Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (215) 21, GJI Geomagnetism, rock magnetism and palaeomagnetism doi: 1.193/gji/ggv3 Motional magnetotellurics by long oceanic waves Hisayoshi Shimizu and Hisashi Utada Earthquake Research Institute, University of Tokyo, Tokyo , Japan. Accepted 215 January 16. Received 215 January 15; in original form 214 June 3 1 INTRODUCTION Magnetic field variations in the sea consist of those due to the external magnetic field and the fields within the sea induced by the external field, as well as those induced in the ocean by the motion of the conducting seawater, known as motional induction (e.g. Chave & Weidelt 212). Recently, observations of electromagnetic field variations due to large tsunamis have been reported by Toh et al. (211) and Sugioka et al. (214) for ocean bottom stations andbymanojet al. (211) and Utada et al. (211) for land stations. Notably, electromagnetic signals of large intensity, several nanotesla or larger, were observed by ocean bottom electromagnetometers in association with the Tohoku tsunami of 211 March 11 (Zhang et al. 214a). This observation motivated us to re-examine the characteristics of motional induction due to oceanic long waves and the possible use of the induced fields for magnetotelluric (MT) sounding. MT soundings typically employ electromagnetic induction by magnetic field variations caused by fluctuations in current systems in the ionosphere and magnetosphere. Motionally induced magnetic fields in the ocean are considered to be noise in electromagnetic soundings (e.g. Chave & Weidelt 212). For example, electromagnetic field signatures, such as those at the period of ocean tides, are often removed from electromagnetic field time-series before the electromagnetic responses are estimated (e.g. Baba et al. 21). However, the signature of a motionally induced magnetic field can SUMMARY The observation of electromagnetic signals by ocean bottom electromagnetometers in association with the Tohoku tsunami of 211 March 11 has raised the opportunity to re-examine the physics of motional induction due to oceanic long waves in the framework of 1-D magnetotellurics (MT). Although a propagating tsunami has a complex structure, the induced electromagnetic field can be simply approximated as a plane wave (though a simple thin-sheet approximation is not valid at higher frequencies). We found that the MT impedance due to a surface gravity wave (or the motional impedance ) is influenced largely by the dispersion of the wave if the period is sufficiently short or the electrical conductivity of the seabed is low. The tipper due to the motional induction (or the motional tipper ) and motional impedance are essentially identical if the underneath structure is 1-D. It would be possible to estimate the motional impedance and tipper from the observed ocean bottom electromagnetic field at the time of passing of a tsunami. The wave amplitude must be much greater than several tens of centimetres for the motional impedance and tipper estimation to be free from the effects of external sources. However, the obtained motional impedance and tipper will mostly represent the property of the wave and use of them may not be suitable to discuss the subseafloor conductivity structure. Key words: Electrical properties; Magnetotellurics; Geomagnetic induction. be used as a source field for electromagnetic sounding if the characteristics of the source field are well known (e.g. Chave & Weidelt 212). For example, Kuvshinov et al. (26) performed a numerical calculation of an electromagnetic field for a realistic distribution of surface electrical conductivity and tidal flow developed by Egbert & Erofeeva (22), and used the relation between the calculated and observed electric field to infer the electrical conductivity of the upper-most mantle. Since the physics of surface gravity waves are well known, the electromagnetic fields they induce may be applied for MT soundings. It is difficult to determine the MT impedance at the seafloor due to external field variations in the period range below several hundred seconds in the deep ocean because of the attenuation of the magnetic field by the relatively large electrical conductivity of the ocean (Section 4, also see Chave et al. 1991; Shimizu et al. 211). In this paper, we examine whether a magnetic field variation generated by the motional induction of a tsunami can be used as a source field for the MT method. The period range of a tsunami is from 1 to 1 s (e.g. Rabinovich et al. 213) and the electromagnetic signature at the ocean bottom due to a tsunami flow in the deep ocean, which is not affected by attenuation in the ocean, can be larger than that from the external field, especially at a period shorter than several hundred seconds. Several theoretical and analytical studies have been made on the electromagnetic field variation by motional induction in the ocean. Sanford (1971) presented a general theory of motionally Downloaded from by guest on 9 October C The Authors 215. Published by Oxford University Press on behalf of The Royal Astronomical Society.

