In order to investigate the induced currents for incident fields typical for lightning discharges, the remote fields associated with the lightning return stroke (RS) have to be obtained in a first step. The geometry of the problem is illustrated in Fig. 3. The objective is to compute the horizontal electric field along the buried cable (z=-d), which will serve as source term in the field-to-cable coupling equations (see next subsection). The lightning return stroke is assumed to be a straight vertical antenna located at 100 km from the cable (typical distance covered by LLS sensors). The average ground conductivity along the propagation path is assumed to be σp, while the local ground conductivity at the sensor site is σloc. This latter will determine the amplitude and waveshape of the horizontal Ex-field driving the cable shield current.

To represent the lightning return stroke channel, the Modified Transmission Line model with Exponential Decay (MTLE; Nucci et al., 1988; Rachidi and Nucci, 1990) was used. The parameters of the model were set to λ = 2 km (exponential decay of the RS current with height), vRS = 1.5 × 10⁸ m s⁻¹ (RS wavefront speed). The channel-base current is represented by the sum of two Heidler's functions, described by the following formula:

I(0,t)=I1ξ1tτ11n1tτ11n1+1e-tτ12+I2ξ2tτ21n2tτ21n2+1e-tτ22,ξi=exp⁡-τi1τi2niτi2τi11/ni. The parameters were chosen to form a channel-base current with characteristics of a typical subsequent RS: I1 = 10.7 kA, τ11 = 0.25 µs, τ12 = 2.5 µs, I2 = 6.5 kA, τ21 = 2 µs, τ22 = 230 µs and n1 = n2 = 2 (see Rachidi et al., 2001). The corresponding subsequent RS-type current waveform, with its short rise time of less than 1 µs, is depicted in Fig. 4.

The vertical electric field and the horizontal magnetic field generated by the return stroke are first computed assuming a propagation over a perfectly electric conducting (PEC) ground. The computation is performed according to Thottappillil et al. (1997), where the contributions of current dipoles along the channel are summed up to obtain the fields at an observation point located on the ground surface.

To account for the attenuation and dispersion that affects the lightning EM fields while propagating above a lossy ground, the fields computed assuming a PEC ground can be corrected by applying specific filters as described, e.g., in Norton (1937), Wait (1953), Shoory et al. (2010) (see also Wait, 1970, for a thorough compendium on wave propagation effects of electromagnetic fields along stratified media). Cross-validations using cylindrically symmetric 2D-FDTD (finite-difference time-domain) simulations have shown the best agreement using the Wait's attenuation function for a stratified ground, as described in details in (Shoory et al., 2010). While the use of Wait's function allows to straightforwardly consider a horizontally stratified ground, the case of a homogeneous (single-layer) ground was considered assuming a very thick upper layer (e.g., 10 km) to account accurately for the attenuation for arbitrary distances without spurious reflection phenomena from the lower layer boundary (for poor ground conductivities). The according equations (see Shoory et al., 2010, for details) can be readily implemented by typical numerical computational libraries. The Wait's expression for the attenuation function is given by 1Fstrpstr=1-jπpstre-pstrerfcjpstr in which pstr=-0.5γ0dΔstr2γ0=jωμ0ε0Δstr=ε0μ0K1K2+K1tanh⁡(u1h1)K1+K2tanh⁡u1h1Ki=ui/σi+jωε0εri,i∈1,2ui=γi2-γ02,i∈1,2γi=jωμ0(σi+jωε0εri),i∈1,2 where d is the propagation distance, h1 is the thickness of the upper layer of the stratified ground. The sub-index i denote the parameters of the respective layer, i∈1, 2, and sub-index 0 denotes parameters of the free space. The ground conductivity of the top layer, is referred to as σp (with index “p” denoting “propagation”) throughout the paper and impacts the rise time of the propagating EM field.

