# Beyond Lithium-Ion Batteries: Are Effective Electrodes Possible for Alkaline
and Other Alkali Elements? Exploring Ion Intercalation in Surface-Modified
Few-Layer Graphene and Examining Layer Quantity and Stages

Yu-Hsiu Lin  Department of Chemical Engineering & Materials Science, Michigan
State University, East Lansing, Michigan 48824, United States.  Jose L.
Mendoza-Cortes<EMAIL_ADDRESS>Department of Physics & Astronomy, Michigan
State University, East Lansing, Michigan 48824, United States.  Department of
Chemical Engineering & Materials Science, Michigan State University, East
Lansing, Michigan 48824, United States.

###### Abstract

In the pursuit of reliable energy storage solutions, the significance of
engineering electrodes cannot be overstated. Previous research has explored
the use of surface modifiers (SMs), such as single-side fluorinated graphene,
to enhance the thermodynamic stability of ion intercalation when applied atop
few-layer graphene (FLG). As we seek alternatives to lithium-ion batteries
(LIBs), earth-abundant elements like sodium and potassium have emerged as
promising candidates. However, a comprehensive investigation into staging
intercalation has been lacking thus far. By delving into staging assemblies,
we have uncovered a previously unknown intercalation site that offers the most
energetically favorable binding. Here we study the first three elements in
both alkali (Li, Na, K) and alkaline (Be, Mg, Ca) earth metals. Furthermore,
the precise mechanism underlying this intercalation system has remained
elusive in prior studies. In our work, we employed Density Functional Theory
(DFT) calculations with advanced hybrid functionals to determine the
electrical properties at various stages of intercalation. This approach has
been proven to yield more accurate and reliable electrical information.
Through the analysis of projecting density of states (PDOS) and Mulliken
population, we have gained valuable insights into the intricate interactions
among the SM, ions, and FLG as the ions progressively insert into the
structures. Notably, we expanded our investigation beyond lithium and explored
the effectiveness of the SM on ions with varying radii and valence,
encompassing six alkali and alkaline earth metals. Additionally, we discovered
that the number of graphene layers significantly influences the binding
energy. Our findings present groundbreaking concepts for material design,
offering diverse and economically viable alternatives to LIBs. Furthermore,
they serve as a valuable reference for fine-tuning electrical properties
through staging intercalation and the application of SMs.

##  I Introduction

The usage of lithium-ion batteries (LIBs) has become omnipresent in our daily
lives, leading to a growing concern about potential supply shortages of
lithium [1, 2, 3, 4]. Consequently, there is a significant focus on developing
batteries that are both more efficient and cost-effective [5, 6, 7, 8, 9]. One
way to address these concerns is by effectively utilizing lithium, such as by
improving its electric capacity through engineering the electrode materials.
Carbon-based materials like graphite and few-layer graphene have shown promise
in enhancing ion intercalation [10, 11, 12, 13, 14]. In addition to optimizing
lithium usage, researchers are exploring alternative intercalants to lithium,
such as sodium and potassium atoms, due to their similar electronic properties
[15, 16, 17, 18, 19]. These alternatives are being investigated as promising
options for intercalation within layer-structured electrodes. Furthermore,
some alternative metals offer more competitive prices compared to lithium. For
example, sodium and magnesium can be economical alternatives based on their
carbonate counterparts [20]. However, these alternative intercalants present
certain challenges that differ from those of lithium. These challenges can
adversely affect the intercalation process and result in reduced storage
capacity [21, 22, 23]. One disadvantage is their larger atomic radius compared
to lithium, which introduces an additional energy barrier for intercalation
and requires a greater degree of geometric transformation to accommodate more
spacious host sites [13, 24].

Figure 1: Key principles explored in this study: (a) The side view and (b) top
view of a Two-Dimensional 50% Fluorinated Monolayer Graphene (2D FSM) Layer;
(c) The side view of an FSM layer showing its intrinsic dipole moment; (d)
Augmented Intercalant Capacity via FSM Insertion, highlighting electron
transfer pathways (shown in the yellow background) facilitating interaction
among various components within the system; (f) The top view of fluorinated
4LG with intercalants at stage 1.

Hence, exploring potential alternatives with comparable atomic radii provides
a new avenue for research. One such alternative is the magnesium atom, which
possesses a slightly smaller radius than lithium and can also act as a
multivalent ion, thereby enhancing battery capacity by carrying more charge
per intercalant. In this study, both electrode engineering and lithium
alternatives were investigated. For the electrodes, the focus was on a
fluorinated surface modifier (FSM) applied to the top of few-layer graphene
(FLG). While graphenes can serve as electrodes and have demonstrated higher
lithium storage capacity than graphite [25, 26, 27, 28, 29, 30], sodium (Na)
intercalation remains unfavorable [31, 32]. Therefore, various modifications
of graphenes, such as defects and surface modifiers, were employed to enhance
intercalation and ion transport [33, 34, 35, 36, 37, 38]. One example is
fluorine-doped FLG, which capitalizes on the high electronegativity of
fluorine to improve capacity [39]. Moreover, fluorinated graphenes can be
easily synthesized and possess unique electronic properties [40, 41, 42].
These considerations led to the development of FLG with FSM. Building upon our
previous work [43], a monolayer of fluorinated graphene on top of four-layer
graphene (4LG) has been shown theoretically to enhance the intercalation of
lithium, sodium, and potassium atoms within the graphene layers, exhibiting
lower binding energies compared to pristine 4LG. This two-dimensional
modification not only reinforces lithium intercalation in FLG but also makes
the substitution of lithium with other atoms more promising.

However, the precise mechanism by which the FSM influences the intercalation
capability in FLG remains unclear and necessitates further investigation.
Considering the high electronegativity of fluorine atoms, we propose that the
electric potential difference induced by the FSM acts as both the driving
force for charge transfer and the stabilizer for the intercalant. To verify
this assumption, we computed the electronic properties (e.g., PDOS, Mulliken
analysis) of ion-intercalated FLG with and without the FSM, in addition to the
relative binding energy. For this purpose, we employed density functional
theory (DFT) with the advanced hybrid functional B3LYP-D3, which enables
accurate predictions of electronic properties, albeit at a higher
computational cost compared to the previously used PBE functional [44].
Moreover, the intercalation mechanism of lithium in carbon-based layers is
known to involve a gradual staging process, resulting in ordered intercalation
at varying concentrations [45, 46, 47]. Similarly, potassium (K) and sodium
(Na) ion intercalation can be described as staging procedures as well [21, 48,
49]. However, FLG with the same ion concentration can exhibit different
assemblies, with ions intercalated within different layers. By introducing a
single-side fluorinated SM, the number of potential assemblies is further
increased due to the breaking of structural symmetry. Unfortunately, limited
research has been conducted on intercalant assemblies in FLG. We believe that
these assemblies can impact thermodynamic stability and electronic properties.
In this study, we not only discuss different staging structures but also
consider how they host intercalants. Interestingly, staging structures with
intercalation sites between the FSM and its neighboring FLG layer are more
likely to be energetically favorable assemblies. This finding underscores the
significance of the FSM, as it not only reduces binding energies but also
creates more stable host sites.

To extend the scope of the FSM’s applicability, we proceeded to study the
intercalation of the first three elements in both alkali and alkaline earth
metals within FLG. Our objective was to unravel the interactions between the
FSM and ions with varying atomic radii and multivalency. The energetic
outcomes revealed that, irrespective of the type of intercalants, all
scenarios involving the FSM exhibited enhanced binding between the
intercalants and the graphene substrate. However, their respective electronic
interactions displayed distinct characteristics. Additionally, the number of
graphene layers emerged as a critical factor affecting the binding energy.
Fewer layers in FLG corresponded to lower binding energies, underscoring the
significance of FLG exfoliation from graphite when considering electrode
materials. In summary, these findings present multiple potential alternatives
to LIBs and shed light on the long-standing ambiguity surrounding surface
modifiers and staging intercalation. Furthermore, the implications of this
research extend beyond the realm of batteries, as it offers valuable insights
for tuning the stability and electrical properties of layered materials
through the application of SMs or by varying the density of intercalants and
the number of layers.

Figure 2: Comparison of binding energies between staging assembles at stage 3
and stage 2, involving ion intercalations of (a) Li, (b) Be, (c) Na, (d) Mg,
(e) K, and (f) Ca within pristine 4LG. The schematic diagrams, inserted for
clarity, correspond to the respective binding energies. Figure 3: Comparison
of binding energies for staging assembles at stage 3 and stage 2, involving
ion intercalations of (a) Li, (b) Be, (c) Na, (d) Mg, (e) K, and (f) Ca within
fluorinated 4LG. The schematic diagrams, inserted for clarity, correspond to
the respective binding energies.

