Down to Earth With Nuclear Electromagnetic Pulse: Realistic Surface Impedance Affects Mapping of the E3 Geoelectric Hazard

An analysis is made of Earth‐surface geoelectric fields and voltages on electricity transmission power‐grids induced by a late‐phase E3 nuclear electromagnetic pulse (EMP). A hypothetical scenario is considered of an explosion of several hundred kilotons set several hundred kilometers above the eastern‐midcontinental United States. Ground‐level E3 geoelectric fields are estimated by convolving a standard parameterization of E3 geomagnetic field variation with magnetotelluric Earth‐surface impedance tensors derived from wideband measurements acquired across the study region during a recent survey. These impedance tensors are a function of subsurface three‐dimensional electrical conductivity structure. Results, presented as a movie‐map, demonstrate that localized differences in surface impedance strongly distort the amplitude, polarization, and variational phase of induced E3 geoelectric fields. Locations with a high degree of E3 geoelectric polarization tend to have high geoelectric amplitude. Uniform half‐space models and one‐dimensional, depth‐dependent models of Earth‐surface impedance, such as those widely used in government and industry reports informing power‐grid vulnerability assessment projects, do not provide accurate estimates of the E3 geoelectric hazard in complex geological settings. In particular, for the Eastern‐Midcontinent, half‐space models can lead to (order‐one) overestimates/underestimates of EMP‐induced geovoltages on parts of the power grid by as much as ± 1,000 volts (a range of 2,000 volts)—comparable to the amplitudes of the geovoltages themselves.

developing such maps, numerical methods pioneered in the 1980s by the Oak Ridge National Laboratory (ORNL) have been widely influential (Barnes, Rizy, et al., 1993;Legro et al., 1985Legro et al., , 1986). ORNL's methods have been used in numerous E3 scenario mapping projects across the conterminous United States (CONUS) (e.g., Electric Power Research Institute, 2017;Electromagnetic Pulse Commission, 2017;Gilbert et al., 2010;International Electrotechnical Commission, 1996;Lee et al., 2019;Rackliffe et al., 1988;Tesche et al., 1991), and these maps have been used in projects for assessing the vulnerability of power grids to the E3 hazard (e.g., Electric Power Research Institute, 2019;Electromagnetic Pulse Commission, 2017;Tesche et al., 1991). The models of E3 geomagnetic disturbance resemble magnetometer measurements made during the high-altitude nuclear tests of the 1960s, and the maps of E3 geoelectric fields are visually compelling, but the surface impedances used to develop those maps commonly assume uniform half-space models of Earth conductivity.
Recognizing that the Earth's conductivity structure is complicated and, generally, three-dimensional across a wide range of spatial scales, and recognizing that this structure distorts the amplitude, polarization, and variational phase of induced geoelectric fields (e.g., Bedrosian & Love, 2015;McKay & Whaler, 2006), we investigate the effects of geologically realistic Earth-surface impedance on the time-dependent mapping of E3 geoelectric fields. Our methods are broadly similar to those used for analyzing geoelectric fields induced in the solid Earth during magnetic storms (e.g., Blake et al., 2016;Kelbert et al., 2017;Love et al., 2019;Lucas et al., 2020;Marshalko et al., 2020;Marshall et al., 2020;Simpson & Bahr, 2020;Torta et al., 2017;Wang et al., 2020). Indeed, comparisons are sometimes drawn between magnetic-storm variation and E3 electromagnetic variation. But important quantitative differences affect hazard and vulnerability assessments and possible mitigation measures (e.g., Meliopoulos et al., 1994;Neal et al., 2011;Rivera et al., 2016). In particular, E3 electromagnetic variation has a different geographic expression than that of magnetic-storm disturbance, and electromagnetic amplitudes are concentrated at higher frequencies than are characteristic of magnetic storms. In developing results that are specific for E3, we use a parameterization of a scenario EMP geomagnetic disturbance similar to that used by ORNL (and other investigators), but instead of using idealized models of surface impedance, we use impedance tensors derived from wideband magnetotelluric measurements acquired during a recent survey of the eastern-midcontinental United States. To our knowledge, our E3 analysis is the first in which realistic impedances are used. We convolve the scenario E3 geomagnetic disturbance with the magnetotelluric impedance tensors to construct scenario E3 geoelectric waveforms across the survey region. These are then interpolated onto power-grid lines and integrated to estimate geovoltages as a function of time. Our results can be used to quantify the errors associated with estimates of the E3 geoelectric hazard obtained using idealized half-space impedances.