The next step is to determine the radial E-field (Er) in the direction of propagation, at the ground level and below the ground surface. To achieve this, the procedure described in Rubinstein (1996) is implemented, in which the so-called “wave-tilt” formula (Rubinstein, 1996) is used to obtain the radial E-field at the ground surface from the horizontal H-field as determined in the previous step, through the surface impedance of the air-ground interface: 2Er(z=0)=-Hφ(z=0)μ0εg+σlocjω with εg being the ground permittivity, σloc the local ground conductivity, μ0 the magnetic permeability of free space and ω the angular frequency. Finally, the electric field at a depth z below the ground level is found using Weyl's formulation, which reads 3Erz=Erz=0⋅ejωμ0εg+σlocjωz(z<0) These three formulations (Eqs. 1, 2 and 3) lead to very accurate results, as confirmed by comparisons with full wave numerical simulations using the FDTD method (see also Sect. 3.1). Finally, the radial E-field Er is projected onto the direction of the cable by multiplying with the cosine of the angle between the cable's direction and Er. These techniques provide us with the means to accurately compute the impinging horizontal electric fields on the cable, which serve as inputs for the field-to-cable coupling equations. Since σloc has a significant influence on the horizontal E-field, the coupling mechanism and, ultimately, the resulting LLS sensor site errors, values on the order of the expected (local) ground conductivities should be assumed when simulating a particular site. Although strong variations in local ground conductivities are generally expected even within small volumes near the cable (see for example Rizki Ramdhani et al., 2020, or Loke, 2001), regional ranges of estimated ground electrical conductivity values are available in the World Atlas of Ground Conductivities (ITU, 2015).

An essential component of the angle and amplitude site error investigations is the field-to-cable coupling model, which uses frequency-domain solutions based on Green's functions. These functions incorporate the coupling equations, while horizontal electric fields act as distributed sources along the cable length, as detailed in Aguet et al. (1980) and Tesche et al. (1997). Various approaches for coupling models have been explored, such as the treatment of bare and insulated wires for infinitely long lines in Bridges (1995), shielded cables with multiple layers and terminations in Aguet et al. (1980), and finite-difference time-domain methods for buried conductors and cable shields subjected to lightning strikes in Petrache et al. (2005). Further discussions on generated electric and magnetic fields in buried cables can be found in Bridges (1992). Bridges derived exact solutions for the induced current on an infinite bare or insulated cable buried in soil due to a transient plane wave (Bridges, 1995) and demonstrated that the transmission line theory provides accurate results for a wide range of cases. An experimental validation for the accuracy of the transmission line theory for field-to-cable coupling computations is presented in Paolone et al. (2005).

The relation describing the induced currents at a point x along a buried cable of length L, using Green's function GI(x,xs), reads (see for example Petrache et al., 2005): 4Ix=∫0LGIx,xsExexs,z=-ddxs where Exe is obtained from Eq. (3) presented earlier and the Green's function reads:

5GIx,xs=e-γL2Zc(1-ρ1ρ2e-2γL)e-γ(xs-L)-ρ2eγ(xs-L)⋅eγx-ρ1e-γxforx<xse-γL2Zc(1-ρ1ρ2e-2γL)e-γ(x-L)-ρ2eγ(x-L)⋅eγxs-ρ1e-γxsforx>xs The equations involve the complex propagation constant γ=Z′Y′, the cable characteristic impedance Zc=Z′/Y′, the line length L and the voltage reflection coefficients ρi=(Zi-Zc)/(Zi+Zc), with i∈1, 2, at the line ends, whereby Z1 and Z2 are the source and load impedances of the cable respectively. Due to the different expressions for x<xs and x>xs in Eq. (5), the integral in Eq. (4) needs to be partially integrated: I(x)=∫0L-x…dxs+∫L-xL…dxs, resulting in an analytical expression that can be straightforwardly calculated.

