An analytical expression for early electromagnetic signals generated by impulsive line-currents in conductive Earth crust, with numerical examples ANNALS OF GEOPHYSICS, 59, 2, 2016, G0212; doi:10.4401/ag-6867 G0212 An analytical expression for early electromagnetic signals generated by impulsive line-currents in conductive Earth crust, with numerical examples Ken’ichi Yamazaki Miyazaki Observatory, Disaster Prevention Research Institute, Kyoto University, Miyazaki, Japan ABSTRACT Changes in the electromagnetic (EM) field after an earthquake rupture but before the arrival of seismic waves (“early EM signals”) have some- times been reported. Quantitative evaluations are necessary to clarify whether the observed phenomena are accounted for by known theories and to assess whether the phenomenon can be applied to earthquake early warning. Therefore, analytical expressions for the magnetic field gener- ated by an impulsive line-current are derived for a conductive half-space model, and for a two-layer model; the somewhat simpler situation of a conductive whole-space is also considered. By analyzing the expressions obtained for the generated EM field, some expected features of the early EM signals are discussed. First, I verify that an early EM signal arrives be- fore the seismic waves unless conductivity is relatively high. Second, I show that early EM signals are well approximated by the whole-space model when the source is near the ground surface, but not when it is at depth. Third, I show that the expected amplitudes of early EM signals are within the detection limits of commonly used EM sensors, provided that ground conductivity is not very high and that the source current is sufficiently in- tense. However, this does not mean that the EM signals are easily distin- guishable, because detector sensitivity does not account for additive noise or false positive detections. 1. Introduction Seismic fault slip mainly generates ground motion, but electromagnetic (EM) variations have been re- ported during and prior to some earthquakes [e.g., El- eman 1966, Iyemori et al. 1996, Honkura et al. 2002, Honkura et al. 2004, Abdul Azeez et al. 2009, Honkura et al. 2009, Okubo et al. 2011, Gao et al. 2014]. The term “early EM signal” means any measurable seismo- genic EM variation that precedes the first seismic phase arrival. A few reports include early EM signals that ap- pear related to earthquakes. For example, Iyemori et al. [1996] observed changes in the geomagnetic field at the time of the 1995 Mw 7.1 Hyogo-ken Nanbu (Kobe) earthquake at two stations nearly 100 km from the source region. The EM variations had maximum am- plitudes of 0.6–1.0 nT, and began ~10 s before the P- wave arrival (~10 s after the estimated seismic origin time). For the 2007 Mw 6.9 Iwate-Miyagi Nairiku earth- quake, Okubo et al. [2011] observed changes in the ge- omagnetic field ~3 seconds before the arrival of seismic waves at a hypocentral distance of 27 km; the maxi- mum signal amplitude was ~0.2 nT. The reliability of other previously reported early EM signals is disputed, as these studies did not sufficiently discuss discrimina- tion between signal and background geomagnetic fluc- tuations (i.e., false positive detections). However, if early EM signals can be reliably observed, then they could be incorporated into early earthquake warning systems, which currently rely solely on seismic obser- vations [e.g., Kanamori 2005]. Various mechanisms have been suggested to ex- plain the conversion of mechanical energy to EM en- ergy. Some examples include the motional induction effect [e.g., Eleman 1966, Gershenzon et al. 1993], the electrokinetic effect [e.g., Ishido and Mizutani 1981, Pride 1994], the piezoelectric effect [e.g., Ogawa and Utada 2000a,b, Huang 2002], and the piezomagnetic ef- fect [e.g., Yamazaki 2011a,b]. To assess the detectability of early EM signals, it is essential to estimate expected amplitudes and to determine the distinguishing signal characteristics (if any) associated with each mecha- nism. The most straightforward way to study early EM signals is, of course, to study the measured EM fluctu- ations themselves; however, strictly observational ap- proaches are difficult because large earthquakes and sensitive geomagnetic recording stations are both rare, meaning the undisputed data set is small. Most obser- vations of early or coseismic EM signals have come from temporary sites designed for EM prospecting, including magnetotelluric (MT) instruments [e.g., Article history Received September 9, 2015; accepted April 4, 2016. Subject classification: Geomagnetism, Earthquake, Subsurface electric current, Early warning. Honkura et al. 2002, Abdul Azeez et al. 2009, Gao et al. 2014]. Successful detections of early EM signals are rather rare at continuous stations; among the only clear examples