Nowcasting and Validating Earth's Electric Field Response to Extreme Space Weather Events Using Magnetotelluric Data: Application to the September 2017 Geomagnetic Storm and Comparison to Observed and Modeled Fields in Scotland

In the United Kingdom, geomagnetically induced currents (GICs) are calculated from thin‐sheet electrical conductivity models. In the absence of conductivity models, time derivatives of magnetic fields are sometimes used as proxies for GIC‐related electric fields. An alternative approach, favored in the United States, is to calculate storm time electric fields from time‐independent impedance tensors computed from an array of magnetotelluric (MT) sites and storm time magnetic fields recorded at geomagnetic observatories or assumed from line current models. A paucity of direct measurements of storm time electric fields has restricted validation of these different techniques for nowcasting electric fields and GICs. Here, we present unique storm time electric field data from seven MT sites in Scotland that recorded before, during, and after the September 2017 magnetic storm. By Fourier transforming electric field spectra computed using different techniques back to the time domain, we are able to make direct comparisons with these measured storm time electric field time series. This enables us to test the validity of different approaches to nowcasting electric fields. Our preferred technique involves frequency domain multiplication of magnetic field spectra from a regional site with a local impedance tensor that has been corrected for horizontal magnetic field gradients present between the local site and the regional site using perturbation tensors derived from geomagnetic depth sounding (GDS). Scatter plots of scaling factors between measured and nowcasted electric fields demonstrate the importance of coupling between the polarization of the storm time magnetic source field and Earth's direction‐dependent deep electrical conductivity structure.


Introduction
The magnetotelluric (MT) sounding technique (Tikhonov, 1950(Tikhonov, , reprinted 1986Cagniard, 1953) involves measuring and cross-correlating natural fluctuations in the electric and magnetic fields that are induced in the Earth due to solar-magnetosphere-ionosphere interactions. As a geophysical technique, MT is primarily used to constrain the conductivity structure of the Earth's crust and upper mantle following computation of period-dependent impedance tensors from cross-correlated horizontal components of the induced electromagnetic fields (e.g., Simpson & Bahr, 2005): where E denotes electric field, B denotes magnetic field, Z ij denotes impedance, and subscripts x and y are orthogonal directions in the horizontal plane at the Earth's surface.
However, the nature of the passive MT source field means that MT also lends itself to the study of space weather by providing information about the electric fields that drive geomagnetically induced currents (GICs) that are potentially hazardous (e.g., Boteler, 2003;Cannon et al., 2013) to power grids (e.g., Bolduc, 2002;Pulkkinen et al., 2017), gas pipelines (Gummow, 2002;Viljanen et al., 2006), and railway networks (Krausmann et al., 2015). From Equation 1, we see that if we already know the MT impedance tensor at a range of periods from previous measurements and have magnetic field data during a magnetic storm from a geomagnetic observatory, we can calculate the electric fields that would be induced by the magnetic storm-that is, we can negate the need for continuous, long-term, electric field measurements once we have determined the impedance tensor at a range of periods. This range of periods is approximately 1-10 5 s (e.g., Simpson & Bahr, 2005). At periods shorter than 1 s, the electromagnetic source field is dominated by ionospheric activity such as lightning rather than solar-magnetosphere-ionosphere interactions; at periods longer than 10 5 s, the source field becomes nonuniform and time dependent even during solar quiet times (e.g., Simpson, 2002), violating the plane wave assumption of the MT method. The plane wave assumption is also more likely to be violated during magnetic storms, when the source field becomes more complex. However, at midlatitudes, the storm time source field is dominated by the ring current, the wavelength associated with the spatial heterogeneity of magnetic storms is an order of magnitude longer than the penetration depth at the longest period and source field heterogeneities are expected to be of second-order importance compared to variations in the Earth's 3-D electrical conductivity structure, which encompass several orders of magnitude (e.g., Simpson, 2000;Simpson et al., 1997;Simpson & Bahr, 2005).