2 induced electromagnetic fields by a quasi-steady ocean flow, assuming a three-layer subseafloor conductivity structure (conductive sediment, insulating crust and somewhat conducting mantle). The theory by Sanford (1971) is not applicable to the tides because the induction cannot be treated as a weak perturbation. Tyler (25) devised a conventional formula to estimate the vertical component of the motionally induced magnetic field above the sea surface by a tsunami flow based on Sanford s theory. Chave & Luther (199) developed a thorough theory of motionally induced fields for a general 1-D subseafloor structure. These studies focused on ocean flows of subinertial frequency range in which the self-induction in the ocean is weak. However, self-induction is important at shorter periods, such as those for surface gravity waves. Larsen (1971) presented an analytical theory of motionally induced electromagnetic fields by a propagating surface gravity wave including self-induction in the ocean, in which a three-layer earth model similar to that of Sanford (1971) was assumed, and derived formulae to calculate the electromagnetic field. The observation of magnetic fields induced by tsunamis has motivated several numerical studies of motionally induced electromagnetic fields. Zhang et al. (214a) developed 3-D motional induction modelling including real bathymetry, a realistic subseafloor conductivity structure and self-induction in the ocean. They were able to reproduce the observed electromagnetic field variations due to the tsunami caused by the 211 Tohoku earthquake using the tsunami flow model obtained by Maeda et al. (211). The possibility of detecting a tsunami-induced electric field using thousand-kilometre scale submarine cables was discussed by Manoj et al. (21) based on induction modelling using a tsunami flow model of the great Indian Ocean tsunami of 24, but the results were rather negative. In this paper, we generalize the study by Larsen (1971) fora case with a general multi-layered 1-D earth model to calculate the motionally induced electromagnetic field by plane surface gravity waves, which is a first-order approximation at a large distance from the wave source. The validity of the approximation is discussed in Section 2.3. Since we consider the period range in which the tsunami is excited, the wave is not influenced by the Coriolis force. This implies that the vertical vorticity is not a factor in the problem and only the poloidal magnetic mode appears in the induced field. The induced electromagnetic field in the Earth is used to calculate an MT response, which we refer to as the motionally induced MT response or the motional impedance. A major difference from the usual MT method is that the source field has a finite length. This may enable us to reveal a shallow structure beneath the seafloor using the impedance. The effect on the MT impedance of the finite wavelength source field, of which the wavelength is related with the frequency through a dispersion relation, is examined to understand the motional impedance in general. The observationally estimated motional impedance and its estimation error are examined to investigate the feasibility of using the motional impedance for conductivity soundings. 2 ELECTROMAGNETIC FIELD INDUCED BY A PLANE SURFACE G R AV I T Y WAV E 2.1 Model and electromagnetic field solutions We consider a horizontally layered electrical conductivity model as shown in Fig. 1. Layers and 1 correspond to the air and seawater, layer layer 1 layer 2... layer j... layer N+1 σ = σ =σ j Motional impedance 391 Air Sea Figure 1. Horizontally layered structure model. z= z=h z=z z=z z=z 2 z=z j respectively. The source of the magnetic field variation is assumed to be motional induction from the flow of seawater in the geomagnetic main field. We employ a local Cartesian coordinate system (x, y, z); the x- andy-axes form a horizontal plane and the z-axis is taken to be the vertically downward direction. z = and h correspond to the sea surface and the seafloor, respectively. We also assume that the flow in layer 1 (the ocean) can be represented by a plane surface gravity wave propagating in the y direction. The assumption is valid when the site is sufficiently far from the origin of the wave (the epicentre in the case of a tsunami; see Section 2.3). The flow velocity of the surface gravity wave, ṽ(y, z, t), is written as ṽ(y, z, t)=re[v(y, z, t)], v(y, z, t)= V y (y, z, t)ŷ + V z (y, z, t)ẑ, (1) V y (y, z, t) = Aω cosh[k(h z)] exp( iky + iωt), (2) sinh(kh) Aω V z (y, z, t) = i sinh[k(h z)] exp( iky + iωt), (3) sinh(kh) where ŷ and ẑ are the unit vectors in the y and z directions, respectively, t is time, k is the wavenumber in the y direction, ω is the angular frequency and A is the amplitude of the sea surface displacement ζ (y, t): ζ (y, t) = Re[ζ (y, t)], ζ(y, t) = A exp( iky + iωt) (4) (see, e.g. Kundu 199). The dispersion relation and phase velocity (c p )ofthiswavearegivenby g ω 2 = gk tanh(kh), c p = tanh(kh), (5) k where g is the acceleration of gravity. In the long wave limit, kh 1, tanh (kh) in eq. (5) can be replaced by kh. It is assumed in deriving the surface gravity wave that the wave amplitude A is much smaller than the wavelength (Ak 1) and ocean depth (A h). The latter condition allows us to treat the sea surface as being flat in calculating the electromagnetic fields induced by the wave. Suppose that the magnetic field B consists of an ambient geomagnetic (primary) field B (x, t) and an induced (secondary) field b(x, t): B(x, t) = B (x, t) + b(x, t), b(x, t) = Re[b(x, t)], (6) where x is the position vector. For the time and spatial scale, we assume that B is constant in time and space so that we have the j-1 N z x y Downloaded from by guest on 9 October 218