Idealized Half-Space Impedance
It is already understood, from analyses of magnetic storms, that induced geoelectric fields can be significantly distorted by heterogeneous solid-Earth conductivity structure. But since many E3 scenario analyses assume half-space models of Earth conductivity, we find it useful to briefly review induction in a simple half-space model. More complicated idealized models can also be considered, such as quarter-space models (e.g., Simpson & Bahr, 2005, their Chapter 2.6; Berdichevsky & Dmitriev, 2008, their Chapter 6), though such models are not specifically needed for our analysis that follows, and, as far as we know, they have not been considered in EMP simulations. For a right-hand geographic coordinate system (north x , east ŷ , down ẑ ), we consider a boundary-value problem with the atmosphere (  0 z ) treated as a vacuum and the Earth's interior (  0 z
In contrast to Equation 2, the elements of a magnetotelluric impedance tensor are fully populated, xx xy yx yy Each magnetotelluric impedance tensor is a nonlinear function of the electrical conductivity  ( ) r beneath and surrounding the survey site, where r is the position vector. These tensors are usually used in inversions for solid-Earth conductivity structure (e.g., Egbert, 2007a;Rodi & Mackie, 2012), but we use them, here, to estimate E3 geoelectric fields. Because subsurface conductivity can be complicated in structure, induced geoelectric fields do not, in general, have a straightforward relationship with geomagnetic field variation; in contrast to the idealized half-space, generally, and    45 .

Geographic Scales
It is important to consider geographic scales in an analysis of E3 geoelectric fields. Fundamentally, for an idealized half-space Earth model, electromagnetic amplitudes are attenuated across a diffusive length scale, a skin-depth, given by For electromagnetic field variation realized across the Earth, at a particular site over conductivity structure that is (possibly) heterogeneous,  in Equation 10 is a bulk-average of the conductivity within a roughly hemispherical volume beneath and surrounding the site with radius  . For either half-space or bulk-average conductivities ranging from 1 S/m for sedimentary rock to 4 10 S/m for metamorphic or igneous rock, and for E3 variation at 1 s (or 1,000 s),  ranges from  0.5 to 50 km (or from  16 to 1,600 km).
Consider, now, the empirical parameterization given by the magnetotelluric Equation 8. This is often a reasonable description of the relationship between ( ) h f B and ( ) h f E -provided geomagnetic field variation across the Earth's surface, for a given variational frequency, has a characteristic lateral length scale ( ) L f that is longer than the corresponding diffusive scale, This is called the plane-wave condition (Wait, 1982, their Chapter VI). Depending on the diffusive length scale, the plane-wave condition can be violated if the source currents generating geomagnetic field variation are both at relatively low altitude and spatially complicated, such as can sometimes be realized with disturbance current systems in the ionosphere (e.g., Jones & Spratt, 2002;Pirjola, 1992). In such circumstances, the patterns of induced currents in the solid Earth can be qualitatively different, and the parameters in Equation 9 can be quantitatively different, each from those for a plane wave.
In terms of the lateral length scale for an E3 EMP waveform, maximum E3A disturbance is realized at the Earth's surface for explosions set at altitudes of 500 km, and maximum E3B disturbance is realized at the Earth's surface for explosions set at altitudes of 150 km (e.g., Rivera et al., 2016, their Figure 23). Therefore, we can plausibly assume that E3 disturbance is generated by electric currents at heights greater than 150 km, and even quite a bit higher. By geometric attenuation, L characterizing the ground-level expression of geomagnetic variation will exceed 150 km and possibly, even, exceed the altitude of the explosion. As discussed in Section 6, we choose an E3 parameterization for which  800 L km. With this, for variation at 1 s and with  ranging from  0.5 to 50 km, the plane wave condition (11) is well satisfied. On the other hand, for variation at 1,000 s and  ranging from  16 to 1,600 km, the plane wave condition (11)  Next, it is important to remain mindful of the effects that geological structure (e.g., Pollard & Fletcher, 2005;Tarbuck et al., 2017) can have on the length scale of induced E3 geoelectric fields. In some geological settings, such as in basins filled by a thick accumulation of sediments, structure can be relatively one-dimensional, depth-dependent. In other settings, tectonic forces have shaped terranes by faulting, deformation, and intrusion, resulting in structure that is fundamentally two-or three-dimensional. In such settings, subsurface conductivity structure and surface impedance can differ significantly across a wide range of geographic scales (e.g., Bahr, 2005;Lovejoy & Schertzer, 2007). Although the shortest of these scales is limited by diffusion, Equation 10, site-to-site differences in impedance strongly affect E3 local geoelectric field, Equation 4. On the other hand, the geomagnetic field is only weakly affected by surface impedance, Equation 6. Therefore, given the possibly shortish length scale of surface impedance, a reasonable method for calculating an E3 geoelectric field is to convolve a broad regional model of E3 geomagnetic field variation with local magnetotelluric estimates of surface impedance as per Equation 8.

Eastern-Midcontinental Tensors
We use 127 wideband magnetotelluric tensors derived from measurements acquired by the U.S. Geological Survey from 2016 to 2019 during a survey of a part of the eastern-midcontinental United States (   35.0 38.5 N    88.0 93.0 W). The study area encompasses the cities of St. Louis, Missouri (MO) and Memphis, Tennessee (TN); it includes parts of Illinois (IL), Kentucky (KY), and Arkansas (AR) . In Figure 1, we show a map of the study region and magnetotelluric survey sites. The Eastern-Midcontinent is underlain by Precambrian basement rock that is, relatively electrically resistive; this is overlain by differing depths of younger and relatively electrically conductive sedimentary rock, the thickness of which is the difference between surface elevation (Danielson & Gesch, 2011) and the Great Unconformity elevation (Marshak et al., 2017). Notable, for our purposes, is the Ozark Dome (e.g., Anderson et al., 1979), where the sedimentary cover is relatively thin and basement granitic and dolomite rocks are exposed in some places. In contrast, the Reelfoot Rift (e.g., Dart & Swolfs, 1998) and the Illinois Basin (e.g., Kolata et al., 2005)  , , , f f f within the band from 3 10 to 3 10 Hz ( 3 10 to 3 10 s). Localized geologic complexities (e.g., DeLucia et al., 2019;Van Arsdale & Cox, 2007;Van Schmus et al., 1993) are manifest in the properties of the impedance tensors.