The magnetic field fluctuations show less spatial variation than the induced electric fields. Therefore, time-dependent magnetic field variations from a geomagnetic observatory can often be used as proxies for magnetic field variations at neighboring sites. However, in the presence of geomagnetic anomalies, the magnetic field will also be modified. In this case, the induced electric fields across a region will not be correctly estimated using the impedance tensor and geomagnetic observatory data only, because it is necessary to correct for horizontal magnetic field gradients between the geomagnetic observatory and neighboring sites where the impedance tensor is known. This correction can be incorporated using transfer functions obtained from geomagnetic depth sounding (GDS). GDS is a technique that is complementary to MT for investigating the electrical conductivity structure of the Earth (Simpson & Bahr, 2005). GDS utilizes the horizontal and vertical components of the fluctuating magnetic field, to obtain a perturbation tensor, W (Schmucker, 1970) that relates the magnetic field components at a local measurement site (subscript L) to those at a simultaneously recording reference or observatory site (subscript O): GDS transfer functions can be computed at no extra cost from MT time series. The impedance tensor correction for the anomalous magnetic field between a geomagnetic observatory and an MT measuring site is derived in the next section (section 2). We note similarities in our approach to Bonner and Schultz (2017). However, our MT array data (Simpson & Bahr, 2020) from the 8 September 2017 magnetic storm provides a unique opportunity to validate the correction procedure.
According to Faraday's law, a time-varying magnetic field induces closed loops of electric fields around an axis oriented parallel to the inducing field: Hence, it has been suggested (e.g., Pulkkinen et al., 2006;Viljanen et al., 2001;Watermann & Gleisner, 2009) that time derivatives of storm time magnetic fields might have a role to play in forecasting GIC-related electric fields. For example, in the idealized scenario of a 1-D Earth, Equation 3 reduces to Therefore, for a 1-D approximation, time derivatives of the horizontally polarized magnetic fields are equal to the vertical gradients of the perpendicularly polarized electric fields. However, the greater regional uniformity of magnetic fields compared to electric fields raises doubts about the efficacy of ∂B/∂t for forecasting GICs.
Increasing awareness of the potential socioeconomic effects (e.g., Eastwood et al., 2018;OECD, 2018) of magnetic-storm-related damage to ground-based infrastructure has spawned a plethora of scientific papers concerned with modeling GICs in different geographic regions (e.g., Viljanen et al., 2012;Bedrosian & Love, 2015;Lucas et al., 2018;Bonner & Schultz, 2017;Espinosa et al., 2019;Love et al., 2019;Marshall et al., 2019;Rosenquist & Hall, 2019). However, because large magnetic storms occur infrequently and site occupation times for most MT surveys targeting crustal structure typically range from 1-2 days to 2-3 weeks, the statistical chances of recording a severe geomagnetic storm during a typical MT survey are low. Therefore, it has not been possible to compare and validate different approaches to generating hazard maps (e.g., Love et al., 2016) and models (e.g., Lucas et al., 2018;Viljanen et al., 2012) of space weather impacts on ground-based infrastructure against empirical measurements of storm time electromagnetic fields. Here, we present an analysis of a unique electromagnetic dataset (Simpson & Bahr, 2020) recorded during the 8 September 2017 magnetic storm at seven MT sites in the Scottish Highlands ( Figure 1) and explore how well we can reproduce the electric fields measured during the storm using different techniques.

Data Acquisition
We recorded five-component (H x , H y , H z , E x , and E y ) continuous, electromagnetic time series at seven sites ( Figure 1) in northern Scotland during the 8 September 2017 geomagnetic storm (Simpson & Bahr, 2020). We employed low-noise, Magson magnetometers and nonpolarizable silver-silver-chloride electrodes (Filloux, 1987) that were buried to a depth of 0.5 m to mitigate against temperature-related fluctuations and drift. Impedance tensors and perturbation tensors were computed (e.g., Egbert & Booker, 1986;Simpson & Bahr, 2005) from MT data collected intermittently over a duration exceeding 3 months at sites chosen for their remoteness from population centers and are, therefore, well constrained to periods up to 30,000 s. The need for such long periods has been discussed by Bonner and Schultz (2017). Although electric and magnetic fields fluctuate with time, for stable tectonic environments, the bivariate impedances calculated from them can be considered stationary (e.g., Hanekop & Simpson, 2006). Therefore, it is not critical which time windows are used to calculate the MT impedances, with the caveat that storm times should be avoided because of an increased likelihood that the plane wave assumption of the MT method (e.g., Simpson & Bahr, 2005) is violated at such times.

