Real-Time 3-D Modeling of the Ground Electric Field Due To Space Weather Events. A Concept and Its Validation
Abstract
We present a methodology that allows researchers to simulate in real time the spatiotemporal dynamics of the ground electric field (GEF) in a given 3-D conductivity model of the Earth based on continuously augmented data on the spatiotemporal evolution of the inducing source. The formalism relies on the factorization of the source by spatial modes (SM) and time series of respective expansion coefficients and exploits precomputed GEF kernels generated by corresponding SM. To validate the formalism, we invoke a high-resolution 3-D conductivity model of Fennoscandia and consider a realistic source built using the Spherical Elementary Current Systems (SECS) method as applied to magnetic field data from the International Monitor for Auroral Geomagnetic Effect network of observations. The factorization of the SECS-recovered source is then performed using the principal component analysis. Eventually, we show that the GEF computation at a given time instant on a 512 × 512 grid requires less than 0.025 s provided that GEF kernels due to pre-selected SM are computed in advance. Taking the 7–8 September 2017 geomagnetic storm as a space weather event, we show that real-time high-resolution 3-D modeling of the GEF is feasible. This opens a practical opportunity for GEF (and eventually geomagnetically induced currents) nowcasting and forecasting.
Key Points
-
We present the formalism of real-time modeling of the ground electric field (GEF) excited by temporally and spatially varying source
-
The formalism relies on the factorization of the source and exploits precomputed GEF kernels
-
Using Fennoscandia as a test region, we show that real-time 3-D modeling of the GEF takes less than 0.025 s
Plain Language Summary
The solar activity in the form of coronal mass ejections leads to abnormal fluctuations of the geomagnetic field. These fluctuations, in their turn, generate so-called geomagnetically induced currents (GIC) in electric power grids, which may pose a significant risk to the reliability and durability of such infrastructure. Forecasting GIC is one of the grand challenges of the modern space weather studies. The critical component of such forecasting is real-time simulation of the ground electric field (GEF), which depends on the electrical conductivity distribution inside the Earth and the spatiotemporal structure of geomagnetic field fluctuations. In this paper, we present and validate a methodology that allows researchers to simulate the GEF in fractions of a second (thus, in real time) irrespective of the complexity of the Earth's conductivity and geomagnetic field fluctuations models.
1 Introduction
As commonly recognized, geomagnetically induced currents (GIC) in electric power grids may pose a significant risk to the reliability and durability of such infrastructure (Bolduc, 2002; Love et al., 2018).
The ultimate goal of quantitative estimation of the hazard to power grids from abnormal geomagnetic disturbances (space weather events) is real-time and as realistic as practicable forecasting of GIC. Under GIC forecasting, we understand the time-domain computation of GIC using continuously augmented data on the spatiotemporal evolution of the source responsible for the geomagnetic disturbances. Specifically, to forecast GIC in the region of interest, one needs: (a) to adequately parameterize the source of geomagnetic disturbances; (b) to forecast the spatiotemporal evolution of the source in the region; (c) to specify/build a three-dimensional (3-D) electrical conductivity model of the Earth's subsurface; (d) to perform real-time modeling of the ground electric field (GEF) in a given 3-D conductivity model, that is, to compute as fast as feasible the spatiotemporal progression of the GEF from continuously augmented data on the spatiotemporal evolution of the forecasted source; (e) to convert the “forecasted” GEF into GIC.
It is well accepted that the decades of satellite observations of the solar wind parameters (plus observations of interplanetary magnetic field) at the L1 Lagrangian point are the most promising data for forecasting the spatiotemporal evolution of the source with algorithms known as neural networks (NN). Despite numerous studies that attempt to forecast the source evolution using different NN architectures quantitatively, the progress here is rather limited. This is, in particular, because the full potential of NN remains unexplored; the reader can find a rather exhaustive review of the literature on the subject in Tasistro-Hart et al. (2021). But even if the source forecasting will be feasible in the future, with the measurements at the L1 point, it is nearly impossible to forecast the source more than an hour in advance. This, in particular, means that forecasting GEF in a given 3-D conductivity model from continuously augmented data on the spatiotemporal evolution of the forecasted source should be performed “on the fly”, that is, within a few seconds, if one wishes to approach an ultimate goal of GIC forecasting in the region of interest—development of trustful alerting systems for the power industry. Note that once the GEF is forecasted, a conversion of the GEF into GIC is rather straightforward (Kelbert, 2020) and requires fractions of seconds provided the geometry of transmission lines and system design parameters are granted by power companies.
This paper presents and validates a methodology that allows researchers to simulate the spatiotemporal progression of the GEF in a 3-D conductivity model “on the fly”. The paper also details how the concept can be exploited for GEF nowcasting and forecasting.
2 Methodology
2.1 Governing Equations in the Frequency Domain