It is useful to plot the frequency dependence of "apparent" resistivity and phase, given, respectively, by and , across a wide range of variational periods (frequencies). In other words, impedance at this site is approximately skew-symmetric, but the off-diagonal apparent resistivities are not constant across variational periods. These observations are consistent with conductivity structure beneath and surrounding the survey site that is close to one-dimensional, depth-dependent. At 1 s, apparent resistivity is 5.5 -m, and, from Equation 10, we estimate that induction occurs across a diffusive depth of 1.2 km. In contrast, RF111 is located atop the Ozark Dome and adjacent to the thick sedimentary fill of the Reelfoot Rift. From Figures 2c and 2d . Furthermore, the non-primary element resistivity  yy is comparable to the primary resistivities  xy and  yx between 0.1 and 1,000 s. Impedance at this site is not skew-symmetric. These observations are consistent with conductivity structure beneath and surrounding the survey site that is far from simply one-dimensional, depth-dependent. At 1 s, apparent resistivity is   4 1.5 10 -m; from this, we estimate that induction occurs across a diffusive scale of 50 km.
Another way to examine impedance tensors is to plot the transfer-function amplitude, 10.1029/2021EA001792 7 of 25 . When plotted as a function of geoelectric polarization, transfer-function amplitude is elliptical on a polar plot. For a given tensor and at a specific variational period, the length of the major (minor) axis of the ellipse corresponds to the maximum (minimum) principal amplitude of the geoelectric field per unit-amplitude geomagnetic field. In contrast, when plotted as a function of geomagnetic polarization, transfer-function amplitude is roughly peanut-shaped on a polar plot. Furthermore, an impedance phase tensor is elliptical when plotted in polar coordinates as a function of geographic direction, ( ( )) D f (e.g., Booker, 2014;Caldwell et al., 2004). Here, the major (minor) axis of a phase ellipse corresponds to the maximum (minimum) principal phase.  Figure 3f, for 1 s variation, the phase ( ) , with maximum along the    126 180 azimuth. As we noted from Figures 2c and 2d, these observations are consistent with conductivity structure beneath and surrounding the survey site that is far from one-dimensional, depth-dependent.
In Figures 4a and 4b, we show a map of transfer-function amplitude ellipses/peanuts for the various magnetotelluric survey sites for 1.0 s variation. These maps demonstrate the relationship between surface impedance and prominent geologic and tectonic structures. Over and surrounding the resistive Ozark Dome in southeast Missouri, transfer functions generally have high amplitude, and some are polarized. We note, in particular, that the transfer function for RF111 has one of the highest maximum amplitudes and is one of the most polarized of all the sites in the study region. In contrast, over the conductive Illinois Basin and Reelfoot Rift, transfer functions, such as that for SFM06, have relatively low amplitude and are generally less polarized. In Figure 4c, we show a map of transfer-function phase ellipses for the various survey sites for 1.0 s variation; here, we see significant differences between the Ozark Dome and the Illinois Basin and Reelfoot Rift. Judging from Figure 4, we can anticipate that the geography of surface impedance will impart significant spatiotemporal distortion to E3-induced geoelectric field variation.

Spatiotemporal Model of Disturbance
Recognizing that nuclear EMP is an extremely complicated phenomenon, and accepting the fact that many numerical EMP models are classified (e.g., Department of Energy, 2017), we appreciate that most publicly published studies of E3 EMP and its effects rely on simplified parameterizations of EMP electromagnetic variation at the Earth's surface. While these parameterizations are physically motivated, they do not represent all of the intricacies and possibilities of an EMP event. They assume that the explosion height (several hundred kilometers) and yield (several hundred kilotons to a few megatons) are such that the E3 signal is of nearly maximum amplitude (e.g., Rivera et al., 2016, their Figure 23), but the assumed explosion altitude and yield are not usually specified. The parameterizations are generic, but they serve as useful benchmarks for comparison. In this regard, ORNL reports provide a parameterization of E3 electromagnetic variation that has had widespread influence. Here, the spatiotemporal form of horizontal-component, ground-level geoelectric field variation is represented by the multiplication of separate geographic and temporal functions for each of the blast A and heave B phases, (Legro et al., 1985, their Equation 6). The time-invariant vector functions , ( , ) A B x y e describe the geographic dependence of the amplitude and polarization of the induced E3 geoelectric field, and the geographically invariant functions t describe the time dependence of E3 geoelectric field variation. Some investigators (e.g., Barnes, Rizy, et al., 1993;Electric Power Research Institute, 2017)   functions describing relative variation in time. Other investigators (e.g., Gilbert et al., 2010) choose a normalization in which the ( ) E t functions have absolute amplitude, and the ( , ) x y e functions serve as basis functions describing relative amplitude in geography.