are Iyemori et al. [1996] and Okubo et al. [2011]. One potential way to capture more early EM signals is to deploy instruments near the epicenter of a large, shallow earthquake, because many aftershocks are expected [e.g., Ujihara et al. 2004, Kuriki et al. 2011]. However, because aftershocks are necessarily smaller than the main shock, and early EM signals related to the main shock are already near the limits of conven- tional detectors, aftershocks are difficult to detect in practice. Consequently, it is difficult to compile a suit- ably large data set of early EM signals. It is therefore ex- pedient to use theoretical models of EM variations to guide observational efforts. In the present work, subsurface electric currents induced by ground motions are examined as possible early EM source mechanisms. The situation is schemat- ically shown in Figure 1. During an earthquake rupture, electric currents can be induced by the motion of con- ductive crustal material in the ambient geomagnetic field [e.g., Gershenzon et al. 1993, Yamazaki 2012, Gao et al. 2014]. The objective is to determine the expected temporal variations in the generated EM field. An electric current in a vacuum generates a magnetic field that can be easily calculated using the Biot–Savart law. Some earlier numerical calculations were performed by assuming that the medium was a vacuum or a per- fect resister [e.g., Taira et al. 2009, Okubo et al. 2011]. However, an electrical current in a conductive medium should be dealt with by solving the time-dependent Maxwell’s equations, to determine, for example, if the conductive medium acts as an insulator against EM propagation. The reminder of this paper is organized as follows. In Section 2, the governing equations are described for specific situations. In Section 3, a set of analytical expres- sions is derived for the EM field generated by an impul- sive line-current in a conductive half space. In Section 4, some characteristics of early EM signals in a conduc- tive medium are discussed, along with some numerical examples. The validity of using a simpler model for eval- uating early EM signals is also considered. Finally, the main conclusions are presented in Section 5. 2. Definition of the problem In this paper, I derive analytical expressions for the early EM signals arising in the two model geometries: a two-layer model comprising a half-space with uni- form conductivity, representing the Earth’s crust, over- lain by a perfect resistor, representing the air (Figure 2a) and a whole-space model consisting of a uniformly conductive infinite medium (Figure 2b), which is the simplest system that can model the effect of electrical conductivity. A line current is considered an approxi- mation of the electric current induced by the coseismic motion of fault-zone material, which is often more conductive than the surrounding crust, as illustrated in Figure 1. Considering the situation whereby the fault length is orders of magnitude greater than the spatial scale of conductivity variation across the fault, it is rea- sonable to assume that the electric current occurs as a line of infinite length, thus reducing the problem to two dimensions. The half-space model (Figure 2a) is the simplest re- alistic approximation of the Earth. A realistic model must involve at least two layers: a conductive ground and resistive air. In contrast, the whole-space model (Figure 2b) is somewhat unrealistic; however, solving Maxwell’s equations in the whole-space model is con- siderably easier than for the half-space model. For this reason, I also consider the whole-space model through- out the following, so that any situations in which the simplified model is valid can be understood. When the electric current density I is given as a function of location (x, z) and time t, the electric and magnetic fields, E and B respectively, are determined via Maxwell’s equations: YAMAZAKI 2 Figure 1. A schematic illustration of the subsurface-current generat- ing mechanism analyzed in this paper. Seismic motion of a fault block containing a zone with high conductivity subject to the ambient geo- magnetic field induces an electric current according to Faraday’s law. Figure 2. The two models for which formulas are derived in this paper. The direction of the y-axis is chosen such that the x-, y- and z-axes form an orthogonal, right-handed, Cartesian system. 