We investigate 1-D models of the westward substorm electrojet using magnetic field observations along a meridian chain of stations. We review two respective linear models from

A ground-based magnetometer is the oldest instrument for space weather research. Data from hundreds of permanent and temporary magnetic stations all over the world are available. Using magnetic records, one can study evolution of the main geomagnetic field,
as well as geomagnetic variations. Most of the latter are driven by the magnetospheric
and ionospheric currents, which ultimately depend on solar activity. In particular,
magnetic records are used to characterize the strength of geomagnetic substorms.
The main substorm characteristic is the amplitude of magnetic variations in the northern auroral zone, which is
summarized using the AE, AU, and AL geomagnetic indices. These variations
are driven primarily by the westward auroral electrojet, which is an
electric current that shortcuts the magnetotail cross-tail current

The goal of a dozen (about 12) AE/AU/AL stations is to catch the global maximum
of magnetic perturbation at all longitudes. To study electrojet and substorm dynamics
in detail, one needs to track at least one meridional profile of auroral
geomagnetic variations with a north–south chain of stations.
The most famous and accessible chains of stations are the Scandinavian IMAGE (International Monitor for Auroral Geomagnetic Effects) chain

While the primary measured parameter is the magnetic field, it needs to be converted to electric current,
which can then be compared with magnetospheric currents and used to quantify substorms as a plasma
phenomenon. Alternatively, one can compute the geoelectric field, which affects pipelines or electric power lines.
Ionospheric parameters in the auroral zone,
such as electron density and conductivity, are also of interest

A number of quantitative and semiquantitative approaches have been
developed to convert the magnetic field to electric current in the auroral zone.
A 2-D model of equivalent ionospheric currents can be implemented if stations are
distributed along both latitude and longitude

Most of these methods, which use instantaneous measurements, require a large number of stations
to discover the electrojet spatial structure. However, in many local time sectors the station
network is sparse. In this report, we develop a simple
model of the westward electrojet and a relevant solution scheme, which can be used with a small number of stations (even with just two or three stations). We also
describe some other useful algorithms. The key to our approach is the essential use of the vertical component
of the geomagnetic field (

For illustration, we use two typical substorms with sudden onsets and clear negative bays that are
gradually moving northward (Fig. 2). The first case was registered on 24 November 1996 by the IMAGE network
and has been widely studied elsewhere

List of the geomagnetic coordinates of magnetometers.

Map of stations on the Yamal Peninsula.

Examples of IMAGE

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

Ground magnetic disturbances can be described as

This method works well for a dense magnetometer chain with a large number of stations.

For magnetometer chains with a small number of stations, we have to use
the simpler method

We formulate the general
maximum likelihood estimation (MLE) solution. We choose the model parameters that maximize the likelihood function

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.

The parameter vector is determined by looking for the minimum of

The errors of the model parameters

The model scheme.

The first model described was suggested by

Regularization, suggested by the authors, is

Event on 24 November 1996 at 23:09:00 UT.

The second model described was suggested by

Regularization, suggested by the authors, is

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.

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 (

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).

Event on 24 November 1996 at 23:09:00 UT. Effect of the station selection.
Panels

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

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

Event on 24 November 1996 at 23:09:00 UT. The map of the initial conditions and the final electrojet boundaries for the case in Fig. 5d, e, and f (blue curve). The initial conditions are shown using black circles, the Starkov model (a center of randomized initials) is shown using the red point, the absolute minimum is shown using the blue filled circle, and the local minima are shown using blue open circles. See the text for further details.

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

Model 3 for Example 2. Panels

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.

Variants of Model 3 for Example 2. Panels

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

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

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

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

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

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

Somewhat counterintuitively, the situation is simpler for
an infinitely thin electrojet. One can force
the current density to be equal to the estimate from

The optimal method to compute electrojet parameters using Model 3 and small number of stations
is summarized below.

Select a substorm interval of interest, preferably with a clear westward electrojet.

Subtract the quiet magnetic field.

Subtract the internal component of the magnetic field using constant coefficients (Eq. 4).

Repeat the following actions for all time instants with 1 min or 5 min cadence.

Create a set of initial latitudes that are normally distributed around the boundaries of the

For each set of initial conditions, solve the minimization problem (Eqs. 6, 7, 12). The solution with the smallest residuals is final.

Check the values of parameters and errors to determine the reliability of individual parameters. If necessary, repeat the computation of the reduced model with a fixed current density, using the estimate of

The proposed 1-D algorithms are computationally simple and efficiently recover the
auroral electrojet parameters in configurations such as that of the westward electrojet developing during the
substorm expansion phase. The possibility of only using a few magnetic stations
substantially increases the span of longitudes at which such modeling is
possible. The electrojet amplitude and location determined can be used for a variety
of studies, including, for example, the comparison of electrojet boundaries
with the oval boundaries, the comparison of electrojet amplitude with
that registered in space using AMPERE project
data

To be fully confident in the reconstructed meridional profile of
the electrojet, one needs the station set to be
dense enough at all of the latitudes in question. A 5

The models described have some natural physical limitations.
First of all, any deviations from 1-D are effectively averaged out. Some issues,
such as the deflection from the latitudinal direction,
can be handled by the reasonable complication of the model (including the

It should be specially noted that the analysis of our test data reveals
frequent apparent inconsistency between the

The usage of

Finally, we solve the considered mathematical problem with a very generic maximum likelihood approach, which allows priors, regularization, comprehensive error-handling, etc. This approach can be used in a variety of other empirical model studies.

In this study, we investigated the models of the westward auroral electrojet using magnetic field observations from sparse meridian chains of ground-based magnetometers.
The model with one current strip works reasonably well, even using only three stations and two magnetic field components (

The

Regression coefficients are shown in Table A1.

Regression coefficients.

The IMAGE network data used in this paper can be downloaded from

MAE performed the data processing, and AAP was responsible for the data analysis and interpretation.

The authors declare that they have no conflict of interest.

The data analysis was funded by the Russian Science Fund (grant no. 18-47-05001). We are grateful to the IMAGE data archive and Aleksandr N. Zaitsev for the Yamal data.

This research has been supported by the Russian Science Foundation (grant no. 18-47-05001).

This paper was edited by Georgios Balasis and reviewed by Vladimir Papitashvili and one anonymous referee.

^{®}engineering magnetometer data, Geophys. Res. Lett., 27, 4045–4048, 2000.