Objectives and Data Analysis
Our objectives in this paper are to compare the measured electric field responses (Simpson & Bahr, 2020) at any particular site during the September 2017 storm with the following:  (Simpson & Bahr, 2020) during the September 2017 magnetic storm.
1. The electric field determined from the local MT impedance tensor (Equation 1) at that site multiplied in the frequency domain with (i) the magnetic field spectra recorded during the storm at the same observation site; (ii) the magnetic field spectra at site RAN; (iii) the magnetic field spectra at RAN with a correction to the impedance tensor that takes into account anomalous geomagnetic fields between RAN and the observation site, as described by the perturbation tensor (Equation 2). We will refer to this approach as "electric fields determined from local impedance tensors." 2. The electric field determined from frequency domain multiplication of the magnetic field spectra recorded during the storm (Simpson & Bahr, 2020) at the MT site of interest with an MT impedance modeled (Siripunvaraporn et al., 2002;Siripunvaraporn et al., 2005) assuming (i) a 100-Ωm half-space; (ii) a 100-Ωm half-space plus 3-D near-surface conductance of the ocean surrounding the United Kingdom; (iii) a 900-Ωm half-space; (iv) a 900-Ωm half-space plus 3-D near-surface conductance of the ocean surrounding the United Kingdom. We will refer to this approach as "electric fields determined from modeled impedance tensors." The resistivity of seawater is assumed to be 0.25 Ωm, and the maximum depth of the modeled ocean is 4 km. 3. Time derivatives of the horizontal magnetic fields measured simultaneously at the same sites (Simpson & Bahr, 2020) during the storm.
Time series analyses were performed by (i) Fourier transformation of the requisite magnetic field into the frequency domain; (ii) multiplication of the magnetic field spectra with the appropriate impedance tensor to calculate electric field spectra; (iii) Fourier transformation of the spectra back to the time domain for plotting as time series that could be compared with the measured time series. We decimated the electromagnetic time series to a cadence of 32 s and used window lengths of 1,024 data points for our Fourier evaluation. We also present frequency domain scatter plots of univariate and bivariate scaling factors between measured and nowcasted electric fields from objectives (1) and (2). Others (e.g., Kelbert et al., 2017;Love et al., 2016) have previously convolved measured impedance tensors with magnetic field spectra to investigate electric field spectra at particular frequencies during magnetic storms, but validation has only previously been undertaken using three long-duration electric field measurements from Japan (Kelbert et al., 2017).

Correction for Horizontal Magnetic Field Gradients
To derive the correction required by the substitution of the measured magnetic field at a site with the magnetic field from a neighboring site (in our case RAN), we consider two sites O and L, where O is a geomagnetic observatory site and L is a local site where the MT impedance is available. In MT, the horizontal electric and magnetic fields measured at the two sites, O and L are related via complex impedance tensors, Z O and Z L : The GDS transfer function between O and L is given by the relation: where, for the purposes of our study, we are only interested in the horizontal magnetic fields-that is, vertical magnetic fields can be neglected.
From Equation 6, the magnetic field at the observation site L is obtained to be The magnetic field at site L can then be substituted into Equation 5a to obtain where W is the perturbation tensor from Equation 2, without the vertical elements w zx and w zy . Therefore, when substituting magnetic fields from a regional site, O (such as a geomagnetic observatory) at a local site, L, where an impedance tensor, Z L , is known, a correction factor of (W + 1) is required to be applied to Z L in order to account for horizontal magnetic fields between O and L before estimating electric fields.
For a larger dataset, covering a larger region, a multistation approach (Zhang & Schultz, 1990;Bonner & Schultz, 2017) might be preferable. Unfortunately, there are gaps in the geomagnetic time series from the nearest geomagnetic observatory at Eskdalemuir (ESK: 55.32°, −3.2°) from 1-9 September 2017. So, our localized, temporary, MT array (Simpson & Bahr, 2020) provides the only measured geomagnetic data available for the 8 September 2017 storm in Scotland.

