Discrimination of ULF signals from an underground seismogenic current

Abstract

A numerical model has been elaborated to calculate ULF electromagnetic fields in the ground-atmosphere–ionosphere system created by an underground horizontal current source of a finite length. The modeling has enabled us to examine in detail characteristic features of ULF response to an underground large-scale emitter that may be used for a search of electromagnetic earthquake precursors. The most promising features that might discriminate signals from an underground source and from a magnetosphere-ionosphere source are (a) the apparent impedance of the electromagnetic field derived from simultaneous observations of horizontal magnetic and electric fields, and (b) the ratio of vertical and horizontal magnetic component amplitudes. Besides that, the amplitude-phase gradients of signals from an underground source differ significantly from those of the magnetospheric source. For the same magnitude of horizontal magnetic disturbance on surface, an underground current source produces in a borehole a much larger vertical electric field E_z than a magnetospheric source does. At the same time, some properties, such as the ratio between the vertical and horizontal electric components, are shown to be ineffective. However, all these differences with ionosphere-magnetosphere source reveal themselves only in a vicinity of lithospheric source (< 30 km for depth 20 km).

Graphical Abstract

1 Introduction

1.1 Observation and modeling of ULF fields of seismic sources

One of the hot topics in modern geophysics is the development of physical foundations for operational (days-weeks) earthquake prediction methods based on the appearance of anomalous disturbances of electromagnetic fields. These electromagnetic methods should supplement standard seismic methods of earthquake prediction. Electromagnetic emissions of different frequency ranges are used in the non-seismic methods. It seems promising to monitor ultra-low-frequency (ULF) broadband emissions and impulses (band 0.01–10 Hz) that are not severely absorbed in a conductive crust. The dynamic processes in an earthquake source accompanied by ULF electromagnetic activity could be the crust micro-fracturing (Molchanov and Hayakawa 1995), irregular flow of underground pore fluids (Fenoglio et al. 1995; Fedorov et al. 2001), acoustic pulses of opening cracks (Surkov et al. 2003), electric currents streaming out of stressed igneous rocks (Freund et al. 2006, 2021), and others. However, on the earth's surface the total ULF electromagnetic field generated by an ensemble of randomly oriented cracks is too small (Surkov and Hayakawa 2006). At the same time, the formation of large-scale electric current systems is possible due to a stick–slip movement of tectonic blocks along faults (Lockner et al. 1983; Gokhberg et al. 1985; Guglielmi and Levshenko 1997) or the motion of charged edge dislocations resulting from stress variation in the lithosphere (Venegas-Aravena et al. 2019) during a seismic activity. Such movements could be monitored by recording ULF pulses and noise on the Earth's surface (Hattori 2004; Bleier et al. 2009; Ohta et al. 2013).

Though, the situation with ULF electromagnetic precursors remains ambiguous to date. Some published results raise doubts about the reliability of the relationship between the detected ULF phenomena and earthquakes (Masci and Thomas 2015; Villante et al. 2001). There are many examples when impulses from distant thunderstorms or magnetospheric pulsations were mistaken for disturbances of a lithospheric nature before an earthquake (Pilipenko and Shiokawa 2024).

Amplitudes of possible electromagnetic noise and impulses generated upon the earthquake preparatory phase are apparently small. For confident discrimination of seismogenic disturbances, the development of special methods for recording and data analyzing is required. To estimate the necessary intensity of a seismic source of anomalous radiation that can be detected on the ground, it is necessary to model the response of the entire lithosphere-atmosphere–ionosphere (LAI) system to a large-scale underground emitter. Theoretical modeling would make it possible to discard unrealistic physical mechanisms, otherwise random coincidences can be perceived as reliable experimental evidence. Thus, the construction of an effective algorithm for separating signals from different LAI sources becomes one of the fundamental problems in the search for seismic-electromagnetic precursors.

In this paper, we propose a model that makes it possible to numerically calculate the 6-component structure of ULF field on the Earth's surface generated by an underground horizontal linear current of a finite length. This model quantifies the relationships between various electric and magnetic components of ULF field produced by an underground current. We have inter-compared characteristic features of ULF emissions produced by sources in the magnetosphere-ionosphere (geomagnetic pulsations), atmosphere (lightning discharges), and underground seismic hypocenter. The problem under consideration cannot be reduced to the classical problems of electromagnetic radiation from a dipole buried in a conducting half-space (Baños 1966), because in this problem the system of oscillating currents in the earth's crust is self-consistently coupled with the electromagnetic fields and currents excited in the atmosphere and ionosphere. Moreover, a finite scale of a seismic source has been considered.

1.2 Numerical model

The model assumes that mechano-electrical transformations create a current $J(t)$, which is then closed by conduction currents in the earth's crust, and partially leaks into the atmosphere. We have to find the electromagnetic fields ${\mathbf{B}}({\mathbf{r}},t)$ and ${\mathbf{E}}({\mathbf{r}},t)$ created by this current system in the entire LAI system. In the considered model, the system of oscillating currents in the earth's crust is self-consistently related to the electromagnetic fields excited by them in the atmosphere and ionosphere. The presented numerical model analyzes the multi-layered horizontally homogeneous medium with a realistic ionospheric vertical profile. We study electromagnetic field only in the vicinity of the epicenter (about several hundred km and less). For such distances, the plane geometry is a good approximation.

The geometry of the model is shown in Fig. 1: the $z$-axis of the Cartesian coordinate system is directed vertically up, setting $z = 0$ at the Earth's surface, the $x$-axis is directed along the current, and $y$-axis is across the current. For simplicity, the geomagnetic field ${\mathbf{B}}_{0}$ is assumed to be vertical. The oscillatory linear horizontal current $J(t) = J_{0} \exp ( - i\omega t)$ with a finite length $L$ is located at the depth $z = h < 0$.