Note also that the form of ji(r) (and their number, L) varies with application. For example, jext(r, ω) is parameterized via spherical harmonics in Püthe and Kuvshinov (2013), Honkonen et al. (2018), Guzavina et al. (2019), and Grayver et al. (2021), current loops in Sun et al. (2015), or eigenmodes from the Principal component analysis (PCA) of the physics-based models in Egbert et al. (2021) and Zenhausern et al. (2021).



2.2 Governing Equations in the Time Domain


Since the radial component of the GEF is negligibly small (due to insulating air) and is not used in GIC calculations (Kelbert, 2020), we will confine ourselves to modeling the horizontal electric field solely; thus, hereinafter, Ei will stand for Ei = (Ex,i Ey,i).
2.3 Real-Time Modeling of the GEF. A Concept

Note that the upper limit in the integrals could be different for different SM, different components of the field, and different locations. However, one can choose a conservative approach, taking a single T irrespective of modes/components/locations as a maximum from all individual upper limit estimates. We will discuss the estimation of T in Sections 3.3 and 3.4.


The reasoning to represent the time-dependent part in Equation 11 in this form is given in Appendix B. Note also that quantities and
are time-invariant, and for the given ji, i = 1, 2, …, L and 3-D conductivity model are calculated only once, then stored and used, when the calculation of E(rs, tk; σ) is required. Actual form and estimation of kernels
and
are also discussed in Appendix B.
Equation 11 is the essence of the real-time GEF calculation, showing that summations and multiplications are required to compute the GEF at a (current) time instant tk plus some overhead to read the precomputed
and
from the disc. Note that Ng is a number of points rs, at which the GEF is computed.
3 Real-Time Modeling of the GEF. Validation of the Concept
The validation of the presented concept will be performed using Fennoscandia as a test region. The choice of Fennoscandia is motivated by several reasons. First, it is a high-latitude region, where GIC are expected to be especially large. Second, there exists a 3-D electrical conductivity model of the region (Korja et al., 2002). Third, the regional magnetometer network (International Monitor for Auroral Geomagnetic Effect (IMAGE; Tanskanen, 2009)), allows us to build a realistic model of the source. Finally, the last but not the least consideration to choose this region is the fact that we have already performed a comprehensive 3-D EM model study in this region (Marshalko et al., 2021).
3.1 Building a Model of the Source




The aim of this section is to obtain and, consequently, ji (using Equation 15). To this end, we apply the SECS method to 10-s vector magnetic field data from all available (38) stations of the IMAGE network during the 7–8 September 2017 geomagnetic storm. Locations of IMAGE sites are shown in Figure 1. Considered (8-hr) time period is from 20:00:00 UT, 7 September 2017, to 03:59:50 UT, 8 September 2017, thus, including the onset and the main phase of the storm. S was estimated at 0.5° × 1° grid of 35.5° × 47° part of a sphere. Coordinates of the region are 47.75°–83.25°N and 0.5°W–46.5°E. This set up, in particular, means that S was computed at M = 72 × 48 = 3,456 grid points and N = 2,880 time instants. Note that the same event, region and grid were considered in our recent study (Marshalko et al., 2021).

