We use the following approximation of the 1-D westward auroral electrojet (Fig. 3): (1) the electrojet flows at a fixed altitude of 110 km above the flat land; (2) the electrojet is infinitely thin vertically; (3) the electrojet flows along in the latitudinal direction; and (4) the electrojet does not vary with longitude.

The magnetic disturbances in question are deviations from the quiet field, which has to be subtracted from the measurements. To determine the quiet level, we average magnetic data from the 5 quietest days of the month, during which the substorm occurred . The model latitudinal range spans ±4∘ from the southernmost and northernmost stations (for the models with many elementary currents). The input magnetic field disturbance is forced to be zero at the edges of this range in order to avoid nonphysical solutions. Ground magnetic disturbances are produced by the ionospheric current (electrojet) and the corresponding induction current inside the Earth. The model latitudinal profile of ionospheric current is reconstructed using the north–south X and the vertical Z magnetic components that are measured at a set of ground observatories (magnetic stations). Currently, we ignore the Y component of the magnetic field.

Ground magnetic disturbances can be described as 1X=Xe+Xi,Z=Ze+Zi, where the “e” and “i” indices denote external and internal components, respectively. According to , the difference between the external and internal components at any point x along meridian can be calculated as follows (here H is horizontal field component): 2He(x)-Hi(x)=-1π∫-∞∞Z(ξ)ξ-xdξ,Ze(x)-Zi(x)=1π∫-∞∞H(ξ)ξ-xdξ. Therefore, the external field components are 3He(x)=12H(x)+IntH(x),Ze(x)=12Z(x)+IntZ(x),IntH(x)=-1π∫-∞∞Z(ξ)ξ-xdξ,andIntZ(x)=1π∫-∞∞H(ξ)ξ-xdξ.

This method works well for a dense magnetometer chain with a large number of stations. H(ξ) and Z(ξ) are obtained using the linear or spline interpolation of the measured magnetic disturbance (forced to zero at the edges of the modeled latitudinal range; see previous subsection). Integrals are calculated over the same latitudinal range.

For magnetometer chains with a small number of stations, we have to use the simpler method that has constant, empirically justified coefficients: 4Xe=23⋅X,Ze=1⋅Z.

We formulate the general maximum likelihood estimation (MLE) solution. We choose the model parameters that maximize the likelihood function L: 5L=∏k=1NPkXk,Zk,Model(p)×PP(p), where N is the number of stations; Xk and Zk represent the disturbance of the magnetic field, caused by the electrojet current, measured at the station k (with the background field and induction field subtracted); Pk is the probability of observing the given magnetic fields Xk and Zk for an electrojet model with the parameter vector p; and Pp represents a priori probabilities for p.

A priori information (also referred to as “priors”) may be predictions from the statistical models or common sense limitations, such as the flatness of the spatial profile. The latter variant is also known as regularization. Regularization might be technically necessary for under-determined problems, when the number of free parameters is larger than the number of degrees of freedom in the sample (the number of independent measurements).

In this investigation, we use one of the simplest MLE variants, assuming a Gaussian distribution of the model residuals and solving the general OLS (ordinary least squares) inverse problem. 6-2ln⁡L=∑k=1N1σX2δXk-δXkmnp2+1σZ2δZk-δZkmnp2+Qr, where n is the model number, Xkmn and Zkmn are calculated model disturbances, Qr denotes possible additional constraints, and σX (σZ) represents standard variations of the measured X (Z) components (at all stations used at a given time).

The parameter vector is determined by looking for the minimum of -2ln⁡L. If the whole model is linear with respect to the parameter vector p, the standard matrix inversion technique is applied to acquire the solution. The nonlinear variants are solved here using the Levenberg–Marquardt algorithm. This method requires the specification of some initial values of the model parameters, and then moves along the gradient of the optimization function towards the minimum. Unlike linear regression, methods such as these for nonlinear problems do not guarantee a unique solution due to existence of local minima.