##  II Methodology

Calculations were carried out using unrestricted periodic DFT with the B3LYP
hybrid functional [50, 51], as implemented in the CRYSTAL17 code [52] with
Gaussian-type basis sets. Since the slab structures with multiple layers are
represented within extended unit cells, the D3 dispersion correction [53, 54,
55], accounting for long-range Coulomb correlation and weak London forces, was
combined with the B3LYP functional. Specifically, Gaussian triple-zeta valence
with polarization (TZVP) basis sets [56] was used to represent the orbitals of
each element, and full optimization was performed to adjust both atom
positions and unit cell parameters. To prevent interactions between cells and
simulate surface conditions, a ${\sim}$500 Å vacuum layer along the z-axis was
employed for the slab structures. For self-consistent field (SCF)
calculations, direct inversion of the iterative subspace (DIIS) [57, 58] was
utilized to achieve an energy convergence criterion of 2.72 $\times$ 10–6 eV.
The convergence criteria for geometry optimization were set to an RMS force of
1.54 $\times$ 10–2 eV/Å, a max force of 2.31 $\times$ 10–2 eV/Å, an RMS
displacement of 6.35 $\times$ 10–4 Å, and a max displacement of 9.53 $\times$
10–4 Å. Subsequently, based on the optimized structures, relative electronic
properties, including density of states (DOS) and Mulliken analysis, were
calculated. To obtain higher-resolution electronic results, the first
Brillouin zone (${\sim}$$2\pi$ $\times$ $1/60$ Å–1) was sampled using a
$k_{a}$ $\times$ $k_{b}$ $\times$ $k_{c}$ Pack-Monkhorst k-mesh grid with
larger shrinking factors ($a\cdot k_{a}$ $\geqq$ 60, $b\cdot k_{b}$ $\geqq$
60, kc = 1) compared to those used for geometric optimization ($a\cdot k_{a}$
$\geqq$ 40, $b\cdot k_{b}$ $\geqq$ 40, kc = 1) [59]. The cohesive energies
were computed using the HSE06-D3 hybrid functional [60] after geometric
optimization with B3LYP-D3. HSE06-D3 incorporates a short-range coulombic
operator and has demonstrated accurate predictions for metallic materials
[61]. The results of cohesive energies based on different functionals were
compared (see Figure S1). It is important to note that the functionals used to
calculate each comparative term of binding energy were unified. The
calculation of binding energy can be referred to a previous study [43].

Cohesive energy ($E_{coh}$) refers to the energy needed to evaporate an
isolated atom from its bulk metal, and it is calculated using equation (1).

$E_{coh}=E_{gas}–\frac{E_{bulk}}{n}$ (1)

Here, $n$ represents the number of atoms in the bulk metal, while $E_{gas}$
and $E_{bulk}$ correspond to the total energy of an isolated atom and the bulk
metal, respectively.

Figure 4: Comparison of binding energies between intercalations with and
without FSM, covering the cases of (a) Li, (b) Be, (c) Na, (d) Mg, (e) K, and
(f) Ca at each stage.

##  III Discussion

The highlights of this study include not only the applicability of FSM to
diverse beyond-Li intercalants but also the scheme behind the consequences.
The main character and the philosophies around it are illustrated in Figure 1
first. Prior to demonstrating the effectiveness of the FSM, it is crucial to
determine the most thermodynamically stable staging assemblies as a basis for
comparison. Since previous studies on staging assemblies were not exhaustive,
this work considers all possible configurations. Figure 2 and Figure 3 present
every assembly at each stage, along with their corresponding binding energies.
Specifically, Figure 2 displays the results for pristine 4LG, while Figure 3
depicts those for fluorinated 4LG. For pristine 4LG at stage 3, structures
with the intercalants located at the edge sites are more likely to form stable
bindings. At stage 2, assemblies with intercalants positioned both at the edge
and middle sites tend to be more energetically favorable. Notably, the
intercalation of lithium and potassium within pristine 4LG can be
thermodynamically stable even without the assistance of the FSM (Figure 2).
This advantage highlights the widespread usage of LIBs in today’s applications
and underscores the potential of potassium-ion batteries. Moving to
fluorinated 4LG at stage 3, structures with the intercalant closest to the FSM
consistently exhibit the lowest binding energies. Interestingly, as the
intercalant moves further away from the FSM, the binding energies gradually
increase. This observation suggests that the FSM can significantly impact
conditions with low intercalant concentrations, as the preferred host site is
just below the FSM, which was not considered in previous work [43]. On the
other hand, fluorinated 4LG at stage 2 shows various assemblies for the most
stable bindings.

Figure 5: Averaged net flow of electrons for (a) fluorine and (b) fluorinated
carbon in the FSM, considering the intercalations of Li, Na, K, Be, Mg, and Ca
within fluorinated 4LG at each stage. The dashed lines represent the averaged
net flow of electrons in the case without any intercalant. Figure 6: Average
valence of intercalants, including (a) Li, (b) Be, (c) Na, (d) Mg, (e) K, and
(f) Ca, when intercalated within fluorinated 4LG at each stage.

With a clear understanding of the dominant assemblies for each element at
different stages, a comparison between the intercalation of pristine 4LG and
fluorinated 4LG becomes applicable. As shown in Figure 4, the binding energies
at each stage decrease significantly with the presence of the FSM, regardless
of the element. Remarkably, while Li and K can spontaneously intercalate into
pristine 4LG, the intercalation of Na, Be, Mg, and Ca becomes
thermodynamically more favorable when fluorinated 4LG is considered. This
finding unveils several potential alternatives to LIBs. Furthermore, the
effectiveness of the FSM is particularly prominent at a low ion concentration
level, as evidenced by the more pronounced decrease in binding energy at stage
3 compared to stages 2 and 1. As the ion concentration in 4LG increases, from
stage 3 to stage 2 or from stage 2 to stage 1, the impact of the FSM weakens.
Nevertheless, regardless of the stage, the theoretical specific capacity of
fluorinated FLG outperforms pristine graphitic anodes. For instance, as
presented in Table S1, the computed capacities of Na (stage 3: 73; stage 2:
137; stage 1: 194 mAhg-1) and Ca (stage 3: 139; stage 2: 252; stage 1: 346
mAhg-1) ion intercalation in fluorinated 4LG are substantially larger than the
experimentally found capacities (Na: 34; Ca: 97 mAhg-1) in graphitic
electrodes [62, 63]. On the other hand, the computed capacity of K-ion
intercalation in pristine 4LG at stage 1, which already exhibits favorable
binding energy, aligns well with the experimental result ($\sim$ 200 mAhg-1)
[64].

Figure 7: The Partial Density of States (PDOS) of ion intercalations within
fluorinated 4LG from stage 3 to stage 1 (from left to right), comprising the
cases of (a) Li, (b) Be, (c) Na, (d) Mg, (e) K, and (f) Ca. The PDOS of
fluorinated 4LG is shown in Figure S3. Figure 8: The comparison of binding
energies for ion intercalation within fluorinated 4LG and 5LG, encompassing
the cases of (a) Li, (b) Na, and (c) Mg at various ion concentrations.

This study not only considers intercalant assemblies but also expands the
range of alternatives to LIBs. However, the role of FSM in this context
remains to be elucidated. Therefore, to deepen our understanding of the
underlying mechanism, we investigate the electronic properties. Specifically,
we focus on the redistribution of electrons and the electron pathways between
each element in the system with FSM. Mulliken analysis was employed to gain
insights into the electron distribution. As depicted in Figure 5, the fluorine
in the FSM gains electrons, while the carbon connected to that fluorine loses
electrons. This phenomenon can be attributed to the ultrahigh
electronegativity of fluorine, which tends to draw electrons. Moreover, in the
presence of intercalants, the fluorine and F-connected carbon show an
increased tendency to gain and lose electrons, respectively, compared to
fluorinated 4LG without any intercalant. However, the net electron of fluorine
and F-connected carbon is negative, leading to the transfer of those missing
electrons to other non-fluorinated carbon atoms, making them positively
charged, especially the carbons in the FSM layer. Regarding the intercalants,
Figure 6 illustrates the averaged valence of intercalants at each stage. The
valence of alkali metals ranges from 0 to 0.45, while that of alkaline earth
metals ranges from 1 to 1.3. Notably, Li shows limited participation in charge
transfer with others in the system, rendering it nearly neutral compared to
other types of intercalants. This observation suggests that the intercalation
of Li within FLG relies on covalent bonds, while the other intercalants
primarily form ionic bonds with negatively charged non-fluorinated carbons.
Covalent bonds, held together by electrostatic forces between shared electrons
and nuclei, require less energy to form than ionic bonds, partially accounting
for the advantage of lithium-ion intercalation. Interestingly, with FSM, other
intercalants at stage 3 tend to retain more electrons (lower valence) compared
to higher intercalant concentration stages (stage 2 and stage 1). This
behavior aligns with the previous finding that the FSM plays a more effective
role at low ion concentration stages, as demonstrated in the comparison of
binding energies in Figure 4. Additionally, a larger difference between
electronegativities leads to a greater amount of transferred electrons. Hence,
compared to either carbon or fluorine, K loses more electrons (higher valence)
than Na does and even more than Li does, as shown in Figure 6. However, this
effect is less pronounced in alkaline earth metals compared to alkali metals.