Location of sites from the International Monitor for Auroral Geomagnetic Effect magnetometer network. Credit: Finnish Meteorological Institute.







Figure 2 presents the cumulative variance for the first 30 PC. Horizontal dashed line allows us to estimate the number of modes needed to explain 99% of the spatial variability of Sm(t). It is seen from the figure that one needs L = 21 PC to explain most (99%) of the variance. This is a dramatic reduction from the original M = 3,456 SECS poles. These 21 PC will be used in the further discussion of the real-time calculation of the GEF. Figure 3 shows ji corresponding to PC of different i, illustrating the fact that the modes with larger i capture smaller spatial structures of the source. The respective time series ci are presented in Figure 4. Figure 5 compares the maps of the original and the PCA-based sources (i.e., external equivalent currents) for two snapshots of the enhanced geomagnetic activity. The original source is built using the SECS method (Equation 14), whereas PCA-based source is calculated using Equations 13 and 15. It is seen that the agreement between the original and PCA-based sources is very good both in terms of the amplitude and spatial pattern. In addition, Figure 6 demonstrates the comparison of the time series of these sources for two exemplary sites (shown in Figure 5 as white circles): one is located in the region where the significant source current is observed (Jäckvik (JCK)), another—aside from this region (Tartu (TAR)). Again, we observe good agreement between the two sources, especially for the site above which the source current is large.

Cumulative variance for the first 30 Principal Components. Red dashed line marks the 99% threshold.

A selection of PCA-recovered ji, i = 1, 7, 14, 21. By color and arrows, the magnitude (in A/m) and direction of the corresponding ji are depicted. See details in the text.

A selection of PCA-recovered ci, i = 1, 7, 14, 21. See details in the text.

Left: the original external equivalent current. Right: the external equivalent current calculated using 21 spatial modes. The results (in A/m) are for two time instants: 23:16:00 (top row) and 23:52:00 (bottom row) UT on 7 September 2017.

Time series of the original external equivalent current (black curves) and external equivalent current calculated using 15 (blue curves) and 21 spatial modes (red curves) above two exemplary sites (Jäckvik (JCK) and Tartu (TAR)). The results are in A/m. Left and right panels show x- and y-components of the currents, respectively. Note different scales in the panels. Locations of the sites are shown in Figure 5 as white circles.
3.2 3-D Conductivity Model of Fennoscandia
We took the 3-D conductivity model of the region from Marshalko et al. (2021), where it was constructed using the SMAP (Korja et al., 2002)—a set of maps of crustal conductances (vertically integrated electrical conductivities) of the Fennoscandian Shield, surrounding seas, and continental areas. The SMAP consists of six layers of laterally variable conductance. Each layer has a thickness of 10 km. The first layer comprises contributions from the seawater, sediments, and upper crust. The other five layers describe conductivity distribution in the middle and lower crust. SMAP covers an area 0°–50°E and 50°–85°N and has 5′ × 5′ resolution. We converted the original SMAP database into a Cartesian 3-D conductivity model of Fennoscandia with three layers of laterally variable conductivity of 10, 20, and 30 km thicknesses (Figures 7a–7c). This vertical discretization is chosen to be compatible with that previously used by Rosenqvist and Hall (2019), Dimmock et al. (2019), and Dimmock et al. (2020) for GIC studies in the region. Conductivities in the second and the third layer of this model are simple averages of the conductivities in the corresponding layers of the original conductivity model with six layers. To obtain the conductivities in Cartesian coordinates, we applied the transverse Mercator map projection (latitude and longitude of the true origin are 50°N and 25°E, correspondingly) to the original data, and then performed the interpolation to a laterally regular grid. Note that a similar procedure was invoked to convert ji from spherical to Cartesian coordinates. The lateral discretization and the size of the resulting 3-D part of the conductivity model of Fennoscandia were taken as 5 × 5 km2 and 2,550 × 2,550 km2, respectively. Deeper than 60 km, we used the 1-D conductivity profile obtained by Kuvshinov et al. (2021) (Figure 7d), which is an updated version of the 1-D profile from Grayver et al. (2017).