3 where f and n are, respectively, the electric and mag- netic permeabilities. It is assumed that n is constant such that n=n0= 4r× 10-7 H/m. It is also assumed that the electric cur- rent density is the sum of an induced current density and a conductive current density; the conductive cur- rent density obeys the isotropic Ohm’s law (3) where v is the electrical conductivity, and Iext is the ex- ternal current density. Additionally, the displacement current term is ignored; that is, the second term of Equation (2). This approximation yields (4) Introducing a vector potential that satisfies B =∇×A, Equations (1) and (4) are combined to form a single ex- pression The two-dimensionality of the problem allows a coordinate system to be chosen such that the y-axis is oriented parallel to the external electric current, and the positive z-direction is vertically downward (see Fig- ure 2). In this coordinate system, only the y-component of Iext is non-zero; the y-component of Iext is denoted as Iy ext. The electric current density of an impulsive line- current at depth d is therefore given by (6) where d is the Dirac delta function. In this situation, we can chose a solution of Equation (5) of the form A = (0, A,0), with the function A satisfying the scalar equation (7) The EM field is then determined by solving Equa- tion (7) under appropriate boundary conditions. The first boundary condition comes from the EM field being zero at infinite distance from the source current: i.e., A(x, z → ±∞, t) = const. When considering the two-layer model, conditions on the continuity of A and ∂A/∂z are also imposed at the boundary z = 0 to ensure the con- tinuity of E and B at any instance. The non-zero com- ponents of the electric and magnetic fields are then given by Ey= ∂A/∂t, Bx= −∂A/∂z, and Bz= ∂A/∂x. 3. Derivation of solutions The governing equations can be simplified by the substitution of regularized variables t*≡n0vt, x *≡n0vx, z*≡n0vz , and d *≡n0vd. Using these variables, the source term is written as (8) The differential operators become ∂/∂t =n0v∂/∂t * and ∇=n0v∇ *. The governing equation including the source term can then be rewritten as (9) in a conductive medium (i.e., v> 0), where I0 *≡n0 2vI0, and in a vacuum (i.e., v= 0). To find the solution, consider the integral expres- sion and the corresponding Fourier expression of A (12) In these identities, g* and ~* represent the wave- number and angular frequency, respectively. Substitut- ing these expressions into Equations (9) and (10), the governing equations for A (g*, z*,~*) are obtained as (13) where Solutions to Equations (13) and (14) are given as E B t #d 2 2 + ,0= B I#d n= + ,E t2 2 nf ,I E I extv= + B E#d nv= .I indn+ , , , ,, , .A A Ix z t x z tx z t t ext2 00d 2 2 n v n- =-Q Q QV V V , , ,VI x z t I x z d texty 0d d d= -Q Q Q QV V V ,V , , , , . A x z t t A x z t I x z d t 2 0 0 0 d 2 2 n v n d d d - = - - Q Q Q Q Q V V V V V , , .I x x z tz t I dexty 0 3V 0n v d d d= -) ) ) ) ) ) )Q Q Q Q QV 3V V V V , , , ,A x z t t A x z t I x z d t 2 0 d 2 2 d d d - = - - ) ) ) ) ) ) ) ) ) ) ) ) Q Q Q Q QV V V V V , ,A x z t2d) ) ) )Q 0=)V dg d ,cos exp x t g x dg i t 2 1 2 0 # d d r ~ ~ = - ) ) ) ) ) ) ) )3 3 3 - # Q Q Q V V V 0 # # , , d dg , , .cos exp A x z t dg A g z g x i t 0 #~ ~ ~ = - ) ) ) ) ) ) ) ) ) ) ) ) 3 3 3 - # Q Q Q V V V# 0 # , ,, , z A g zA g z u z d 2 1 2 2 2 22 2 ~ ~ r d- =- -) ) ) ) ) ) ) ) )Q Q QV V V , , , , z A gQ z A gQ zg 022 222 ~ ~- =) ) ) ) ) ) ) )gQ gQV V , .Reu ug i 02/ $~-) ) ) )Q V EARLY EM SIGNALS BY LINE CURRENTS (1) (5) (10) (11) (14) and (2) (15) follows. In the case of the whole-space model (Figure 2b), which is described solely by Equation (13), the so- lution is given by In the case of the two-layer model (Figure 2a), which is described by Equation (13) for z > 0 and by Equation (14) for z < 0, upward- and downward-propa- gating terms should be added to Equation (16) [e.g., Stoyer 1977]. These additional terms are determined such that they satisfy the boundary conditions, and the following solution is obtained: (17) Observations of variations in the EM field are usu- ally obtained near to the ground surface (i.e., z ≅ 0). Be- cause of the continuity of the EM field at ground level (z = 0), it is sufficient to consider A either for z > 0 or for z < 0. For this reason, only the time-domain solution for A at z > 0 is considered. To evaluate the integral in Equation (12) over ~, I refer to the formulas for the inverse Fourier transforms and where u is defined by Equation (15), and g is a positive real number. The function erfc represents the comple- mentary error function, The result shown in Equation (18) can be derived by changing the integration variable from ~* to u and choosing an appropriate integral path over which to apply the Cauchy integral theorem (see Figure 3). Equation (19) can then be derived from Equation (18) after some algebraic manipulation. In the following, I consider phenomena only for t > 0 because, from Equa- tions (18) and (19), it is clear that the solution is trivial at negative times. To evaluate the integration of Equation (12) over the variable g*, I use the easily derived results formulas: (21) (22) Here, erfi is the imaginary error function defined by Equation (21) is derived by integration by parts using the