As discussed following Equations 4 and 6 and in Section 4, geoelectric field variation is a strong function of Earth conductivity, while geomagnetic field variation is not. Therefore, to estimate E3 geoelectric induction across heterogeneous Earth structure, we use a model of geomagnetic field variation and magnetotelluric impedance tensors to estimate E3 geoelectric field variation. Instead of Equation 15, we use the following (very similar) parameterization of horizontal-component, ground-level geomagnetic field variation which can be motivated on physical grounds similar to those for Equation 15. We use a normalization in which the ( ) B t functions have absolute amplitude, and the ( , ) x y b functions serve as basic functions describing the geographic dependence of relative amplitude.
In considering the idealized constitutive functions in Equation 16, let us begin with the function describing the geography of the E3 disturbance geomagnetic field generated by the blast-phase plasma bubble. It expands very rapidly, and in a matter of seconds, it attains a lateral dimension of thousands of kilometers. Electric currents J on the underside surface of this bubble, but in the atmosphere, are assumed to be horizontal, uniform, and eastward directed   Figure   3b; Gilbert et al., 2010, their Figures 2-4). By Ampère's law,     0 J B , these currents generate a horizontal, uniform, and northward-directed disturbance geomagnetic field (e.g., Gilbert et al., 2010, their Figures 2-3), so that Given our chosen normalization, we take  1 A b ; this means that ( ) A B t is a positive function. While this geomagnetic field component increases in intensity, a westward geoelectric field is induced in a half-space Earth model, Equation 15, and, so, ( ) A E t is a negative function. Technically, A b should be parallel to the local horizontal component of the Earth's main geomagnetic field, which has a non-zero declination almost everywhere over the Earth's surface. However, over most of CONUS, geomagnetic declination is relatively small, and the zero declination line passes through the middle of the eastern-midcontinental study region. We note, furthermore, that other investigations of E3 (e.g., Barnes, Rizy, et al., 1993;Gilbert et al., 2010) do not make a correction for local geomagnetic declination.
We also need to prescribe the function ( , ) B x y b describing the geography of the E3 disturbance geomagnetic field generated by the heave phase. For a buoyant plasma bubble rising upward with velocity u through the ambient geomagnetic field, a westward dynamo electric field  u B is generated. This drives a westward current that, outside of the dynamo, closes through two oppositely circulating current gyres    r, colatitude ˆ, longitude λ ), specified by the Earth's rotational axis. The basis in the heaving region at (, ) and at height e r is given by where the gyres are spherical elementary currents,  (Amm & Viljanen, 1999;Rigler et al., 2019;Vanhamäki & Juusola, 2020), defined in spherical coordinates (   ,λ ) specified by the focal point of each gyre, and where 0 I is a scaling factor. The halfway point between the northern and southern focal points of the two gyres is ( e ,  e ); this is, itself, the geographic epicenter of the heaving plasma. The colatitudinal separation of the gyres is , and The ground-level horizontal-component geomagnetic field generated by each current gyre is poloidal and can be derived using the Biot-Savart law, Given our chosen normalization, we take . Since the geomagnetic field beneath this current system and between the two gyre focal points is southward directed, ( ) B B t is a negative function. While this geomagnetic field component increases in intensity between the focal points, an eastward geoelectric field is induced in a half-space Earth model, Equation 15, and, so, ( ) B E t is a positive function.
In Figures 5a and 5b, we show the two ground-level E3 geographic basis functions for the blast phase ( , ) A x y b and for the heave phase ( , ) B x y b centered on our chosen explosion epicenter (36.75N, 90.50W, which is also shown in the center of Figure 1). We choose  800 km so that the pattern of E3B disturbance resembles Barnes, Rizy, et al. (1993, their Figure 3b) and Gilbert et al. (2010, their Figures 2-10). With this, and in consideration of the discussion in Section 4, we recognize that the lateral scale of these geographic functions satisfy the plane-wave condition given by Equation 11 for all but the longest spatial scales at the lowest E3 frequencies (longest E3 periods). In Figure 5, we also show as a box the study region where we map E3 geoelectric fields. As idealized functions, the forms of ( , ) far outside this box are immaterial to our analysis-over spatial scales broader than those considered here, realistic treatment of E3 fields should include an attenuation term so that E3 amplitudes properly diminish with increasing distance from the explosion epicenter.
Next, we consider the waveform functions in Equations 15 and 16. The International Electrotechnical Commission (1996, IEC) has published reference waveforms of the horizontal-component EMP geoelectric field, including for E3. The IEC E3 standard geoelectric waveform is a scalar time series describing ground-level variation for a uniform half-space conductivity of    4 10 S/m. The applicable latitudes are given in the IEC document as between 30° and 60°; no specification is given of explosion parameters. The exact physical foundation of the geoelectric waveforms is not described in the IEC document, but mention is made of reports from high-altitude nuclear tests conducted over the South Pacific during the early 1960s LOVE ET AL.  . The study region is in the red square. (International Electrotechnical Commission, 1996), which, presumably, includes the Starfish Prime test summarized in our introduction.