The errors of the model parameters p are calculated as the inverted Hessian of ln⁡L: 7cov(p)=∂2ln⁡L∂pi∂pj-1.

The first model described was suggested by . It includes a large number of the infinitely thin, fixed wires with unknown currents. The wires are evenly distributed within the modeled latitudinal range, which is ±4∘ from the stations closest to the Equator and the pole-most stations. The magnetic field at the edge wires is set to zero. 8δXkm1=μ0h2π∑j=1MIjh2+Δxjk2,δZkm1=μ02π∑j=1MIjΔxjkh2+Δxjk2. Here, h is the height of the wires, M is the number of the wires, Ij represents currents, and j=1…M and Δxjk=xj-xk are the difference in the coordinates of the wire j and the station k along the magnetic meridian, respectively. The model magnetic disturbances δXkm1 and δZkm1 depend on the unknown model parameters Ij in a linear fashion.

Regularization, suggested by the authors, is 9Qr=α∑j=1MIj-Iaj2+q∑j=2MIj-Ij-12, where Iaj is the current at the previous time step; coefficient α does not allow currents to change too quickly (controls smoothness in the time domain), and q controls smoothness of the current profile along the meridian. Regularization is necessary, as the number of wires (of the model parameters) can be larger than the number of stations (50 wires were proposed in the original paper). Still, the number of stations should be large enough (e.g., as in the IMAGE chain) to provide enough information on the spatial inhomogeneity of the current.

The second model described was suggested by . It is fundamentally similar to Model 1, except it consists of evenly distributed strips with an unknown current density. 10δXkm2=μ02π∑i=1Mjiarctan⁡Δxik+dh-arctan⁡Δxik-dh,δZkm2=μ04π∑i=1Mjiln⁡h2+Δxik+d2h2+Δxik-d2, where d is the half-width of the strip, and Δxjk=xj-xk is the difference in coordinates of the strip center j and the station k. The positions of the strips are fixed. Disturbances δXkm2 and δZkm2 depend on the unknown model parameters ji in a linear fashion.

Regularization, suggested by the authors, is 11Qr=q∑i=2Mji-ji-12+β∑i=2Mji2. Here, the coefficient q is responsible for the smoothness of the current profile along the latitude, and β limits the maximal current amplitude. Regularization is necessary, as a large number of strips is proposed in the original paper.

For a small number of stations, a simpler model with one electric current element is required. Model 1 is inconvenient, as a single infinitely thin current will return an unphysical magnetic profile. We use a version of Model 2 that has one current strip with floating borders. The optimal unknown model parameters are as follows: the current density and the low-latitude and high-latitude electrojet boundaries (explained in the following section). This model is nonlinear. 12δXkm3=μ02πjarctanxk-xlh-arctanxk-xhh,δZkm3=μ04πjln⁡h2+xk-xh2h2+xk-xl2. Here, j is the current density in a strip; xhandxl are coordinates of the high-latitude and the low-latitude current borders, respectively; and xk is the coordinate of the station k.

In Model 1, each infinitely thin wire creates a characteristic spatial peak of the magnetic field with a latitudinal scale approximately equal to the height of the wire. As height (∼100 km or ∼1∘ of latitude) is much smaller than the typical electrojet width and the modeled latitudinal domain, a small set of wires will generate an unphysical magnetic profile with several sharp minima (for the westward electrojet). Figure 4a, b, and c present such Model 1 runs, using Example 1 with 8 and 15 wires (with no regularization). Both variants return oscillating magnetic profiles, indicating that the number of wires is insufficient. Note that the case with 15 wires also exhibits another problem, which is typical for models with too many parameters: some wires are attributed positive currents, creating positive excursions of the magnetic field between stations, which are not supported by any evidence (measured field).

The linear model with 15 wires becomes underdetermined, as the number of independent inputs (the double number of stations) is comparable to or smaller than the number of unknowns. An underdetermined solution usually results in physically unrealistic, large, and very variable values of (here) elementary currents that ideally cancel each other out at the magnetic stations, where measurements are available (Fig. 4d, e, and f; the model with 50 wires, red curves).