Conductivity distribution [S/m] in the model of Fennoscandia: (a–c) Plane view on three layers of the 3-D part of the model; (d) global 1-D conductivity profile from Kuvshinov et al. (2021) used in this study. Locations of geomagnetic observatories Abisko (ABK), Uppsala (UPS), and Saint Petersburg (SPG) are marked with circles in plot (a).
Note that the lateral discretization and the size of the conductivity model of Fennoscandia imply that the GEF is calculated at a grid comprising Ng = 512 × 512 points.
3.3 Computation of Ei(rs, ω; σ)
To implement the real-time modeling concept one needs—as is seen from Equations B13 and C2—to compute Ei(rs, ω; σ) at a number of frequencies, or, in other words, to solve Maxwell's equations (Equations 6 and 7). These equations are numerically solved using the 3-D EM forward modeling code PGIEM2G (Kruglyakov & Kuvshinov, 2018), which is based on a method of volume integral equations with a contracting kernel (Pankratov & Kuvshinov, 2016). PGIEM2G exploits a piece-wise polynomial basis; in this study, PGIEM2G was run using the first-order polynomials in lateral directions and third-order polynomials in the vertical direction.
Figures 8-10 demonstrate Ei(rs, ω; σ) at locations of observatories Abisko (ABK), Uppsala (UPS), and Saint Petersburg (SPG), respectively. The results are for the excitations corresponding to the first, seventh, fourteenth and twenty-first SM and are shown for the frequency range from 10−5 to 1 Hz. From these figures, a few observations can be made. The behavior of Ei (with respect to frequency) varies with location and mode. Real and imaginary parts of Ei are comparable in magnitude. As expected, Ei are smooth functions with respect to the frequency; apparent non-smoothness of the results in some plots is due to the fact that absolute values of real and imaginary parts are shown.

From left to right: absolute values of the real part, the imaginary part and the magnitude of Ei(rs, ω; σ) with respect to frequency, and for a number of spatial modes. Results are for observatory Abisko (ABK) located near the seashore (Figure 7a). Top and bottom rows show the results for Ex,i and Ey,i components (in mV/km), respectively.

The same caption as in Figure 8 but for inland, Uppsala (UPS), geomagnetic observatory.

The same caption as in Figure 8 but for Saint Petersburg (SPG) geomagnetic observatory.
Finally, it is important to note that Ei decrease—irrespective of the mode and location—as frequency decreases; specifically, the magnitude of Ei drops down more than two orders of magnitude as frequency decreases from 1 Hz down to 10−3 Hz. These plots suggest a value for T in Equation 10; recall, that useful rule of thumb is that the value for T corresponds to the inverse of frequency at which the field becomes small compared to the higher frequencies. Following this rule, T = 1,000 s seems to be a reasonable choice which will be further justified in the next section.
3.4 Model Study to Justify a Value for T
-
The source jext(t, r) is transformed from the time to the frequency domain with a fast Fourier transform (FFT).
-
Frequency domain Maxwell's equations 1 and 2 are numerically solved using PGIEM2G at FFT frequencies between
and
where K is the length of the event, and Δt is the sampling rate of the considered time series. In this study, Δt is 10 s, and K is 8 hr.
-
E(t, r) is obtained with an inverse FFT of the frequency domain field.


Electric field components at the Abisko (ABK) geomagnetic observatory location calculated using 3-D electromagnetic modeling with 21 spatial modes for the whole 8 hr time interval (from 20:00:00 UT, 7 September 2017, to 03:59:50 UT, 8 September 2017) (red curves) and electric field components at the same observatory calculated using real-time 3-D ground electric field modeling approach with T = 15 min (blue curves) and T = 1 hr (green curves). The results are in mV/km.

