 Express Letter
 Open Access
 Published:
Tippers at island geomagnetic observatories constrain electrical conductivity of oceanic lithosphere and upper mantle
Earth, Planets and Space volume 71, Article number: 17 (2019)
Abstract
Geomagnetic field variations as recorded at geomagnetic observatories are important for global electromagnetic studies. However, this data set is rarely used for studying the local electrical conductivity at depths \(<\,200\) km. The main reasoning being that given a single geomagnetic observatory, one can at most constrain the onedimensional (1D) conductivity structure beneath it. At the same time, tippers, magnetic transfer functions resolving these depths, are zero for any 1D conductivity distribution. We show that the ocean induction effect alleviates these limitations for observatories on islands and develop a method to invert tippers for a 1D conductivity profile in the presence of threedimensional conductivity structure due to bathymetry. This allows to recover 1D upper mantle conductivity profiles at remote oceanic locations where little or no knowledge is available and that would otherwise be difficult to access. We apply the method to Gan in the Indian Ocean and to Tristan da Cunha in the South Atlantic, and the obtained conductivity profiles indicate a normal oceanic mantle and elevated conductivities, respectively, which fits well with their geological settings.
Introduction
More than 150 INTERMAGNET geomagnetic observatories around the world readily provide highquality magnetic field data with a sampling rate of up to 1 Hz (StLouis 2012). In principle, the recorded field variations with periods up to a few hours would allow to resolve the electrical conductivity distribution in the Earth’s lithosphere and upper mantle. However, there are two limitations impeding such studies. First, geomagnetic observatories are sparse, hence for studies of the lithosphere and upper mantle only individual observatories can be considered. In consequence, one can at best resolve a onedimensional (1D) conductivity structure beneath specific locations. Second, the source of field variations under consideration is seen at midlatitude observatories as a timevarying vertically incident plane wave (Chave and Jones 2012). With such a source field, tippers are the only response functions that might be estimated from single observatories recording only geomagnetic data. However, tippers are zero for a purely 1D conductivity distribution (Simpson and Bahr 2005; Berdichevsky and Dmitriev 2008), and therefore do not allow a meaningful interpretation with a single station.
Still, the situation is different for island and coastal observatories, where lateral conductivity contrasts between conductive sea water and more resistive island rock result in lateral conductivity variations and therefore large tipper amplitudes even if the underlying structure is essentially 1D (Samrock and Kuvshinov 2013). This is referred to as the ocean (or coast) induction effect (Parkinson and Jones 1979). Berdichevsky and Dmitriev (2008) argued that tippers are also sensitive to vertical variations in conductivity. This was confirmed by Samrock and Kuvshinov (2013) using realistic threedimensional (3D) bathymetry and 1D conductivity profiles that significantly differ within the lithosphere and upper mantle (\(< 200\) km). In their model study, the resulting variability in the modeled tippers exceeded the uncertainties in the observed tippers.
In this paper, we go a step further and develop a methodology to invert island tippers for a 1D conductivity distribution in the presence of known 3D bathymetry. Using this methodology, we first perform realistic synthetic tests and demonstrate that tippers from a single island observatory are indeed able to recover the synthetic 1D conductivity profile. Then, we invert tippers obtained at two island observatories located in different tectonic environments, namely Gan (GAN) in the Indian Ocean and Tristan da Cunha (TDC) in the South Atlantic. Finally, we interpret the recovered 1D profiles in terms of their regional geology and discuss the caveats of the 1D assumption at remote oceanic islands.
Methods
Tippers
Tippers \(\mathbf {T}\) are the only singlesite geomagnetic response functions within the planewave paradigm. They relate the horizontal (\(B_x\), \(B_y\)) and the vertical (\(B_z\)) magnetic field components as
Here \(\omega = 2 \pi /p\) is the angular frequency of geomagnetic field variations, where p is a period, z points positive downward, and x and y point to geographic North and East, respectively. As a consequence of the planewave excitation, variations in \(B_z\) (and thus \(\mathbf {T}\)) are zero above 1D conductivity structures (Simpson and Bahr 2005; Berdichevsky and Dmitriev 2008). Tippers are often represented in terms of real and imaginary induction arrows. In what follows, real induction arrows point away from conductive zones (Wiese convention, Wiese 1965) and are given by \((\mathfrak {R}{T_x}, \mathfrak {R}{T_y})\). We estimated tippers from timeseries of geomagnetic observatory data using robust processing methods (Appendix A). Additionally, univariate coherencies as well as multivariate coherencies were calculated, i.e., \({\mathrm{coh}}(T_{zx})={\mathrm{corr}}(B_z,T_{zx}B_x)\) and \({\mathrm{coh}}(\mathbf {T})={\mathrm{corr}}(B_z,T_{zx}B_x+T_{zy}B_y)\), respectively. Here, \({\mathrm{corr}}\) refers to the Pearson correlation coefficient. In consequence, and as \(B_z\) is a linear combination of \(B_x\) and \(B_y\) (Eq. 1), univariate coherencies \({\mathrm{coh}}(T_{zx})\) and \({\mathrm{coh}}(T_{zy})\) cannot simultaneously have values close to one. Note that this is also true for any univariate coherencies contributing to the same output variable (such as, for example, \({\mathrm{coh}}(Z_{xx})\) and \({\mathrm{coh}}(Z_{xy})\) for impedances).