Measured Electromagnetic Fields
Horizontal magnetic and electric fields recorded (Simpson & Bahr, 2020) during the 8 September 2017 storm are shown in Figures 2 and 3. Comparison of these fields confirms that the magnetic fields show significantly less spatial (i.e., site-to-site) variability than the electric fields. In particular, the north-south (E x ) component of the electric fields at site LEA is significantly smaller than the electric fields at other sites, while the east-west (E y ) component is larger than at all other sites with the exception of ACH. This diminution of the E x electric fields at LEA is attributable to strong polarization of the electric fields in one direction due to current channeling (Bahr, 1991). The difference in spatial variability between the magnetic fields and induced electric fields demonstrates the significant effect of the Earth's 3-D electrical conductivity structure on the form of the storm time electric fields. However, some differences in magnetic field magnitude are evident in Figure 2. For example, magnetic fields at ACH are larger than at RAN.  ) site, because the MT impedances indicate that the electrical conductivity structure is less complex with regard to current channeling and dimensionality in the vicinity of RAN than at other sites.
Measured storm time electric fields (Simpson & Bahr, 2020) are significantly larger at ACH than at other sites ( Figure 3 and Table 1). There is no obvious explanation for this, but we speculate that it could be due to current channeling along a nearby railway line. We also observe that the electric fields at site LEA are strongly polarized (Bahr, 1991), with weak E x fields and large E y fields ( Figure 3 and Table 1). We attribute this strong polarization to the proximity of site LEA to the Great Glen Fault (Kennedy, 1946) that can be seen as a green valley cutting diagonally across the Highlands on Figure 1. Figure 6 compares the electric fields obtained from Method (1) (see section 2.2). At MAR, there is a qualitatively good agreement between the measured fields and those computed using the local impedance tensor and either local magnetic fields or those from RAN ( Figure 6a). However, from Table 1, we see that the nowcasted electric fields are mostly smaller than the measured ones. This diminution in maximum peak-to-peak magnitude of the electric field is most pronounced in E x when magnetic fields from RAN are substituted for those at MAR without making a correction for anomalous magnetic fields. Application of the perturbation-tensor correction in this case results in a significant improvement in the agreement between the maximum peak-to-peak electric field magnitudes determined from the magnetic fields at RAN and MAR. However, the nowcasted fields for both E x and E y polarizations are only about 75% of the measured ones. At TAN, nowcasted E x fields are consistently smaller than the measured ones, whereas nowcasted E y fields are consistently larger than the measured ones (Table 1). However, the time series exhibit similar forms ( Figure 6b). The discrepancy between measured and nowcasted electric fields at TAN is most pronounced when magnetic fields from RAN are substituted for the local magnetic fields at TAN with no perturbation-tensor correction. In this case, application of the perturbation-tensor correction improves the agreement between the maximum peak-to-peak electric field magnitudes nowcasted from the local magnetic fields at TAN and substituted fields from site RAN. At site LEA, nowcasted E x fields are significantly larger than the measured ones, whereas E y fields nowcasted from the local magnetic field at LEA are 85% of the measured ones (Table 1). We attribute this to strong polarization of telluric currents at this site as discussed in section 3.1. The characteristic peak oscillation between 01:00 and 02:00 hr (just under half way through the recording window shown) seen in all other time series traces ( Figure 5) is not pronounced in the E x fields measured at LEA. Again, application of the perturbation-tensor correction improves the agreement between the maximum peak-to-peak E y electric field magnitudes nowcasted from the local magnetic fields at LEA and substituted fields from site RAN. However, there is a poor agreement between nowcasted and measured E x fields at this site.

Time Domain Analyses 3.2.1. Electric Fields Nowcasted From Local Impedance Tensors
The measured electric ( Figure 3) and magnetic ( Figure 2) fields at ACH are larger than at any other site. E x fields nowcasted at ACH are significantly smaller than the measured ones (Table 1). Substitution of the magnetic field from RAN for the local magnetic field from ACH produces maximum peak-to-peak E y magnitudes that are in excellent agreement with the measured ones, whereas maximum peak-to-peak E y magnitudes nowcasted from the local magnetic field spectra at ACH are significantly overestimated compared to the measured ones (Table 1). Application of the perturbation-tensor correction at ACH has a less significant effect than at other sites ( Table 1), suggesting that these observations are not due to anomalous regional magnetic field gradients present in the perturbation tensor.