To ensure a sufficiently flat electrojet profile, a denser current network with the separation much smaller than the height is required; however, overly sharp variations between stations need to be damped. The standard way to solve this problem is to use the so-called regularization procedure, which penalizes variability and/or amplitude of the model parameters. The introduction of the regularization term in Model 1 with a reasonable coefficient q∼1 effectively reduces unwanted variations of currents, while still preserving reasonable complexity of the latitudinal profile (Fig. 4d, e, and f; the green curve compared with the blue and red curves).

Here, as seen in Fig. 4, it is important to note several aspects related to the applicability of such 1-D models. First, Model 1 reconstructs the X component reasonably well: the calculated values of the fields in Fig. 4d, e, and f always correspond to the measured data (black stars). However, the flattened Model 1 (with regularization) often fails to reproduce extreme Z values (such as at a latitude of 75∘).

Second, when the station coverage is sparse (which for IMAGE is in the Norwegian/Barents Sea – with stations only on the mainland and Svalbard), even the model with sufficient regularization may return positive currents (Fig. 4d, e, and f, above 75∘, green curve). This positive current results in positive model X values in the gaps between stations. However, all available stations only measure negative X; thus, the presence of a positive current cannot be directly confirmed. These issues are further elaborated upon in Sect. 5. With many elementary electric currents, it is possible to describe a relatively complex spatial profile of an electrojet without the need to explicitly define the nonlinear latitudinal profile. Elementary currents can be placed at evenly spaced fixed positions; thus, the only free model parameters are the electric current amplitudes in the numerator of the functional form (Eq. 8), and the model therefore remains linear. The spatial inhomogeneity of an electrojet is well described by these changing amplitudes.

The most natural variant for a case with a small number of stations is to use one strip from Model 2. The free parameters are then current density, center, and the half-width of the strip. However, this variant has several drawbacks.

The current density and width of the electrojet are strongly anticorrelated in the model with one strip and two to three magnetic stations. Almost the same magnetic field can be produced with a variety of strips with a different width and current density, but the same total current. The correlation of parameters complicates the error analysis, as the standard error bars are produced by the diagonal elements of the error matrix (Eq. 7). The correlation of the parameters creates large non-diagonal elements, which often avoid sufficient attention.

The second drawback is related to the definition of electrojet boundaries. For example, if there is no station in a relevant position to catch a poleward boundary, the corresponding error will be propagated to both parameters: the electrojet center and width.

Thus the optimal Model 3 has three parameters: the current density, and the poleward and equatorward boundaries. All parameters are defined almost independently. The current density mostly depends on the largest observed X component disturbance, and the boundaries depend on the sign of the Z component at the nearest station.

When the number of stations is small, the stations might quite often not be optimally located relative to a specific electrojet. To illustrate how this problem is handled using Model 3, we resurrect one latitudinal profile from Example 1 (Fig. 5). Figure 5a, b, and c show the model for the case with all stations, whereas the right panels show two variants. The red curve corresponds to the case with three stations, two southward and one northward electrojet, and the model electrojet is identical to that in the left panel (only the current density error is larger). However, the case with four stations (blue curve), all equatorward of the electrojet, results in a substantially different model with a shifted poleward border. This border is also defined with a substantially larger error. To get this particular solution, the local minimum must also be avoided; this issue is described in the next subsection.

Contrary to with linear regression, the determination of the right nonlinear solution is not guaranteed. All algorithms are sequential and may lead to a local, rather than a global, minimum of the target function (Eqs. 5, 6). The result may depend on the initial approximation of the model parameters, which needs to be specified to start the search. There are several standard ways to avoid local minima in a more or less automatic fashion.

