideration of the polarization (Eq. 12) in the association measure results in a general reduction of 262 the FAC spatial scales by ∼ 40 km toL n = 227 km compared to the "scalar" (L s ) and "vector only" (L v ) 263 cases (Fig. 5g, gray vs. blue & red). As expected, this is significantly less thanL i = 656 km obtained 264 from the j 12 based preprocessing (black). The resolution power of the latter is limited to L i ≥ 150 km 265 because, in contrast to VALOR, the underlying magnetic field residuals are low-pass filtered (cf., Fig. 1b).

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The obtained statistics depend on the chosen association measure (Fig. 5h, Eq. 4) in that different FAC the medium scales fall in-between (Fig. 6b, g, h). In that sense the sample event is a typical medium to 272 large scale hybrid.

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However, the amplitude of the association measure at the selected center points does not readily lend 274 itself to a quantitative assessment of the results' validity, i.e., large-scale structures are not per se less 275 eligible for the VALOR analysis. In fact, one possibility to assess the fundamental assumption of sheet-like 276 current geometries is via the "planarity" measure N , i.e., the mean ratio of variances of b projected onto 277 thep 0 andn 0 directions (Fig. 5i). Truly planar structures should give planarity measures significantly 278 larger than one which holds for the majority of cases (Ñ = 18.6; 20% ≤ 5). Here, it is the large-scale 279 fraction that scores best although by a smaller margin (Fig. 6i). sensitivity to aligned small-scale amplitude fluctuations inḃ σ (large P (t, τ 0 ) ) and the fact that one 296 position is calculated in the presence of two current sheets. Actually, if one applies VALOR separately to 297 the periods before and after t 0 the resulting sheet centers (and widths) are consistent with the reference 298 analysis (Fig. 8). This experiment also shows that different optimum spacecraft lags (−4 s and −10 s) 299 can be estimated for adjacent FAC crossings (τ 0 = −11 s). The ability to adapt to different crossing 300 geometries (cf., Fig. 2) is a profound advantage of VALOR compared to the reference technique which, given that it operates on scalar j σ , assumes that all FACs are crossed perpendicularly (α = 0) so that the 302 differences in SwA and SwC equatorial crossing times τ eq are used as proxies for τ 0 (Forsyth et al 2017, 303 see also Fig. 5c, black). Here, we used the VALOR-determined τ 0 for the reference analysis in order to 304 facilitate the comparison. (α = 19 • ). Also, the MVA eigenvalue ratio is much higher (λ max /λ min =100) than the planarity measure 311 we've constructed (N , Fig. 5i). We don't think that MVA's negligence of the ambient magnetic field 312 variability along the orbits (∡(B,B * ) = 6.6 • ± 3.8 • ) is the sole cause for these differences. For now, 313 we attribute them to the spatial resolution power of the respective methods with MVA targeting the 314 orientation of the entire sheet structure while VALOR extracts the local geometry around t 0 .

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The purpose of the statistical study is to demonstrate that VALOR can be readily applied to a vast 316 range of auroral oval crossings in a meaningful and largely automated way. We find that the presented

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Regarding the event selection process it is worth to deliberate on the suitability of j 12 as decisive measure.

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While its smoothness compared to the single-spacecraft estimates helps the automated determination of 328 the analysis windows, it is not available in the so-called exclusion zones near the geographic poles (lat.

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> 86 • ) due to the degeneracy of the underlying virtual four-point configuration to a quasi-linear array.

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"Loosing" auroral oval crossings due to the exclusion zones could be avoided by estimating current density 331 based on a least-squares approach (e.g., Blǎgǎu and Vogt 2019) that is conveniently formulated in the 332 RASADA framework . The other critical point is the absence of a minimum amplitude 333 requirement for j 12 which exposes the results to potentially unreliable contributions from signals just 334 above instrument noise. We've tested the influence of FAC density on the results and conclude that low 335 density FACs preferably occur at dawn/dusk with large characteristic scales while high density FACs 336 are prevalent at noon/midnight with smaller scales, respectively (additional file 1, Fig. 2). Excluding center time 02:01:34 UT at which the polarization helps to constrain the lag while the vectorial nature of the magnetic input data helps to constrain the center position (Fig. 3).

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• The center position falls in between the individual FAC traversals and within the field of view of 362 the Kuujjuaq all-sky camera that concurrently observes a relatively stable auroral arc (southward 363 drifting at ∼10% of spacecraft velocity, Fig. 4). Given the bi-modal nature of the pattern function 364 an interpretation of the center position as the auroral oval center is justified.

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• The reference analysis, i.e., linear correlation & regression according to Forsyth et al (2017), does 366 not indicate large overall similarity at t 0 but rather small-to meso-scale (< 305 km) fluctuations of 367 matching amplitude (Fig. 7) to which we ascribe VALOR's choice of t 0 . The VALOR determined 368 characteristic length scale L n = 345 km is corroborated by the reference analysis.

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• The sample event presents a typical case of a low-latitude, evening-sector, large-scale auroral current The datasets used and/or analysed during the current study are available from the following repositories.
The MSDs of X, Y on the one hand and X c , Y c on the other are related as follows: Given that r(X, Y ) can also be interpreted as the standardized slope m of the linear regression line we can rewrite MSD(X, Y ) as:  (magenta). Additional step functions in black, blue and red (not stacked) refer to results obtained 569 either in the preprocessing or from the "scalar" and "vector only" methods, respectively (cf., Fig. 3a Please see the Manuscript PDF le for the complete gure caption Figure 3 Please see the Manuscript PDF le for the complete gure caption Figure 4