Fig. 1

Schematic representation of the underground current source geometry in the presence of a conductive atmosphere and ionosphere. The source current $J_{0}$ is oriented along the X-axis

We search the electromagnetic fields ${\mathbf{B}}({\mathbf{r}},t)$ and ${\mathbf{E}}({\mathbf{r}},t)$ as a time harmonic $\propto \exp ( - i\omega t)$. For that purpose the Maxwell’s equations with a source current density ${\mathbf{j}}$

$$ \nabla \times {\mathbf{B}} = \mu_{0} ( - i\omega \varepsilon_{0} {\hat{\mathbf{\varepsilon }}}_{r} {\mathbf{E}} + {\mathbf{j}}),\qquad \nabla \times {\mathbf{E}} = i\omega {\mathbf{B}} $$

(1)

are to be solved. Here ${\hat{\mathbf{\varepsilon }}}_{r} = {\hat{\mathbf{\varepsilon }}}/\varepsilon_{0}$ is the tensor of relative dielectric permeability in the LAI system, $\varepsilon_{0}$ and $\mu_{0}$ are the dielectric and magnetic constants of vacuum, $c$ is the light velocity. In the atmosphere, the relative permeability tensor diagonal elements are $\varepsilon_{xx} = \varepsilon_{yy} = \varepsilon_{zz} = 1 + i\sigma_{a} /(\omega \varepsilon_{0} )$, and inside the homogeneous Earth they are $\varepsilon_{xx} = \varepsilon_{yy} = \varepsilon_{zz} = {\text{Re}} \varepsilon_{g} + i\sigma_{g} /(\omega \varepsilon_{0} )$, where $\sigma_{a}$, $\sigma_{g}$ are the atmosphere and Earth conductivities. In the ionosphere $(80\;{\text{km}} < z < 1000\;{\text{km}})$ the vertical profile of the ionized and neutral particle parameters (used for permeability tensor elements calculation) is derived from the International Reference Ionosphere (IRI) model. Thus, the model under consideration is like the model with a layered horizontally homogeneous ionosphere with a realistic vertical profile used in (Fedorov et al. 2023a, b). The mathematical and computational details of the numerical model are presented in the Appendix. The calculated normalized fields correspond to the source current $J_{0} = 1\;{\text{A}}$.

Input parameters of the IRI model have been chosen to correspond to summer local noon(${\text{LT = 12}}\,{:}\,{00}$) at latitude of Kamchatka Peninsula(Petropavlovsk-Kamchatsky ${53}{\text{.0}}^\circ \;{\text{N}}$, ${158}{\text{.6}}^\circ \;{\text{E}}$). Under these conditions the maximum plasma density $N_{m} F2 = 2.5 \cdot 10^{5} \;{\text{cm}}^{ - 3}$ is located at altitude $h_{m} F2 = 230\;$ km. Atmospheric conductivity $\sigma_{a}$ is assumed to increase exponentially $\sigma_{a} (z) = \sigma_{a}^{(0)} \exp (z/z_{a} )$ till the altitude of 80 km. The scale $z_{a}$ is chosen in such a way to match $\sigma_{a} (z)$ the conductivity provided by the IRI model. The atmospheric conductivity near the ground is taken to be $\sigma_{a}^{(0)} = 10^{ - 14} \;{\text{S/m}}$, and the scale of the conductivity profile has turned out to be $z_{a} = 4.2\;$ km.

All reported ULF electromagnetic precursors were found at relatively small distances from the epicenter, so the horizontal distances of no more than 200 km have been considered. The calculations have been carried out for a source with scale $L = 20\;{\text{km}}$ at depth $h = - 20\;{\text{km}}$. The conductivity and resistivity of the Earth is chosen to be $\sigma_{g} = 10^{ - 3} \;{\text{S/m}}$ and $\rho_{g} = 10^{3} \;\Omega$, correspondingly, and the dielectric constant ${\text{Re}} \varepsilon_{g} = 10$. The source frequency is supposed to be $f = 0.1\;{\text{Hz}}$ (period $T = 10\;$ s). The corresponding skin-depth $\delta_{g}^{{}} = \sqrt {2/\mu \omega \sigma_{g} }$ ~ 50 km. The surface impedance $Z_{g}$ of homogeneous crust is determined by the crust conductivity $\sigma_{g}$ as $Z_{g} (\omega ) = \sqrt { - i\omega \mu_{0} /\sigma_{g} }$. For the chosen frequency and conductivity $Z_{g} = {0}{\text{.028 }}\Omega$. The modeling result on σ_g = 10^–1 S/m is shown on additional file 1: ULF fields on the ground for a high conductivity”.

1.3 Electromagnetic fields on the Earth's surface excited by a seismogenic current

To estimate the seismotelluric current in the region of the earthquake hypocenter, it is necessary to model electromagnetic signals observed on the Earth's surface. The use of analytical solutions for an underground dipole as a source provides only qualitative estimates. Under real conditions, the length of the fault, and hence the scale of the hypothetical seismogenic current, can reach several tens of km. The point dipole model does not allow one to estimate the change in the ground disturbance depending on the source scale, but such an estimate can be made by the proposed model of a source of finite length. The geometrical factor is significant for interpretation of realistic observations in the vicinity of an epicenter.

As characteristic features, we will analyze the relationships between vertical, $E_{z}$ and $B_{z}$, and horizontal, ${\mathbf{E}}_{{\text{h}}}$ and ${\mathbf{B}}_{{\text{h}}}$, electric and magnetic components from various sources—atmospheric, ionospheric, and underground. Characteristic features of the field of an underground source at the earth's surface ($z = 0$) can be seen from Figs. 2 and 3, which show the spatial distribution of the amplitudes of horizontal and vertical magnetic and electric components in the direction along the source current (axis $X$), and across it (axis $Y$).

Fig. 2

The spatial structure of the amplitude of the magnetic component $|B_{y} (x)|$ on the earth's surface $(z = 0)$ in the direction along the source current (solid line) and across the current for magnetic components $|B_{y} (y)|$ and $|B_{z} (y)|$(dashed line). The parameters are $L = 20\;{\text{km}}$, $J_{0} = 1\;{\text{A}}$, $f = 0.1\;{\text{Hz}}$, $\sigma_{g} = 10^{ - 3} \;{\text{S/m}}$, and $h = - 20\;{\text{km}}$. These values of parameters are used for all other figures

Fig. 3

The spatial structure of the amplitude of the electric components $|E_{x} (x)|$ and $|E_{z} (x)|$ on the earth's surface in the direction along the source current (solid line). The components $|E_{x} (y)|$ and $|E_{z} (y)|$ in the direction across the source current (dashed line)

Along the current direction ($y = 0$) the components $B_{x} (x) \,$ and $B_{z} (x)$ are small and are not shown. The amplitude of the horizontal component $|B_{y} (x,0)|$ has maximum ~ 2 pT above a source (x = 0), and at large distances decays according to the power law $x^{ - 3}$. Across the current ($x = 0$) $|B_{y} (0,y)| \,$ has a maximum ~ 2 pT directly above the source ($y = 0$) and at large distances also obeys the power law decay $\sim y^{ - 3}$. A local minimum of $|B_{y} (y)| \,$ is formed at $y \approx 25\;{\text{km}}$. A reason of its occurrence will be discussed further in section on location of a source. The component $B_{x} (y) \,$ is small and is not shown.

The spatial structure of the horizontal electric field on the Earth's surface created by an underground current is shown in Fig. 3. The maximal amplitude of the horizontal component $\max |E_{x} (x){ |} \sim {0}{\text{.3}}\;\mu {\text{V/m}}$ is reached above the source. The component $E_{y} (x) \,$ is small and is not shown. A local minimum of $E_{x} (x) \,$ is formed at x ~ 18 km, and its formation will be discussed further.