The longitudinal impedance Z' and transverse admittance Y' involve the calculation of the per-unit-length ground impedance Zg′ and ground admittance Yg′ (see in Fig. 5). In the present study, Theethayi's ground impedance formulation was used (see Eq. 9 in Theethayi et al., 2007). Detailed discussions on ground impedance models are available in Petrache et al. (2005) and Theethayi et al. (2007), while advanced formulations that account for the soil parameter frequency dependence are found in Visacro and Alipio (2012) and Duarte et al. (2021). For the convenience of the reader, the expressions for the distributed parameters (Fig. 5), as described in Theethayi et al. (2007), are reproduced as follows. 6Z′g=jωμ02πln⁡1+γgRabγgRab+2e-2dγg4+γg2Rab27Y′g≈γg2Zg′ in which γg=jωμ0(σg+jωεg) is the complex propagation constant in the ground. 8L′=μ02π⋅ln⁡ρbρa(foraninsulatedwire)9C′=2πεins/ln⁡(ρbρa)(foraninsulatedwire) where,

for insulated wires, ρa is the inner wire radius, ρb is the outer radius (including the dielectric jacket of permittivity εins), Rab=ρb;

All the equations were implemented in the frequency domain and applied under the assumption that both the input signal and the resulting outputs are real, causal signals. The input signal (horizontal magnetic field Hφ above PEC) spectrum was obtained using Fast Fourier Transform (FFT), with frequencies considered up to half the Nyquist frequency. To reconstruct real, causal signals, the upper half of the frequency spectrum (from half the Nyquist frequency to the Nyquist frequency) was completed by appending the complex conjugate of the computed results from the lower half of the spectrum. The final time-domain signal was then obtained by performing an inverse FFT on the completed frequency-domain data.

As shown in Bridges (1992), the calculated currents along the line can be straight-forwardly used to compute the nearby magnetic fields, as they are not strongly impacted by the air-earth interface. Consequently, the scattered magnetic field is computed using Biot-Savart's law, integrating the contributions of the current elements, obtained in the previous step (Sect. 2.2), along the nearest 50 m to the magnetic field sensor for each time instant. To solve the spatial integral by summing the contributions of the small current elements, it is important to have an accurate spatial current function. This can be readily achieved using a quadratic interpolation function. Moreover, due to the 1/r2 distance dependency, contributions beyond 50 m are assumed to be negligible. Figure 6 illustrates the mechanism of site errors. The scattered field is denoted as Herr(Ish,x), exhibiting y- and z-components in the given geometrical arrangement. The cable is oriented in the x-direction, as the field depends on the induced shield, respectively conductor current Ish,x, which is aligned with the cable's direction.

This field adds a spurious term to the incident field Hφ (horizontal, thus purely in the xy-plane) resulting in a sampled field Hsampled that exhibits an altered angle and magnitude in comparison to the true incident field Hφ. As a consequence of Ampere's law, the error magnetic field vector Herr is azimuthal around the power supply cable, which is assumed to be straight. The vertical component of the head of the supply cable below the sensor, and thus the corresponding H-field, is not considered, as it is aligned axially with the sensor and is assumed to have a negligible impact. For more complex shapes of the power supply cable paths, including corners and bends), the scattered (error) field vector Herr may have arbitrary orientations. Further, it must be noted that Fig. 6 must be understood as a snapshot at a specific sampling instant, where all vector lengths and angles are time-dependent according to the incident EM wave and the induced currents. A typical LLS sensor samples the field when the vector Hsampled reaches its maximum value, referred to as maximum sampled magnitude in the present study. At this instant, the difference between the true incident field angle φ and the sampled angle, φsampled, computed using the arc tangent of the output voltage ratio of the x- and y-component of the crossed loop antenna, is defined as the angle error αerr. The amplitude error (sometimes also called “signal error”) is denoted as serr and defined as serr=|Hsampled||Hφ|.