identity: (24) Using Equations (18), (19), (21) and (22), the inte- gration of Equation (12) using the function given by Equation (17) can be evaluated to yield an analytical ex- pression for the vector potential in the time domain, specifically (25) where z*+= z *± d*. In cases where the source current persists for a fi- nite amount of time, the corresponding vector poten- tial can be determined by integrating A over time t. For example, in the situation where a source electric cur- rent exists over the time period 0≤t≤T with constant , , .expA gQ z I u u z d 2 1 1 2V 0~ r= - - ) ) ) ) ) ) ) )gQ Q RV 2V W , , exp exp exp exp A g z I g u u d g z u u z d g u u u z d z z 2 1 2 1 2 1 0 0 < > 2V 0 #~ r= + - - - + + - - + ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) S Q R Q Q Q Q V X 2V W V V V Z] [] \ ]] Z]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]][] ][]]]]]]]]]]]]]]]]]]]]]]]]]]]]]] \ ]]]] \ E H dexp exp exp exp u u t t t t i t g- t 1 0 0 2 1 4 0 > 2 2 # g ~ ~ r r g - - = g- - ) ) ) ) ) ) ) ) ) ) ) ) ) 3 3 - # Q Q Q T V V V Y Z] [] \ ]]] Z]]]]]]]]]]]]]][] ][]]]]]] \ ]]]]]] \ # d erfcfc exp exp exp u u i t gr gQ gT t t t t g 1 0 2 0 0 2 > # g ~ ~ gr g g + - - = + ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 3 3 - # gQ Q gT Q V V V Y Z] [] \ ]]] Z]]]]]]]]]]]]]][] ][]]]]]] \ ]]]]]] \ # erfcfc d .expw ww 2 w # 2/ r - 3 l lQ QV V w # dgexp cos exp t t t t g- g x dg x1 2 1 40 # 2 2 r g- = -) ) ) ) ) ) ) ) )3 Q SV X 0 # erfcfc dg erfcfc exp 4 1 erfifii 1 exp cos Im g g g t t g x dg x ix t ixix t x t x 2 1 2 2 0 # 2 2 + + 2 2 g g g g g g + = - + + - + ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 3 Q T T T T V Y Y Y Y # &G J 0 # erfifi .expw ww d 2 w 2 0 #/ r l lQ V 0 # / .cos Im expg g gi xx gi x2 2=) ) ) ) ) )Q V , , erfcfc erfifi exp exp Im exp exp A x z t I t t x z t t x z x ixix z t z ixix z t x z ixix z i t x z t x 2 1 2 1 4 2 1 4 2 1 2 1 4 4 2 0 2 2 2 2 2 2 2 2 2 2 r = + + + + - + + + + + + + ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) + + + + + + + Q T T T U T T V Y Y Y Z Y Y # & G J YAMAZAKI 4 (18) (19) (20) (23) and (16) 5 intensity I0, the vector potential, denoted by A STEP, is calculated as and the corresponding EM field is given by derivatives of ASTEP. Note that in this calculation evaluated in the next section, we must ensure that the differences be- tween regularized and non-regularized variables (i.e., t and t*) are correctly accounted for. 4. Discussion and numerical examples This section considers three points based on the analytical expressions derived for early EM signals. First, the onset times of early EM signals are consid- ered, in order to impose principal limitations on the de- tectability of early EM signals. Second, the validity of using vacuum and conductive whole-space models is considered, in order to assess the viability of further simplifying the calculations. Third, the detectability of early EM signals is discussed for a few specific cases. The numerical examples below use the following parameters. The current intensity I0 and hypocentral distance d are set to 50 A and 10 km, respectively. For the proposed current generation mechanism (i.e., in- duction due to coseismic fault motion; Figure 1), I0 = |SvV × Bamb|, where S, V, and Bamb represent the cross-sectional area of the conductor, the ground mo- tion velocity, and the ambient geomagnetic field, re- spectively. An example situation for which I0 = 50 A is |V| = 0.1 ms−1, S = 10 km2, v = 0.1 Sm−1, and |B| = 50000 nT, with V perpendicular to B. Given the nu- merical considerations of Iyemori et al. [1996], these values might be typical for a Mw 7.1 earthquake. How- ever, for simplicity the following calculations are per- formed by approximating that the electric currents ap- pear on a line, as in Equation (6); the discussions below still hold under this approximation. The value of I0 does not become considerably larger, even for earthquakes with much higher mo- ment magnitudes. As long as we assume that induc- tion is due to ground motion (i.e., Figure 1), the generated current intensity depends on the ground ve- locity, not the seismic moment. In addition, early EM signals do not scale with seismic moment, but instead scale with seismic moment released at the first stage of the rupture. Yet initial seismic moment release is never particularly large, even at Mw ≥ 9 [e.g., Ammon et al. 2005, Koketsu et al. 2011]. Therefore, the ampli- , , , , d ,A x z t x z t tASTEP ,m # ax t T t 0 = - l lQ QQV VVm# EARLY EM SIGNALS BY LINE CURRENTS (26) Figure 3. Integral paths in the complex ~-plane used to evaluate the integral in Equation (18). When t < 0, we can use the path colored green; integration along C– converges to zero for this case. When t > 0, we adopt the black integral contour. Integration over C+1 and C+3 converges to zero, and C+2 becomes Gauss’s integral in this case. Figure 4. An illustration of the two source-receiver geometries used to test the validity of the analytical solutions developed for the two mod- els shown in Figure 2. tudes of early EM signals are not expected to increase, even for great earthquakes. 