3 392 H. Shimizu and H. Utada following set of basic equations for the secondary field within layer j ( b j ): b j μ σ j = μ δ 1 j j s + 2 b j, (7) t b j =, (8) where μ is the magnetic permeability of vacuum, σ j is the electrical conductivity of layer j, andδ 1j is the Kronecker delta. j s is the motionally induced electric current due to the flow. Since b B (see, e.g. Larsen 1971; Chave1983), j s can be approximated as j s (x, t) = σ 1 v(x, t) B, (9) where v is the flow velocity in layer 1. The time derivative of the secondary field (LHS of eq. 7) must be included in the equation because self-induction in the ocean and mutual induction with the conducting Earth cannot be neglected for the period range of a tsunami in the deep ocean (Larsen 1971; see also Chave & Luther 199). We suppose that the ambient magnetic field can be written as B = B H cos θ ˆx + B H sin θŷ + B z ẑ, (1) where B H and B z are the horizontal and vertical ambient magnetic field components, respectively, and θ is the magnetic azimuth relative to the x axis (perpendicular to the wave propagation direction; see Fig. 2). Because the forcing term depends on time as exp (iωt), the solution to eq. (7) has the form b exp (iωt). Let b = [b x (x, ω), b y (x, ω), b z (x, ω)]exp (iωt); then the general solution for (7) for layer j can be written as b m = P m j exp( iky + γ j z) + Q m j exp( iky γ j z) + δ 1 j p m (y, z), (11) where m is x, y or z, γ 2 j = k 2 + iωμ σ j, (12) p x (y, z) =, (13) p y (y, z) = (α H cosh[k(h z)] + α z sinh[k(h z)]) exp( iky), (14) z x B z θ B H wave propagation Figure 2. Relationship between the wave propagation direction and the horizontal component of the ambient field. The angle between the positive x direction and the horizontal magnetic field is θ. y p z (y, z) = i (α z cosh[k(h z)] + α H sinh[k(h z)]) exp( iky), (15) α H = ku B H ω sin θ, α z = ku B z iω, (16) U(kh,ω,A) = Aω (17) sinh(kh) and Pj m and Q m j are integration constants. Here we suppose that Re[γ j ] without loss of generality. Solutions are also valid for σ j =. The continuity of the magnetic flux density (8) requires P z j = ik γ j P y j, Qz j = ik γ j Q y j (18) for j =, 1, 2,..., N + 1. The horizontal electric field is calculated using Faraday s law as E x = ω k b z, (19) and the horizontal components are found by using Ohm s law and Ampère s law. At the boundaries, the horizontal components of the electric and magnetic field are continuous. Also, the vertical components of the magnetic flux density and the electric current are continuous. The induced electromagnetic field tends to zero as z approaches ±. Note that the conditions for b z and E x are redundant through eq. (19). By applying the boundary conditions, the coefficients for b y in the sea are obtained as P y 1 = (Ɣ 11 + δ N Ɣ 12 )(λ Q + ξ 1 ) (Ɣ 21 + δ N Ɣ 22 )λ P (Ɣ 21 + δ N Ɣ 22 ) δ 1 (Ɣ 11 + δ N Ɣ 12 ) (2) Q y 1 = δ (Ɣ 11 + δ N Ɣ 12 )(λ Q + ξ 1 ) (Ɣ 21 + δ N Ɣ 22 )λ P 1 + ξ 1, (21) (Ɣ 21 + δ N Ɣ 22 ) δ 1 (Ɣ 11 + δ N Ɣ 12 ) where [ ] N 1 Ɣ11 Ɣ 12 Ɣ = = Ɣ 21 Ɣ 22 j=1 [ (M z j j ) 1 ] z L j j, (22) [ ] exp(γ M z j z) exp( γ j z) j =, (23) γ j+1 exp(γ j z) γ j+1 exp( γ j z) [ ] exp(γ L z j+1 z) exp( γ j+1 z) j =, (24) γ j exp(γ j+1 z) γ j exp( γ j+1 z) [ λp λ Q ] = ( ) M h 1 α H 1 γ 1γ 2 α z k, (25) δ 1 = γ 1 γ γ 1 + γ, δ N = γ N+1 + γ N γ N+1 γ N exp(2γ N z N ), (26) ξ 1 = γ [( 1 α H + γ ) α z cosh[kh] (γ 1 + γ ) k ( + α z + γ ) ] α H sinh[kh], (27) k Downloaded from by guest on 9 October 218

4 Motional impedance 393 depth [km] (a) electrical conductivity [S/m] depth [km] (b) electrical conductivity [S/m] Figure 3. Supposed electrical conductivity model beneath the seafloor. (b) is a magnification of the electrical conductivity profile near the seafloor. Using the obtained coefficients calculated by eqs (18), (2) and (21), we can evaluate the magnetic and electric fields at an arbitrary position between the sea surface and seafloor from the information of the ocean current (surface gravity wave) and the electrical conductivity structure. by bz amplitude [nt] Ex amplitude [micro V/m] Evaluation of the electromagnetic field at the seafloor due to motional induction We suppose a four-layer electrical conductivity model beneath the seafloor to examine the typical features of a motionally induced electromagnetic field at the seafloor (Fig. 3). A sediment layer that contains a large amount of seawater is supposed as the first layer. The thickness of the layer is assumed to be 4 m and its electrical conductivity is taken to be 1.1 S m 1 (e.g. Flosadóttir et al. 1997; Bourlange et al. 23; Shankar & Riedel 211), which is one-third of that of seawater. The second layer consists of the oceanic crust with a conductivity of 1 3 Sm 1. The layer is assumed to extend down to 6 km deep from the seafloor (e.g. Cox et al. 1986). The third layer is the lithospheric mantle. Its conductivity is expected to be very low (e.g. Larsen 1971; Coxet al. 1986); we suppose here the conductivity to be 1 5 Sm 1 (Cox et al. 1986). The thickness of this low-conductivity layer increases with increasing age of the seafloor (Filloux 198; Babaet al. 21, 213), but we suppose here the bottom depth of the layer to be 7 km. The layer beneath the resistive layer is the conductive mantle (asthenosphere) and its electrical conductivity is taken to be Sm 1 (see, e.g. Baba et al. 21, 213). The depth of the sea is assumed to be 4 m. Fig. 4 shows the amplitude and phase of b y, b z and E x at the seafloor for the conductivity model shown in Fig. 3. The phase is determined with respect to that of the surface gravity wave. The vertical displacement amplitude of the surface gravity wave is assumed by bz phase [degree] Ex phase [degree] Downloaded from by guest on 9 October 218 Figure 4. Relative amplitude and phase of the induced magnetic field (b y and b z, top panels) and electric field (E x, bottom panels) due to a surface gravity wave of 1 cm amplitude for the conductivity models shown in Fig. 3. Theb y and b z components are shown by blue and black lines in the top panels, respectively. The ambient field was B H = 3 nt (θ = 9 )andb z = nt (dashed lines) or B H = ntandb z = 3 nt (solid lines). For comparison, those with the case in which subseafloor is insulator are shown by red lines.