In section 3.3.1, we will perform univariate and bivariate analyses in the frequency domain to further investigate the discrepancies between measured and nowcasted electric fields evident in Table 1. This will allow us to decompose the time series data into its constituent frequencies and compute average scaling factors between measured and nowcasted electric fields from the resulting scatter plots. This will provide insight into the processes involved.

Electric Fields Modeled From Synthetic Impedance Tensors
We investigated the effect of ocean bathymetry on the MT impedances using the 3-D forward modeling code of Siripunvaraporn et al. (2005). Synthetic impedances for individual site locations in models consisting of either (i) a 100-Ωm half-space; (ii) a 900-Ωm half-space; (iii) 3-D ocean bathymetry underlain by a 100-Ωm half-space; (iv) 3-D ocean bathymetry underlain by a 900-Ωm half-space were multiplied with measured magnetic field spectra from the relevant site in the frequency domain to obtain electric field spectra. The resistivity of seawater was assumed to be 0.25 Ωm and the maximum depth of the modeled ocean was 4 km. The modeled electric field spectra were Fourier transformed into the time domain to allow for comparison with measured electric field time series. Comparing measured electric fields at MAR and ACH (Simpson & Bahr, 2020) with those estimated from modeled impedances, we find a generally poor agreement between measured and modeled maximum peak-to-peak electric field magnitudes (Table 2) and the forms of the time series traces (e.g., Figure 7). This is in agreement with a synthetic modeling study performed by Bedrosian and Love (2015). Maximum peak-to-peak electric field magnitudes are approximately 3 times greater for the case of a 900-Ωm half-space than a 100-Ωm half-space. This is to be expected since apparent resistivity ρ a is dependent on |Z ij | 2 according to (e.g., Simpson & Bahr, 2005): where ω is angular frequency.
Comparing results in Table 2, we find that the effect of ocean bathymetry is greater for the model containing a 900-Ωm half-space compared to the model containing a 100-Ωm half-space. This is to be expected since a subsurface of higher resistivity provides less shielding from lateral conductivity anomalies, resulting in a Note. Errors are less than 3% based on error propagation from measured impedances. Percentages shown in brackets are nowcasted magnitudes relative to measured magnitudes.
larger horizontal adjustment length (Ranganayaki & Madden, 1980) for electromagnetic fields of a given wavelength.
In a previous study aimed at modeling GICs in the U.K. power network, Beggan et al. (2013) used a 900-Ωm half-space to represent the conductivity of the upper crust, lower crust and mantle below a thin sheet representing the conductance of an ocean up to 4-km deep surrounding the United Kingdom. However, below the upper crust, electrical resistivity is generally significantly less than 900-Ωm (e.g., Simpson, 1999;Simpson, 2002;Simpson & Warner, 1998) and our results (Figure 7 and Table 2) show that the magnitudes and orientations of electric fields and therefore GIC magnitudes are likely to be misrepresented by Beggan et al.'s model. Inclusion of near-surface conductance due to ocean bathymetry does not improve the agreement between the measured and modeled electric field responses at our sites. For the model containing a 100-Ωm half-space, inclusion of 3-D ocean bathymetry has no significant effect on modeled E x at MAR, TAN, and RAN or modeled E y at any of the five sites. We therefore consider the effect of ocean bathymetry to be insignificant compared to the Earth's deep electrical conductivity structure embodied in the MT impedance tensor for the sites studied. As shown by other MT studies (e.g., Kelbert et al., 2012;Sakkas et al., 2002;Simpson, 2000;Simpson, 2002), the electrical conductivity of the Earth can vary by several orders of magnitude, not just at the interfaces between continents and oceans but also across tectonic boundaries and mantle Figure 6. Comparison of measured (Simpson & Bahr, 2020) and nowcasted north-south (E x ) and east-west (E y ) electric fields during the 8 September 2017 storm at (a) MAR; (b) TAN; (c) LEA. Subscript "meas" denotes measured electric fields, site name as subscript denotes the electric fields nowcasted using the site's magnetic fields and impedance tensors, subscript "RAN" denotes electric fields nowcasted using the magnetic fields recorded at RAN and the site's impedance tensor, subscript "RANW" denotes electric fields nowcasted using the magnetic fields recorded at RAN and the site's modified impedance tensor that includes the perturbation-tensor correction for horizontal magnetic field gradients between RAN and the site.