The results of Mulliken analysis provide insight into the electron
distribution in the system, but the mechanism of electron transfer remains a
question. If there is no or insufficient channel for electrons between each
element, the flow and drainage of electrons can exhibit distinct behaviors. To
shed light on this aspect, electron density of states (DOS) can offer valuable
information by indicating the presence of adequate electron channels based on
the overlap of DOS. Figure S2 and 7 represent the elemental PDOS, illustrating
the DOS of each element. In the case of pristine 4LG, Figure S2 reveals a
clear overlap between carbon and each intercalant, and as the system
approaches higher intercalant concentrations from stage 3 to stage 1, this
overlap gradually increases. A similar phenomenon is observed in the
intercalation of fluorinated 4LG, as shown in Figure 7, indicating the
existence of electron channels between carbon and intercalants. However,
fluorinated 4LG offers additional channels, not limited to intercalant-carbon
interactions, but also open to intercalant-fluorine and carbon-fluorine
interactions. This suggests that, with the assistance of the FSM, electron
transfer can exhibit greater flexibility, leading to a more stable ground
state and lower binding energy. Specifically, the overlap of DOS with FSM
around the Fermi energy level, which is more likely to be occupied by
electrons, tends to be greater compared to that of pristine 4LG.

Furthermore, the positive impact of FSM on ion intercalation in FLG is not the
only contributing factor; the dimension of FLG also plays a significant role.
To illustrate this, a comparison of binding energies for Li, Na, and Mg
between 4LG and 5LG with FSM is shown in Figure 8. Given that the
configurations of each stage differ between 4LG and 5LG, the intercalant
concentration was considered instead of the stage. Similar to 4LG, FSM in 5LG
proves to be more effective at lower ion concentrations and in staging
assemblies with intercalants positioned close to it, as depicted in Figure 8
and Figure S4, respectively. Additionally, the binding energy in 5LG gradually
increases as the intercalant concentration rises. However, it is noteworthy
that the intercalation within fluorinated 4LG outperforms that of fluorinated
5LG, as it requires less energy for binding at each stage. This finding
provides a valuable clue for practical energy storage applications, suggesting
that a higher degree of graphitic exfoliation for graphene could further
enhance the capacity of FLG electrodes.

##  IV Conclusion

In conclusion, this comprehensive study thoroughly explores the application of
fluorinated FLG in diverse ion intercalations (Li, Na, K, Be, Mg, and Ca) by
considering staging assemblies, electronic properties, and dimensions. By
comparing the binding energies of staging configurations, we found that FSM
proves to be remarkably effective in facilitating intercalation at low ion
concentrations. Notably, the most preferable host layer at stage 3 is located
beneath FSM, and as the intercalant moves farther away from FSM, the binding
energy gradually increases. This trend applies to all six types of ions. Most
importantly, FSM was observed to either reverse nonspontaneous intercalations
(Na, Be, Mg, and Ca) or further facilitate spontaneous interactions (Li and K)
at each stage, opening up possibilities for multiple potential alternatives to
LIBs. Moreover, our electronic analysis sheds light on the mechanism of FSM-
assisted intercalation. Specifically, the Mulliken analysis suggests that Li-
ion intercalation may involve covalent bonds, while other intercalants tend to
undergo ionic interactions with significant charge transfer. Additionally,
based on the PDOS, FSM creates additional channels for flexible electron
transfer, enabling the system to reach a more stable ground state.
Furthermore, we found that the number of layers can influence the theoretical
capacity of FLG. Intercalation in fluorinated 4LG was observed to be more
thermodynamically stable than in fluorinated 5LG. In summary, this work not
only highlights the advantages of applying FSM in beyond-lithium-ion batteries
but also provides dimensional and electronic perspectives on electrode
engineering. These findings contribute valuable insights to the development of
advanced energy storage systems.

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##  V Supporting Information

Table S 1: The maximum theoretical specific capacity of each intercalated system. Model | Stage 3 / $mAh~{}g^{-1}$ | Stage 2 /$mAh~{}g^{-1}$ | Stage 1 $mAh~{}g^{-1}$
---|---|---|---
P 4LG-Li | 90.82 | 177.46 | 260.22
F 4LG-Li | 76.12 | 149.30 | 219.71
P 4LG-Na | 86.14∗ | 160.42∗ | 225.15∗
F 4LG-Na | 72.80 | 137.05 | 194.17
P 4LG-K | 81.90 | 146.32 | 198.32
F 4LG-K | 69.75 | 126.63 | 173.88
P 4LG-Be | 180.37∗ | 350.13∗ | 510.18∗
F 4LG-Be | 151.35 | 295.20 | 432.08
P 4LG-Mg | 171.55∗ | 318.34∗ | 445.37
F 4LG-Mg | 145.09 | 272.27 | 384.67
P 4LG-Ca | 163.31∗ | 291.08 | 393.78
F 4LG-Ca | 139.15 | 252.08 | 345.57
F 5LG-Li | 63.19 | 124.35 | 240.94
F 5LG-Na | 60.80 | 115.73 | 210.57
F 5LG-Mg | 121.41 | 230.16 | 416.84
*Notes:
1\. ∗ represents the system with thermodynamically unfavorable binding energy.
2\. P stands for pristine, while F means fluorinated.
Figure S 1: The comparison of cohesive energy calculated in different
functionals, including B3LYP-based geometric optimization, B3LYP-based
geometric optimization with HES06-based singlepoint calculation, and
HES06-based geometric optimization. Figure S 2: The Partial Density of
States (PDOS) of ion intercalations within pristine 4LG from stage 3 to stage
1 (from left to right), comprising the cases of (a) Li, (b) Be, (c) Na, (d)
Mg, (e) K, and (f) Ca. Figure S 3: The PDOS of fluorinated 4LG. Figure S
4: The comparison of binding energies of each staging assembles at stage 3
(left) and stage 2 (right) in the case of Na ion intercalation within
fluorinated 5LG. The inserted schematic diagrams are aligned with the
corresponding binding energies.


11institutetext: Astronomy & Astrophysics Section, DIAS Dunsink Observatory,
Dublin Institute for Advanced Studies, Dublin, Ireland 22institutetext: School
of Physics, Trinity College Dublin, Dublin, Ireland 33institutetext: South
East Technological University, Carlow, Ireland 44institutetext: Geological
Survey Ireland, Dublin, Ireland 55institutetext: Department of Geophysics,
University of Oslo, Oslo, Norway 66institutetext: Geophysics Section, Dublin
Institute for Advanced Studies, Dublin, Ireland

# Nowcasting Geoelectric Fields in Ireland using Magnetotelluric Transfer
Functions

John Malone-Leigh*<EMAIL_ADDRESS>1122 Joan Campanyà 3344 Peter T.
Gallagher 11 Maik Neukirch 55 Colin Hogg 66 Jim Hodgson 44

(Received 02 Aug 2022; accepted 07 Feb 2023)

###### Abstract

Geomagnetically induced currents (GIC) driven by geoelectric fields pose a
hazard to ground-based infrastructure, such as power grids and pipelines.
Here, a new method is presented for modelling geoelectric fields in near real
time, with the aim of providing valuable information to help mitigate the
impact of GIC. The method uses magnetic field measurements from the
Magnetometer Network of Ireland (MagIE; www.magie.ie), interpolates the
geomagnetic field variations between magnetometers using spherical elementary
current systems (SECS), and estimates the local electric field using a high
density ($<~{}40~{}km$) network of magnetotelluric transfer functions (MT-TF)
encompassing the island. The model was optimised to work in near real time,
with a correction curve applied to the geoelectric field time series. This
approach was successfully validated with measured electric fields at four
sites for a number of geomagnetic storms, providing accurate electric fields
up to a 1-minute delay from real time, with high coherence (0.70 – 0.85) and
signal-to-noise ratio (SNR; 3.2 – 6.5) relative to measured electric field
validation time series .This was comparable to a standard non real-time
geoelectric field model (coherence = 0.80 – 0.89 and SNR = 4.0 – 7.0). The
impact of galvanic distortion on the model was also briefly evaluated, with a
galvanic distortion correction leading to a more homogeneous representation of
the direction of the electric field, at a regional scale.