4.1. Characteristics of the solution An important feature of temporal variations in early EM signals is the “delay” of the signal. First, let us analyze the form of the analytical expression for EM potential given in Equation (23). The arguments of the exponentials and error function terms in Equation (23) are proportional to r*2/t*=nvr2/t or its square root, where r represents a value with the same order of mag- nitude as x and z. Therefore, it is expected that Equa- tion (23) will have its extremal value at around t*= r*2, which is equivalent to t =nvr2 ≅ 10−6 ×vr2. The time at which the vector potential takes this extremal value is regarded as the delay time of the signal arrival. For the case of r = 10 km = 104 m, for example, the delay time assuming v =10–1, 10–2, and 10–3 Sm−1 is of the order of 10, 1, and 0.1 s, respectively. A typical value for the arrival time of a seismic wave at a hypocen- tral distance of 10 km is about 2 seconds. These results imply that arrivals of early EM signals will precede those of seismic waves unless conductivity is rather high (i.e., 10–2 Sm−1). 4.2. Effect of the ground surface Let us consider the temporal variations in the early EM signals for the two source geometries shown in Figure 4 using the analytical solutions developed above. I consider the case whereby the temporal variation of the source current is given by the Heaviside step function H(t), which is zero for t < 0 and unity for t > 0. The intensity of the current I0 and hypocentral distance d are assumed as 50 A and 10 km, respectively. This in- tensity value is realistic for the proposed current-gen- eration mechanism (i.e., induction due to coseismic fault motion generation; see Figure 1), and is approxi- mately equivalent to that expected for a ground mo- tion velocity, fault width, thickness of the conductor around the fault, conductivity of the conductor, and ambient magnetic field intensity of 0.1 ms−1, 10 km, 1 km, 0.1 Sm−1, and 50000 nT, respectively. We first consider the case whereby the source cur- rent is located directly beneath the observation point (“case 1”, shown in Figure 4a). In this situation, the x- component of the magnetic field Bx can be written as and the z-component is zero. Plots of Bx for v= 1.0 × 10−1, 10–2, 10–3, and 10–4 Sm−1 are shown in Figure 5. Note that, regardless of the conductivity, this function converges to 1/2rz as expected from the Biot–Savart law for a steady current. For small values of v(i.e., 10–3 or 10–4 Sm−1), Bx rapidly converges to its final value Bx (0,0,t =∞). In contrast, for large values of v, the con- vergence is rather slow. These results demonstrate that predictions made using the Biot–Savart law are good approximations of the time-varying magnetic field for small v, but are poor for large v. It should be noted that magnetic field variations calculated for the whole-space model shown in Figure 2b are also considerably different from those calculated for the two-layer model (Figure 2a). When we consider the whole-space model, the temporal variation of Bx is given by Expanding the exponential, we can confirm that its convergence speed is of the order of t –1. In contrast, from Equation (25) we see that the convergence speed of Bx for the two-layer model is of the order of t –1/2. These differing convergence rates are apparent in Fig- ure 6, which compares Bx for the two models assuming v= 10−2 Sm−1. For comparison, we also consider the situation where both the source current and observation site are located on the boundary of the layer (“case 2”, shown in Figure 4b). In this case, the z-component of the mag- netic field Bz for the two-layer model can be expressed as , , erf ,exp expB t d t t d d t t d d t t d d I d t 0 0 4 4 4 2 4 2 1 4 2 2 x 2 2 2 2 0 2r r + - =- - + - - ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Q T S SV Y X XG J , , .expB t d d I t 0 0 2 1 4x 2 0r =- -) ) ) )Q SV X , , .expB t t d I d t 0 0 1 2 4z 2 0 3r = -) ) ) ) )Q SV XG J YAMAZAKI 6 (27) Figure 5. A comparison of the temporal variation in the horizontal component of the magnetic field for the source-receiver geometry shown in Figure 4a for four conductivity values. (28) (29) 7 Numerical examples of the temporal evolution of Bz are shown in Figure 7 for various conductivity values. In contrast to the results obtained for the buried line-source, the whole-space solution provides a rea- sonable approximation to that of the two-layer model for the situation shown in Figure 4b when the source and receiver are located at the same depth. In this geometry, the convergence speed of Bz is proportional to T –1 for both the two-layer and whole-space models. This result implies that the temporal variation in Bz ob- tained using the whole-space model can be considered a good approximation to that predicted by the two- layer model, although there are some small differences. The level of agreement between the two models is apparent in Figure 8, which compares the temporal evolution of Bz computed for half-space, conductive whole-space, and resistive whole-space models assum- ing v= 10−2 Sm−1. 