1.4 The polarization ratio of vertical and horizontal magnetic components

Magnetospheric ULF disturbance with amplitude $B_{y}^{(M)}$ and transverse scale $\Delta_{M}$ are mainly transported to the ionosphere by field-aligned current $J_{z}^{(M)}$. This current excites in the conductive E-layer of the ionosphere a system of Hall and Pedersen currents, which produce a magnetic response on the ground $B_{x}^{(A)} = B_{y}^{(M)} (\Sigma_{H} /\Sigma_{P} ) = \mu_{0} \Delta_{M} J_{z}^{(M)} (\Sigma_{H} /\Sigma_{P} )$. Ionospheric currents excite in the atmosphere predominantly magnetic (TE) mode with vanishing $E_{z}$ component. An occurrence of weak vertical magnetic component may be caused either by a lateral inhomogeneity of the crust conductivity or by a finite horizontal scale of an incident disturbance λ. The ratio between the vertical and horizontal magnetic components above a homogeneous crust can be estimated from the boundary condition $B_{z} = (i/\mu_{0} \omega )Z_{g} \nabla_{ \bot } {\mathbf{B}}_{h}$. At typical horizontal scales $\lambda$ of magnetospheric-ionospheric disturbances (at least few hundred km) and conductivity of the underlying earth's surface, the vertical component of the field of such disturbances is small compared to the horizontal one: $|B_{z} |/|B_{h} | \sim \delta_{g} {/}\lambda < < {1}$. This conclusion can be violated only above a high-resistive crust, where $\delta_{{\text{g}}} { > }\lambda$.

For an underground source, the relationship between the vertical and horizontal magnetic components can be seen in Fig. 2. The amplitude of the vertical component $|B_{z} (y)| \,$ above the source passes through zero and reaches a maximum at a distance of $y \approx 15\;{\text{km}}$. At the same time, near the source (10 km < y < 70 km) the vertical magnetic component prevails the horizontal component, $|B_{z} |\, > \,|B_{y} | \,$. However, far from it (y >  > 70 km) the vertical component becomes small as compared with the horizontal one, $|B_{z} |\, \ll \,|B_{y} |$.

Thus, we can point to a characteristic feature of the field of an underground source in comparison with the field of magnetospheric-ionospheric disturbances incident from above. This feature is a relatively large $B_{z}$ component, which occurs, however, only at small distances from the source “hypocenter”, whereas at large distances the difference between both sources vanishes.

1.5 Apparent impedance of an underground emitter field

For the magnetosphere-ionosphere source, the relationship between the horizontal electric and magnetic components on the ground of the magnetosphere-ionosphere disturbance is determined by the impedance condition (Kaufman and Keller 1981)

$$ {\mathbf{E}}_{{\text{h}}} = \mu_{0}^{ - 1} Z_{g} [{\mathbf{n}} \times {\mathbf{B}}_{{\text{h}}} ]\,, $$

(2)

where ${\mathbf{n}}$ is the normal to the ground surface. Thus, the ratio between horizontal electric and magnetic components is determined by the surface impedance $|{\mathbf{E}}_{{\text{h}}} |/|{\mathbf{B}}_{{\text{h}}} |\; = \mu_{0}^{ - 1} |Z_{g} |$.

For the underground source, the apparent impedance of seismogenic disturbances $Z_{xy} = \mu_{0} E_{x} /B_{y}$ has been calculated from the model values of the magnetic and electro-telluric fields (Fig. 4). The calculation shows that in a vicinity of an underground source (x and y < 50 km) the apparent impedance is much higher than the impedance of the Earth $Z_{g}$. However, at large distances (> 100 km) in both directions the apparent impedance tends to the ground impedance $Z_{g}$.

Fig. 4

Profile of apparent impedance $|Z_{xy} (x)|$ in the directions along (black line) and across (blue line) the source current. The Earth impedance $Z_{g}$ is shown by horizontal dotted line

The key problem in the search for electromagnetic precursors is the feasibility to discriminate magnetospheric and seismogenic disturbances. Here we highlight a feature that can be used to identify signals from underground sources—the apparent impedance $Z_{xy}$. In a vicinity of hypothetical underground source, the estimated apparent impedance $Z_{xy}$ must be larger than the crust impedance $Z_{g}$ determined by the magnetotelluric sounding with magnetospheric ULF waves. The same relation must hold for other Earth’s conductivity and source depth, though the actual values of the apparent impedance and Z_g will be different.

1.6 Vertical electric field

Even a distant lightning produces a noticeable ULF impulse (Schekotov et al. 2011). The vertical atmospheric discharge excites predominantly the electrical (TH) mode. The vertical component $E_{z}$ of the electric field in the atmosphere can be found from Faraday equation $[{\mathbf{n}} \times {\mathbf{B}}_{{\text{h}}} ]_{z} = - i\omega \mu_{0} \varepsilon_{0} E_{z}$ and the impedance condition (2). As a result, we obtain $E_{z} = - ik_{0}^{ - 1} (Z_{0} /Z_{g} )(\nabla \cdot {\mathbf{E}}_{{\text{h}}} )$, where $Z_{0} = 120\pi$ is the impedance of vacuum. These relationships show that E_z/E_h = k_g/k₀ >  > 1, where $k_{0} = \omega /c$ is the vacuum wave number, and $k_{g}$ is the wave number in the crust, $k_{g}^{2} = i\mu \omega \sigma_{g}$. Thus, the vertical electric component excited by a lightning discharge greatly exceeds the horizontal one.

For an underground source, the spatial structure of the vertical component of the electric field on the Earth's surface is shown in Fig. 3. The maximal amplitude of vertical component is about the same as the amplitude of horizontal component max|E_x(x)|~ max(E_z(x)|~ 0.3 μV/m. In the direction across the current, right above the source, $E_{z}$ is very small because of symmetry. In the vicinity of a source (< 40 km) both vertical and horizontal components of electric field are of the same order, |E_z| ~|E_h|. However, at larger distances (> 100 km) the vertical component dominates, |E_z|> >|E_h|. Thus, at such distances the electric field polarization structure in the vertical plane reminds the field produced by an atmospheric source.