This section presents the simulation results of lightning incident electric fields following the procedure described in Sect. 2.1, considering a channel-base current waveform that exhibits characteristics that are typical of subsequent return strokes (in particular, characterized by a short rise time), as depicted in Fig. 4. All results are obtained for a distance to the lightning discharge of 100 km. Due to the linearity and time-invariance of the equations utilized in this paper, the amplitude of the channel-base current was kept constant throughout all computations. Variations of the E-fields used as input for the coupling analyses were solely the result of the assumed ground parameters along the propagation path (see Fig. 7). The main results of this paper, namely the angle and amplitude site errors, are independent of the selected channel-base current amplitude; that is, they are unaffected by any scaling of the waveform.

The vertical electric field above the PEC is shown in Fig. 7 (blue curve). Note that the vertical E-field (Ez) and the horizontal H-field (Hφ, which is sensed by an MDF sensor) above a PEC ground are related through Hφ=c0/μ0⋅Ez. Simulation results using Wait's formalism, accounting for the attenuation of the fields due to the propagation over a lossy ground, are also shown in Fig. 7 for different ground conductivities. As can be seen, the higher the conductivity, the lower the attenuation and dispersion. Lower values for the ground conductivity lead to more attenuated and dispersed fields with longer rise times (about 2, 4 and 10 µs for the orange, green and red curve, respectively). It is worth noting that the frequency-dependent attenuation function (Eq. 1) is also a function of distance: the farther the field propagates, the greater the attenuation and dispersion. Thus, for closer lightning strikes, the fields retain more of the high-frequency content and exhibit shorter rise times, e.g., at 50 km compared to those depicted in Fig. 7 for 100 km.

The radial E-fields at the ground level and below ground are obtained after applying Eqs. (2), and (3) to the horizontal magnetic field above lossy ground. The results are shown in Fig. 8.

The impact of the burial depth for higher ground conductivities, namely σloc = 10 × 10⁻³ S m⁻¹ and σloc = 50 × 10⁻³ S m⁻¹, for an incident field according to Fig. 7 (σp = 10 × 10⁻³ S m⁻¹, orange curve), is shown in Fig. 8a and b. As can be seen, a significant reduction in the amplitude is already observed within the first 5 m below the ground level.

Since these horizontal E-fields serve as the input for the cable current coupling model, ensuring their validity is crucial for achieving accurate results in subsequent computational steps. Thus, the fields were validated through cross-comparison with results obtained from a cylindrical-symmetry 2D-FDTD solver (Oskooi et al., 2010; NanoComp, 2025). The results for both approaches are found to be in excellent agreement. A validation example is shown in Fig. 9. The depicted scenario represents very low ground conductivities (0.1 × 10⁻³ and 0.01 × 10⁻³ S m⁻¹) and large depths (d = -50 m) below ground. The difference in the amplitudes between the fields at ground level and those below ground is very small, indicating that attenuation in low-conductivity ground is negligible for typical burial depths of power supply cables, which range from a few tens of centimeters to about 1 m.

Figure 10 illustrates the general dependencies of the horizontal electric fields, both at the ground level and below ground, on the propagation path ground conductivity. Figure 10a shows the ratio of the peak value of the horizontal electric field at ground level, Ex,peak (d = 0 m) to that of the vertical E-field, Ez,peak, at the ground level. This ratio can be as high as 30 % for very low conductivity (σloc = 0.01 × 10⁻³ S m⁻¹) and drops to 2.5 % for high ground conductivity (σloc = 50 × 10⁻³ S m⁻¹). Due to the frequency dependence of the physical mechanisms governing the local field configuration, these ratios depend on the frequency content of the incident field. To account for attenuation and dispersion along the 100-km propagation path, different lossy grounds with ground conductivity values ranging from σp = 100 × 10⁻³ S m⁻¹ to σp = 0.1 × 10⁻³ S m⁻¹ have been investigated.

As previously shown in Fig. 7, propagation over a highly conducting ground (ideally PEC) preserves the high-frequency content of the propagating fields. This results in incident fields exhibiting fast transients and corresponding short risetimes. In contrast, propagation over less conductive ground attenuates the high-frequency content and causes dispersion, leading to incident fields with slower transients and longer risetimes. Examination of Fig. 10a now reveals two key aspects.