4.3. Detectability of early EM signals To assess the detectability of early EM signals, we examine expected amplitudes and shapes for a few spe- cific cases. For earthquake early warning, the situation demonstrated in case 1 (Figure 4a) is most important, be- cause seismic P-waves cannot be detected at observation sites before reaching at the ground surface in this situ- ation. For this reason, we focus primarily on the situation in Figure 4a, and specifically on the generated magnetic field Bx (Figure 5). Note, however, that many of the fol- lowing characteristics are unchanged for the geometry in Figure 4b and the resultant Bz field in Figure 7. The numerical examples show that a high resistiv- ity is required for the signal to be detected. In resistive cases (i.e., v= 10–4 or 10–3 S/m), an early EM signal ap- pears almost at the seismic origin time, and the tem- poral pattern is not considerably different from a step function. Recall that this result was obtained by assum- ing that temporal variations in the source current were also given by a step function. The results for the resistive case therefore imply that the temporal variations in B largely reflect temporal variations in the source current. It was assumed that the source current is generated by EARLY EM SIGNALS BY LINE CURRENTS Figure 6. A comparison of the temporal variation in the horizontal component of the magnetic field for the source-receiver geometry shown in Figure 4a for a conductive whole-space (solid black line) and two-layer model (dotted line). For reference, the solution for a resistive whole-space (i.e., a vacuum) is also shown. Figure 7. A comparison of the temporal variation in the vertical component of the magnetic field for the source-receiver geometry shown in Figure 4b for four conductivity values. Figure 8. A comparison of the temporal variation in the vertical component of the magnetic field for the source-receiver geometry shown in Figure 4b for whole-space (solid black line) and two-layer (dotted line) models. For reference, the solution for a resistive whole-space (i.e., a vacuum) is also shown. motional induction, so that the intensity of the source current is proportional to the slip velocity. Therefore, it can be concluded that temporal variations in Bx repre- sent temporal variations in the slip velocity (i.e., the earth- quake source-time function) when v≤ 10–3 S/m. In contrast, in conductive cases, temporal variations in Bx are quite emergent, and appear considerably different from the source-time function. Smoothed variations in the magnetic field are rather difficult to distinguish from background geomagnetic variations; hence, de- tection of early EM signals is expected to be difficult in a conductive medium. Concerning the amplitudes of early EM signals, the present technique yields estimates of ~0.5–1.0 nT. Whether or not signals of these amplitudes can be re- solved is primarily dependent on the type of instru- ment used. Most devices that have observed a coseismic or pre-seismic EM signal are either flux-gate magne- tometers [e.g., Iyemori et al. 1996] or induction coils [e.g., Ujihara et al. 2004, Abdul Azeez et al. 2009]. Their minimum resolutions are 0.1 nT or finer; however, con- sidering fluctuations in (or stability of ) the data, their practical resolution limit is ~1 nT. Finer-scale instru- ments, such as SQUID magnetometers [e.g., Katori et al. 2013], have better resolution; however, these sensors are costly, and dense installations are thus infeasible at present. Taking background noise into account, we can estimate the general detection limit of early EM signals at ~0.5–1.0 nT. We can therefore tentatively conclude that signals are detectable in principle. Numerical examples demon- strate that 0.5–1.0 nT is the expected range of early EM signal amplitudes, assuming that the current intensity (I0 = 50 A) is realistic. As mentioned at the beginning of this section, the current intensity (I0 = 50 A) corresponds to a Mw 7 earthquake. Therefore, a Mw 7 or larger earthquake could be detectable before the P arrival if EM signals could be observed at distances of ~10 km. It must be noted, however, that the above discus- sion only provides one possible scenario in which early EM arrivals can be detected. This is by no means com- prehensive. A wide