The same caption as in Figure 11 but for the Uppsala (UPS) geomagnetic observatory.

The same caption as in Figure 11 but for the Saint Petersburg (SPG) geomagnetic observatory.
ABK | UPS | SPG | |
---|---|---|---|
2017/09/07 20:00:00–2017/09/08 03:59:50 | |||
corr(Ex,15 min, Ex,ref) | 0.984 | 0.991 | 0.989 |
corr(Ex,1 hr, Ex,ref) | 0.984 | 0.995 | 0.995 |
corr(Ey,15 min, Ey,ref) | 0.985 | 0.993 | 0.983 |
corr(Ey,1 hr, Ey,ref) | 0.979 | 0.997 | 0.992 |
2015/03/17 00:00:00–2015/03/18 23:59:50 | |||
corr(Ex,15 min, Ex,ref) | 0.986 | 0.992 | 0.988 |
corr(Ex,1 hr, Ex,ref) | 0.986 | 0.996 | 0.995 |
corr(Ey,15 min, Ey,ref) | 0.984 | 0.993 | 0.983 |
corr(Ey,1 hr, Ey,ref) | 0.980 | 0.997 | 0.992 |
2003/10/29 00:00:00–2003/10/31 23:59:50 | |||
corr(Ex,15 min, Ex,ref) | 0.983 | 0.991 | 0.989 |
corr(Ex,1 hr, Ex,ref) | 0.984 | 0.994 | 0.994 |
corr(Ey,15 min, Ey,ref) | 0.986 | 0.995 | 0.989 |
corr(Ey,1 hr, Ey,ref) | 0.985 | 0.997 | 0.994 |
- Note. The results are shown for three time intervals: from 20:00:00 UT, 7 September 2017, to 03:59:50 UT, 8 September 2017; from 00:00:00 UT, 17 March 2015, to 23:59:50 UT, 18 March 2015; from 00:00:00 UT, 29 October 2003, to 23:59:50 UT, 31 October 2003.
ABK | UPS | SPG | |
---|---|---|---|
2017/09/07 20:00:00–2017/09/08 03:59:50 | |||
nRMSE(Ex,15 min, Ex,ref) | 0.237 | 0.167 | 0.181 |
nRMSE(Ex,1 hr, Ex,ref) | 0.233 | 0.128 | 0.128 |
nRMSE(Ey,15 min, Ey,ref) | 0.227 | 0.147 | 0.228 |
nRMSE(Ey,1 hr, Ey,ref) | 0.238 | 0.112 | 0.161 |
2015/03/17 00:00:00–2015/03/18 23:59:50 | |||
nRMSE(Ex,15min, Ex,ref) | 0.217 | 0.157 | 0.179 |
nRMSE(Ex,1 hr, Ex,ref) | 0.211 | 0.122 | 0.122 |
nRMSE(Ey,15 min, Ey,ref) | 0.232 | 0.143 | 0.214 |
nRMSE(Ey,1 hr, Ey,ref) | 0.233 | 0.112 | 0.158 |
2003/10/29 00:00:00–2003/10/31 23:59:50 | |||
nRMSE(Ex,15 min, Ex,ref) | 0.231 | 0.164 | 0.175 |
nRMSE(Ex,1 hr, Ex,ref) | 0.224 | 0.136 | 0.128 |
nRMSE(Ey,15 min, Ey,ref) | 0.215 | 0.130 | 0.188 |
nRMSE(Ey,1 hr, Ey,ref) | 0.213 | 0.114 | 0.145 |
- Note. The results are shown for three time intervals: from 20:00:00 UT, 7 September 2017, to 03:59:50 UT, 8 September 2017; from 00:00:00 UT, 17 March 2015, to 23:59:50 UT, 18 March 2015; from 00:00:00 UT, 29 October 2003, to 23:59:50 UT, 31 October 2003.
3.5 Computational Loads for the Real-Time GEF Calculation
Once and
are computed and stored on the disc, GEF at a grid
and time instant tk is computed using Equation 11. In accordance with this equation, the GEF calculation requires forecasting/nowcasting the L × Nt array c, reading the L × Nt × Ng array
and L × Ng array
, and performing
summations and multiplications. For our problem setup with Ng = 512 × 512, Nt = 90 and L = 21, the calculation of E(rs, tk; σ) takes from 0.00625 to 0.025 s, depending on the computational environment. Note that to store arrays for this setup one needs 7.25 Gigabytes of disc space.
4 Discussion
4.1 Further Justification of the Concept
So far, we have demonstrated the concept's validity on an example of a single space weather event. However, one can argue that ji(r) obtained for a specific event could be non-adequate for other events. To address this question, we performed the following modeling experiment. First, we built the “sources” (i.e., external equivalent currents) for two other space weather events—St. Patrick's Day geomagnetic storm on 17–18 March 2015 and Halloween storm on 29–31 October 2003—and then approximated corresponding sources using ji(r) obtained for the 7–8 September 2017 storm (Equation 13). Note that for each new event, ci(t) in Equation 13 are calculated using Equation 19, where are taken from the PCA of 7–8 September 2017 data, but Sm(t) are formed using new event data. To ensure that the spatiotemporal structure of the source for new events is different from that of the (reference) 7–8 September 2017 event, we took the new events' lengths as 48 and 72 hr, respectively; recall that the duration of 7–8 September 2017 event was taken as 8 hr. Figure 14 shows snapshots of the original and SM-based sources for St. Patrick's Day (top panels) and Halloween (bottom panels) geomagnetic storms. It is seen from the figure that the SM-based source (with SM obtained from another event) approximates very well the source of the other two events. Figure 15 confirms this inference by showing the agreement between the time series of the original and SM-based sources above exemplary site JCK, again, for Halloween (top panels) and St. Patrick's Day (bottom panels) storms. These results suggest that irrespective of the event (which corresponds to sources of different geometry), the spatial structure of these sources is well approximated by a finite number of SM obtained from the analysis of some specific event. The prerequisite to getting adequate set of SM is that the event to be used for SM estimation should be long enough and sufficiently energetically large and spatially complex.