5 394 H. Shimizu and H. Utada to be 1 cm. The applied ambient magnetic field is B H = 3 nt (θ = 9 ; the horizontal component of the ambient field is in the y-direction) and B z = nt for the cases shown by dashed lines, and B H = ntandb z = 3 nt for those shown by solid lines. The vertical field of 3 nt is a typical value around the midlatitude and the same value is assumed as the horizontal ambient field strength. If the resolution of the magnetic and electric field measurements are.1 nt and.1 µv m 1 or better, we can detect the electromagnetic signature of the surface gravity wave with a surface vertical displacement amplitude of several centimetres or larger for a period larger than about 1 s. Fig. 4 also shows the motionally induced electromagnetic field when the subseafloor is insulator. The difference between the electromagnetic fields is more evident at a long period, and the signature of induction in the supposed conductivity structure is seen at a period longer than 5 s. The induced magnetic and electric fields are stronger when the ambient field is vertical than when it is horizontal if the intensity of the two is the same. This is a consequence of the surface gravity wave, which generally has a larger horizontal velocity than vertical velocity. However, the part induced by the horizontal magnetic field will be comparable to or larger than that induced by the vertical magnetic field if motional induction occurs at low latitude where the ambient field is dominated by the horizontal component. This implies that the induced electromagnetic field can depend on the angle between the direction of the surface gravity wave propagation and magnetic azimuth (i.e. dependence on θ), particularly at low latitude. To examine the propagation direction and latitude dependence of the motionally induced field, we suppose that the ambient field is approximated by that of an axial dipole placed at the centre of the Earth with an intensity corresponding to 3 nt at the equator on the Earth s surface. Thus, the ambient field components at the surface are written in terms of the latitude λ as B z = 6 sin λ nt, B H = 3 cos λ nt. (28) The amplitude and phase of the induced b y with respect to θ at various latitudes at a period of 1 s are shown in Fig. 5, because the influence of the horizontal ambient field component on b y is largest around this period (Fig. 4). Only b y is shown here; the ratios b amplitude [nt] y high low latitude b phase [degree] b z /b y and E x /b y do not depend on θ, so that the dependency of b z and E x on θ is the same as that of b y. The amplitude of b y at 1 latitude and θ = 9 is twice as large as that at θ =.Onthe other hand, the difference in the amplitude is about 25 per cent at 2 latitude, and decreases as the latitude increases. The phase has a more marked dependence on θ; the arrival time of the induced magnetic field signal with respect to the wave crest significantly depends on the wave propagation direction. At 1 latitude, the phase is negative for smaller θ and it increases to become positive, that is, the phase relation between the wave crest and induced field has a different sign due to the wave propagation direction. The phase delay at 2 latitude changes to about 3 as θ changes from to9.theθ-dependence of the amplitude and phase on the induced magnetic field implies that their dependence on the wave propagation direction is significant at latitudes lower than 2,but the effect can be negligible at a higher latitude. On the other hand, the amplitude and phase of b y at a period of 1 s (Fig. 6) does not show much dependence on θ, because the field induced by B H is an order of magnitude smaller than that induced by B z (see Fig. 4). Therefore, this result suggests that consideration must be given to the relation between the magnetic azimuth and the direction of the wave propagation for an induced electromagnetic field at around a period of 1 s. 2.3 Validation of the plane-wave approximation for modelling tsunami-induced electromagnetic fields at the seafloor Observations of the electromagnetic fields at the seafloor in the northwestern Pacific were being undertaken by the Normal Mantle Project when the tsunami due to the 211 Tohoku earthquake occurred (Baba et al. 213). Zhang et al. (214a) examined the observed electromagnetic field induced by the tsunami and the field modelled by a 3-D induction calculation. In the modelling, the tsunami flow model by Maeda et al. (211) was used to calculate the motional induced electromotive force in the ocean. A realistic bathymetry based on ETOPO1 (Amante & Eakins 29) and a distribution of the geomagnetic main field calculated from IGRF11 (Finlay et al. 21) were used for the modelling. The supposed electrical conductivity beneath the seafloor was based on y low latitude high Downloaded from by guest on 9 October theta [degree] theta [degree] Figure 5. Relative amplitude and phase of induced b y with respect to the angle between the horizontal field and wave propagation direction θ. The amplitude of the surface gravity wave was assumed to be 1 cm, and the period was 1 s. The ambient field was assumed to be an axial dipole given by eq. (28). Each line corresponds to the amplitude and phase at a different latitude: the dashed lines show those at latitude and the solid lines show those at 1 9 latitudes. The difference in latitude for adjacent lines is 1. The assumed conductivity model is shown by the black line in Fig. 3. The ocean depth was taken to be 4 m.

6 Motional impedance b amplitude [nt] y.1.5 b phase [degree] y theta [degree] Figure 6. As Fig. 5 but for a wave of period of 1 s. Baba et al. (213). The observed and modelled electromagnetic fields agreed well with each other except the relatively short period fluctuations following the main phase. Zhang et al. (214a) mentioned in reference to Saito & Furumura (29) that the highfrequency dispersion, which was not included in the tsunami flow model, was the cause of the difference between the observed and calculated electromagnetic fields. Here, we examine how well the motionally induced field obtained by a plane-wave approximation can represent that by a more realistic flow, of which wave front is not a plane and direction of wave propagation is not uniform in space and time. For this purpose, we compare the electromagnetic field, at locations shown in Fig. 7(a), induced by the flow by Maeda et al. (211) modelled by Zhang et al. (214a) and that calculated using a plane-wave approximation in this paper. To calculate the electromagnetic field based on the plane-wave approximation, we first calculate the wave height of the tsunami from the flow model by Maeda et al. (211) using the continuity equation of the fluid. The amplitude and phase of the wave height are evaluated in the frequency domain from the 18-min section of (a) A B C D E F (b) normalized misfit theta [degree] the wave height, and they are used to evaluate the induced electromagnetic field in the time domain. Because the tsunami propagated almost eastward at the supposed locations, we assume that y is in the east direction. The geomagnetic field at a site calculated from IGRF11 is employed to calculate the induced electromagnetic field at the site. The electrical conductivity model assumed in the previous section is employed for the calculation (also see, Baba et al. 213). Fig. 8(a) shows the estimated wave height ( ζ ) at a Normal Mantle Project site, NM4 (38 deg min N, 154 deg 11.4 min E, 5947 m depth). The geomagnetic main field at NM4 in March 211 is calculated to be X = nt, Y = nt and Z = nt from IGRF11. The horizontal intensity and declination of the field at the site are H = nt and D = 4.322, respectively. For the estimation of the induced magnetic field by the plane wave, we assume that B H = 2814 nt, B z = 3485 nt and θ = 4. The depth of ocean is assumed to be h = 6 m. Figs 8(b) (d) show the modelled electromagnetic field in this study (blue) and that by Zhang et al. (214a) (red). The modelled fields are similar to each other. Possible causes of the difference between Downloaded from by guest on 9 October km distance [km] Figure 7. (a) Locations at which the electromagnetic fields induced by the tsunami flow due to the 211 Tohoku earthquake are calculated. The solid circle shows the site NM4 and the open circles are supposed sites. The star represents the epicentre of the earthquake. (b) Normalized misfit defined by eq. (29) between the calculated magnetic field component by this study and Zhang et al. (214a). The open and closed symbols represent the eastward and downward (b z ) components, respectively.