variety of V, S, and v values are pos- sible in the scenario depicted in Figure 1. Exact values depend on the electrical structure of the fault and indi- vidual earthquake source processes. In this sense, the detection limit is not rigid. The ability to distinguish sig- nal from noise is another limiting factor, though address- ing this challenge falls outside the scope of this work. 5. Conclusions In this paper, I have derived analytical expressions for the time-varying EM field generated by an impul- sive line-current in both a two-layer and whole-space model. Some quantitative features of the predicted early EM signals are demonstrated through analysis of the expressions obtained. Concerning the arrival times of early EM signals, an important result is that the ar- rivals of early EM signals precede the arrivals of the first seismic waves by a few seconds unless conductiv- ity is high (i.e., 10−2 Sm−1). Consequently, with a low conductivity, early EM signals could be detectable in principle. Concerning the viability of the simplified model used in the calculations, the temporal evolution of early EM signals is well approximated by a whole- space model when the source is near the surface; for a source at depth, this approximation breaks down and it is necessary to consider a two-layer model instead. Concerning the detectability of early EM signals, we expect amplitudes of >0.5 nT if ground conductivity is not very high and the source current is sufficiently in- tense. Given that this exceeds the noise floor of most common types of magnetometers, early EM signals could be detected in principle. However, accurate real- time detection, and discrimination between signal and noise, are outstanding challenges that will be explored in future work. Acknowledgements. This work was supported by JSPS KAK- ENHI Grant Numbers 24740304 and 26800233. Figures 5-8 were generated using Generic Mapping Tools [Wessel et al. 2013]. Com- ments by two anonymous reviewers helped to improve an early ver- sion of the manuscript. References Abdul Azeez, K.K., C. Manoj, K. Veeraswamy and T. Harinarayana (2009). Co-seismic EM signals in mag- netotelluric measurement - a case study during Bhuj earthquake (26th January 2001), India, Earth Planets Space, 61, 973-981. Ammon, C.J., H.-K. Thio, D. Robinson, S. Ni, V. Hjor- leifsdottir, H. Kanamori, T. Lay, S. Das, D. Helm- berger, G. Ichinose, J. Polet and D. Wald (2005). Rupture process of the 2004 Sumatra–Andaman earthquake, Science, 308, 1133-1139. Eleman, F. (1966). The response of magnetic instru- ment to earthquake waves, J. Geomag. Geoelectr., 18, 43-72. Gao, Y., S. Chen, H. Hu, J. Wen, J. Tang and G. Fang (2014). Induced electromagnetic field by seismic wave in Earth’s magnetic field, J. Geophys. Res. Solid Earth, 119, 5651-5685. Gershenzon, N.I., M.B. Gokhberg and S.L. Yunga (1993). On the Electromagnetic Field of an Earthquake Focus, Phys. Earth Planet. In., 77, 13-19. Honkura Y., M. Matsushima, N. Ohshiman, M.K. Tuncer, S. Baris, A. Ito and A.M. Isikara (2002). Small electric and magnetic signals observed before the arrival of YAMAZAKI 8 9 seismic wave, Earth Planets Space, 54, e9-e12. Honkura, Y., H. Satoh and N. Ujihara (2004). Seismic dynamo effects associated with the M 7.0 earth- quake of 26 May 2003 off Miyagi Prefecture and the M 6.2 earthquake in northern Miyagi Prefecture, NE Japan, Earth Planets Space, 56, 109-114. Honkura, Y., Y. Ogawa, M. Matsushima, S. Nagaoka, N. Ujihara and T. Yamawaki (2009). A model for ob- served circular polarized electric fields coincident with the passage of large seismic waves, J. Geophys. Res., 114; doi:10.1029/2008JB006117. Huang, Q. (2002). One possible generation mechanism of co-seismic electric signals, Proc. Jpn. Acad., Ser. B, 78, 173-178. Ishido, T., and H. Mizutani (1981). Experimental and theoretical basis of electrokinetic phenomena in rock-water systems and its applications to geo- physics, J. Geophys. Res., 86, 1763-1775. Iyemori, T., T. Kamei, Y. Tanaka, M. Takeda, T. Hashimoto, T. Araki, T. Okamoto, K. Watanabe, N. Sumitomo and N. Oshiman (1996). Co-seismic geo- magnetic variations observed at the 1995 Hyogo- ken-Nanbu earthquake, J. Geomag. Geoelectr., 48, 1059-1070. Kanamori, H. (2005). Real-time seismology and earth- quake damage mitigation, Annu. Rev. Earth Planet. Sci., 33, 195-214. Katori, Y., K. Okubo, T. Hato, A. Tsukamoto, K. Tan- abe, N. Onishi, C. Furukawa, S. Isogami and N. Takeuchi (2013). Development of High Tempera- ture Superconductor Based SQUID (HTS-SQUID) Magnetometer System for Super-sensitive Observa- tion of Geomagnetic Field Changes, paper pre- sented at EGU General Assembly 2013, Vienna, Austria (Geophys. Res. Abstr., EGU2013-8632-1). Koketsu, K., Y. Yokota, N. Nishimura, Y. Yagi, S. Miyazaki, K. Satake, Y. Fujii, H. Miyake, S. Sakai, Y. Yamanaka and T. Okada (2011). A