Fields with shorter rise times (fast transients) produce larger Ex-field peaks (as evidenced by the bold blue curve with the thin red curve at a given local ground conductivity σloc) and

The penetration of the horizontal E-field, Ex (d = 0 m) into various depths below ground level is shown in Fig. 10b, illustrating the ratio of the Ex-field peaks at the surface and below ground. The figure clearly shows that low ground conductivities allow for deep penetration, with minimal attenuation of the field peak over depth. Conversely, for high conductivities such as σloc = 50 × 10⁻³ S m⁻¹, the attenuation with depth below ground is more significant. Notably, e.g., at d = 1m below ground, attenuation ranges between 13 % (dashed, thickest blue line) to 20 % (solid, thickest blue line) – a critical observation discussed further in Sect. 4. The discrepancy between the solid and dashed lines again stems from the fact that waveforms with a higher frequency spectrum (i.e., fast transients) are better preserved during propagation along a medium with high conductivity σp during propagation, are more significantly attenuated through the ground at the sensor site, irrespective of the local ground conductivity σloc.

Hereafter in this section, the field-to-cable coupling computations described in Sect. 2.2 are performed assuming a propagation ground conductivity σp = 10 × 10⁻³ S m⁻¹ and a local ground conductivity σloc = 10 × 10⁻³ S m⁻¹, unless stated otherwise. The conductor radius is ρa = 10 mm, while the cable jacket is 5 mm thick, resulting in an outer radius of ρb = 15 mm. The cable jacket has a relative permittivity of εr,d = 3. The conductor can be regarded as a cable shield with an equivalent outer radius typical of power supply cables. The impact of a slightly higher electrical resistance due to a thin screen, as opposed to a solid conductor, is negligible in the coupling analyses that follow.

Two examples considering different termination impedance scenarios are considered. The results for the induced currents are shown in Fig. 11. Figure 11a presents the results for a cable terminated at its ends with an impedance of Z1 = Z2 = 10 Ω, a value typically achievable at sites with σloc = 10 × 10⁻³ S m⁻¹. The peak current at the line end (at x = 450 m, thickest lines in Fig. 11) reaches a value of about 100 mA. Figure 11b shows the simulation results assuming that the line end is disconnected (shield not connected to ground modeled through a large impedance value of Z2 = 10 MΩ). In this configuration, the current at x = 450 m is naturally zero. However, within the first 50 m away from the line end (see x = 400 m), the current gradually increases, reaching a peak value of about 60 % of the overall peak current, which occurs near the middle of the line. Since a completely ungrounded shield (with no connection to ground on either side of the cable) is an uncommon installation practice for shielded cable, this scenario is not considered. Such a setup would also result in zero current at the start of the line (x = 0 m).

The effect of the line length on the induced current is shown in Fig. 12. As the line length decreases, a corresponding reduction in the induced shield current peak is observed. A significant increase of the induced shield current peak current would be observed when the local conductivity is lower. This observation is aligned with the findings of Sect. 3.1, where a reduction in the local conductivity results in an increased horizontal electric field and decreased attenuation during ground penetration. However, in scenarios with reduced conductivities, the smallest achievable line termination impedance also increases, compensating the current increase and reducing the currents near the line ends. The largest shield current amplitudes would in turn be closer to the midsections of the cable.

The induced currents also depend on the conductor radius, generally exhibiting higher amplitudes for larger radii. However, these differences are in the order of +10 % when the conductor radius is doubled. Therefore, the influence of conductor radius on the results is not further analyzed in this study.

In this section, the waveshape of the scattered magnetic field Herr (se Fig. 6) is examined more in detail, and the related angle and amplitude site errors that would, at least theoretically, be observed by the MDF sensor used in LLSs is explored. The impact of several parameters, such as line length, ground conductivity, burial depth and the vertical separation between the MDF antennae and the supply cables is investigated.