Numerical model
As is shown in Fig. 1a, we parameterized the model by dividing the subsurface into M “layers” with unknown homogeneous conductivities \(\sigma _i\). Depending on bathymetry, these layers may not extend throughout the modeling domain and can be interrupted by regions of seawater conductivity. Further, conductivities of the air and sea remained fixed, and the sea conductivity values were obtained from the World Ocean Atlas (available as supplementary material in Grayver et al. 2016).
The conductivity contrast between rocks and saline seawater results in a complex 3D conductivity structure. In order to model it accurately, a 3D forward solver with adaptive mesh refinement is used that solves Maxwell’s equations. The numerical mesh used for Tristan da Cunha is shown in Fig. 1b, and more details on the numerical scheme and solver can be found in Grayver and Kolev (2015) and in Appendix B.
Synthetic test
Samrock and Kuvshinov (2013) demonstrated that tippers are sensitive to different 1D conductivity profiles in the presence of the ocean induction effect. Here, we perform synthetic tests to show that these differences are sufficient to infer the conductivity distribution with depth. For this purpose, we use the same bathymetry and numerical setup as for the real data from TDC (c.f. "Inversion of TDC data" section) and evaluate tippers at the location of the TDC observatory (see red diamond on inlet of Fig. 3b) for different profiles of electrical conductivity.
First, we illustrate the significance of the ocean induction effect by calculating tippers for a model with ocean plus a homogeneous Earth model (HOM) of \(\rho =100\) \(\Omega {\text{m}}\) (dashed gray line in Fig. 2a). Without ocean induction effect, such a model would result in zero tippers. However, modeled tippers are significant as shown by dashed blue and red lines in Fig. 2b. Next, we choose a 1D conductivity profile that is varying with depth, as indicated by the red line in Fig. 2a. The crosses in Fig. 2b show the resulting tippers which differ significantly from those corresponding to the HOM. This result confirms that tippers are sensitive to the underlying 1D conductivity structure, as has been shown by Samrock and Kuvshinov (2013). Next, we add 3 % Gaussian noise to the synthetic tippers, as indicated by the error bars, and invert them for a 1D conductivity profile. The synthetic tippers can be well fit (crosses and solid lines in Fig. 2b), and the obtained conductivity profile (black solid line in Fig. 2a) matches the true synthetic profile (red line) within its error limits (black dashed lines) as obtained from the diagonals of the model covariance matrix (i.e., only uncorrelated errors).
Observed tippers
Gan observatory (GAN)
The GAN geomagnetic observatory (Velimsky et al. 2014) is located on the Addu Atoll at the southern end of the Maldives archipelago, at \(0.6946^\circ\)S and \(73.1537^\circ\)E (red diamond in Fig. 3a). The Maldives archipelago consists of corals on top of the Maldive Ridge which was formed by the Réunion hot spot about 50–55 Ma ago (Fontaine et al. 2015, Fig. 1).
The top panel of Fig. 3c displays the observed induction arrows and the middle panel shows the real and imaginary parts of the corresponding tipper values. As expected, the real (blue) arrows point toward the more resistive island chain of the Maldives in the North (cf. Fig. 3a). Further, univariate coherencies (bottom panel) are significantly lower for \({\mathrm{coh}}(T_{zy})\) than for \({\mathrm{coh}}(T_{zx})\). This is due to the fact that both \({\mathrm{coh}}(T_{zy})\) and \({\mathrm{coh}}(T_{zx})\) cannot be simultaneously large ("Tippers" section), and that tippers are mainly oriented along the S–N axis which results in a better signaltonoise ratio for \(T_{zx}\). Multivariate coherencies reach values \(> 0.8\) (black line), except for periods \(< 50\, {\text{s}}\) where the sensitivity of the fluxgate magnetometer starts to decrease.