In more detail, the IEC E3 geoelectric waveform is (International Electrotechnical Commission, 1996). The first function on the right can be identified with the E3A blast phase and the second function with the E3B heave phase, each of which are double exponential functions of a form commonly used in EMP analyses (e.g., Oughstun, 2009, their Chapter 11.2.3), The eight constants , , and in Section 4, we recognize that a geoelectric waveform depends on surface impedance. Since the IEC geoelectric waveform ( ) E t pertains to a uniform half-space impedance for a particular conductivity, it cannot, in general, be applied in regions of different conductivities, let alone any region of complicated geology. We also understand that direct measurements of geoelectric fields made during EMP tests in particular geological settings (above particular subsurface conductivity structures) (e.g., Bomke et al., 1964;Burch & Green, 1963;Gill, 1962;Poletti & Gadsden, 1963)    On the other hand, and in light of the discussion following Equation 6 and in Section 4, we understand that a geomagnetic waveform ( ) B t is insensitive to geological setting. Although an E3 geomagnetic waveform is not given in the IEC document, for the assumed uniform half-space conductivity, we can recover it using Equation 1 and taking an inverse Fourier transform to obtain the convolution (see also, Gilbert et al., 2010); the minus sign in this equation assumes a westward (eastward) oriented electric field, which is induced by a northward (southward) oriented geomagnetic field of increasing amplitude. Performing the integration, we obtain and where ( )  10 -4 10 Hz (10-4 10 s). Note that, for clarity, time is shown on a logarithmic axis. The moment of the electromagnetic pulse explosion is at  1.0 t s.
In Figures 6a and 6b, we show the IEC E3 geomagnetic ( ) B t waveform, Equation 26, and the geoelectric ( ) E t waveform, Equation 23, together with their individual blast E3A-phase and heave E3B-phase components. For clarity, time is given on a logarithmic axis, with the moment of the explosion at  1.0 t s. Immediately after the explosion, the amplitudes of the geomagnetic and geoelectric E3 waveforms increase rapidly. This is followed by a more gradual decline and a small overshoot of the opposite sign. In detail, the E3 geomagnetic waveform has a rise time of 20.9 s from its start to its maximum of 1,458 nT and a full-width at half-maximum of 65.8 s. The induced geoelectric waveform has a rise time of 2.1 s from its start to its maximum and a full-width at half-maximum of 19.9 s; for the assumed uniform half-space conductivity of    4 10 S/m, the maximum value attained is 38.7 V/km. By  1000.0 t s, the E3 signals have mostly faded. Qualitatively, the IEC geomagnetic waveform resembles direct geomagnetic recordings made during tests in the 1960s (Chavin et al., 1979;Dyal, 2006, their Figure 12; Electromagnetic Pulse Commission, 2017, their Chapter 3; Legro et al., 1985, their Figure 1); the IEC geoelectric waveform resembles those in other influential reports    ; (e) Geovoltage on power grid, given by Equation 28, for the half-space model; (f) Geovoltage V on power grid derived using magnetotelluric impedance tensors. Explosion epicenter (red x).

Frequency/Period Bandpass
Before proceeding, recall that the magnetotelluric impedance tensors we use are wideband limited from 10 3 to 3 10 Hz ( 3 10 to 3 10 s). Recalling that the Fourier ingredients of a (non-periodic) impulse cover a broad range of frequencies (e.g., Bracewell, 2000), we need to establish that the magnetotelluric wideband is sufficient to resolve the E3 geoelectric impulse. In Figure 7a, we compare both the complete geomagnetic ( ) B t IEC waveform and the wideband-limited counterpart, which generally follows the complete waveform. The offset in the wideband-limited waveform is due to lack of resolution at the low-frequency (long period) end of the signal spectrum,   3 10 Hz ( 3 10 s). This offset does not significantly affect the IEC (half-space) geoelectric field ( ) E t ; as we see in Figure 7b, the wideband-limited waveform almost perfectly matches the complete waveform. From this, we understand that wideband magnetotelluric tensors, such as used here, are sufficient for mapping E3 geoelectric fields. For comparison, and in support of discussion in Section 10 concerning the possibility of using magnetotelluric impedance tensors for E3 mapping in other parts of CONUS, in Figure 7, we also show band-limited geomagnetic and geoelectric waveforms, 1 10 -4 10 Hz (10-4 10 s). In this case, the band-limited ( ) E t IEC waveform does not match the complete waveform-significant Gibbs ringing is seen for the early blast phase from  1.0 t to 10.0 s; we note that the long-period band-limited ( ) E t is acausal, commencing before the explosion,  1.0 t s. From this, we understand that long-period magnetotelluric tensors are inadequate for detailed mapping of E3 geoelectric fields. LOVE ET AL.