###### keywords:

Keywords: geoelectric fields – nowcasting – magnetotellurics– GIC – geohazard
– galvanic distortion

##  1 Introduction

Magnetic variations originating from currents in the magnetosphere and
ionosphere are responsible for induced geoelectric fields at the Earth’s
surface. Geoelectric fields are a significant hazard to ground-based
technology, being responsible for geomagnetically induced currents (GIC) in
pipelines and power grids. Many examples exist of damage to transformers and
disruption to power networks, such as the well known example of Quebec, Canada
in 1989, as well other examples such as New Zealand in November, 2001 (Allen
et al., 1989; Béland and Small, 2004; Bolduc, 2002) and South Africa and
Sweden in October, 2003 (Gaunt and Coetzee, 2007; Koen and Gaunt, 2003;
Pulkkinen et al., 2005). In Ireland, Blake et al. (2016) and Blake et al.
(2018) evaluated the consequences of severe geomagnetic storms on the Irish
power grid. They predicted a potentially large GIC in the south-west of the
island. Accurate real-time assessment and forecasting of geoelectric fields
could aid in reducing the hazard posed by some of these storms.

####

To monitor the potential impacts caused by geomagnetic storms on ground-based
infrastructure, knowledge about the intensity and direction of the geoelectric
fields is required. This can be achieved by either measuring or modelling the
geoelectric fields. Modelling geoelectric fields is the practical approach for
two main reasons: 1) Geological variations resulting in substantial
differences in geoelectrical properties at lithospheric depth (the outermost
shell of the Earth made up of the crust and a portion of upper mantle) and can
lead to significant differences in the geoelectric fields between nearby
stations. The complex distribution of the geoelectrical properties of
Ireland’s lithosphere requires many measuring electrometers, a network which
would be challenging and expensive to operate. 2) Real-time measurements of
geoelectric fields are highly sensitive to localised noise from electrical and
ferrous sources (Schmidt et al., 2020). While modelling is preferred for real-
time monitoring of geoelectric fields, measurements of geoelectric fields are
key to validating the model.

####

Magnetotelluric transfer functions (MT-TF) are an effective approach for
modelling geoelectric fields, suitable to the complexity of the geoelectrical
structures beneath the island of Ireland (Rao et al., 2014). MT-TF relate
magnetic variations to electric field variations using MT-TF at individual
sites, related to the subsurface resistivity of the lithosphere. The use of 3D
resistivity models and 3D MT-TF produced highly accurate results, when
compared to other common geoelectric field models, such as simpler 1D
resistivity profiles and the thin sheet model (Weigel, 2017; Beggan et al.,
2021). MT-TF were chosen for this study over traditional 3D lithospheric
models, because of their computational efficiency, as they only use a transfer
function and do not require a full resistivity model of the lithosphere. This
is important when nowcasting, as a time efficient method is required to ensure
the model is truly real time. However, recent research by Kruglyakov et al.
(2022) presents a possible alternative to this problem, by using a memory-
based method to significantly reduce computation time of 3D resistivity
models.

####

Magnetic field data are monitored at specific geomagnetic observatories, and
so, there is need for an interpolation approach to constrain the magnetic
field variations at the MT sites used to model the geoelectric fields. A
Spherical Elementary Current Systems (SECS) interpolation was developed to
model magnetic field variations. The SECS method (Amm, 1997; Amm and Viljanen,
1998) is commonly used to estimate magnetic field variations at sites without
magnetic field data, and uses nearby magnetometers as inputs (Marshalko et
al., 2021; Beggan et al., 2021; Bosse et al., 2022) to construct a plane of
equivalent sheet currents in the ionosphere, for a set altitude. More
computationally efficient approaches were considered for estimating the
geomagnetic field, such as magnetic perturbation functions (Simpson and Bahr,
2020). Given that Ireland is a small region, a simpler approach may produce
accurate results and would speed up computation time. However, significant
differences were observed between the north, south east and west of Ireland,
in terms of amplitude and local effects during geomagnetic storms (Blake
(2017), example in Fig. 2) and as such, a more complex interpolation method
was required to obtain reliable results. The SECS method works reliably for
sites at higher latitudes, including similar latitudes to Ireland (McLay and
Beggan, 2010). It must be noted, however, that the accuracy of SECS
interpolations fall off at middle and low latitudes (Torta et al., 2017) and
may require separate interpolation methods, or a modification of the SECS in
these regions, to return accurate results (Vanhamäki et al., 2020).

####

Here, we combined the SECS-MT approach with the real-time FFT modelling method
of Kelbert et al. (2017) and expand on it by applying an additional real-time
correction. The nowcast geoelectric field was compared with and optimised
using a standard non real-time model (Section 3.5), using MT-TF from Space
Weather Electromagnetic Database for Ireland (SWEMDI; Campanyà et al., 2018).
The optimisation involved obtaining a correction curve, which was used to
correct the electric field time series near real time, to provide the most
accurate electric fields at each instance, for generation of a real-time
movie. The nowcasting approach, based on the SECS-MT modelling technique, was
validated against measured geoelectric field data (Section 3.6). In addition,
the impact of galvanic distortions on modelling geoelectric fields was
evaluated, comparing results with and without correcting the MT impedance
tensors for galvanic distortion (3.8) following the approach of Neukirch et
al. (2020).

####

##  2 Methods

###  2.1 Geomagnetic Fields: SECS

Magnetic field variations are monitored in Ireland by the Magnetometer Network
of Ireland (MagIE, www.magie.ie) at Armagh, Birr and Valentia observatories
(Fig. 1). Magnetic time series (1 – second resolution) from the magnetic
observatories were used to estimate the magnetic field (Fig. 2) at each of the
MT sites in Figure 1 using a SECS interpolation. The SECS interpolation was
implemented across Ireland using a 0.5${}^{o}~{}\times~{}0.5^{o}$ grid and
assumed currents originate 110 km above the surface. When applied to different
region, this altitude should be tested to ensure accurate results. The
accuracy of SECS in modelling magnetic field variations between magnetic
observatories in Ireland and the UK was evaluated by Campanyà et al. (2019)
using nine magnetometers, with respect to distance from the nearest
magnetometer observatory. They highlighted that coherence and SNR (signal-to-
noise ratio) fall off the further away the MT site was from the closest
magnetic observatory, SNR of above 9 and coherence above 0.9 were observed for
MT sites that are at less than 200 km from a magnetic observatory, which is
the current situation in Ireland. However, the relative position of the site
relative to the magnetometers and can also have an important bearing on the
accuracy of the interpolation (i.e. an output site in between input sites will
work well, but a site not between will not work as well). The presented
nowcasting methodology is focused on only using magnetometers from Ireland
(the three sites in Fig. 1), thus the performance of SECS interpolation was
re-assessed for the current set-up. This was considered to be important due to
the relative position of the MT sites towards the magnetic observatories, as
in some cases SECS will work more like an extrapolation rather than an
interpolation method. The assessment of the potential for SECS was validated
using a) Measured time series at the MT sites (Figure 1), b) Modelled time
series with differing numbers of magnetometers (nine, three and two),
expanding on validation by Campanyà et al. (2019) by accounting for relative
position of output sites to magnetometers.

Figure 1: MagIE (ARM, BIR, VAL) magnetometer observatories (blue) and the 43
MT site locations (red) used in this study. Example MT sites (A, B, C, D, E)
used subsequently, are marked with red text. Site A was an example site used
when optimising a correction for the nowcast geoelectric field. Sites B - E
were all used to validate the nowcast model.

Figure 2: (top) An example of the input magnetic fields at each magnetometer
for the 08 September 2017, during an increase in geomagnetic activity.
(bottom) The estimated magnetic field at MT sites using a SECS interpolation.
At (a) no activity is present. Over time, activity increases from north-east
to south-west, with the direction of magnetic fields changing.

###  2.2 Geoelectric Fields: MT-TF

Geoelectric fields are generated by the interaction of magnetic field
variations originating from the magnetosphere and ionosphere with the
electrically resistive Earth. The dominant layer of the Earth, in terms of
generating geoelectric fields, is the lithosphere, as it is the closest layer
to the surface and is usually more resistive than the layers below. Varying
magnetic fields ($B$) induce electric fields ($E$) according to Faraday’s law.
In the case of the Earth’s subsurface, geoelectric fields are dependent on a)
the local strength of magnetic variations and b) the electrical resistivity
properties of the Earth, expressed by a complex impedance tensor Z,

$E(\omega)=Z(\omega)\cdot B(\omega).$ (1)

When the impedance tensor $Z$ (Eq. 3), is known, the electric field can be
modelled using the magnetic field, or vice-versa (Tikhonov, 1950; Cagniard,
1953) with MT-TF (Eq. 1). To retrieve the estimated electric field time series
(Eq. 2), an inverse Fast Fourier transform is performed:

$E_{est}(t)=FFT^{-1}[Z(\omega)\cdot FFT(B_{SECS}(t))].$ (2)

where $B_{SECS}$ is the magnetic field estimated after using a SECS
interpolation. The complex impedance tensor Z, can be expressed by:

$Z(\omega)=\left[{\begin{array}[]{cc}Z_{xx}&Z_{xy}\\\ Z_{yx}&Z_{yy}\\\
\end{array}}\right].$ (3)

Note that each component of $E~{}(E_{x},E_{y})$, $B_{SECS}~{}(B_{x},B_{y})$
and $Z$ are complex numbers. The MT-TF across Ireland were obtained from
SWEMDI. The database contains MT-TF derived from simultaneously measured
electric and magnetic field time series at 43 sites. The MT-TF were derived
focusing on periods between 20 - 32,000 s, which are appropriate for modelling
geoelectric fields caused by geomagnetic storms (Grawe et al., 2018; Oyedukun
et al., 2020; Trichtchenko, 2021). Although MT-TF have been proved to provide
accurate results when modelling geoelectric fields in Ireland during
geomagnetic storms (Campanyà et al., 2019), it is important to remember that
this approach is limited by the plane-wave approximation, which is not always
true during geomagnetic storms and can introduce errors to the modelled
geoelectric fields (Ernst and Jankowski, 2005).