7 396 H. Shimizu and H. Utada (a) 2 (b) 4 wave height [m] 1 b_east [nt] 4 (c) b [nt] z : 7: 8: 6: 7: 8: time [UT] (d) E_north [microv/m] : 7: 8: 6: 7: 8: time [UT] Figure 8. Wave height and electromagnetic field at NM4 for three hours from the origin time of the 211 Tohoku earthquake (5:46, 211 March 11 UT) at the ocean bottom site NM4. (a) Wave height calculated from the tsunami flow model by Maeda et al. (211). (b) Eastward component of the calculated magnetic field. (c) Vertically downward component of the calculated magnetic field. (d) Northward component of the calculated electric field. In (b), (c) and (d), the modelled electromagnetic fields by Zhang et al. (214a) are shown by red lines, and those modelled using the formulation shown in Section 2.1 are shown by blue lines. them include the effect of non-uniform bathymetry, a non-uniform geomagnetic main field, and the actual form of the wave front. Nevertheless, we can say that the electromagnetic field calculated using the assumptions employed in this paper can reproduce the observed electromagnetic field reasonably well. For a more quantitative discussion, the normalized misfit between the induced magnetic field calculated by the method presented in this paper [bm p (t)] and that by Zhang et al. (214a) [bz m (t)] defined as 18 [ b p m (t i) b z m (t i) ] 1/2 2 i=1 nms m = 18 [ b z m (t i) ] 2 i=1, (29) are evaluated at the stations shown in Fig. 7(a), where t i is the discretized time at which the induced magnetic fields are calculated. The supposed ambient magnetic field, ocean depth, and θ at the assumed sites to calculate the motionally induced magnetic field are shown in Table 1. The normalized misfit (Fig. 7b) for both components shows a decreasing tendency with the distance from the epicentre up to 6 km. This seems to show that the induced magnetic field calculated using the plane-wave approximation of the surface gravity wave is more valid at a large distance from the epicentre, if the distance is less than 6 km. However, the normalized misfit stays at a similar level for the stations further than 6 km, although the forms of the induced magnetic field calculated by the two methods at each station are similar to each other. This Table 1. Longitude of the supposed stations and NM4, ambient magnetic field, ocean depth and θ (=declination) supposed for the calculation of the induced magnetic field. The latitude of the stations was 38.21N ( ). Station Longitude B H B z Depth θ ( ) (nt) (nt) (m) ( ) A E B E C E D E E (NM4) E F E tendency of the validity of the plane-wave approximation is also seen when we evaluate it using the mutual correlation between the two methods. 2.4 Vertical uniformity of the induced electromagnetic field in the ocean Electromagnetic induction within vertically thin but laterally heterogeneous layers has sometimes been modelled using a thin-sheet approximation (e.g. Price 1949; Vasseur & Weidelt 1977; Fainberg & Zinger 198; Ranganayaki & Madden 198; Fainberg et al. 1993; Schmucker 1995; Sun & Egbert 212). The electromagnetic induction in the ocean may be reproduced well by a thin-sheet approximation when the lateral scale of the electromagnetic field is much larger than the ocean depth. In general, a thin-sheet approximation Downloaded from by guest on 9 October 218

8 Motional impedance 397 (a) b /b amplitude thin thin z z y y (c) b /b amplitude b /b phase [degree] (b) thin y y (d) b /b phase [degree] thin z z Figure 9. Amplitude ratio and phase difference of numerically evaluated exact magnetic field components (b y and b z ) and thin sheet approximated field components (b thin y and bz thin ). The black and blue lines indicate the cases when the ambient field was in the y and z directions, respectively. The dispersion relation of eq. (5) is applied for the cases shown by solid lines, and the long wave approximation was used for those shown by dotted lines. is valid when the exponential terms in eq. (11) are written as linear functions of z, thatis,when γ 1 h 1. We can evaluate the magnetic field at the seafloor using this approximation as, using the coefficients obtained in Section 2.1, b thin m = Pm 1 (1 + γ 1h) + Q m 1 (1 γ 1h) + p m (y, h). (3) Fig. 9 shows the amplitude and phase of the ratio b m /bm thin at tends to b m as the period becomes large, except the seafloor. b thin m b y induced from the horizontal magnetic field. The approximation seems to hold at a period longer than 1 s for b y induced from B z, and longer than 4 s for b z. On the other hand, the amplitude ratio is less than.5 even at a long period for b y from B H and the thin-sheet approximation does not hold for the entire period range in this case. A significant phase variation of the component with depth caused the thin-sheet approximation to be invalid (Fig. 1). However, this invalidity causes problem only at low latitude where B z is much smaller than B H. The induction from B H and B z becomes comparable if B z /B H r b (ω), where r b (ω) is the ratio of the induced field from horizontal and vertical ambient magnetic fields of the same intensity (the ratio of the values shown by the dashed line and solid line in Fig. 4). At a period of 5 s or longer, r b.1 for 4 m in the deep ocean. If we assume the ambient field distribution as in eq. (28), the induction from B H becomes comparable to that from B z only at a latitude lower than 3. The induced magnetic field time-series calculated by the method in this paper and by the thin sheet approximation are compared in Figs 11(a) and (b). The ambient magnetic field at NM4 and the wave height shown in Fig. 8(a) were assumed in calculating the magnetic field. The thin-sheet approximation introduces more highfrequency components in the calculated time-series with increasing depths, as is expected from the amplitude ratios shown in Fig. 9. Fig. 11(c) shows the normalized misfit between the two time-series. To avoid the obvious discrepancy in the time-series at high frequencies before the main phase of the magnetic field variation, a 6-min window from the arrival of the tsunami at the 63rd minute is used to calculate the normalized misfit. In addition to the tendency of the misfit to increase with depth, we can see that the gradient of the misfit with depth is larger for the b y component than for the b z component. 3 MOTIONAL IMPEDANCE BY SURFACE GRAVITY WAVES 3.1 Motional impedance The electromagnetic field at the seafloor induced by the surface gravity wave contains information on the electrical conductivity beneath the seafloor. In this section, we present a formulation to calculate an MT impedance and tipper due to the electromagnetic field (or motional impedance and motional tipper ). Downloaded from by guest on 9 October 218