In the geophysical literature one may encounter the statement that an underground source creates vertical electric field of significant magnitude because of the vertical current continuity on the Earth's surface (e.g., Finkelstein and Powell 1970). The jump of $E_{z}$ can be estimated from the relationship $|\hat{\sigma }_{a} E_{z} (z = + 0)|\; = \;|\sigma_{g} E_{z} (z = - 0)|$, where $\hat{\sigma }_{a} = \sigma_{a} - i\omega \varepsilon_{0}$ is the complex conductivity. From this relationship it seems that the earth's surface could be imagined as a powerful “amplifier” of $E_{z}$ with the amplification coefficient given by the ratio of the conductivities of the earth's crust and air $|\sigma_{g} /\hat{\sigma }_{a} |$ (Alekseev and Aksenov 2003). This question is of fundamental importance, since it determines the strategy for searching seismic-electromagnetic effects. Our simulations have shown that the idea of the Earth-atmosphere interface as an “amplifier” of $E_{z}$ is incorrect. The vertical current flowing from the depths of the earth's crust, approaching the Earth-atmosphere boundary, sharply deviates from the vertical, because it “does not want” to flow into a low-conductive medium. As a result, a significant part of the current is deflected in the horizontal direction, and only a small part of the vertical current leaks into the atmosphere. Although $E_{z}$ associated with this current experiences a jump at the boundary, in the general structure of the field the vertical field turns out to be approximately the same as the horizontal telluric field.

1.7 Vertical electric field in a borehole

There were suggestions to measure the vertical electric field associated with seismic activity inside the earth's crust using borehole observations (Tsutsui 2002). Such observations may be more promising to reveal signals from seismogenic sources and discriminate them from magnetosphere-ionosphere disturbances. The vertical structure of the electric field around the ground-atmosphere interface generated by an underground source has been numerically calculated with our model at x = 10 km (Fig. 5).

Fig. 5

The $|E_{z} (z)|$ and $|E_{x} (z)|$ profiles across the ground surface at x = 10 km: a in the range $- 10\;{\text{km}} < z < 3\;{\text{km}}$; b in the range $- 200\;{\text{m}} < z < 100\;{\text{m}}$

Upon crossing down the Earth’s surface the vertical electric field $|E_{z} (z)|$ drops significantly to very low values $\sim {10}^{{ - 9}} \, \mu {\text{V/m}}$. However, inside the Earth the amplitude $|E_{z} (z)|$ increases almost linearly with depth from very small values at $z = 0$ to $|E_{z} (z_{0} )| = {7} \cdot {10}^{ - 3} \, \mu {\text{V/m }}$ at $z_{0} = - 200\;{\text{m}}$ according to the law $|E_{z} (z)/E_{z} (z_{0} )| \approx z/z_{0}$ (Fig. 5b). This behavior is caused by the current declination from the vertical direction near the ground-atmosphere boundary. The component $E_{z} (z)$ increases significantly while submerging deep underground and at z < − 4 km the vertical component exceeds the horizontal component $E_{x}$ (Fig. 5a). At depths > 6 km $E_{z}$ even exceeds the value of $E_{z}$ above the ground-atmosphere boundary. In the considered case, the horizontal components of the magnetic and electric fields on the ground surface are B_y(x = 10) ~ 2 pT and E_x(x = 10) ~ 0.15 μV/m (see Figs. 2 and 3). Thus, 1-nT B_y disturbance on surface correspond to the vertical electric field disturbance E_z ~ 3.5 μV/m at a borehole of 200 m depth.

Numerical modeling of the atmospheric electric mode excitation by an incident ULF Alfven wave (Pilipenko et al. 2021) showed that 1-nT magnetic disturbance from the magnetospheric Alfvenic disturbance with $f = 0.1\;{\text{Hz}}$ is to be accompanied by the $E_{z}$ burst of the order of ${10}^{{ - 2}} {\text{ V/m}}$ above the Earth’s surface. The corresponding vertical current density under typical $\sigma_{a}^{(0)} = 10^{ - 14} \;{\text{S/m }}$ must be $j_{z} = \sigma_{a}^{(0)} E_{z} = 10^{ - 16} \;{\text{A/m}}^{{2}}$. The vertical current is continuous, so the same current must be observed in a borehole. Therefore, the electric field $E_{z}$ in the borehole under $\sigma_{g} = 10^{ - 3} \;{\text{S/m}}$ is to be $E_{z} { = 10}^{{ - 7}} \, \mu {\text{V/m}}$. From these estimates it follows that the same magnetic disturbance from an underground source produces about 7 orders of magnitude larger response in $E_{z}$ in the borehole than the magnetosphere-ionosphere source does.

1.8 Location of the underground source “epicenter”

For the location of anomalous ULF emission source, the use of multi-station magnetic observations was suggested (Surkov et al. 2004; Schekotov et al. 2008). For that one should know the 2D spatial pattern of the anomalous magnetic vector field. The magnetic effect on the surface of the earth is created by the source current and spreading currents in the crust.

The structure of horizontal currents near the earth's surface can be represented by a 2D picture of the transverse electric field unit vectors ${\mathbf{E}}_{{\text{h}}} { / |}{\mathbf{E}}_{{\text{h}}} {| }$(Fig. 6). Spreading currents are closing currents flowing out from one end of the underground linear current into the other end. At large distances (x ~ 17 km) along X-axis the spreading current changes its direction. This direction reversal is responsible for the minimum of $|E_{x} (x)|$ amplitude (Fig. 3).

Fig. 6.

2D picture of the horizontal electric field unit vectors ${\mathbf{E}}_{{\text{h}}} { / |}{\mathbf{E}}_{{\text{h}}} {| }$(the field of directions). Thick red line shows the source current projection to the earth surface

2D snapshot of the horizontal magnetic disturbance vector ${\mathbf{B}}_{{\text{h}}}$ produced by this current system on the ground is shown in Fig. 7. The pattern is shown separately for a “near-field” zone ($\rho < 25\;{\text{km}}$), where the field is produced predominantly by a source current (left-hand panel), and “far-field” ($\rho > 30\;{\text{km}}$) (right-hand panel). In a close vicinity of a source, the magnetic disturbance vector is perpendicular to the source current (Fig. 7, left-hand panel). At a greater distance, the magnetic disturbance is orthogonal to radius-vector from the source in the sector along X-axis. The orientation of magnetic field vectors changes direction upon transition from near-field to far-field in the direction across the current ($|x| < 10$ km). The change of the magnetic field pattern is caused by the fact that in vicinity of a source the magnetic disturbance is produced by the linear source current, whereas the magnetic disturbance far from the source is mainly produced by spreading currents. This reversal of the magnetic field direction is responsible for the minimum of $|B_{y} (y)|$ amplitude (Fig. 2).

Fig. 7

Snapshot of the horizontal magnetic disturbance vector on the ground. a Close vicinity ($\rho < 25\;{\text{km}}$) of the line current projection to the earth surface (near-field zone); b far-field zone ($\rho > 30\;{\text{km}}$). Red line—the same as in Fig. 6

2 Amplitude-phase gradients

The ULF electromagnetic disturbance inside the homogeneous conductive space is characterized by the dispersion equation that describes the ordinary skin-effect—exponential amplitude damping with the wave number $k_{g} = (1 + i)\delta_{g}^{ - 1} = \sqrt {i\mu_{0} \omega \sigma_{g} }$. Besides the amplitude damping, phase variation occurs which correspond to the apparent phase velocity $U_{g} = \;|\omega /k_{g} |\; = \sqrt {2\omega /(\mu \sigma_{g} )}$ (Ismagilov et al. 2006; Kopytenko et al. 2000). However, to imagine the signal transmission from underground source to an observation site as a wave propagation with phase velocity U_g and damping factor k_g would be oversimplification.