###  2.3 Galvanic Distortion

The impedance tensor $Z$ can, in certain cases, be affected by local galvanic
distortion. Galvanic distortions are real distortions in the regional electric
field measurements caused by local, near surface inhomogeneities, commonly
observed in MT surveys (Ledo et al., 1998; Rao et al., 2014; Delhaye et al.,
2017). Galvanic distortion can affect the impedance tensor Z, and has a
relevant impact on MT geophysical studies as it can make accurate modelling
difficult to impossible, and, if ignored, can lead to erroneous conclusions
(Chave and Jones, 2012). In terms of GICs studies, galvanic distortions may
locally modify the geoelectric fields at a particular site, producing results
not representative of the surrounding area between the site and the
surrounding sites.

####

Electric distortion is described by a real-valued, frequency independent
tensor $C$ which relates the observed MT impedance tensor $Z_{d}$ to the
regional MT impedance tensor $Z$ in the absence of distortion:

$Z_{d}(\omega)=CZ(\omega),$ (4)

where $\omega$ denotes frequency dependence (Bahr, 1987; Groom and Bailey,
1989). Algebraically, the distortion matrix mixes the complex-valued impedance
components and therefore affects the amplitude and phase information of
individual impedance components, which can lead to the aforementioned
difficulties. The impedance tensor, Z, can be decomposed into a phase ($\Phi$,
Eq. 5) and an amplitude tensor (Neukirch et al., 2019). The impedance phase
tensor, the inverse imaginary part multiplied with the real part of the
impedance, is shown to be unaffected by distortion (Caldwell et al., 2004):

$\Phi=Re(Z_{d})^{-1}~{}Im(Z_{d})=C^{-1}~{}Re(Z)^{-1}~{}C~{}Im(Z)=Re(Z)^{-1}~{}Im(Z).$
(5)

This has been used extensively to interpret MT data without most of the
challenges pertained to galvanic distortion by disregarding the distorted
amplitude tensor (Patro et al., 2012; Tietze et al., 2015; Bakker et al.,
2015; Samrock et al., 2018)). While the phase tensor only holds information
about subsurface induction processes, the amplitude tensor describes galvanic
and inductive effects that complement the phase tensor information (Neukirch
et al., 2019). Both the amplitude and phase tensors describe the same physical
induction processes in the subsurface and it has been hypothesised that they
should reflect similarly the present subsurface geometry, i.e. electric
strike, dimensionality and anisotropy, allowing computation of an optimal
distortion matrix that maximises this geometric similarity (Neukirch et al.,
2020). While empirical, this method of estimating present electric galvanic
distortion has been shown to be effective. Here, we use this method to reduce
the impact of galvanic distortion and compare results for nowcasting
geoelectric fields with and without distortion correction.

###  2.4 FFT Analysis for Nowcasting

Before computing the geoelectric fields, the input magnetics at MT sites were
manipulated for nowcasting. FFT (Fast Fourier Transform) derived time-series
suffer from issues at the edge of the time series (and hence at real time) due
to the lack of frequency information past the edge of the time series. Zero-
padding is the standard tool for improving the performance of the FFT close to
the edge of a time series which involves adding an array of zeros to the end
of the input signal, which reduces errors in the output when the FFT is
applied. Kelbert et al. (2017) successfully implemented a padding approach to
estimate geoelectric fields using an adapted form of zero-padding, where the
last recorded values were used instead of zeroes (Fig. 17). Both padding
approaches, and the combination of both, were analysed.

####

The performance of the FFT at real-time was evaluated by comparing padded near
real-time (or nowcast) modelling of geoelectric fields with standard non real-
time modelling of geoelectric fields (where electric fields time series do not
suffer from edge issues). Several storms (Table 1) were considered during the
assessment. A single rectangular window function was used over other FFT
window functions as this window worked best closer to real time (compared to a
Hanning and Tukey windows), indicating frequency resolution is more important
than spectral leakage.

Table 1: Storms used to optimise the model together with their corresponding
Kp, local K-index maxima in Ireland during each storm. Storms of varying
strength were studied to see whether storm strength influenced the padding or
correction curve. The magnetometer sites (see Fig. 1) used for the SECS
interpolation are given in the right column.

Storm | Local K Maxa | Kp Maxb | Local stations |
---|---|---|---|---
17-18 March 2015 | 7 | 8- | BIR, VAL |
22-23 June 2015 | 8 | 8+ | BIR, VAL |
08-09 May 2016 | 6 | 6+ | BIR, VAL |
07-08 September 2017 | 7 | 8+ | ARM, BIR, VAL |
27-28 September 2017 | 5 | 6+ | ARM, BIR, VAL |
26-27 August 2018 | 7 | 7+ | ARM, BIR, VAL |

  * a

Derived from local magnetometers.

  * b

From German Research Centre for Geosciences (GFZ,
https://www.gfz-potsdam.de/en/kp-index/).

Length and type/shape of the padding were evaluated to select the parameters
that provide more accurate geoelectric fields. Potential parameters were
analysed over several spatio-temporal scenarios, considering all 43 sites over
Ireland (Fig. 1). Storms of varying strength (Table 1) were chosen to see
whether the model performed similarly for strong and weak storms. Real time
electric field modelling, nowcasting, was compared to the modelled electric
field (non real-time), to evaluate the performance of padded FFT. This was
implemented as follows:

  1. 1.

The real-time/nowcast model is calculated between the timeframe [t - 48 hours
: t], where t denotes real time (i.e. the last point in the magnetic field
time series, the nowcast magnetic field). The non real-time synthetic model
(standard model) is calculated between [t - 43 hours: t + 5 hours] (data past
the defined real time for the non real-time model, as for the non real-time
model, we have the future magnetic field time series).

  2. 2.

The points between [t - 5 hours: t] (at the edge for nowcast, away from the
edge for standard) are compared between each model.

  3. 3.

The relative error (Eq. 7), root-mean-squared error (RMS, Eq. 8), signal-to-
noise ratio (SNR, Eq. 9) and coherence (Eq. 10) were calculated to compare the
performance at different delays from real time (between [t - 5 hours: t]) and
are recorded. Here, we define delay ($\Delta{t}$) as the interval of time
between real-time/now ($t$), and the time series at an earlier point ($T$),
$\Delta{t}~{}=~{}t~{}-~{}T$. The purpose of calculating these metrics at each
delay between [t - 5 hours: t] is to improve the accuracy not only at t, but
previous times close to real time, to provide the most accurate results at
each delay for the generation of a real-time movie.

  4. 4.

Steps 1) to 3) were recalculated, as the time series was moved along a sliding
window for the course of the storm.

  5. 5.

The mean of metrics from step 3) (relative error, root-mean-squared error
(RMS), signal-to-noise ratio (SNR) and coherence, see Equation 12 in the
Appendix) are calculated at each delay.

Figure 3: The main steps for computing geoelectric fields using our real-time
method. White indicates the steps involved in preprocessing/manipulating the
magnetic and electric field time series, grey indicates inputs, while black
indicates the final electric field ouput. The method was calibrated using MT-
TF corrected for galvanic distortion and validated against measured
geoelectric time series, which were not corrected for galvanic distortion.
Steps in the implementation are labeled based on their corresponding sections
3.1, 3.2,…

##  3 Implementation and Validation

The nowcast approach, Figure 3, was implemented as follows:

####

3.1: The geomagnetic field was measured in real time with magnetometers,
filtered for noise. (Section 3.1).

3.2: SECS are implemented to estimate magnetic field variations across Ireland
(Section 3.2).

3.3: The magnetic time series were padded and an FFT was performed to convert
to the frequency domain (Section 3.3).

3.4: Geoelectric fields were estimated using MT-TF (following the approach of
Campanyà et al. (2019)) with padded magnetic time series inputs (considering
MT-TF corrected and non-corrected for galvanic distortion) and an inverse FFT
converts back to the time domain (Section 3.4).

3.5: Amplitudes of the geoelectric field time series close to real time are
multiplied by a correction curve (Section 3.5).