9 398 H. Shimizu and H. Utada 1 18 amplitude [nt] phase [degree] depth [m] depth [m] Figure 1. Depth profile of the amplitude and phase of the induced magnetic field by a surface gravity wave of 1 cm amplitude for the conductivity models showninfig.3. The period of the wave is 3174 s. The b y and b z components are shown by blue and black lines, respectively. The ambient field was B H = 3 nt (θ = 9 )andb z = nt (dashed lines) and B H = ntandb z = 3 nt (solid lines). (a) b y [nt] 2 m 4 m 6 m time [min] 4 nt (b) b [nt] z 2 m 4 m 6 m time [min] 4 nt (c) normalized misfit depth [m] Figure 11. Calculated time-series of the induced magnetic field and the normalized misfit between the time-series. (a) b y component and (b) b z component, where the exact solution and thin sheet approximated solutions are shown by black and blue lines, respectively. The ambient magnetic field at NM4 and the wave height shown in Fig. 8(a) were assumed in calculating the magnetic field. The ocean depth was taken to be 2, 4 and 6 m to show its effect on the thin sheet approximated solution. (c) Normalized misfit for b y (open circle) and b z (solid circle). For wave propagation in the y-direction, we consider the component of the motional impedance, defined as Z (ω) = E x(ω) b y (ω). (31) This response has the unit of velocity because the magnetic flux density is used in the definition. The information on the amplitude and phase of the source field cancel in taking the ratio because it is common between b y and E x. The wavenumber of the source magnetic field induced by the surface gravity wave corresponds to a frequency based on the dispersion relation. A response in the frequency domain contains information on the spatial scale of the source field for the case considered, and we do not have to know the spatial structure of the source field as independent information. The motional tipper estimated using the motionally induced field is directly related to the motional impedance in the 1-D case, that is, it can be written as The tipper for a uniform source field is not sensitive to the 1-D conductivity structure. However, the motional tipper can sense a 1-D structure because the source field has spatial dependency. To calculate the motional impedance at the seafloor, we use the impedance at each layer and a boundary condition that requires the impedance to be continuous across each boundary (impedance matching, see, e.g. Schmucker 197; Simpson & Bahr 25; Constable 29). After some calculations, we obtain G Z j 1 j tanh(γ j d j ) + Z j (z j 1 ; ω) = G (z j; ω) j G j + Z(z j j ; ω)tanh(γ j d j ), (33) where Z j (z; ω) = E x j (z; ω) by(z; j ω), Z N iω (z; ω) =, γ N+1 and d j is the thickness of the jth layer, that is, G j = iω γ j, (34) d j = z j z j 1. (35) Downloaded from by guest on 9 October 218 M zy (ω) = b z(ω) b y (ω) = k E x (ω) ω b y (ω) = 1 Z (ω). (32) c p The response at the seafloor can be obtained regressively using the impedance of deeper layer boundaries by applying (33).

10 3.2 Effect of wavelength and dispersion relation on the motional impedance The motional impedance due to the electromagnetic field induced by a surface gravity wave depends not only on the underground electrical conductivity structure but also on the wavelength (wavenumber) of the source field. The wavenumber and frequency of the surface gravity wave are related by the dispersion relation (5). Therefore, the effect of the finite-length-scale source field appears in the motional impedance systematically. The asymptotic behaviours of the motional impedance in the short and long period limits facilitate understanding the motional impedance. Here we consider G j, which is equivalent to the motional impedance for a uniform half-space when the conductivity is σ j. The size of the two terms in γ j determines the behaviour of G j. The absolute value of the ratio of the two terms, ωμ σ j /k 2,isa decreasing function of ω by the dispersion relation. Hence, at the short period limit (high-frequency limit, ωμ σ j /k 2 1), we obtain lim G iω j = lim ω ω k 1 + (iωμ σ/k 2 ) ic p, (36) where indicates asymptotic equality. This implies that the effect of dispersion on the motional impedance is significant in the limit where the imaginary part approaches the phase velocity of the wave. No information on the subseafloor conductivity is contained in the motional impedance in this limit to dominant order. In the long period limit (low frequency, 1 ωμ σ j /k 2 ), on the other hand, the impedance approaches the usual MT response for which the source field does not have spatial dependence (e.g. Simpson & Bahr 25; Constable 29); ( ) iω 1/2 lim G j. (37) ω μ σ j The behaviours of the motional impedance in the short and long period limits imply that there is a certain period below which the impedance cannot be used for conductivity sounding. Fig. 12(a) shows the curves at which μ ωσ/k 2 = 1 for various ocean depths. If the period and electrical conductivity are below the curve of the corresponding ocean depth, the short period limit (36) applies. If the conductivity is sufficiently low that ωμ σ j /k 2 1 is satisfied, Z is as shown in Fig. 12(b). The corresponding phase of the period [s] (a) Z ~ icp Motional impedance 399 impedance is 9. The curves in the figure purely represent the dispersion relation, and Z deviates from the curves if it contains information on the subseafloor conductivity, which is proof that the signal is due to motionally induced fields. 3.3 Motional impedance for several simple conductivity structures The motional impedance with respect to several assumed conductivity models is examined to understand its characteristics and potential for use in sounding of the subseabed conductivity. A discussion on the feasibility of the use of motional impedance for conductivity soundings, based on observed impedance, is presented in the next section (Section 4). Fig. 13 shows the motional impedance Z when the subseabed structure is a uniform half-space and the electrical conductivity is 1 4,1 3,1 2 and 1 1 Sm 1. In addition to the motional impedance, the usual MT response with a uniform source excitation (dotted lines) is presented in the figure for comparison. The ocean depth was taken to be 4 m. The effect of dispersion is significant for shorter periods, as was expected from the discussion in the previous subsection. Although Z for a uniform source is sensitive to the half-space conductivity for the entire period range, the imaginary part of the motional impedance approaches the phase velocity of the surface gravity wave for a short period. Z contains information on the half-space conductivity if the period is larger than 4 or 4 s for a half-space conductivity of 1 1 or 1 2 Sm 1, respectively. For a conductivity of less than 1 3 Sm 1, Z cannot sense the subseafloor conductivity in the tsunami period range. On the other hand, the phase deviates from 9 at a shorter period than that for Z for the same half-space conductivity. In Fig. 13, we can see that the amplitude of usual MT response is larger than that of the motional MT response at a short period. This holds when a more realistic conductivity is assumed as in the following. This implies that the size of the electric field at the seafloor due external field excitation tends to be larger than that due to the oceanic wave, if amplitudes of magnetic field variations by the two excitations are the same. This issue on the expected size Z [m/s] (b) Downloaded from by guest on 9 October electrical conductivity [S/m] 1 1 Figure 12. (a) Curves of ωμ σ j /k 2 = 1. (b) Amplitude of Z when the subseafloor is an insulator. The red, black, and blue lines correspond to ocean depths of 2, 4 and 6 m, respectively.