As explained in Sect. 2.2, once the spatiotemporal behavior of the conductor currents is determined, the scattered magnetic field can be computed by applying Biot-Savart's law. Given the short distances of the relevant current elements impacting the sensed magnetic field (less than 50 m), time retardation can be disregarded. Fig. 13 presents the results for four local ground conductivity (σloc) scenarios as described in the figure caption. To get reasonable results, the termination impedances Z1 and Z2 were assumed to be conductivity- and frequency-dependent, considering a 10 m long, 3 cm thickness vertical grounding rod (see for example Grcev, 2009). Their DC-values are given in the sub-figures of Fig. 13. Otherwise, the shield currents would reach unrealistically high amplitudes due to a grounding impedance value which would be unattainable at a site with, for example, a very low conductivity. The graphs show the main lightning field to be detected, Hφ (black dotted line), incident with an angle of φ = 30° relative to the cable (aligned with the x-axis). They also show the x- and y-components (orange and green) of Hφ, the scattered field Herr, which is a y-component that adds to Hy, due to a current Ish oriented along the x-axis, and finally the resulting total field Hsampled, which the sensor samples at the instant of its maximum magnitude. The scattered field Herr is responsible for a distortion of the H-field vector of the true incident lighting EM field, resulting in angle and amplitude errors, αerr and serr, respectively. Note that the estimated time of arrival is not significantly affected by the addition of the Herr field, as it is determined as close as possible to the onset of the waveform's rising edge. Thus, the LLS location results obtained using the ToA technique remain unaffected by the phenomena illustrated in Fig. 13.

The differences in site errors shown in Fig. 13a–d is attributed to the characteristics of the scattered fields (Herr) impacting the site errors at the sampling instant. For high conductivity values (e.g., σloc = 10 × 10⁻³ and 1 × 10⁻³ S m⁻¹, see Fig. 13a and b), the maximum of the scattered field Herr nearly coincides with the maximum of Hsampled. However, for very low ground conductivity (0.1 × 10⁻³ S m⁻¹ and smaller, see Fig. 13c, d), the induced current wave on the cable shield experiences minimal attenuation as it propagates along the shield. This leads to pronounced reflections and resonances along long lines. As a consequence, oscillations arise in the induced currents, producing fast ringing effects in the scattered field Herr. These oscillations exhibit a frequency that depends on the line length and can arbitrarily impact the sampling instant – and thus the site errors potentially causing sampling to occur when Herr is zero or even of opposite polarity. It must be noted that the sharper the impinging transients, the more pronounced the oscillations of the induced current response. While incident fields with very high-frequency content (i.e., short rise times), combined with very low local ground conductivity σloc and long cables, may occur in reality, such scenarios are rare. Nevertheless, this possibility should not be overlooked, because, as explained in Sect. 3.1, lightning discharges occurring close to the sensor also contain high frequency content, and thus short measured rise times can be expected.

If an insulated power supply cable were hypothetically replaced by a bare conductor of the same length, parameters, and termination conditions, simulations indicate that the induced conductor currents near the line end would be moderately reduced. The associated scattered field Herr would, in turn, be reduced as well, resulting in reductions in site errors. Specifically, the angle site error decreases by 33 %, and the amplitude site error by 37 %, as shown in the example presented in Fig. 14a and b and shows the result for a cable of length L = 100 m. The results of currents and H-fields were compared to results obtained from fully integrated 3D-FDTD simulations, using the open source FDTD-solver Elecode (see Kuklin, 2021, 2022), which supports modeling insulated conductors. The comparisons showed very good agreement.

Next, after thoroughly addressing the physical quantities, including the electric fields below ground, coupled currents and scattered magnetic fields, that contribute to the LLS sensor site errors, it remains to finally investigate the angle and amplitude site errors over a full 360° azimuth rotation. Hereby, key parameters impacting the results will be highlighted. The H-field sensor is assumed to be located at a height of z = 2 m above ground.