The modeled spatial structure of the local gradient of the amplitude and phase of the largest horizontal magnetic component is shown in Fig. 8. Logarithmic derivative gives local value of a characteristic scale of spatial variability of ULF disturbances, $\delta_{x}^{ - 1} = d\ln |B_{y} (x)|/dx$. According to modeling results, in the vicinity of a source (< 30 km) is $\delta_{y}$ ~ 2–10 km in the $y$-direction and is $\delta_{x}$ ~ 20 km in the $x$-direction. This value can be compared with typical scale of the ULF disturbance on the ground produced by an ionospheric source, $\Delta_{g} = \Delta_{m} + h$, where Δ_m is the latitudinal scale of ULF resonant peak above the ionosphere, and h is the height of the ionospheric conductive layer. Typically, at low latitudes $\Delta_{g}$ is about few hundred km for ULF pulsations. Thus, the characteristic scale of an underground source is nearly order of magnitude less than that of the magnetosphere-ionosphere source. However, this contrast holds only in the vicinity of an underground source.

Fig. 8

From top to bottom: normalized gradients of amplitude of $B_{y}$ component; phase gradients ${\text{Arg}}(B_{y} )$; apparent phase velocities $U_{y}$. Left panels correspond to the direction along the source current, right panels—the same, but across the current. For comparison the bottom panels show the velocity $U_{g}$ (dashed line)

The phase gradient $d{\text{Arg}}\,B_{y} /dx$ of the horizontal component increases with distance from the source and reaches an extreme value of ~ 0.8 deg/km at ~ 30 km (Fig. 8, left middle panel). The corresponding apparent phase velocity $U_{y} (x)$ drops from ~ 10³ km/s above the source ($x = 0$) to ~ 40 km/s at distances $x > 30\;{\text{km}}$ (Fig. 8, left bottom panel). The phase gradient $d{\text{Arg}}\,B_{y} /dy$ of the horizontal component reaches maximal value ~ 34 deg/km at y ~ 23 km (Fig. 8, right middle panel). The corresponding apparent phase velocity $U_{y} (y)$ drops in this region down to ~ 1 km/s. However, at larger distances ($y > 40\;{\text{km}}$) the apparent phase velocity of $B_{y}$ component $U_{y} (y)$ tends to $U_{g}$ (Fig. 8, right bottom panel). For the chosen crust and source parameters, U_g ~ 32 km/s.

3 Discussion

This paper describes the results of the model for calculating electromagnetic fields in the LAI system created by an underground current. As compared with a model of underground dipole (Baños 1966; Honkura and Kuwata 1993; Bortnik et al. 2010; Dong et al. 2005), our model takes into account a finite scale of an underground source. In contrast with previous studies, we are more interested not in the amplitude of a seismogenic signal on the ground, but in its spatial and polarization structure. Moreover, our model self consistently accounts for the influence of the ionosphere. A separate examination of the modeling results obtained with and without account of the ionosphere has shown that the ionospheric influence on the surface electromagnetic field is weak, although is not negligible. Mostly it becomes visible for components with small amplitudes, and for vertical $E_{z} (y)$ component. A similar modeling with the same program proved that it is unrealistic to associate ULF bursts with amplitudes about several mV/m recorded by low-orbiting satellites above upcoming earthquakes with a direct emission from an underground source (Mazur et al. 2024).

Analytical calculations of the electromagnetic field of an underground electric dipole were performed within the framework of two-layer (King et al. 1992) or three-layer (Pan and Li 2014) models. The exact formulas have the form of very complex Sommerfeld integrals, and the approximate relationships derived from them are valid only at a sufficiently large distance from the source $\rho = \sqrt {x^{2} + y^{2} } \ge 5|h|$. The complete solutions are extremely cumbersome, and therefore their use does not in any way facilitate the calculation compared to numerical models. The use of analytical relationships is justified only for elucidating the asymptotic behavior at large distances, while for the problem under consideration it is most important to estimate the field structure near the source. Therefore, we prefer to obtain quantitative results by numerically solving differential equations and perform the integral Hankel transform numerically. This approach is much less labor-intensive and more reliable. Moreover, the numerical method is easily generalized to the case of an arbitrary vertical distribution of parameters in LAI system, when one cannot even talk about explicit analytical formulas.

Characteristic features of the relationships between components vary significantly with distances from epicenter. The propagation of electromagnetic field from an underground source to an observation site on the Earth’s surface can be qualitatively imagined in the following way. At distances less than several depths or skin-depths, the electromagnetic field leaks upward from an underground source. At larger distances, $\rho > h$ and $\rho > \delta_{g}$, this field “propagates” to an observation site not as a direct wave inside the crust, but as a leaking mode along the Earth’s surface. The properties of electromagnetic disturbances are different in those regions. The characteristic features in the near-field zone (|B_z/B_h|> 1, |E_z/E_h|~ 1, Z >  > Z_g, and U >  > U_g) are very different from those of magnetosphere-ionosphere sources. However, in the far-field zone, the properties of signals “from below” becomes very similar to properties of a signal “from above” ((|B_z/B_h|< < 1, |E_z/E_h|> > 1, Z → Z_g, and U → U_g).

Let us summarize briefly what peculiarities of ULF field may be promising to reveal signals of lithospheric origin, and what kind of difficulties may be encountered upon in-field observations. However, we cannot validate the modeling predictions by existing reported ULF observations in seismo-active regions, because information on crust conductivity, spectral content, source depth and orientation, distance to observation site, is not fully available.

The finite magnitude of the vertical magnetic component was used to isolate seismogenic signals before strong earthquakes (Hayakawa et al. 1996; Hobara et al. 2004; Li et al. 2011; Kanata et al. 2014). Anomalous enhancements of spectral density ratio $S_{z} /S_{h}$ at the frequency range of 0.01 ± 0.003 Hz based on wavelet transform were found about one week before earthquake (M = 7.5) in Indonesiawithin 160 km from the station (Febriani et al. 2014).

The synchronous electro-telluric and geomagnetic observations, which make it possible to determine the impedance of the recorded disturbances, can effectively isolate disturbances of lithospheric origin in a vicinity of a source. However, this technique was not applied yet in seismo-electromagnetic studies.

The orientation of ULF emission polarization ellipse was used to determine the direction towards the epicenter of forthcoming earthquake (Du et al. 2002). However, our modeling has shown that polarization ellipse cannot be a universal recipe. Only in the sector along the current the small axis is directed towards a source, whereas in the sector across the source current the disturbance orientation may vary with distance.