E3 EMP Scenario
We calculate E3 geoelectric fields for an EMP explosion with an epicenter (36.75  N, 90.50  W) centered on the eastern-midcontinental study region. Using the magnetotelluric Equation 8, magnetotelluric tensors   , , f x y Z derived from the survey measurements, Section 5, and Fourier transformation of the model E3 geomagnetic field variation, ( , , ) h f x y B , given by Equation 16, we obtain the frequency domain expression of the geoelectric field ( , , ) h f x y E at each survey site; inverse Fourier transformation gives ( , , ) h t x y E . These calculations are similar to those that we (e.g., Bedrosian & Love, 2015;Kelbert et al., 2017;Love et al., 2019) and others (e.g., Dimmock et al., 2019;Marshalko et al., 2020;Marshall et al., 2020;Wang et al., 2020) use (at lower frequencies) for analysis of magnetic-storm induction of geoelectric fields in the three-dimensional solid Earth. For comparison with previously published EMP results (e.g., Barnes, Rizy, et al., 1993;Electric Power Research Institute, 2017;Electromagnetic Pulse Commission, 2017;Gilbert et al., 2010;Legro et al., 1985;Siebert & Witt, 2019), we also calculate the E3 geoelectric fields that would be induced in a hypothetical Earth with a half-space impedance  ( ) HS f Z , using Equation 1 and    3 10 HS S/m (the referenced authors use a conductivity that is a factor of 10 higher than that used for the IEC geoelectric benchmark).
In the dynamic-content that the time-dependent movie-map given as Supporting Information S1 a scenario simulation is depicted of ground-level E3 electromagnetic field variation (and geovoltages on power-grid lines, discussed in Section 9) across the study region, covering 60 s of time after the moment of the explosion at  1.0 t s, derived for both the half-space Earth model and for the magnetotelluric tensors. In Figure 8, we show a snapshot of this movie-map at the moment of the explosion, but before any E3 effects have been realized. In Figure 9, we show a snapshot from the movie-map at time  1.5 t s, just half a second after the explosion. At this instance in time, the geomagnetic field, Figures 9a and 9b, is primarily ( , , ) uniform northward, and its amplitude is 170 nT. The geoelectric field for the half-space model, Figure 9c, is close to uniform westward, with an amplitude of 8.2 V/km. In contrast, the geoelectric field for the magnetotelluric tensors, Figure 9d, is far from uniform. Geoelectric amplitudes across the study region range from 0.8 V/km at one location to 29.0 V/km at another location; the median amplitude is 4.6 V/km, and the 68% interval is [1.2, 10.4] V/km.
In Figure 10, we show a snapshot at time  3.0 t s, two seconds after the explosion. At this instance, the geomagnetic field, Figures 10a and 10b, is still primarily ( , , ) , still close to uniform northward, but its amplitude has grown to 609 nT. The geoelectric field for the half-space model, Figure 10c is still close to uniform westward, but its amplitude has grown to 12.2 V/km. The geoelectric field for the magnetotelluric tensors, Figure 10d, while still generally westward, has grown in amplitude, now ranging from 1.7 V/km at one location to 33.2 V/km at another location; the median amplitude is 6.2 V/km, and the 68% interval is [2.7, 12.3] V/km. In Figure 11, we show a snapshot at time  20.0 t s. At this instance, the geomagnetic field amplitude is 1,455 nT; deviation from northward polarization is due to an increased contribution from ; related deviation from westward polarization in the half-space geoelectric field is seen in Figure 11c. The decrease in the rate of change of the geomagnetic field corresponds to a decrease in geoelectric amplitudes for both the half-space model and for the magnetotelluric tensors, Figure 11d. The geoelectric field for the magnetotelluric tensors has lost much of its westward polarization.
Recall from Section 5 that the magnetotelluric impedance for site SFM06 is nearly skew-symmetric, something seen in geological settings that are close to one-dimensional, depth-dependent. As a result of the site's simple impedance, and given that the E3 geomagnetic field waveform is approximately northward, Figure 12a, the geoelectric waveform is approximately westward, Figure 12b. The geoelectric field attains its LOVE ET AL. maximum value of 2.2 V/km at  6.7 t s. The geoelectric field for the half-space model is, not surprisingly, also westward, but its amplitude is much higher than the magnetotelluric amplitude since SFM06 impedance is significantly lower than that of the half-space. At this site, the half-space geoelectric field attains its maximum value of 12.2 V/km at  3.1 t s, or 3.6 s before that for the SFM06 magnetotelluric tensor. The difference in amplitude could have been anticipated from Figure 2a and Equations 2 and 12, which indicate that  / 1 HS Z Z .