####

The nowcast model was then validated (Section 3.6), the geoelectric fields
were then visualised across the entire grid using a contour map (Section 3.7)
and the effect of using a galvanic distortion correction was examined (Section
3.8).

###  3.1 Measurement , Preprocessing and Quality Control

Magnetic field measurements were taken at individual magnetometer
observatories (note, magnetic field data from the individual MT sites is not
used here). Data for the last 48 hours (172,800 seconds) was used to ensure
enough information about the largest periods of analysis (32,000 seconds, $>$
5 wave cycles) was retained (after testing we found 48 hours to be most
suitable for our purposes). This data was linear detrended and then treated
for any high frequency noise. A simple $dB/dt$ threshold was employed in real
time to filter noise from moving ferrous sources (i.e., cars). This filter
removes any signal above a 10 nT/sec $dB/dt$ threshold, which is sufficiently
sharp and large to ensure no real signal is removed. Periods outside the range
of interest (20 s – 32,000 s) were not considered.

###  3.2 SECS

The potential of SECS approach to interpolate between the MagIE monitoring
stations to MT sites (Fig. 1) was assessed by:

  1. 1.

Comparing the SECS-derived magnetic fields with measurements at time of
acquisition of MT sites.

  2. 2.

Quantifying the performance of SECS, comparing the results of using
magnetometers from Ireland, UK, France, Belgium and Germany (the nearest 6
INTERMAG observatories) with results using only the available magnetometer
data from Ireland, considering that 3 or 2 of the magnetometers are available.

The analysis was performed assuming that the three magnetometers are working,
but also with the hypothetical case where the northern magnetometer stops
working. SECS interpolations are more accurate during greater levels of
magnetic activity, which is when we are more interested in the geoelectric
time series. However, not all MT sites have data acquired at storm times,
therefore we chose a magnetic activity of Kp4 as the minimum threshold for the
site to be evaluated (extra analysis in Fig. 18).

####

Figure 4 compares measured data at the time of calculation of the MT sites
with SECS derived magnetic time series using magnetic time series from: a)
nine magnetic observatories (the three from MagIE (ARM, BIR, VAL) with the
nearest six INTERMAG observatories (HAD, ESK, LER, CLF, DOU, WNG), b) three
magnetic observatories (VAL, BIR, ARM), and c) two observatories (VAL, BIR).
The analysis was performed looking at coherence and SNR. Performance decreases
with a lower number of magnetometers, in particular when going from three to
two, missing the magnetometer in the north (Arm), but no significant
difference were observed between using nine or three magnetometers, which
suggest that the current configuration is reasonable, although it could
improve if more data was available in real time. Sites with the strongest
decrease in performance are the two sites further west.

####

The impact of using two or three magnetometers was also evaluated by comparing
with interpolating with nine magnetometers (Fig 5). The performance of the
SECS method decreases . The performance of the SECS method decreases when
using only three sites, however it still performs well with high coherence
($>$ 0.875) and SNR ($>$ 4). The performance of the SECS method decreases
significantly for northernmost sites, when the ARM magnetometer is removed,
with high coherence ($>$ 0.8) but low SNR (0 - 4). This highlights the
importance of using the ARM magnetometer for accurate magnetic fields at
northern sites. Despite the lack of a magnetometer in the south-east, these
sites still perform well. This is likely due to the magnetometers retaining
the regional effects, with weak local effects in the south-east, due to the
lack of strong influence from the auroral electrojet, as opposed to the north-
west, which is more strongly influenced.

Figure 4:  The coherence (top) and SNR (bottom) between the measured magnetic
field at MT sites and modelling magnetic fields (SECS interpolation with nine,
three and two magnetometer sites) for all MT sites with available data.

Figure 5: The coherence (left) and SNR (right) between modelled magnetic field
using SECS with nine sites is compared to modelled magnetic field using SECS
with three magnetometers (top) and two magnetometers (bottom) for the 7-8
September 2017 storm (Kp8). Positions of the MagIE magnetometers are denoted
by red crosses.

###  3.3 Padding

To optimise padding, the performance metrics (relative error, SNR, Coherence
and RMS) were analysed comparing the a padded electric field time series at
real-time to an unpadded time series away from real-time (see Section 2.4 for
the method used). Electric fields below 2 mV/km were discarded for this
analysis to help avoid division/multiplication by 0 for SNR and relative
errors. Signals below 2 mV/km also pose a negligible risk in comparison to
larger electric fields.

####

The best length of the padding was evaluated by using lengths between 0 and
180 min with 5 min intervals. The types of padding considered included zero
padding, end padding and end-zero padding (Fig. 17). The best overall
performance was obtained with a length of 105 min using either end padding or
end-zero padding. The ratio of pad to time series (48 hours, 2880 min) is
0.036.

####

Firstly, padding type was analysed. Figure 6 compares the performance of the
FFT with three different types of padding (using 120 min of padding, an
estimate). End and end-zero padding outperform zero padding in each metric.
The difference between end and end-zero padding is more subtle. Both perform
at a similar level in each metric. The most significant difference between the
two is in coherence, where end-zero padding is marginally better across all
delays. For this reason, we chose to continue with end-zero padding as the
padding method.

####

Figure 7 expands on the analysis in Figure 6 by comparing the performance of
end-zero padding using three different padding lengths. If the padding length
was set too small, the coherence was not accurately re-produced by the model.
If the padding was set too large, the SNR and RMS worsened (although coherence
improved marginally). Note, despite the latter three storms in Table 1 missing
ARM data for the interpolation, this had little impact on the performance
metrics. This is likely due to the same input MT-TF being compared, so while
the magnetic field will be more erroneous, this has little effect on the
comparison between different padding types.

Figure 6: A comparison of the performance of “end-zero padding”, “end padding”
and “zero padding”. Figure 7: A comparison of the performance of end-zero
padding with different lengths of padding (in minutes).

###  3.4 MT-TF - Geoelectric Field Calculation

Geoelectric fields were modelled across Ireland using the MT-TF approach for
all sites from SWEMDI (Figure 1). Padded magnetic field time series were
converted to to frequency domain using an FFT, and then the geoelectric field
was calculated at each site using Equation 2. For the implementation, we
focused on sites with corrected galvanic distortion, and kept the sites not
corrected for galvanic distortion for the validation. Note that despite the
use of padding, the 1 minute delayed/nowcast geoelectric field at this point
underestimates peaks when compared with standard modelling (non-real time),
but retains coherent structure (Fig 8). This underestimation is due to the
inability of the FFT to reproduce all of the long period information at the
edge of the time series. However, short periods are retained and give spatial
coherence to the results. In the following section we discuss how we attempt
to account for this effect.

Figure 8: An example of the modelled time series of the electric field at a
1-minute delay (nowcast) plotted against a time series of the standard non
real-time model for $Ex$ (top) and $Ey$ (bottom) for the 30 October 2003 for
Site A (Fig. 1).

###  3.5 Correction Curve

A correction curve was considered to correct the underestimation of the
nowcast geoelectric fields. The correction curve applies a scalar correction
factor on the amplitude of the nowcast geoelectric fields for each time delay.
This correction factor has been optimised for, by comparing the real-time and
non real-time models (explanation of real-time vs non real-time in Section
2.4) for 5 storms (Table 1) and for all MT sites, by calculating the average
ratio (example at 1-minute delay, Fig. 9).

Figure 9: The amplitude of the standard electric field model (both $Ex$ and
$Ey$) are plotted against the corresponding 1-minute delayed from real time
(nowcast) electric fields, for all sites in Figure 1 and storms in Table 1.
The ideal correction factor (in red) is the best fit slope between the two
models and is compared to a slope of unity (in blue). Figure 10: The corrected
time series of the electric field at a 1-minute delay (nowcast) against the
standard model for $Ex$ (top) and $Ey$ (bottom) from Figure 8, with a pre-
optimised correction factor of 1.61 (Fig. 9).

A performance score (Eq. 11) was used to optimise the model’s correction
factor. The correction curve was defined by considering correction factors
between 0.5 and 2 with 0.01 increments, constraining median, min and max
correction factors for each time delay between 1 and 200 min. Figure 11 shows
a graphical summary of the role of the correction curve. Figure 12 shows the
median, minimum and maximum values of the correction curve for each time delay
between each site and storm.

Figure 11: An example of the change in the time series near real time when the
electric field time series is multiplied by the correction curve. The
uncorrected time series (top left) are multiplied by a correction (bottom
left), to give the corrected time series (right). Figure 12: The median
correction factor calculated at each time delay (the correction curve). The
maximum (green) and minimum (orange) extent of the curve at each delay are
also marked.