The numerical modeling has shown that the vertical electric component is nearly vanishing beneath the Earth’s surface, because the conductivity spreading current flows near surface nearly horizontally, and only a small part leaks into the low-conductive atmosphere. Thus, the amplification factor concept is misleading. The underground source produces a disturbance in the electric field |E_z|–|E_h|. Monitoring the atmospheric electric field is a promising direction in the search for earthquake precursors since it is an indicator of the release of radioactive emanations and the formation of aerosols during the development of the seismic process. However, one cannot count on detecting intense bursts of $E_{z}$ from underground sources.

At the same time, the vertical component of the electric field in a borehole should be examined as a possible discrimination factor between the magnetospheric and seismogenic disturbances. Because the magnetospheric $E_{z}$ disturbances attenuate severely upon transmission into the ground, the $E_{z}$ measurements in a borehole may be effective tool for the discrimination of signals from an underground source (Tsutsui 2005). However, this conjecture was put forward based on the idealistic model. Upon realistic observations in boreholes, the ground inhomogeneities may significantly distort the vertical current and make measurements of $E_{z}$ ambiguous.

There were proposals to use gradient measurements with a small baseline which would suppress the contribution of large-scale disturbances of the ionospheric origin (Ismaguilov et al. 2002; Surkov et al. 2004). Using two near-by sites, separated by some distance (not more than a few km), the amplitude gradient and phase velocity can be measured. The measured amplitude gradient and phase velocity in the band $0.1 - 0.4\;{\text{Hz}}$ of expected lithospheric signals were typically around ~ 0.1 pT/km, and $U \approx 20\;{\text{km/s}}$ (Kopytenko et al. 1994). Our calculations have shown that noticeable amplitude and phase gradients can be recorded only in the immediate vicinity of the source, no more than 30 km. These characteristics are distance-dependent, therefore the gradient method should be applied with caution for the discrimination of signal from underground source.

Surely, the exact numerical results depend on the choice of the model parameters—source frequency and length, and crust conductivity. The crust conductivity within the fault may be much bigger than that of surrounding area. In the paper supplement we demonstrate the influence of the crust conductivity on the properties of ULF signal on the ground. We have used the same parameters, but the crust conductivity is chosen to be σ_g = 10^–1 S/m. The skin-depth corresponding to these parameters is δ_g ~ 5 km, instead of ~ 50 km used here. The comparison of the key plots for two values of crust conductivity shows that within a radial scale of about few skin-depths a general structure of the electromagnetic disturbance on the ground surface remains qualitatively similar.

As a first step, here we have considered the source as a harmonic oscillator. In reality, the seismogenic source may be an impulsive one, accompanying stick–slip movement along a fault. The ULF signal propagation inside the Earth crust occurs in diffusive manner, but no as a wave (Losseva and Nemchinov 2005). So, a characteristic scale with disturbed field should expand with time $t$ as $\Delta \rho \approx \sqrt {Dt}$, where $D = \left( {\mu_{0} \sigma_{g} } \right)^{ - 1}$ is the magnetic diffusion coefficient. Analytical relationships for electromagnetic signals generated by impulsive line-currents in a conductive space (Yamazaki 2016) show that maximal response at a distance $R$ from a source is observed at delay time $\tau \sim R^{2} /D$. The apparent propagation group velocity $V_{g}$ of impulsive disturbance measured as a time delay between signal maxima probably must be $V_{g} = D/R$. This relation is very different from the relationship of (Ismagilov et al. 2006) assumed that inside the crust the electromagnetic disturbance generated by harmonic seismic source may be treated as a wave process with phase velocity $U_{g}$.

Of course, the assumption about the source of seismogenic disturbances as a horizontal large-scale current is very speculative. Other types of ELF-ULF underground sources were considered: vertical electric current dipole (Tian and Hata 1996), surface charges at edges of deformed blocks (Losseva et al. 2012), etc. In the future, only laboratory and field studies will be able to determine possible types of mechanical-electromagnetic converters in the earth's crust. However, the qualitative features of the field of an underground source described in this work may remain the same for other possible types of lithospheric sources.

4 Conclusion

A numerical model has been developed for calculating ULF electromagnetic fields in the LAI coupled system, generated by an underground current source of a finite length. This numerical model is used to calculate the spatial structure of the field on the Earth's surface. The modeling demonstrates that in the source vicinity a number of characteristic features of such field (an impedance largely exceeding the Earth's impedance, a finite value of the ratio $|B_{z} |/|{\mathbf{B}}_{h} |$, small lateral gradient scale, high apparent phase velocity) can be used to reveal specific disturbances from seismogenic sources.

Appendices

Appendix

Mathematical and computational features of the algorithm

The first step is to find a solution of Maxwell’s equations with a source as a point horizontal current (current dipole). Having found a solution for such an elementary source, at the next step it is easy to calculate the total field, excited by a horizontal current of a finite length, just summarizing the fields generated by elementary dipoles. We use the representation of electromagnetic field via the wave potentials. The driving current is split into a curl-free and divergence-free (vortex) components, generating TE (transverse electric) and TM (transverse magnetic) modes. This division is fundamental in solving the problem under consideration. The initial problem is axially non-symmetric, while the curl-free and vortex components of the driving current separately have axial symmetry. Thanks to this, it becomes possible to separate variables applying Hankel's transform and come to boundary problem for a system of ordinary differential equations (ODEs) separately for curl-free and vortex components.

The electric and magnetic fields are represented through scalar $\Phi$ and vector ${\mathbf{A}}$ potentials as follows: ${\mathbf{B}} = \nabla \times {\mathbf{A}}$ and ${\mathbf{e}} = - \nabla \Phi + ik_{0} {\mathbf{A}}$. The transverse component of the vector potential can be represented in the form ${\mathbf{A}}_{ \bot } = (ik_{0} )^{ - 1} \nabla \times \Psi {\hat{\mathbf{z}}}$, where $\Psi$ is the scalar magnetic potential. Thus, we arrive at the following representation of the electromagnetic field via 3 potentials $A = A_{z}$, $\Phi$, and $\Psi$:

$$ \begin{gathered} {\mathbf{B}} = \nabla A \times {\hat{\mathbf{z}}} + (ik_{0} )^{ - 1} \nabla_{ \bot } \partial_{z} \Psi - (ik_{0} )^{ - 1} \nabla_{ \bot }^{2} \Psi {\hat{\mathbf{z}}}, \hfill \\ {\mathbf{e}} = - \nabla \Phi + ik_{0} A{\hat{\mathbf{z}}} + \nabla \Psi \times {\hat{\mathbf{z}}}. \hfill \\ \end{gathered} $$

(3)