We begin by examining the impact of the burial depth of the power supply cable on the site errors. The simulation results are presented in Fig. 15 and cover two distinct scenarios:

Scenario 1: As the burial depth increases, the distance between the cable to the sensor head also increases, reflecting the most realistic scenario. In this case, the site error reduction is influenced by a combined effect of increasing distance between the cable to the H-field sensor and the field attenuation by the ground (solid lines in Fig. 15).

The results presented in Fig. 15 were obtained for a local ground conductivity σloc = 10 × 10⁻³ S m⁻¹. They reveal a significant finding: The site errors are very strongly impacted by the (vertical) distance of the cable to the H-field sensor, as indicated by the solid-line curves. In contrast, the dashed-line curves, representing the scenario with a fixed 2 m distance, exhibit only a minor reduction in site errors with increasing burial depth. Specifically, at a burial depth of 1.5 m in Scenario 2, the angle site error αerr is reduced by only 8.5 %. However, in Scenario 1, where the cable-to-sensor distance increases with burial depth, the reduction reaches 46 %. This finding is consistent with results presented in Fig. 10b which suggests the same effect based on the attenuation caused by the ground penetration of the Ex-field for the assumed parameters. The amplitude site errors serr exhibit a similar trend, decreasing by comparable amounts.

Next, the impact of a significantly higher local ground conductivity σloc is investigated. As shown previously in Fig. 10b, higher conductivity increases the attenuation of the illuminating Ex-field as it penetrates to ground. Additionally, Fig. 10a demonstrated that higher σloc leads to smaller site errors due to the reduced horizontal Ex-field illuminating the cable shield. To account for this effect, a new baseline angle site error was calculated for a cable placed at ground level (d = 0 m) and a sensor located 2 m above, assuming a value for the local ground conductivity of σloc = 50 × 10⁻³ S m⁻¹. The angle site error in this case drops to 3.86°, compared to 7.5° for σloc = 10 × 10⁻³ S m⁻¹ at an azimuth of 130°, for example. Using this new baseline angle site error, the impact of ground attenuation for a buried cable is re-evaluated. For Scenario 2 (only the effect of ground attenuation), the angle site error is reduced by 20 % at a burial depth of d = 1.5 m, compared to just 8.5 % for the lower conductivity case σloc = 10 × 10⁻³ S m⁻¹. In Scenario 1 (which includes both ground attenuation and increased distance to the sensor), the reduction reaches 54 %, compared to 46 % for σloc = 10 × 10⁻³ S m⁻¹.

Thus, while the attenuation-caused reduction is greater for higher σloc (20 % vs. 8.5 %), the dominant factor contributing to the total site error reduction in Scenario 1 remains the increased vertical distance between the sensor and the cable. It is important to note that these findings are independent of the significant overall decrease in site error of almost 50 % (for σloc = 50 × 10⁻³ S m⁻¹ in contrast to σloc = 10 × 10⁻³ S m⁻¹) that results directly from the reduced Ex-field strength at high local ground conductivity.

At this point, one further investigation naturally presents itself: examining the impact of the increasing sensor's vertical distance to the cable, as the sensor may be mounted on top of a high mast. This installation type has been employed at some LLS sensor sites, where it has been associated with minimal site errors. It is expected that the observed behavior will approximate an inverse relationship to the distance 1/r, with r being the vertical distance from the cable to the sensor. This expectation aligns with Ampere's law, according to which the magnetic field of an infinitely long cable is H=I/(2πr). This relationship is confirmed in Fig. 16, which shows the angle site error decreasing from ±6.7° at a vertical distance of 2.5 m from the cable to ±1.8°. Although slightly less pronounced, this reduction closely follows the expected 1/r relationship. One important comment must be added: further induction phenomena are expected due to the prevailing vertical E-field (Ez) impinging on the high mast and the cable. This could lead to potential additional induced currents in the grounding of the mast, which could impact the shield currents of the supply cable and, consequently, the behavior illustrated in Fig. 16. The results match well with real-world experience of this type of installation, which was previously employed by the Austrian LLS operator ALDIS (Austrian Lightning Detection and Information System).