Left: the original external equivalent current. Right: the external equivalent current calculated using 21 spatial modes. The results are for two time instants: 23:45:40 UT on 17 March 2015 (top panels) and 20:08:20 UT on 30 October 2003 (bottom panels). Note that Jäckvik (JCK) site became a part of the International Monitor for Auroral Geomagnetic Effect network on 1 September 2010. Thus, its data were not used for the equivalent current calculation in case of the 29–31 October 2003 geomagnetic storm. The results are in A/m.

Time series of the original external equivalent current (black curves) and external equivalent current calculated using 21 spatial modes (red curves) above Jäckvik magnetometer (JCK) for two time intervals: from 00:00:00 UT, 17 March 2015, to 23:59:50 UT, 18 March 2015 and from 00:00:00 UT, 29 October 2003, to 23:59:50 UT, 31 October 2003. The location of JCK is shown in Figure 14 as a white circle. Note that JCK magnetometer became a part of the International Monitor for Auroral Geomagnetic Effect network on 1 September 2010. Thus, its data were not used for the equivalent current calculation in case of the 29–31 October 2003 geomagnetic storm. The results are in A/m.
The linked question we also address is whether T = 15 min is a valid choice for the real-time modeling of the GEF during the above-discussed events. As in Section 3.4, we calculate electric fields using Equation 11 with T = 15 min and with T = 1 hr and compare them with the reference fields. Two top panels in Figures 16 and 17 show the comparison of electric field time series at the location of the ABK observatory for Halloween and St. Patrick's Day events, respectively. Similar to the 7–8 September 2017 event results, both “real-time” electric fields agree well with the reference electric field. Besides, bottom panels in Figures 16 and 17 show the difference between “real-time” electric fields calculated using T = 15 min and T = 1 hr. It is seen that this difference is small compared to signals themselves (cf., top panels in the same figures), once again confirming the fact that T = 15 min can be used for the real-time GEF modeling.