Using this representation Maxwell’s equations are transformed to the system of partial differential equations (PDEs) for these potentials (Fedorov 2023a; b):

$$ \begin{array}{*{20}c} {\nabla_{ \bot }^{2} \partial_{z} A - ik_{0} \varepsilon_{ \bot } \nabla_{ \bot }^{2} \Phi - k_{0} g\nabla_{ \bot }^{2} \Psi = \mu_{0} {\text{Div}}{\mathbf{j}}_{ \bot } ,} \\ { \, \nabla_{ \bot }^{2} (\partial_{z}^{2} + \nabla_{ \bot }^{2} + k_{0}^{2} \varepsilon_{ \bot } )\Psi + ik_{0}^{2} g\nabla_{ \bot }^{2} \Phi = ik_{0} \mu_{0} {\text{Curl}}{\mathbf{j}}_{ \bot } ,} \\ { \, \partial_{z} \Phi - ik_{0} [1 + (k_{0}^{2} \varepsilon_{\parallel } )^{ - 1} \nabla_{ \bot }^{2} ]A = - (ik_{0} \varepsilon_{\parallel } )^{ - 1} \mu_{0} j_{\parallel } .} \\ \end{array} $$

(4)

Here 2D operators ${\text{Div}}{\mathbf{j}}_{ \bot } = \nabla_{ \bot } {\mathbf{j}}_{ \bot } = \partial_{x} j_{x} + \partial_{y} j_{y}$ and ${\text{Curl}}{\kern 1pt} {\mathbf{j}}_{ \bot } = (\nabla_{ \bot } \times {\mathbf{j}}_{ \bot } )_{z} = \partial_{x} j_{y} - \partial_{y} j_{x}$ have been used. The system (A2) can be applied for the excitation by a horizontal current ($j_{\parallel } = 0$) or vertical current (${\mathbf{j}}_{ \bot } = 0$).

Now we study the field produced by a horizontal current dipole with the moment $M_{0} = J_{0} d$ (with the current magnitude $J_{0}$ and infinitesimal length $d$) which is positioned at the point with the coordinates $x = 0,\;y = 0,\;z = h$. The source terms in (A2) in the cylindrical coordinates $\rho ,\,\varphi ,\,z\,$ comprise the curl-free part of the driving current ${\text{Div}}{\mkern 1mu} {\mathbf{j}}_{ \bot } = q(\rho )\delta (z - h)\cos \varphi$ and its vortex component ${\text{Curl}}\,{\mathbf{j}}_{ \bot } = - q(\rho )\delta (z - h)\sin \varphi$, where $q(\rho ) = M_{0} \partial_{\rho } [(2\pi \rho )^{ - 1} \delta (\rho )]$. The solution of the system (A2) with such specific form of driver terms is searched as follows $F(\rho ,\varphi ,z) = F_{c} (\rho ,z)\cos \varphi + F_{s} (\rho ,z)\sin \varphi$, where $F = A,\;\Phi ,\;\Psi$. Substituting these combinations into (A2) and grouping terms with factors $\cos \varphi$ and $\sin \varphi$, the following system is obtained:

$$ \begin{array}{*{20}l} {\partial_{z} A_{c,s} = ik_{0} \varepsilon_{ \bot } \Phi_{c,s} + k_{0} g\Psi_{c,s} + a_{c,s} {\mkern 1mu} ,} \hfill \\ {\partial_{z} B_{c,s} = - k_{0} g\Phi_{c,s} - (ik_{0} )^{ - 1} (k_{0}^{2} \varepsilon_{ \bot } + {\mathbf{R}})\Psi_{c,s} - b_{c,s} {\mkern 1mu} ,} \hfill \\ {\partial_{z} \Phi_{c,s} = ik_{0} [1 + (k_{0}^{2} \varepsilon_{\parallel } )^{ - 1} {\mathbf{R}}]A_{c,s} {\mkern 1mu} ,} \hfill \\ {\partial_{z} \Psi_{c,s} = ik_{0} B_{c,s} {\mkern 1mu} .} \hfill \\ \end{array} $$

(5)

Here the operator ${\mathbf{R}} = \rho^{ - 1} \partial_{\rho } \rho \partial_{\rho } - \rho^{ - 2}$, the inhomogeneous terms are $a_{c} = b_{s} = \mu_{0} {\mathbf{R}}^{ - 1} q(\rho )\delta (z - h)$ and $a_{s} = b_{c} = 0$. For symmetry the additional unknown function $B_{c,s} = (ik_{0} )^{ - 1} \partial_{z} \Psi_{c,s}$ has been introduced.

The system (A3) of PDEs has a cylindrical symmetry. Thanks to this, it may be reduced to the system of ODEs in respect to the variable $z$ using the Hankel’s transform of the 1-st order. This transform ${\mathbb{K}}_{1}$ reduces the differential operator ${\mathbf{R}}$ to a simple multiplication by $- k^{2}$. Applying the transform ${\mathbb{K}}_{1}$ to the system (A3) and introducing new variables $\tilde{A}_{c,s} = k{\mathbb{K}}_{1} [A_{c,s} ]$, $\tilde{B}_{c,s} = k{\mathbb{K}}_{1} [B_{c,s} ]$, $\tilde{\Phi }_{c,s} = k{\mathbb{K}}_{1} [\Phi_{c,s} ]$ and $\tilde{\Psi }_{c,s} = k{\mathbb{K}}_{1} [\Psi_{c,s} ]$ we arrive at the system of ODEs

$$ \begin{array}{*{20}l} {\partial_{z} \tilde{A}_{c,s} = \alpha \tilde{\Phi }_{c,s} + \beta \tilde{\Psi }_{c,s} + \tilde{a}_{c,s} {\mkern 1mu} ,\qquad \partial_{z} \tilde{\Phi }_{c,s} = \lambda \tilde{A}_{c,s} {\mkern 1mu} ,} \hfill \\ {\partial_{z} \tilde{B}_{c,s} = \gamma \tilde{\Phi }_{c,s} + \delta_{k} \tilde{\Psi }_{c,s} - \tilde{b}_{c,s} {\mkern 1mu} ,\qquad \partial_{z} \tilde{\Psi }_{c,s} = ik_{0} \tilde{B}_{c,s} {\mkern 1mu} ,} \hfill \\ \end{array} $$

(6)

where $\alpha = ik_{0} \varepsilon_{ \bot }$, $\beta = k_{0} g$, $\gamma = - k_{0} g$, $\delta_{k} = ik_{0} (\varepsilon_{ \bot } - k^{2} /k_{0}^{2} )$, $\lambda = ik_{0} [1 - k^{2} /(k_{0}^{2} \varepsilon_{\parallel } )]$, $\tilde{a}_{c} = \tilde{b}_{s} = S_{0} \delta (z - h)$ and $\tilde{a}_{s} = \tilde{b}_{c} = 0$. The factor $\delta (z - h)$ in the inhomogeneous terms of (A4) makes this system equivalent to a homogeneous system with the matching condition at the source level $z = h$. This condition can be obtained by integration of (A4) across the source. The homogeneous system corresponding to the system (A4) can be presented in a vector form as

$$ \partial_{z} {\mathbf{u}} = {\mathbf{Sv}},\qquad \partial_{z} {\mathbf{v}} = {\mathbf{Tu}}, $$