discontinuities. These electrical conductivity variations have been shown to be of first-order importance in space weather studies conducted in the United States (e.g., Meqbel et al., 2014). As the ocean surrounding the United Kingdom is shallow compared to the thickness of the crust and mantle, variations in the conductivity of the Earth's crust and mantle, rather than the conductance of the ocean will dominate the long-period MT response.
Comparison of the model results in Table 2 also demonstrates sensitivity to the magnetic field spectra chosen for frequency domain multiplication with impedance, since for our half-space model, the impedance used for the calculation is the same at each site. For example, nowcasted peak-to-peak electric field magnitudes for E y are approximately twice as big at ACH compared to TAN, which is consistent with the larger B x fields measured at ACH compared to TAN (Figure 2). These examples show the necessity of ensuring that assumptions related to spatial variations in magnetic source fields are valid before a regional magnetic field is used for nowcasting electric fields at any given location. We will further investigate the scaling factors between  Simpson & Bahr, 2020) and modeled, north-south (E x ) and east-west (E y ) electric fields during the 8 September 2017 storm at (a) MAR and (b) ACH. Modeled electric fields were obtained using the magnetic field recorded (Simpson & Bahr, 2020) at MAR or ACH, respectively, during the September 2017 storm and the impedances modeled at MAR or ACH, respectively, for the cases of models containing either a 100-Ωm half-space (blue), a 100-Ωm half-space plus ocean bathymetry (green), a 900-Ωm half-space (red), and a 900-Ωm half-space plus ocean bathymetry (pink). Notice that there are significant discrepancies between features seen in the measured and modeled time series. measured and modeled electric fields using univariate and bivariate frequency domain analyses in section 3.3.2.

Are Time Derivatives of Magnetic Fields Effective Proxies for Electric Fields?
We calculated the time derivatives of the horizontally polarized magnetic fields (Figure 8). Comparing the times series in Figure 8 with the electric fields for the same storm time window shown in Figure 3 demonstrates that peak-to-peak electric field magnitudes are substantially more spatially variable than would be inferred from the time derivatives of the magnetic fields.
We, further, find that substantially different scaling factors are evident between the peak values of the horizontal time derivatives of the magnetic fields and the corresponding measured peak-to-peak electric field magnitudes (Table 1) for the north-south and east-west polarizations at individual sites and that scaling factors for neighboring sites, particularly for the north-south polarization are significantly different (Table 3). We attribute these different scaling factors to multidimensionality of the subsurface electrical conductivity structure in Scotland leading to spatial gradients and direction dependence of the induced electric fields that can only be adequately investigated if MT impedances are available. As the field costs of an MT survey are similar to the costs of magnetovariational surveys, we suggest that the ∂B/dt technique (and other techniques relying purely on magnetic variations) should be considered obsolete for space weather forecasting in favor of the direct, simultaneous measurements of electric and magnetic fields provided by the MT technique. Similar ambiguity is present when using horizontal magnetic gradients, which provide relative but not absolute information about electric field scaling factors and conductivity variations-that is, magnetovariational techniques cannot substitute the need for electric field measurements.

Univariate and Bivariate Analyses in the Frequency Domain 3.3.1. Comparison of Measured and Nowcasted Electric Field Spectra
The nowcasted peak-to-peak electric fields in Table 1 are mostly underestimates of the measured values. Therefore, we investigate the ratio of nowcasted to measured electric fields in the frequency domain at frequencies Figure 8. Time derivatives of (a) the north-south (B x ) magnetic field; (b) the east-west (B y ) magnetic field at four sites for the same time window as the measured electric fields shown in Figure 3. between 0.015 and 0.000124 Hz (64 to 8,200 s). We use MT processing techniques that average neighbored raw spectra (Simpson & Bahr, 2005) to estimate the transfer functions a and b between the nowcasted electric fields (subscript "(now)") and the measured electric fields (subscript "(meas)"): Figure 9. Scatter plots of univariate and bivariate scaling factors for sites (a) ELC and (b) LEA between measured electric fields (Simpson & Bahr, 2020) and electric fields obtained from nowcasting using (i) local magnetic fields; (ii) magnetic fields from RAN; (iii) magnetic fields from RAN with perturbation-tensor correction. Solid horizontal lines show averages for the scatter plots of corresponding color. Dashed horizontal lines (purple E x ; yellow E y ) show the combined averages of the univariate and bivariate analyses.