The same caption as in Figure 11 but for a 48 hr time interval from 00:00:00 UT, 17 March 2015, to 23:59:50 UT, 18 March 2015. The bottom panel demonstrates absolute differences between ground electric field (GEF) components calculated using real-time 3-D GEF modeling approach with T = 15 min and T = 1 hr.

The same caption as in Figure 11 but for a 72 hr time interval from 00:00:00 UT, 29 October 2003, to 23:59:50 UT, 31 October 2003. The bottom panel demonstrates absolute differences between ground electric field (GEF) components calculated using real-time 3-D GEF modeling approach with T = 15 min and T = 1 hr.
Tables 1 and 2 quantify the agreement for these two events at three geomagnetic observatories, as it was already done for 7–8 September 2017 event. It is seen that the agreement between results for two new events is as good as for the 7–8 September 2017 geomagnetic storm. Notably, throughout all three events, the correlation coefficient is lower, and nRMSE is larger for ABK than for UPS and SPG. This is probably because ABK is located in the region with a large lateral contrast of conductivity (Figure 7), where modeling results are expectedly less accurate.
4.2 Nowcasting and Forecasting GEF Using the Proposed Concept
-
Using magnetic field data collected at an observational network for historical (past) event/several events one obtains
at rm. This is done by exploiting the procedure described in Section 3.1. These
allow us to represent the source at any time instant t and at any position r via Equations 13-15. In this paper, we used IMAGE network of magnetic field data to obtain L = 21
and further ji(r). Using IMAGE data, we confine ourselves to Scandinavian region. If Canada, for example, is a region of interest, one would use the data from the Canadian networks of magnetic field observations, like CARISMA (Mann et al., 2008) and AUTUMNX (Connors et al., 2016).
-
Once
and, subsequently, ji(r) are obtained and stored, one estimates electric field at the current time instant, tk, using Equation 11. This, in particular, requires knowledge of coefficients ci at time instant tk and at a number of time instants in the past, tk − Δt, tk − 2Δt, …, tk − NtΔt. The coefficients at these instants are obtained by reusing Equations 16 and 19, namely.