(7)

using 2D vectors ${\mathbf{u}}$ with components $\tilde{A}_{c,s}$, $\tilde{B}_{c,s}$, ${\mathbf{v}}$ with components $\tilde{\Phi }_{c,s}$, $\tilde{\Psi }_{c,s}$, and $2 \times 2$ matrices ${\mathbf{S}}$ and ${\mathbf{T}}$ of the coefficients in right-hand parts of (A4). To avoid numerical instabilities caused by growing exponential solutions of the system (A5), we introduce the matrix ${\mathbf{Y}}(z)$, which transforms the vectors ${\mathbf{v}}(z)$ into ${\mathbf{u}}(z)$ as follows: ${\mathbf{u}}(z) = {\mathbf{Y}}(z){\mathbf{v}}(z)$. This matrix is analogous to the frequently used admittance matrix relating electric and magnetic components. The matrix ${\mathbf{Y}}(z)$ obeys the matrix Riccati equation $\partial_{z} {\mathbf{Y}} = {\mathbf{S}} - {\mathbf{YTY}}$.

The system (A5) is augmented with boundary conditions at $z \to \pm \infty$, assuming that the electromagnetic field tends to zero upon an uplift into the ionosphere and immersion into the ground. These boundary conditions determine unambiguously the limiting values of the matrix ${\mathbf{Y}}$ at $z \to \pm \infty$. We calculate ${\mathbf{Y}}(z)$ by numerical solution of the matrix Riccati equation from above and from below to the source level and obtain different matrices ${\mathbf{Y}}(h \pm 0)$ there. Combining this discontinuity with matching conditions at the source altitude $z = h$, we find the vectors ${\mathbf{u}}(h \pm 0)$ and ${\mathbf{v}}(h \pm 0)$. Using them as the boundary values for a numerical solution of Cauchy problem for the system (A5) upwards and downwards from the source level we get a solution of (A5) at any height $z$ (Fedorov, 2023a,b). The Cauchy problems both for matrix Riccati equation and for the system (A5) were numerically solved using the standard Runge–Kutta method. The stability of the numerical solution is significantly improved by step-by-step corrections using the pre-computed matrix ${\mathbf{Y}}(z)$.

Finally, when the solution of (A5) has been obtained (which gives simultaneously the solution of (A4) as well), we apply to it the inverse Hankel’s transform and use the relationships (A1) transformed into cylindrical coordinates. As a result, the expressions for the field components produced by the horizontal current dipole are obtained:

$$ \begin{gathered} B_{{\varrho }} = \cos \varphi \left( {{\varrho }^{{ - 1}} (A_{s} - B_{c} ) + {\mathbb{K}}_{0} [\tilde{B}_{c} ]} \right) - \sin \varphi \left( {{\varrho }^{{ - 1}} (A_{c} + B_{s} ) - {\mathbb{K}}_{0} [\tilde{B}_{s} ]} \right) \hfill \\ B_{\varphi } = \cos \varphi \left( {{\varrho }^{{ - 1}} (A_{c} + B_{s} ) - {\mathbb{K}}_{0} [\tilde{A}_{c} ]} \right) + \sin \varphi \left( {{\varrho }^{{ - 1}} (A_{s} - B_{c} ) - {\mathbb{K}}_{0} [\tilde{A}_{s} ]} \right) \hfill \\ B_{z} = (ik_{0} )^{{ - 1}} \left( {\cos \varphi {\mathbb{K}}_{1} [k\tilde{\Psi }_{c} ] + \sin \varphi {\mathbb{K}}_{1} [k\tilde{\Psi }_{s} ]} \right) \hfill \\ e_{{\varrho }} = \cos \varphi \left( {{\varrho }^{{ - 1}} (\Phi _{c} + \Psi _{s} ) - {\mathbb{K}}_{0} [\tilde{\Phi }_{c} ]} \right) + \sin \varphi \left( {{\varrho }^{{ - 1}} (\Phi _{s} - \Psi _{c} ) - {\mathbb{K}}_{0} [\tilde{\Phi }_{s} ]} \right) \hfill \\ e_{\varphi } = \cos \varphi \left( {{\varrho }^{{ - 1}} (\Psi _{c} - \Phi _{s} ) - {\mathbb{K}}_{0} [\tilde{\Psi }_{c} ]} \right) + \sin \varphi \left( {{\varrho }^{{ - 1}} (\Psi _{s} + \Phi _{c} ) - {\mathbb{K}}_{0} [\tilde{\Psi }_{s} ]} \right) \hfill \\ \end{gathered} $$

The field-aligned electric component is negligibly small in the ionospheric plasma, $e_{z} \propto (k_{0} \varepsilon_{\parallel } )^{ - 1}$. With the electromagnetic components of the elementary dipole found, we further find the total field of the linear current of a finite length. The line current with amplitude $J_{0}$ and length $L$ is divided into a large number $N$ of equal parts with length $d = L/N$ and moment $m_{0} = J_{0} d$. The total moment corresponding to a system of dipoles is $Nm_{0} = NJ_{0} d = J_{0} L$ that is equivalent to the considered current $J_{0}$ with length $L$. A discrete structure of such system of elementary currents reveals itself only at small distances $\le d$. Upon calculations we have placed one dipole per kilometer.

Now we find the fields ${\mathbf{B}}(x,y,z)$ and ${\mathbf{E}}(x,y,z)$ produced by this set of horizontal current dipoles. To obtain the components of the field produced by $n$-th dipole we substitute the cylindrical coordinates $\rho_{n} (x,y)$ and $\varphi_{n} (x,y)$ of the observation point $(x,y,z)$ with respect to $n$-th dipole to the formulas for field components of the elementary dipole presented above, namely $B_{\rho ,\varphi ,z}^{(n)} (x,y,z) = B_{\rho ,\varphi ,z} (\rho_{n} (x,y),\varphi_{n} (x,y),z)$, and $E_{\rho ,\varphi }^{(n)} (x,y,z) = E_{\rho ,\varphi } (\rho_{n} (x,y),\varphi_{n} (x,y),z)$. The horizontal components additionally are to be transformed into the Cartesian coordinate system $B_{\rho ,\varphi }^{(n)} (x,y,z) \to B_{x,y}^{(n)} (x,y,z)$ and $E_{\rho ,\varphi }^{(n)} (x,y,z) \to E_{x,y}^{(n)} (x,y,z)$. Finally, these fields from all dipoles are summed up to determine the total field generated by the finite-length line current.