11 4 H. Shimizu and H. Utada Z [m/s] Z phase [degree] 6 3 real Z [m/s] imag Z [m/s] Figure 13. Impedance due to motional induction (solid lines) and due to a uniform source induction (dotted lines). The assumed conductivity was 1 4 Sm 1 (green), 1 3 Sm 1 (blue), 1 2 Sm 1 (black) and 1 1 Sm 1 (red). The ocean depth was taken to be 4 m. of the electric field variation at the seafloor is discussed more in Section 4. It is difficult to estimate the usual MT response for periods shorter than several hundred seconds at the bottom of a deep sea because of the attenuation of the electromagnetic signal through the conducting seawater. Determining a very shallow structure, such as the thickness and conductivity of the sediment layer, is difficult using the usual MT response. Motional impedance may present a better method to determine shallow structures because the motionally induced field is not affected by the attenuation. The effect of the sediment conductivity on the motional impedance was examined by varying the conductivity of a 4-mthick subseafloor layer, which covers a half-space of conductivity 1 2 Sm 1, from.11 to 3.3 S m 1 (Fig. 14). The ocean depth was taken to be 4 m. The amplitude of the motional impedance is not sensitive to the sediment layer, but its phase has the potential to sense the conductivity of the sediment layer in the period range from 1 to 1 s. The effect of the half-space conductivity beneath a 4-m-thick sediment layer of 1.1 S m 1 is shown in Fig. 15. The sounding curves are similar to those for the half-space case (Fig. 13). Also, that of the top depths of the conductive mantle (Fig. 3) isshown in Fig. 16. The phase has a sensitivity approximately the same as the phase of the usual MT response. However, the amplitude of the motional impedance is much less sensitive to the depth compared to that of the usual MT response. Employing the motional impedance does not have an advantage over the use of the usual MT response for the determination of the deep structure. 4 USE OF MOTIONAL IMPEDANCE FOR ELECTRICAL CONDUCTIVITY SOUNDINGS In this section, we examine whether the motional impedance can be applied for practical electrical conductivity sounding for a shallow structure. To estimate the motional impedance, the amplitude of the motionally induced electromagnetic field must be larger than the detection level of the ocean bottom instruments. As was mentioned in Section 2.2, we expect that the signature of the surface gravity wave at a period longer than 1 s can be detected using an ocean bottom electromagnetometer if the amplitude of the wave is larger than several centimetres (Fig. 4). The external field variations attenuate in the sea due to induction in a relatively highly conducting ocean. Fig. 17 shows the seafloor to sea surface horizontal magnetic and electric field ratios for an ocean of 2, 4 and 6 m depth calculated based on Chave and Filloux (1984) due to a horizontally uniform source external field. The conductivity of the ocean was assumed to be 3.3 S m 1, and that beneath the seafloor was supposed to be as shown in Fig. 3. The external magnetic field variation amplitude must be larger than several nano-tesla to be detected at the seafloor of a deep ocean of 4 m depth or deeper if the period of variation is less than 1 s. Note that the amplitude is about the same as that expected for the field variation (see Filloux 1987; also, Constable 29). Fig. 18 shows the ranges of the magnetic and electric field variations at the seafloor due to the external field and motional induction by the surface gravity wave. The magnetic field variation induced by Downloaded from by guest on 9 October 218

12 Motional impedance Z [m/s] phase [degree] Figure 14. Impedance due to motional induction (solid lines) and uniform source (dotted lines) for various values of sediment conductivity. The assumed conductivity structure was a 4-m-thick sediment layer of.11 S m 1 (green),.33 S m 1 (blue), 1.1 S m 1 (black) and 3.3 S m 1 (red) and a half-space beneath the layer with a conductivity of 1 2 Sm 1. Z [m/s] phase [degree] Figure 15. Impedance due to motional induction (solid lines) and uniform source (dotted lines) for two-layer conductivity models with a sediment layer and half-space beneath it. The conductivity of the sediment layer was fixed as 1.1 S m 1, and that of the half-space was 1 4 (green), 1 3 (blue), 1 2 (black) and 1 1 (red). Z [m/s] phase [degree] Downloaded from by guest on 9 October Figure 16. Impedance due to motional induction (solid lines) and uniform source (dotted lines) for various depths of a conducting layer in the conductivity model shown in Fig. 3. The top depth of the conductive layer from the seafloor was 4 km (red), 7 km (black) and 1 km (blue).