The correction curve is dependant on the length of time series used (48 hours
in this case) and can be given by the conditional equation:

$f(t)\approx\begin{cases}-(3.0\times 10^{-6}t^{3})+(5.3\times
10^{-4}t^{2})-(3.0\times 10^{-2}t)+1.61&{\text{if }x/n\leq 0.0024}\\\
1.00&{\text{if }x/n>0.0024}\end{cases}$ (6)

where n is the length of the time series used, t is the time delay, 0.0024 is
derived from the length of the time series 2880, divided by delay 70, the edge
of where a correction is needed (see Figure 12). As the amplitude of the
correction factor reduces, it gets further than real time, which is expected,
as more longer period variations are present closer to the centre of the time
series, so the underestimation becomes less and less. At approximately 70
minutes the correction factor approaches unity, indicating that any loss in
long period information here and at larger delays is negligible.

####

Applying the correction at each delay is performed to construct an nowcast
movie of geoelectric fields (to visualise how the electric fields develop up
until real time). The correction curve multiplies the electric field time
series by a correction factor at each time delay, close to real time, to
account for lost longer period magnetic variations (i.e. the electric field
time series is multiplied by 1.61 at 1 minute, 1.58 at 2 mins, etc.). Rather
than re-calculate the correction at each delay individually, the curve can be
used to account for the error at each delay. This significantly reduces
computation time.

###  3.6 Validation

The presented approach for nowcasting geoelectric fields was validated with
clean measured electric field data (band pass filtered between 20 - 32,000s)
from the SWEMDI database (Campanyà et al., 2018). During validation galvanic
distortion was not corrected as it is present in the measured electric field
data. Four measurement sites were used for validation 2. Validation results
are presented in Figure 13 and summarised in Table 3. Note that data measured
at these sites during the selected storms were not used for
implementing/optimising/calibrating the system for nowcasting geoelectric
fields.

Figure 13: The measured and modelled E are compared for four example sites
(Fig. 1), (top left) 17-18 March 2015 (Kp8) for site B, (top right) 07
November 2021 (Kp7) for site C, (bottom left) 21 November 2017 (Kp5) for site
D and (bottom right) 05 May 2018 (Kp5) for site E. The top subplot compares
the measured electric field (gold) to the nowcast model at a 1-minute delay
(blue), the middle subplot the measured electric field versus the standard
model (black), with the bottom comparing the RMS (10 minute bins) between the
models and the measured data.

.

Table 2: The storms used to validate the model with their corresponding Kp and
local K-index maxima in Ireland during each storm. Storms of varying strength
were studied to see whether storm strength influenced the padding or
correction curve. The magnetometer sites (see Fig. 1) used for the SEC
interpolation are given in the right column.

Storm | Local K Maxa | Kp maxb | Local stations | MT Site |
---|---|---|---|---|---
17-18 March 2015 | 7 | 8- | BIR, VAL | B |
22 November 2021 | 6 | 8- | ARM, BIR, VAL | C |
21 November 2017 | 5 | 5 | ARM, BIR, VAL | D |
07-08 September 2017 | 5 | 6- | ARM, BIR, VAL | E |

  * a

Derived from local magnetometers.

  * b

From German Research Centre for Geosciences (GFZ,
https://www.gfz-potsdam.de/en/kp-index/).

Table 3: The storms used to validate the model with their corresponding Kp index value and max local K-index in Ireland during each storm. Storms of varying strength were studied to see whether storm strength influenced the padding or correction curve. The magnetometer sites (see Fig. 1) used for the SEC interpolation are given in the right column. MT Site | Model | Coherence | SNR | RMS |
---|---|---|---|---|---
B | Standard | 0.80 | 4.3 | 1.3 |
B | Nowcast | 0.75 | 3.2 | 1.7 |
C | Standard | 0.89 | 5.4 | 17 |
C | Nowcast | 0.81 | 6.5 | 15 |
D | Standard | 0.90 | 7.0 | 3.9 |
D | Nowcast | 0.84 | 4.0 | 3.8 |
E | Standard | 0.83 | 3.8 | 4.0 |
E | Nowcast | 0.89 | 4.5 | 3.3 |

####

Each site in Figure 13 is ordered in terms of distance to magnetometer
observatories (Figure 1, B is the closest, E is the furthest). In terms of
overall accuracy compared to measured data, site C and D perform best in terms
of the metrics (Table 3). Site B is more erroneous as the MT site is located
at the coast, with extra noise present in the electric field due to tidal
effects. For site E, while coherent shape is accurate, the electric field is
underestimated, due to the underestimation in prediction of magnetic fields
with SECS, related to distance and position relative to the magnetometers
(Figures 4, 5). Comparing between the standard model and nowcast model, a
small decrease in accuracy (drop in coherence and SNR), is present in general,
due to the loss of high frequency information near real time, when the FFT is
performed. However, some discrepancy is observed. At some sites (C, E), SNR is
higher for the nowcast, as the nowcast overestimates the standard model, which
itself underestimates the measured data. And at site E, coherence of the
nowcast is greater, indicating that there may have been noise present at
longer periods, for measurements of this site.

###  3.7 Visualisation

####

To plot the real-time electric fields across the entire grid, a contour map
was used. The map uses modelled electric field, amplitude and direction, for
each MT site in Figure 1. The electric field is then estimated across 10 km
$\times$ 10 km ($0.1^{o}~{}\times$ $0.1^{o}$) grid encompassing the island of
Ireland . The direction of the electric field at each impedance tensor site is
also displayed. A cubic spline interpolation is then performed to create the
contour and a mask is then applied to the sea.

Figure 14: An example of the real-time model operating for the 8-9 September
2017 storm. The model treats data like it is in real time. The nowcast
electric fields were plotted across Ireland at the top, with colour indicating
magnitude and the arrows indicating direction on the left and the standard
deviation error of the model on the right. A cubic interpolation used to
estimate the electric fields between impedance tensor sites. The horizontal
(H) component of the Valentia magnetometer is on the bottom to illustrate the
changing magnetic field. (An animation of the storm is attached in the
supplementary materials).

####

The real-time map of geoelectric fields is available at the MagIE website
(www.magie.ie/geoelectrics). The standard deviation map considers the range of
error in the geoelectric field measurements and SECS interpolation, they are
proportional to amplitude and usually vary depending on the quality of the MT
data and distance from nearest magnetic observatory.

###  3.8 Galvanic distortion

The regional effects of the model with and without galvanic distortion were
examined:

Figure 15: (Top) The model with undistorted electric field (left) is compared
to the model with distorted electric field (right). (Bottom) The normalised
direction of the vectors above are compared for significantly strong
($>~{}5~{}mV/km$) electric fields at the same origin.  Figure 16: The
frequency of amplitudes ($50~{}mV/km$ bins) at each site, for the entire storm
in Figure 15 are compared using a bar chart with undistorted electric fields
(blue) and distorted electric fields (green).

####

Galvanic distortion was corrected for using the method from Neukirch et al.
(2020) (explained in Section 2.2). Two main differences between the models are
present: 1) the local direction of the electric field between sites is less
homogeneous with galvanic distortion (Fig. 15), 2) larger electric fields are
generally observed when no correction is applied for galvanic distortion (Fig.
16). One would also expect there to be more volatile peaks in the model with
galvanic distortion in Figure 15, however the cubic spline interpolation
cleanses this effect. Locally in Figure 15, some patterns are also apparent,
a) sites in the centre of Ireland have comparably weak electric fields
compared to the sites closer to the coasts and the South-West, which could be
explained by underlying lithospheric features in their respective regions. b)
galvanic distortions appear to have the greatest effect near the coast, often
being orders of magnitude stronger than the galvanic corrected electric field,
which matches results from previous findings (Chave and Jones, 2012) .

##  4 Discussion

A framework has been developed for nowcasting geoelectric fields in Ireland,
caused by space weather events. The framework was calibrated using modelled
geoelectric fields, and validated against measured geoelectric fields,
producing reasonable, albeit slightly worse results, when compared to the
standard non real-time model, which is to be expected given the loss in
frequency information. The framework also considers correction for galvanic
distortion of the MT-TF and evaluated its impact on modelled geoelectric
fields across the region. Spatial resolution is comparable to other national
scale attempts to model geoelectric fields (i.e. NOAA for the US 111NOAA
Geoelectric Field 1-Minute, available at
https://www.swpc.noaa.gov/products/geoelectric-field-1-minute as of
12/01/2023), with no site in Ireland less than 200 km from magnetic
observatories, and MT-TF at around 40 km distance between sites on average.

####

This framework was built in a way that it could be implemented outside
Ireland, providing that real time magnetic field variations and MT-TF for the
area of study are available. However, it is important to highlight that the
framework was calibrated using data from Ireland, that no major changes are to
be expected, it is recommended that the calibration/optimisation process is
repeated if implemented in a new region. The framework represents a key step
for space weather and GIC studies in Ireland provides for the first time
geoelectric field data in near real time for the whole island. Data will be
stored at www.magie.ie where it can be used to assess the impact of past
events.