-
One obtains
and, subsequently, ji(r) in a similar manner as it is done in case of the GEF nowcasting.
-
One trains the neural network (NN) using as input data the time series of solar wind parameters collected by satellite(s) at the L1 Lagrangian point and ci(t) as output data. Time series ci(t) for the training period are obtained using Equations 16 and 19. There is a common understanding that the longer time series are used for the training phase, the better the quality of the forecasted results. Therefore, this period preferably should include multiple years of the L1 and ground magnetic field data; recall that ci(t) during the training phase are obtained from the ground magnetic field data.
-
One forecasts GEF using the trained NN. Ideally, one has to forecast well ahead. However, given observations made at the L1 point, a geomagnetic disturbance is seen on the ground as fast as an hour ahead. This time latency can be further shrunk to half an hour or so, depending on the solar wind speed. This, in particular, advocates real-time modeling of the GEF which is a topic of this paper.
5 Conclusions
In this paper, we presented a formalism for real-time computation of the GEF in a given 3-D Earth's conductivity model excited by a continuously augmented spatially and temporally varying source responsible for a space weather event.
The formalism relies on the factorization of the source by SM and time series of respective expansion coefficients, and exploits precomputed GEF kernels generated by corresponding SM.
To validate the formalism, we invoked a high-resolution 3-D conductivity model of Fennoscandia and considered a realistic source built with the use of the SECS method as applied to magnetic field data from the IMAGE network of observations. Factorization of the SECS-recovered source is then performed using the PCA. Eventually, we show that the GEF computation at a given time instant on a 512 × 512 grid requires at most 0.025 s provided that GEF kernels due to the pre-selected SM are computed in advance. This opens a practical opportunity for GEF nowcasting, using ground magnetic field data, or even forecasting, using both ground magnetic field and L1 data.
We illustrate the concept on a Cartesian geometry problem setup. Global-scale implementation is rather straightforward; for this scenario, the source could be obtained either using magnetic field data from the global network of geomagnetic observatories or exploiting the results of the first-principle modeling of the global magnetosphere-ionosphere system.
Acknowledgments
M. Kruglyakov was supported by the New Zealand Ministry of Business, Innovation, and Employment through Endeavour Fund Research Programme contract UOOX2002. A. Kuvshinov was supported in the framework of Swarm DISC activities, funded by ESA contract no. 4000109587, with the support from EO Science for Society. E. Marshalko was supported by Grant 21-77-30010 from the Russian Science Foundation. The authors thank the institutes that maintain the IMAGE Magnetometer Array: Tromsø Geophysical Observatory of UiT, the Arctic University of Norway (Norway), Finnish Meteorological Institute (Finland), Institute of Geophysics Polish Academy of Sciences (Poland), GFZ German Research Center for Geosciences (Germany), Geological Survey of Sweden (Sweden), Swedish Institute of Space Physics (Sweden), Sodankylä Geophysical Observatory of the University of Oulu (Finland), and Polar Geophysical Institute (Russia). The authors thank INTERMAGNET (www.intermagnet.org) for promoting high standards of magnetic observatory practice.
Appendix A: Properties of Transfer Functions and Impulse Responses
- 1.
Linearity allows us to define the response, ζ(t), of the medium at time t to an extraneous forcing as

-
where χ is the extraneous forcing that depends on time t′ and
is the medium Green's function.
- 2.
Stationarity implies that the response of the medium does not depend on the time of occurrence of the excitation. In this case
and Equation A1 is rewritten as a convolution integral

-
where f(t) represents the impulse response of the medium. In the frequency domain, the convolution integral degenerates to

-
where
is called the transfer function and we use tilde sign
to denote complex-valued quantities. Equations A2 and A3 are related through the Fourier transform

- 3.
Since we work in the time domain with a real-valued forcing, the impulse response is also real. To see implications of this, let us define the inverse Fourier transform of
as

-
For an impulse response to be real, the last term in the integral (Equation A5) has to vanish. This is possible only if fR(ω) and fI(ω) are even and odd functions of ω, respectively. Therefore, Equation A5 reduces to

- 4.
Impulse response is causal. This property implies that

-
Under this assumption, the convolution integral (Equation A2) is recast to

-
Further, using the fact that cos(ωt) and sin(ωt) are odd and even functions with respect of t, one obtains for t > 0

-
Using the latter equation and Equation A6 one can state that the impulse response is determined by using either only real or imaginary part of
:

Appendix B: Details of the Numerical Computation of the Real-Time GEF






The integrals can be computed using the digital filter technique (see Appendix C), whereas first term in the RHS of Equation B5 is estimated as follows.











An important note here is that, according to Equation B13, one does not need to compute Ei(rs, ω; σ) for . This may be obvious, however, this is not the case if one uses piece-wise constant (PWC) approximation of ci(t) as it is done, for example, in Grayver et al. (2021). With PWC approximation, one is forced to compute the fields at very high frequencies irrespective of Δt value; this can pose a problem from the numerical point of view.
Appendix C: Computation of 






Appendix D: Formulas for P and Q





From Equations D1 and D2, it is seen that P(r, rm) and Q(r, rm) tend to infinity as r tends to rm. The simplest way to deal with this issue is, as mentioned in Vanhamäki and Juusola (2020), to consider the grids for r and rm that are shifted with respect to each other. This approach is used in the current paper.
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Data Availability Statement
The SMAP model is available via the EPOS portal (http://mt.bgs.ac.uk/EPOSMT/2019/MOD/EPOSMT2019_3D.mod.json).