Very different from induction at site SFM06 is that at RF111. Recall from Section 5 that, as a result of complicated Earth structure, the magnetotelluric impedance for RF111 is far from skew-symmetric. From Figure 3d, we note that the geomagnetic polarization inducing the largest amplitude geoelectric field at RF111 (and at 1.0 Hz; 1.0 s) is along the    37 180 azimuth. But from Figure 13a, we see that the E3 geomagnetic field is approximately northward. Despite this, the E3 geomagnetic field is sufficient to induce high-amplitude E3 geoelectric fields. From Figure 13b, the geoelectric field attains its maximum amplitude of 37.3 V/ km at  2.1 t s. This exceeds the 15.0 V/km maximum value used by the U.S. Defense Threat Reduction Agency (Siebert & Witt, 2019). It exceeds the 24.0 V/km maximum value from an ORNL report (Barnes, Rizy, et al., 1993) that EPRI has used in an analysis of EMP effects on the U.S. power grid (e.g., Electric Power Research Institute, 2017). It is less than the 84.6 V/km maximum value given by the Electromagnetic Pulse Commission (2017). The geoelectric vector at RF111 is directed along the    110 180 azimuth, very close to the dominant polarization (at 1.0 Hz; 1.0 s) of the transfer function, along the    119 180 azimuth, seen in Figure 3c. In contrast, in Figure 13b, we see that the half-space model does not support west-northwest geoelectric polarization. At this site, the half-space geoelectric field attains its maximum value of 12.2 V/km at  3.1 t s, or 1.0 s after that for the RF111 magnetotelluric tensor. The difference in amplitude could have been anticipated from Figure 2c and Equations 2 and 12, which indicate that  In comparing Figures 1 and 4 with the movie-map and with the snapshots, such as Figure 10d for  3.0 t s, we note correlation between high (low) E3 geoelectric amplitudes and magnetotelluric sites with the high (low) impedance of the Ozark Dome (Illinois Basin). Furthermore, the orientation of the most polarized impedance tensors, seen as narrow ellipses in Figure 4a, roughly correspond to the polarizations of induced geoelectric vectors at those sites. These results are a clear demonstration that, should an EMP event actually occur over the eastern-midcontinental United States, localized differences in E3 geoelectric amplitude, polarization, and variational phase would result from geographically complicated surface impedance. Recalling from Section 3 that magnetotelluric impedance is, itself, a function of subsurface three-dimensional conductivity structure, we understand that simple half-space or one-dimensional, depth-dependent models of surface impedance will not, in general, provide accurate estimates of the E3 geoelectric hazard in complex geological settings.

Voltages on Power-Grid Lines
The integrated projection of a geoelectric field onto a power-grid line gives the geovoltage on that line, where G is the grid path between grounding points (e.g., Molinski, 2002;Pirjola, 2007). Using transmission power-grid topological information from the Department of Homeland Security (DHS), our estimated E3 geoelectric fields, and numerical interpolation methods given by Lucas et al. (2018Lucas et al. ( , 2020, we estimate time-dependent geovoltages on power-grid lines across the eastern-midcontinental study region. We show a map of the grid geovoltages in the movie-map of S1; we show snapshots of the grid geovoltages in Fig highest (lowest) for lines oriented east-west (north-south) but, otherwise, relatively uniform across the region. In contrast, in Figure 10f, we see that geovoltages derived using magnetotelluric tensors MT V show significant geographically localized differences-notably, geovoltages are high (low) across the Ozark Dome (Illinois Basin). For a given inducing E3 geomagnetic field, geovoltages can, in principle, be close to zero due to grid-line orientation and due to the local impedance.
In Figure 14, we show a map of the geovoltage difference  MT HS V V at  3.0 t s. Importantly, here, we see that geovoltages for the half-space model are generally lower (higher) than for the magnetotelluric tensors by as much as 1000 V across the Ozark Dome (Illinois Basin)-the peakto-peak range in error is 2,000 V. As we noted for the E3 geoelectric field, the geographic granularity in grid geovoltage is due primarily to the three-dimensional surface impedance of the solid Earth. Other candidate half-space conductivities, such as    4 10 HS S/m or    2 10 HS S/m, either of which might be reasonably used for hypothetical type calculations, would result in systematic errors in grid voltages (compared to magnetotelluric tensors) that would be even larger than shown in Figure 14.
In more detail, in Figure 15a, we show geovoltage waveforms for line 42,687 (the identification given in the DHS files), which runs 93 km approximately east-west across southern Missouri and across the Ozark Dome. The geovoltage obtained using magnetotelluric tensors attains a maximum value of 1,960 V at  2.1 t s. The waveform for the half-space model generally gives a lower geovoltage, and it is notably different in shape from the waveform for the magnetotelluric tensors. Geovoltage for the half-space model attains a maximum value of 1,042 V at  3.1 t s, after the maximum obtained using the magnetotelluric tensors. In contrast, in Figure 15b, we show geovoltage waveforms for line 34,886, which runs 64 km approximately east-west across the border between southern Illinois and Missouri. The geovoltage obtained using magnetotelluric tensors reaches a maximum value of 298 V at  2.5 t s. Here, the geovoltage waveform for the half-space model is generally higher, and it is also different in shape from the waveform for the magnetotelluric tensors. Geovoltage for the half-space model attains a maximum value of 731 V at  3.2 t s, also after the maximum obtained using the magnetotelluric tensors. Under Ohm's law, and assuming a typical line resistance of (say) one or a few ohms, E3 induced quasi-direct currents of hundreds of amps could plausibly be realized on power grid lines in the study region. Such currents would be sufficient to cause grid system failures and blackouts, though, because of the short duration of the E3 pulse, it is not certain that high-voltage transformers would be damaged (e.g., Electric Power Research Institute, 2019).