Whereas for the estimation of impedance tensors, spectra from multiple time windows are stacked, the 8 September 2017 storm covers only one time window (Figures 2 and 3). Therefore, the numbers of degrees of freedom (numbers of independent information) for each evaluation frequency is smaller for a than for Z. However, the average scaling factors obtained using univariate analysis (Figure 9 and Table 4) are similar to the underestimated scaling factors that we obtained in the time domain (Table 1). We, therefore, also calculated scaling factors for a bivariate analysis: The bivariate scaling factors, e and f, tend to be slightly larger than unity and therefore yield nowcasted electric fields that are overestimates of the measured ones. The averages of the scaling factors a and e or b and f obtained from the univariate and bivariate analysis is close to unity, with the remaining discrepancies-that is, departures from a scaling factor of 1-attributable to the limited numbers of degrees of freedom.
A possible source of the existence of a correlation between orthogonal, nowcasted and measured, electric field components lies in differences between the polarizations of the magnetic fields used in the analysis: whereas the impedance tensors are estimated from the Fourier coefficients obtained from several time windows with arbitrary polarizations of the magnetic field, the storm time magnetic field is strongly polarized. For example, the magnetic field of the classic ring current model has only a B x (and B z ) component but no B y component. This is not the case for the 8 September 2017 storm in Scotland, for which the magnetic fields ( Figure 2) are NNW-SSE polarized. When this polarization interacts with the direction-dependent conductivity of the mantle, nonvanishing transfer functions (b 2 and d 2 ) between orthogonal components of the nowcasted and measured electric fields are created in addition to the dominant transfer functions (a 2 and c 2 ) describing the scaling factors between the parallel components of the nowcasted and measured electric fields. From the summary of the frequency domain estimation of the scaling factors between nowcasted and measured electric fields in Table 4, we see that the scaling factors are closest to unity when the local magnetic fields measured at the MT sites are used. For the case that the magnetic field of site RAN is substituted for the local magnetic field at any particular site, the perturbation-tensor correction mostly produces scaling factors closer to the ones using local magnetic fields than those derived from uncorrected impedance tensors.

Comparison of Measured and Modeled Electric Field Spectra
We conducted similar univariate and bivariate analyses to those described in section 3.3.1 on electric fields determined using impedances synthesized using the models incorporating the 3-D conductance of the ocean described in section 3.2.2. For models containing a 100-Ωm half-space, the E x -polarized electric fields are significantly too small (Figure 10b). For models containing a 900-Ωm half-space, the E y -polarized electric fields are significantly too large ( Figure 10c) and the scatter of the data points is significantly greater than observed in the measured data ( Figure 10a).

Discussion and Conclusion
Severe magnetic storms are infrequent and short-lived events. The magnetic fields associated with them can be analyzed quantitatively using data from a global network of geomagnetic observatories, but long-term, continuous measurements of induced electric fields are less readily available. This means that models and techniques used to infer the magnitudes of induced electric fields and GICs associated with magnetic storms have not been extensively validated against measured electric field data from the region to which they are applied. Without adequate validation (see also Kelbert et al., 2017), we cannot be sure that hazard maps (e.g., Love et al., 2016) showing the risk that GICs present to technological infrastructure are correct. MT data recorded in Scotland before, during, and after the September 2017 magnetic storm (Simpson & Bahr, 2020) have provided a unique opportunity to compare measured storm time electromagnetic time series with electric field time series synthesized using different techniques. We have compared electric fields measured during the 8 September 2017 storm with those determined by multiplying electromagnetic transfer functions and magnetic field spectra, considering both impedance tensors and perturbation tensors derived from MT data and synthetic impedance tensors derived using 1-D and 3-D modeling.
