Altitude effects of localized source currents on magnetotelluric responses

The effects of localized source currents on Earth's magnetotelluric (MT) responses have been evaluated in terms of the changes in period and subsurface structure. The focus is on the bias within the MT responses arising from variations in altitude of the source current. The MT responses at 20 and 200 s are calculated at various vertical distances of the source current. A slight change in the source's altitude causes a shift in the MT responses, and the bias is large especially over the vertical distances explored in the MT data analysis (i.e., 100-150 km). This shift due to the source field should be considered in a real-data analysis because the distribution of conductivity with altitude in the ionosphere and the region controlling the ionospheric electrical process change temporally.


Introduction
In magnetotelluric (MT) surveys, the primary electromagnetic fields arising from source fields are assumed horizontally uniform. The effects of localized source currents on the MT responses have been discussed in the literature (Madden and Nelson, 1964;Schmucker, 1970;Hermance and Peltier, 1970;Häkkinen et al., 1989;Pirjola, 1992), where impedances at long periods and at sites above structures of high resistivity are biased. For example, Pirjola (1992) reported that the apparent resistivity of 100 Ωm and at periods larger than 60 s were clearly affected. Their studies focused on period-dependent subsurface bias. However, the bias stemming from the variation in altitude of localized currents was not discussed in detail. In this study, the electromagnetic fields and MT responses were calculated by increasing the vertical distance of the source current (100, 105, …, 595, and 600 km). The study revealed i) the numerical examples of the bias in the MT responses because of the variation in vertical distance of the source field, ii) implications from these examples, iii) the mathematical underpinning of this bias, and iv) the mathematical conditions for upholding the plane-wave assumption.

Electromagnetic fields above Earth's surface
We chose a Cartesian coordinate system, where the , , and axes are northward, westward, and downward positive, respectively, with = 0 at Earth's surface. Ignoring the displacement current and using the SI system, Maxwell's equations in the frequency domain are where , , and are the electric field, magnetic induction, and source current, respectively; , , and are the electrical conductivity, the magnetic permeability of free space, and the angular frequency, respectively. Introducing the vector potential and the scalar potential , the electromagnetic fields are Applying the Lorenz gauge, the and the must satisfy Considering the electromagnetic fields above Earth's surface (i.e., ≤ 0), may be taken as denoting the electrical conductivity of free space. As in Hermance and Peltier (1970), we consider a wire at an altitude < 0 carrying an electric current ; the current density is We focus on only , the component of , and . For this study, the horizontal Fourier transforms (FTs) are defined as Eq. (10) enables us to transform Eq. (6) into where and are and in the Fourier domain, respectively, and = ( + ) + . Eq. (11) is the Helmholtz equation and for which its Green's function satisfies As shown in Arfken et al. (2012), the solution of ( , ) is where is a constant required to uphold the boundary condition at = 0 . Consider a structure beneath Earth's surface ( > 0 ) having a half-space of conductivity . The continuity of the electromagnetic fields parallel to the boundary yields where = ( + ) + . Applying the FT, Eq. (9), in Eq. (11) becomes Using Green's function, the solutions for are Applying the inverse FT, Eq. (10), and considering = 0, is written as The scalar potential is ignored because both sides in Eq. (7) are equal to zero. From Eqs. (4) and (5), the magnetic induction and the electric field at Earth's surface ( = 0) is written as: Taking their ratio / gives the impedance , For this study, the apparent resistivity is given by where is defined in Eq. (20). Note that hereafter the distance unit used is "km" instead of "m" when stating horizontal/vertical distances ( , ) although all the above values are calculating based on the SI system of units.

MT responses biased by line current
Here, the subsurface resistivity and the time period is set to 1000 Ωm (i.e., = 10 S/m) and 20 s, respectively. The subsurface resistivity has the same value as that used for the crust in Hermance and Peltier (1970). By changing the altitude of the source current from -100 to -600 km in increments of 5 km and the horizontal distance ( =1, 10, 100, 1000 km), the variation in the field

Discussion
A discussion is presented next of i) the mathematical basis of the bias on the MT responses due to the source field, ii) the mathematical condition upholding the plane-wave assumption, and iii) the implication arising from the numerical examples performed in this study.
The electromagnetic fields [Eqs. (18) and (19)] generated by the line current have an attenuation term and a term conveying information regarding the subsurface structure Substituting 1/s, 1.26 • 10 H/m, and 0.001 S/m for , , and , respectively, the apparent resistivity and phase are plotted (Fig. 1). When the wavenumber | | is greater than 2.0 • 10 , the weight of is greater than . When ( ) in Eq. (22) is smaller than 0.01, the effect of | | is assumed negligible. To uphold this assumption, | | should be smaller than −4.6, and when = −100 km, | | should be greater than 4.6 • 10 . Therefore, the integrands in Eqs. (18) and (19) are biased by wavenumber | | at least within the interval 2.0 • 10 < | | < 4.6 • 10 .
On the other hand, given that = −600 km, the wavenumber effect is small enough to be negligible, and as a result, the apparent resistivity approaches a constant value of 1000 Ωm, i.e., the subsurface resistivity. As expected from the above discussion, the MT responses at 200 s are biased more than those at 20 s.
Both responses at 20 and 200 s have the same polarity; that is, they shift depending not only on the vertical ( ) but also on the horizontal distance ( ) between the site and the source current. The integrands in Eqs. (18) and (19) where = . Focusing on the term with ( ) and integrating by parts, we can obtain, The triangle inequality and the inequality ( ) ≤ ( ∈ ℝ) enable the integrand of the last term in Eq. (27) to be examined, Eqs. (28) and (29) The right-hand sides of Eqs. (30) and (31) Using Eqs. (32) and (33), in Eq. (20) is written as In the limit → ∞, the plane-wave assumption is established which is also upheld in the limit → −∞. This means that if either the horizontal or vertical distance of the localized current is large enough, the plane-wave assumption remains valid. If this condition is not established, the MT responses would be biased by the source field (see Figs. 1 and 2).
In this study, the focus was on the relationship between the MT responses and the altitude of the source current. The altitude distribution of the conductivity in the ionosphere changes temporally/seasonally (Sheng et al., 2014), and the region (i.e., E/F) controlling the ionospheric electrical processes also changes temporally (Du and Stening, 1999). Therefore, the altitude of the source current may be considered to vary temporally. Temporal/seasonal changes in the MT responses have been recently well reported (Brändlein et al., 2012;Romano et al., 2014;Vargas and Ritter, 2016;Murphy and Egbert, 2018). These reports are explained by the results in Figs. 1 and 2 because the altitude of the source current varies temporally/seasonally. For example, Romano et al. (2014) reported that, for time periods 20-100 s, the apparent resistivities have a negative correlation with geomagnetic activity. This case indicates an identical polarity with the result in Fig. 1 ( = 1 or 10 km). When the altitude of the source current decreases, the magnetic amplitude increases and the apparent resistivity decreases. As a result, both have a spurious negative correlation.

Summary
In this study, the focus was on the bias in the MT responses due to the variation in altitude of a localized