With Kirchhoff's circuit laws, our estimates of E3 geovoltages could, conceivably, be used to estimate the induced currents on the eastern-midcontinental power grid. To do that, we would need several types of grid parameters: the connectivity of the grid, line resistances, and network grounding resistances (e.g., Boteler & Pirjola, 2017;North American Electric Reliability Corporation, 2013). Unfortunately, these parameters are not generally publicly available, and, at least in the United States, grounding resistances are often either unknown (e.g., Bui et al., 2017) or difficult to obtain (e.g., Overbye et al., 2013, their Section VI). Indeed, in an analysis of magnetic-storm induced currents on the grid of the eastern United States, Overbye et al. (2012) used grounding resistivities that they describe as being "ballpark" estimates. Our review of the literature found published reports using hypothetical grounding resistivities for the United States grid ranging from 0.1  (e.g., Horton et al., 2012;Pulkkinen et al., 2012) to 2  (e.g., Overbye et al., 2012)-a range factor of 20 that would certainly affect quantitative estimates of E3 induced currents. Until accurate grid-system parameters become available, we choose to focus our analysis on calculations we can make with confidence, such as the E3 geovoltages depicted in the movie-map of S1 and the snapshots of

Conclusions and Discussion
From this study, we learn that E3 EMP geoelectric fields, generated by a nuclear explosion in the near-Earth space environment above our heads, can be strongly distorted by the geography of the Earth's surface impedance, a tensor that is a function of three-dimensional geological structures beneath our feet. This qualitative point might have been anticipated from previous analyses in which synthetic magnetic signals (e.g., Bedrosian & Love, 2015;McKay & Whaler, 2006) and measured magnetic-storm variation (e.g., Cuttler et al., 2018;Kelbert et al., 2017;Lucas et al., 2018) are convolved with long-period (   1 10 Hz;  10 s) magnetotelluric impedance tensors. Local surface impedance can have a significant effect on the amplitude, polarization, and variational phase of local induced geoelectric fields. In our analysis of E3, based on the convolution of a standard parameterization of E3 geomagnetic field variation with wideband magnetotelluric impedance tensors, 3 10 -3 10 Hz ( 3 10 -3 10 s), we draw a similar set of general conclusions. But our conclusions are not merely qualitative, we also arrive at important quantitative conclusions that are specific for the E3 hazard. In particular, just two seconds after a hypothetical nuclear explosion above the eastern-midcontinental United States, Section 8, geoelectric amplitude ranges from 1.7 V/km at one location to 33.2 V/km at another location, with a median amplitude of 6.2 V/km and a 68% interval of [2.7, 12.3] V/km. Generally, sites with a high degree of geoelectric polarization tend to have high geoelectric amplitude. We furthermore find, Section 9, that E3 geovoltages on power-grid lines can, for a hypothetical nuclear explosion over our chosen study region, be as high as 1,960 V; for other grid lines, due to localized small-amplitude geoelectric fields, and due to the orientation of a grid line relative to the orientation of the geoelectric field, geovoltages can, in principle, be close to zero. But using the half-space model leads to overestimation of geovoltages on some lines and underestimation on other lines-errors have a range LOVE ET AL.  of 2,000 V. Overestimating an E3 hazard might motivate implementation of overly protective (and possibly expensive) mitigation strategies. Underestimating an E3 hazard might end up leaving a power grid vulnerable.
Given the results of our E3 EMP scenario analysis for the Eastern Midcontinent, it is reasonable to envision performing similar scenario analyses for other places. A long-period magnetotelluric survey, with sparse 70 km station spacing, has been performed for most of CONUS (Schultz et al., 2006(Schultz et al., -2018Schultz, 2010) as part of the National Science Foundation's EarthScope project (Williams et al., 2010). Magnetic-storm geoelectric hazards have been mapped using these long-period tensors and data time series from magnetic observatories-the eastern United States and the northern Midwest exhibit both high long-period surface impedance and high magnetic-storm geoelectric hazard (e.g., Love et al., 2016;Lucas et al., 2020). But as demonstrated in Section 7, long-period tensors do not adequately resolve E3 geoelectric signals-wideband impedance tensors are needed. Still, the long-period survey serves as useful reconnaissance-we might reasonably expect the eastern United States and the northern Midwest to each exhibit high E3 geoelectric hazards. Targeted wideband magnetotelluric surveying and E3 scenario analyses in the eastern United States would support quantitative E3 vulnerability estimates for power-grid systems serving many of the nation's largest cities.
The methods used here might be further developed for more detailed studies in the future. To date, realistic estimates of magnetic-storm induced geoelectric fields have been obtained by convolving geomagnetic field variation, such as recorded at magnetic observatories, either directly with the magnetotelluric survey tensors (an approach similar to that used for this report) or with surface impedance derived from models of Earth electrical conductivity structure that are, themselves, derived through inversion of magnetotelluric survey tensors (models like those constructed for traditional solid-Earth investigations) (e.g., Kelbert, 2020;Marshalko et al., 2020;Simpson & Bahr, 2020). In extending the latter modeling approach to a study of the E3 geoelectric hazard, joint inversion might be made for Earth models using both geographically sparse long-period survey tensors and a denser distribution of wideband tensors. Collaboration is needed between different research groups to enable physics-based modeling of the EMP source and, just as importantly, proper treatment of three-dimensional Earth-surface impedance. Such a modeling project would provide improved understanding of the geographic-temporal nature of the EMP hazard for different scenarios and in different geological settings. Then, realistic assessments could be made of the vulnerability of power-grid systems to a realistic E3 geoelectric hazard.