13 42 H. Shimizu and H. Utada 1 1 Bseafloor/Bsurface Eseafloor/Esurface Figure 17. Ratio of the seafloor to the sea surface magnetic field (a) and electric field (b). The dotted, solid and dashed lines show the cases with ocean depth of 2, 4 and 6 m, respectively. The conductivity profile beneath the seafloor was taken to be the same as that shown in Fig. 3. Bhorizontal [nt] (a) Ehorizontal [micro V/m] (b) Figure 18. Amplitude range of the horizontal magnetic field (a) and electric field (b) due to the external field (grey) and the motionally induced field by a surface gravity wave (black) at the bottom of a 4-m-deep ocean. Only the part induced by the ambient vertical field (B z ) is shown for the motionally induced field. The amplitude of the external field variation was assumed to be from 1 to 5 nt. The amplitude of the surface gravity wave was assumed to be 1 cm, and the range of B z was taken to be from 3 to 5 nt. The conductivity profile beneath the seafloor was taken to be the same as that shown in Fig. 3. the surface gravity wave in mid-to-high latitudes can have a larger amplitude than that from the external field in the period range from 1 to 1 s if the wave height is more than 1 cm. On the other hand, the size of the electric field by the two sources is of the same order of magnitude in the period range for the considered parameters. To observe the electric field due to a tsunami to estimate the motional MT response, the tsunami wave height has to be sufficiently larger than 1 cm and/or the part induced by the external field variation has to be removed from the observed electric field, which may be done by using electromagnetic transfer functions between two stations (e.g. Utada et al. 211). The magnetic and electric fields observed at NM4 at the time of the 211 Tohoku earthquake show clear signature of motionally induced field by the tsunami (Zhang et al. 214a). The MT impedance and tipper at the time of passing of the tsunami due to the 211 Tohoku earthquake was evaluated by Zhang et al. (214b). They applied a five-point average for the ratios of E north /B east and B z /B east in the frequency domain to calculate the MT impedance and tipper for the electromagnetic field observed at NM4 at the time of the tsunami. The MT impedance and tipper are shown in Fig. 19. The MT impedance was clearly different from the ordinary MT impedance by Baba et al. (213), which is also shown in the figure. The obtained MT impedance has the characteristics of the motional impedance shown in Section 3; the amplitude is close to the phase speed of the surface gravity wave and the phase is about 9 (see eq. 36). The tipper multiplied by c p also shows the same characteristics as the MT impedance (see eq. 32). It is likely that the estimated Z and M yx represent the motional impedance and tipper. The estimated error for the tipper is slightly smaller than that for the motional impedance. The vertical component due to the external Downloaded from by guest on 9 October 218

14 Motional impedance Z [m/s] phase [degree] 9 6 Mzy.cp [m/s] Mzy phase [degree] Figure 19. MT impedance Z and tipper M yz (multiplied by c p ) at NM4 (solid circles) obtained by Zhang et al. (214b) using electromagnetic fields observed at the site during the passing of the Tohoku tsunami on 211 March 11. The dashed lines are the motional impedance and tipper when the subseafloor is an insulator (see Fig. 12b). The MT impedance from the time when there was no tsunami (open circles, from Baba et al. 213) is also plotted. field is usually very small in the seafloor and the observed b z during the time was mostly due to the tsunami induced field. On the other hand, attenuation of the horizontal electric field in the ocean is not as severe as that of the horizontal magnetic field in a period from 1 to 1 s (see Fig. 17), and therefore the effect of the external field variations would have caused the larger estimation error in Z than in M yz. It was possible to obtain the motional impedance in such a simple way partly because the electromagnetic field induced by the tsunami was large. The maximum amplitude of the surface displacement above the station was estimated as more than 1 m (Zhang et al. 214a), which is extremely large in the open ocean. Also, the simple analysis works because the electromagnetic field due to the surface gravity wave can be considered as a linear-polarized wave. If it had contained more complicated polarization as the external field variations, a more sophisticated statistical treatment (e.g. Chave & Thomson 24) would have been required to obtain the impedance. Although the motional impedance could be estimated using the observed electromagnetic field, it would not be very practical in general to use it for an electrical conductivity sounding for two reasons. One is that the motional impedance almost equals ic p in the tsunami period range irrespective of the assumption of the subbottom conductivity distributions. This means that there is very little information on the subseafloor conductivity in the motional impedance. The second reason is the large estimation error of the impedance. For example, the estimated error of the phase of the impedance at a period of 1 s was about ±15 (Zhang et al. 214b). The phase of the motional impedance has a slight dependence on the subseafloor conductivity. However, with this error, the obtained motional impedance does not have sufficient resolution to distinguish the representative conductivity models shown in Section 3. 5 CONCLUSIONS A formulation to calculate the electromagnetic field driven by motional induction due to a surface gravity wave and a geomagnetic main field is obtained for a general plane-layered Earth. The dominant induction comes from the vertical main magnetic field component. The motionally induced electromagnetic field can be detected by present-day ocean bottom instruments if the amplitude of the wave is several centimetres or more in the mid-to-high latitudes. Also, the motional impedance and tipper can be estimated by a rather simple method if the wave amplitude is larger than 1 cm. A comparison with the induced magnetic fields calculated by employing a tsunami flow model showed that the induced field calculated using a plane surface gravity wave is a reasonable approximation for the tsunami-induced field, although the tsunami has a complex structure. On the other hand, a simple thin-sheet approximation is not valid at higher frequencies. The motional impedance is affected largely by the dispersion of the long wave. The contribution from the subseafloor conductivity on the impedance decreases at shorter periods. The amplitude of the motional impedance is very close to the phase speed of the wave in Downloaded from by guest on 9 October 218