The final evaluation aims to provide a comprehensive overview of the influence of the local ground conductivity, length and termination impedance – factors previously illustrated by means of time-domain graphs of coupled currents and magnetic fields – on the site errors, summarized in a single figure. All simulations were performed considering an insulated wire with an outer cable radius of ρb = 7.5 mm and core radius of ρa = 5.8 mm, a jacket dielectric relative permittivity of εr,d = 3, buried at a depth of 50 cm. Two different sets of ground parameters are considered to examine the impact of different (short vs. long) rise times of the incident lightning fields: (i) εr,g = 10, σp = 1 × 10⁻³ S m⁻¹, and, (ii) εr,g = 10, σp = 0.1 × 10⁻³ S m⁻¹. The distance to the lightning strike was assumed to be 100 km. Both line ends are assumed to be grounded with a vertical grounding rod (see Grcev, 2009), resulting in termination impedances which are frequency- and conductivity-dependent. This approach provides a more realistic representation compared to a constant grounding impedance, which may not appropriately represent the prevailing local grounding conditions. Having both ends grounded, where the largest currents can flow, represents the worst-case scenario for LLS sensor site errors. This analysis deliberately focuses on this worst-case scenario with the aim of understanding the primary interrelation between the influencing parameters. In this context, the interpretation of this significant, yet very isolated and hypothetical scenario should be taken with care and regarded solely as a rough estimate for the maximum expected site errors. The discussion section is dedicated to further considerations and to the elaboration on more special or real-world case studies.

Figure 17a–d (σp = 1 × 10⁻³ S m⁻¹) and Fig. 18a–d (σp = 0.1 × 10⁻³ S m⁻¹) depict the maximum observable angle site errors αerr (occurring at an incident angle of about φ = 45°, refer to Fig. 6 for comparison) and amplitude site errors serr (occurring at φ = 0°) for different incident fields – one with faster rise times (Fig. 17) and one with slower rise times (Fig. 18). Figures 17a, c and 18a, c show αerr and serr for insulated cables, Figs. 17b, d and 18b, d for bare wires. The graphic also shows the direct current impedance value ZDC, which the line termination impedances Z1 and Z2 assume for a vertical grounding rod of 10 m length, on the right ordinate. The frequency dependence was taken into account as well, although the impact is negligible for the frequency range of the induced currents.

The most important observations are summarized in the following bullet points, based on the use of “fast fields” for the case with σp = 1 × 10⁻³ S m⁻¹ (i.e., Fig. 17), with rise times of about 4 µs (Fig. 7, green curve), due to the higher frequency components present in the incident field, and “slow fields” for σp = 0.1 × 10⁻³ S m⁻¹ (i.e., Fig. 18) with rise times of about 10 µs (Fig. 7, red curve), reflecting the low frequency content of the incident field:

Decreasing the supply cable length decreases site errors but causes a slight shift in the maximum site errors towards higher conductivity values. This trend is consistent for incident fields of different rise times.

The graphs in Figs. 17a, c and 18a, c have practical application. For a given site provisioned for sensor installation, the LLS operator can easily estimate the expected maximum site errors. These graphs represent the worst-case scenario, where the cable shield of a supply cable (insulated conductor scenario) is grounded at both sensor ends. For a given cable length L and a vertical cable-to-sensor distance of 2.5 m, the maximum angle error αerr or amplitude error serr can determined based on the local ground conductivity σloc, and the sensor grounding impedance ZDC. If the sensor grounding impedance is lower than the ZDC value (blue ordinate in Figs. 17 and 18) for the given conductivity, the maximum site errors will exceed those shown in the graphs (due to higher currents at lower impedance). Conversely, if the grounding impedance is higher, the site errors will be smaller.