###  4.1 Nowcasting Geoelectric Fields

The nowcast geoelectric fields method used the FFT-based method of Kelbert et
al. (2017) as its foundation and built upon it. Firstly, the optimisation of
the method used by Kelbert et al. (2017) was elaborated upon, including a more
detailed description of how to calibrate an ideal padding length. In addition,
a correction curve was added to account for the underestimation at modelling
geoelectric fields near real time. The considered correction curve approach is
similar to the method used in Simpson and Bahr (2021), where they correct the
individual MT-TF, instead of correcting the electric field time series.
However, we found the correction curve to be more practical for our purposes
(i.e. for a real-time movie), when correcting multiple delays from real time,
as separate MT-TF would need to be made for each separate delay, which would
reduce computation time and would be tedious to perform for many MT sites. The
correction curve improves the RMS of the modelled geoelectric field without
affecting coherence. The simulation of the storms provides both intensity
information as well as orientation at each individual site, adding $\approx
10-20~{}\%$ additional relative error/RMS compared to a standard non real-time
method, which is still sufficient to monitor geoelectric activity (Fig. 13).
The choice of ideal correction curve varies slightly from site to site and
storm to storm, depending on what frequencies the site is most sensitive to.
Figure 12 illustrates this difference in the correction curve obtained for
each MT site, between the minimum and maximum extents. Due to this
variability, the correction curve applied for the real-time model is not
perfect, but does improve upon no correction (Fig. 9). However, almost no
additional error is added, despite increasing the amplitude of the electric
fields. Hence, the correction curve approach proved successful.

###  4.2 Galvanic Versus Non-Galvanic Distortion

This study is the first attempt to apply a galvanic distortion correction, for
space weather, across an entire region. Previous work by Murphy et al. (2021)
indicated that disregarding local information can misrepresent the regional
effects and can lead to an over/underestimation of the electric field in a
local area. Here, we reach a similar conclusion, observing that galvanic
corrections lead to more homogeneity in the direction of the electric field,
which is likely to be more representative of the regional lithospheric
structure. The distorted MT-TF also contains local information about the
smaller near surface structures, which also generate electric fields and for
certain sites can be up to an order of magnitude greater than the regional
geoelectric field generated in the lithosphere. Individually, local near-
surface effects are likely too weak to drive GIC on their own. As such,
despite the relatively high density of MT sites used here (compared to other
studies), one would still expect a model with a galvanic distortion correction
to produce more temporally coherent results. However, a combination of local
inhomogeneities, related to a real geological feature (i.e., a fault), could
have a relevant role on induced GIC for a particular power-line or pipeline,
given that near surface inhomogeneities can group together (in our case, we
see these inhomogeneities are more common along the coastline and in some
mountainous regions like the north-west, while the midlands remained very
homogeneous). For sparse MT networks, using galvanic corrections would most
likely improve GIC modelling, however this is less clear if a high density of
MT-TF were available and further study is needed to clarify this point. While
galvanic corrections provide a better regional picture, and are likely to be
more representative and provide more coherent results in the long term,
uncorrected galvanic distortion also needs to be considered to highlight
potentially larger geoelectric fields in certain areas, than what is implied
by the corrected galvanic distortion.

###  4.3 Limitations

  1. 1.

The plain wave approximation is likely to be more erroneous when local effects
are present in geomagnetic storms. This could be solved by using a 3D model of
the lithosphere in Ireland and its interaction with the ionosphere and
magnetospheric currents. However, this approach would be more computationally
expensive and would require modification. New research by Kruglyakov et al.
(2022) presents a possible solution to this problem, by using a memory based
method to reduce computation time.

  2. 2.

At present the magnetometer density is low. This leads to errors in the
calculation of SECS at more isolated sites (such as site E, Fig. 1). This, in
turn, can lead to under/over estimations in the amplitude of the modelled
geoelectric fields. To mitigate this we have installed one new magnetometer in
the east of Ireland in Dunsink Observatory, Co. Dublin, and we plan to install
two more, one in the north-west and one in the south-east. In particular, a
new site is required in the north-west, demonstrated by Figures 4, 5.

  3. 3.

The correction for galvanic distortion considered in this study does not
account for the scalar factor. This means that although the impact of galvanic
distortion is largely reduced, some distortion may still be present, affecting
the amplitude of the electric field.

##  5 Conclusions

  1. 1.

An operational/automated geoelectric field monitoring approach has been
successfully implemented to accurately estimate amplitude, orientation and
uncertainties of near real time (1-minute) geoelectric fields in Ireland.

  2. 2.

The nowcast model adds a greater uncertainty than standard modelling, with a
coherence loss of $\approx~{}5~{}\%$ and an additional $5-10~{}\%$ in the
uncertainties of the total amplitude.

  3. 3.

Optimising the padded magnetic time series and applying a correction factor
were crucial for real-time modelling with MT-TF, in particular the correction
factor improved upon the previous state of the art for near real-time
modelling of geoelectric fields.

  4. 4.

The effect of galvanic corrections on electric fields across the entire
network was investigated. The correction led to lower amplitudes in electric
field strength and more homogeneity in electric field direction. Using
galvanic correction for MT-TF would likely lead to improved temporal coherence
in GIC modelling.

  5. 5.

The real-time model can be used as a proxy to highlight regions or power-lines
that are more likely to be affected by GIC in near real time, which is more
relevant than simply looking at magnetic variations as a proxy.

  6. 6.

Modelled real-time electric field data is stored at www.magie.ie, where it is
made freely available to the public.

## Data Availability and Supplementary Materials

  1. 1.

Magnetometer data used in this project is publicly available and found at
https://data.magie.ie.

  2. 2.

The MT-TF from the SWEMDI database are currently available upon request to
Geological Survey Ireland<EMAIL_ADDRESS>and will be made open access in
the future.

  3. 3.

Live plots of nowcast geoelectric fields are available at
https://www.magie.ie/geoelectrics/.

  4. 4.

The scripts for the nowcast model are located at
https://github.com/TCDSolar/NowcastGeoelectrics/.

  5. 5.

Example animations of the nowcast model are available at …

     * •

The nowcast model – nowcast_model.mp4

     * •

The standard deviation of the nowcast model – nowcast_model_SD.mp4

     * •

The nowcast model with galvanic distotion – nowcast_model_galvanic_dist.mp4

###### Acknowledgements.

We would like to thank the Irish Research Council, Geological Survey Ireland
and Dublin Institute for Advanced Studies for supporting this work as part of
an IRC Enterprise Partnership Scheme. We also acknowledge Armagh Observatory
and Trinity College Dublin for hosting magnetometers which contributed to this
work, to Met Éireann for the Valentia Observatory magnetic measurements and
INTERMAGNET for provided access to the other magnetometers used in this study.

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## Appendix

###  5.1 Performance Metrics

The performance of the both the nowcast model was compared to the standard
models results/measurements using relative error, root-mean-squared (RMS)
error, signal-to-noise ratio (SNR) and coherence, for multiple time series.

A performance score or parameter P (first used by Torta et al. (2015)) was
used to optimise the best correction curve for single time series. A
performance score worked best when optimising a real-time correction factor,
when compared to optimisation using solely RMS, SNR or coherence.

$Rel=\frac{1}{N}\sum_{i=1}^{N}\frac{E_{data}-E_{mod}}{E_{data}}$ (7)

$RMS=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(E_{data}-E_{mod})^{2}}$ (8)

$SNR=10log_{10}\frac{\sum_{i=1}^{N}E_{data}^{2}}{\sum_{i=1}^{N}(E_{data}-E_{mod})^{2}}$
(9)

$Coh=\frac{\sum_{i=1}^{N}(E_{data}-\bar{E}_{data})(E_{mod}-\bar{E}_{mod})}{\sqrt{{\sum_{i=1}^{N}}(E_{data}-\bar{E}_{data})^{2}(E_{mod}-\bar{E}_{mod})^{2}}}$
(10)

$P=1-\frac{RMS}{\sigma_{0}}$ (11)

where $N$ is the length of the time series and $\sigma_{0}$ is the standard
deviation

The mean of each metric was also calculated across multiple time series, when
comparing performance of the model at separate delays from real time. For
example the mean of the RMS can be given by

$\overline{RMS}=\frac{\sum_{i=1}^{n}{RMS}}{n}$ (12)

where $n$ is the number of time series used.

###  5.2 Padding Types

Figure 17: An example of the different padding types applied to the magnetic
field time series, zero-padding, end-padding and end-zero padding.

###  5.3 Performance of SECS with Kp Index

SECS-derived magnetic field data were compared with measured magnetic data at
MT sites across varying magnetic activity. Figure 18 gives a summary of the
accuracy of the SECS interpolation using coherence and SNR.

Figure 18: The mean of coherence (left) and SNR (right) comparing measured
magnetic field data at newer SWEMDI MT sites with SECS interpolated data
(using 3 sites). The SNR and coherence of this comparison is plotted against
kp value, highlighting the increase in these metrics with greater magnetic
activity. The SNR and coherence of each 3 hour Kp Index bin was recorded, with
the mean value plotted.