unified source model for the 2011 Tohoku earthquake, Earth Planet. Sci. Lett., 310, 480-487. Kuriki, M., M. Matsushima, Y. Ogawa and Y. Honkura (2011). Spectral peaks in electric field at resonance frequencies for seismically excited motion of ions in the Earth’s magnetic field, Earth Planets Space, 63, 503-507. Ogawa, T., and H. Utada (2000a). Coseismic piezoelec- tric effects due to a dislocation 1. An analytic far and early-time field solution in a homogeneous whole space, Phys. Earth Planet. In., 121, 273-288. Ogawa, T., and H. Utada (2000b). Electromagnetic sig- nals related to incidence of a teleseismic body wave into a subsurface piezoelectric body, Earth Planets Space, 52, 253-260. Okubo, K., N. Takeuchi, M. Utsugi, K. Yumoto and Y. Sasai (2011). Direct magnetic signals from earth- quake rupturing: Iwate-Miyagi earthquake of M 7.2, Japan, Earth Planet. Sci. Lett., 305, 65-72. Pride, S. (1994). Governing equations for the coupled electromagnetics and acoustics of porous media, Phys. Rev. B, 50, 15678-15695. Stoyer, C.H. (1977). Electromagnetic fields of dipoles in stratified media, IEEE Trans. Antennas Propagat., 25, 547-552. Taira, K., T. Iyemori and D. Han (2009). Geomagnetic variations observed at the arrival of seismic wave of Sumatra Earthquake, Paper presented at the 126th Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS) fall meeting, Kanazawa, Japan, abstract A003–02. Ujihara, N., Y. Honkura and Y. Ogawa (2004). Electric and magnetic field variations arising from the seis- mic dynamo effect for aftershocks of the M 7.1 earth- quake of 26 May 2003 off Miyagi Prefecture, NE Japan, Earth Planets Space, 56, 115-123. Yamazaki, K. (2011a). Piezomagnetic fields arising from the propagation of teleseismic waves in magnetized crust with finite conductivity, Geophys. J. Int., 184, 626-628. Yamazaki, K. (2011b). Enhancement of co-seismic piezomagnetic signals near the edges of magnetiza- tion anomalies in the Earth’s crust, Earth Planets Space, 63, 111-118. Yamazaki, K. (2012). Estimation of temporal variations in the magnetic field arising from the motional in- duction that accompanies seismic waves at a large distance from the epicenter, Geophys. J. Int., 190, 1393-1403. Wessel, P., W.H.F. Smith, R. Scharroo, J.F. Luis and F. Wobbe (2013). Generic Mapping Tools: Improved version released, EOS Trans. AGU, 94, 409-410. * Corresponding author: Ken’ichi Yamazaki, Miyazaki Observatory, Disaster Prevention Research Institute, Kyoto University, Miyazaki, Japan; email: kenichi@rcep.dpri.kyoto-u.ac.jp. © 2016 by the Istituto Nazionale di Geofisica e Vulcanologia. All rights reserved. EARLY EM SIGNALS BY LINE CURRENTS << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles false /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Warning /CompatibilityLevel 1.3 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.1000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams true /MaxSubsetPct 100 /Optimize false /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments false /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile (None) /AlwaysEmbed [ true /AndaleMono /Apple-Chancery /Arial-Black /Arial-BoldItalicMT /Arial-BoldMT /Arial-ItalicMT /ArialMT /CapitalsRegular /Charcoal /Chicago /ComicSansMS /ComicSansMS-Bold /Courier /Courier-Bold /CourierNewPS-BoldItalicMT /CourierNewPS-BoldMT /CourierNewPS-ItalicMT /CourierNewPSMT /GadgetRegular /Geneva /Georgia /Georgia-Bold /Georgia-BoldItalic /Georgia-Italic /Helvetica /Helvetica-Bold /HelveticaInserat-Roman /HoeflerText-Black /HoeflerText-BlackItalic /HoeflerText-Italic /HoeflerText-Ornaments /HoeflerText-Regular /Impact /Monaco /NewYork /Palatino-Bold /Palatino-BoldItalic /Palatino-Italic /Palatino-Roman /SandRegular /Skia-Regular /Symbol /TechnoRegular /TextileRegular /Times-Bold /Times-BoldItalic /Times-Italic /Times-Roman /TimesNewRomanPS-BoldItalicMT /TimesNewRomanPS-BoldMT /TimesNewRomanPS-ItalicMT /TimesNewRomanPSMT /Trebuchet-BoldItalic /TrebuchetMS /TrebuchetMS-Bold /TrebuchetMS-Italic /Verdana /Verdana-Bold /Verdana-BoldItalic /Verdana-Italic /Webdings ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 150 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.10000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.10000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.08250 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /Unknown /CreateJDFFile false /SyntheticBoldness 1.000000 /Description << /ENU (Use these settings to create PDF documents with higher image resolution for high quality pre-press printing. The PDF documents can be opened with Acrobat and Reader 5.0 and later. These settings require font embedding.) /JPN /FRA /DEU /PTB /DAN /NLD /ESP /SUO /NOR /SVE /KOR /CHS /CHT /ITA >> >> setdistillerparams << /HWResolution [2400 2400] /PageSize [595.000 842.000] >> setpagedevice