Although horizontal magnetic field gradients in Scotland are low (e.g., Figure 5), we find that application of a perturbation-tensor correction to the measured impedance tensor to account for horizontal magnetic field gradients when using magnetic field spectra from a site other than the local site where the impedance tensor has been measured improves the agreement between the measured and the nowcasted electric fields. We, therefore, suggest that perturbation tensors should be calculated routinely alongside impedance tensors for the purpose of investigating space weather impacts. This is particularly desirable for regions (unlike Scotland) with strong horizontal magnetic gradients.
Comparing measured and nowcasted, peak-to-peak, electric field values from local impedance tensors in Table 1, we find that some nowcasted values are overestimates, whereas others are underestimates of the measured values. Scatter plots of scaling factors between measured and nowcasted electric fields from univariate and bivariate analyses in the frequency domain ( Figure 9) have enabled us to attribute some of the discrepancies observed to coupling between the polarization of the storm time magnetic source field and the direction-dependence of Earth's deep electrical conductivity structure. Other possible causes of misfit between measured and nowcasted electric fields have been discussed by Kelbert et al. (2017). These include cultural noise, filtering effects, storm time departures from the plane wave assumption of the MT method, induction in the oceans (particularly tidal harmonics), current channeling, multidimensionality of Earth's subsurface conductivity structure and static shift (e.g., Simpson & Bahr, 2005). However, the average nowcasted peak-to-peak magnitude is close to the average measured value. Therefore, it is desirable to calculate geomagnetic induced currents (GICs) from electric fields averaged over a regional array of MT sites to mitigate statistical biases due to local effects that commonly distort electric fields. As determination of voltages from which GICs are calculated involves integrating electric fields over transmission line lengths, interpolation across an array of sites is anyway desirable.
We find that 1-D and 3-D modeling techniques based on assumed models (e.g., Kappenman, 2004) that ignore the Earth's deep electrical conductivity structure (e.g., Simpson, 2002;Simpson & Warner, 1998) do not provide an acceptable degree of agreement between nowcasted and measured electric fields. We conclude that techniques of this kind are significantly inferior to techniques that rely on measured impedance tensors that contain all information about Earth's 3-D conductivity structure. For our particular sites located inland from shallow seas, coast effects are found to be less significant than Earth's deep conductivity structure.
We have also demonstrated that time derivatives of magnetic fields are of limited value compared to direct measurements of electric fields. As has also been discussed by Bonner and Schultz (2017), only MT impedances, uncorrected for static distortion effects (e.g., Simpson & Bahr, 2005) can provide information about absolute scaling factors for local electric fields.
Long-duration electromagnetic time series and long-period MT impedances are necessary prerequisites for any approach to nowcasting and validating models of extreme space weather effects on ground-based technological infrastructure. If the ultimate aim is to forecast these effects, then the models on which forecasts are based must first be validated. However, the database of storm time electric field measurements is limited. Based on the results of our validation study using electric and magnetic fields measured during the 8 September 2017 magnetic storm (Simpson & Bahr, 2020), we recommend basing risk assessments for vulnerability to extreme space weather events on measured impedance tensors, perturbation tensors, and magnetic field spectra. We suggest that this approach is likely to be more robust than techniques relying on indirect inferences about storm time electric fields from overly simplistic electrical conductivity models or time derivatives of magnetic fields.
Our bivariate, frequency domain analyses have also highlighted the importance of the polarization of the storm time electric field. This demonstrates that for the purpose of forecasting the effects of extreme space weather events, it is not sufficient to forecast magnetic field magnitudes-two storms with similar magnitudes, but different polarizations could pose very different hazards to ground-based infrastructure in a given region. The analyses that we have presented are only possible because of the availability of electromagnetic data recorded during a magnetic storm (Simpson & Bahr, 2020). Therefore, longer-term, continuous electromagnetic recordings are recommended in order to enhance the likelihood of capturing further spatially and temporally coincident electric and magnetic field data during future magnetic storms.

Data Availability Statement
Electromagnetic data recorded by the authors before, during, and after the September 2017 storm are available in Simpson and Bahr (2020).