The first approach is to introduce a prior – some a priori information on the location of the electrojet boundaries or electrojet amplitude. The a priori boundaries can be taken, e.g., from the Starkov model shown in Appendix A of this paper. As an input for Starkov model one can take either the AL index or the local maximal negative X component (from the modeled magnetic chain data). In Eq. (6), one may then define Qr=wd(d-d0)2, where d is a parameter, d0 is an a priori value, and wd is a weight. This form penalizes any strong deviations from the a priori value. Thinking about a solution process as a descent along the local gradient in a landscape of the minimized function, the introduction of a prior modifies this landscape, removing the local minima. However, although effective in some cases, this approach is very sensitive to the selection of weights, which have to be specified manually for each model run.

The second approach is to use a so-called multi-start algorithm. We generate a normally randomized set of initial conditions around a Starkov model solution, run Model 3 several times, and choose a result with the minimal residuals (Eq. 6). We show the map of 50 initial conditions (for the boundary locations only) in Fig. 6 for the case of Fig. 5 (c, d, e, blue curve). In Fig. 6, the Starkov model is shown using the red point, and the solutions (starting from the filled black circles) lead to the absolute minimum (shown using the filled blue point). The empty black circles lead to the local minima (blue open circles). As Model 3 is computationally simple, the method works well, and it is not necessary to densely fill the parameter space during randomization.

We illustrate the operation of Model 3 by running it for the whole of Event 2 (Fig. 7). In Fig. 7a, b, and c, the time profiles of the magnetic field, current density and electrojet boundaries are shown, respectively. This was a rather strong substorm with a negative bay of almost -1000 nT. Generally, Model 3 returns reasonable results for magnetic profiles (Fig. 7a), but the electrojet boundaries are somewhat different from the statistical Starkov model (Fig. 7c). During the growth phase (16:00–16:45 UT) the real electrojet is more poleward, which may be related to the absence of a station at a sufficiently southward location. During the extended recovery phase (after 18:00 UT), the electrojet is consistently more southward. However, a detailed analysis of this substorm is beyond the scope of this report.

Besides these easily interpretable results, at some moments the model reports definitely unphysical electrojet parameters, which appear as spikes in Fig. 7b and c. We highlight the four time instants with problems of various kinds (shown using colored vertical lines). The detailed model results for these instants are shown in Fig. 7d, e, and f.

The black vertical line (at 18:15:30 UT in Fig. 7a, b, and c) and the corresponding black curves (Fig. 7d, e, f) show fully reliable result with small errors. The blue lines and curves for 17:37:30 UT show the case with an unreliable poleward border, which is even above 90∘. Here, all three stations fall on the more equatorward side of the electrojet (all Z values are negative). The uncertainty interval for the poleward border is very large and extends down to a very reasonable latitude of 75∘. The corresponding equatorward border and current density are well defined, as expected. A similar error, but for the equatorward border, occurs at 19:50:30 UT (red color). Here all three stations have a positive Z.

A more serious problem arises if all three model parameters are physically incorrect, which is the case at 17:55:30 UT (orange). Here the model returns an electrojet with a zero width and a very high current density amplitude. The model X component profile shows a very narrow dip between the stations with an amplitude 1.5 times larger than the actual observed field (green stars). The model current density amplitude is very large and is therefore not shown.

The problems described are features of the true global maximum in the mathematical solution and cannot be resolved within the core model algorithm. They have to be removed using some additional physical considerations. In a case with one unreliable border, one can fix the troubled parameter at limiting values, e.g., 55 and 85∘; however, these numbers are not still justified by any observations.

Somewhat counterintuitively, the situation is simpler for an infinitely thin electrojet. One can force the current density to be equal to the estimate from (see Appendix B). The model then returns a more reasonable, but still rather narrow (2∘ wide), electrojet (Fig. 7, green line). The green model in Fig. 7d corresponds to this adjusted solution. A substantial X value at the station closest to the Equator at 62∘ still suggests that the real electrojet is wider than the result, but the solution here balances both the X and Z residuals.