Tristan da Cunha observatory (TDC)
The TDC magnetic observatory (Matzka et al. 2009, 2010) is located on Tristan da Cunha island at \(37.067^\circ\)S and \(12.315^\circ\)W (red diamond on inlet of Fig. 3b) with seafloor ages of 20–25 Myr (Müller et al. 2008). Tristan da Cunha is an active volcano in the South Atlantic Ocean, located at the western end of the Walvis Ridge that marks the TristanGough hotspot track (Rohde et al. 2013).
The observed induction arrows, real and imaginary values of tippers, and coherencies are shown in Fig. 3d. As expected, the real arrows point toward the more resistive island of Tristan in the southsoutheast (Fig. 3b), at least for periods up to 500 s. For longer periods, they tend to be influenced by the presence of the islands Nightingale and Inaccessible to the southsouthwest (Fig. 3b). Again, univariate coherencies are lower for \({\mathrm{coh}}(T_{zy})\) than for \({\mathrm{coh}}(T_{zx})\), especially at longer periods where tippers are almost perfectly aligned with the S–N axis. Multivariate coherencies for tippers at TDC have values \(> 0.8\), except for periods \(< 20\, {\text{s}}\). Compared to GAN, multivariate coherencies at short periods are higher as more data were available.
Results and interpretation
Inversion of GAN data
The modeling domain covers an area of \(936 \times 936\) km with the GAN observatory at the center (Fig. 3a). Tippers were inverted for periods ranging between 40 and \(10{,}000\,\hbox {s}\), and a residual weighted RMS of 2.2 was achieved. The resulting bestfit tippers agree well with observed tippers, except for the real part of \(T_{zy}\) (Fig. 4b). The poor fit to \(T_{zy}\) may be related to the low coherency of \(T_{zy}\) (c.f. Fig. 3c). Indeed, additional induction coil data, shown by crosses in Fig. 4b, show a better agreement with modeled tippers (solid lines). Alternatively, the misfit may result from 3D conductivity structures that are incompatible with the assumed 1D model, for example conductive seafloor sediments at some distance from the station, or from inaccuracies of the bathymetry model. Disregarding \(T_{zy}\) slightly improves the fit to \(T_{zx}\) without corrupting the fit to \(T_{zy}\), as shown in Fig. 4b by comparing the observed (open circles) and modeled (solid lines) tippers.
We validate the robustness of our model by additionally inverting \(T_{zx}\) for a homogeneous Earth model (HOM), resulting in an electrical conductivity of \(\sigma =24\) mS/m, and we note that an inversion to all the data results in a very similar conductivity of \(\sigma =25\) mS/m. The model is shown by the gray dashed line in Fig. 4a and resembles the conductivity of the 1D model at depth. The corresponding tippers are shown by the dashed lines in Fig. 4b and fit the data with an RMS of 3.8 as compared to an RMS of 2.2 for the 1D model. Clearly, the HOM Earth model is not able to reproduce all of the data. Therefore, additional variations in conductivity are required by the data, and here we assume that these additional variations are only depth dependent.
The conductivity model resulting from a fit to \(T_{zx}\) only (black line in Fig. 4a) is characterized by a thin layer of conductive sediments, a thick layer of resistive crust and lithosphere, and a subsequent increase in conductivity with depth. Compared to the model resulting from a fit to all data, it is only slightly more resistive in the crust and lithosphere (not shown). Along with the model, the resulting 95% confidence limits are provided, and we note that these depend on the data errors which in turn depend on the assumed error floor (see Appendix B).
The upper resistive layer probably characterizes volcanic and basement rock (Aubert and Droxler 1996), and the obtained upper mantle conductivity at depths \(>110\) km agrees with the conductivity of dry olivine (brown line in Fig. 4a) as calculated for a normal mantle geotherm with \(T = 1360^{\circ }\) C at the lithosphereasthenosphere boundary (Katsura and Yoshino 2015). For comparison, the corresponding conductivity of wet olivine is shown by the blue line (Katsura and Yoshino 2015). Overall, the recovered 1D conductivity profile resembles the conductivity of resistive 125–150 Myr old Pacific upper mantle (Baba et al. (2010), red line in Fig. 4a), and differences at depths of 110–170 km possibly result from lower sensitivity at these depths and smoothing regularization that penalizes any structure. In conclusion, our model suggests no thermal or compositional anomalies at the base of the lithosphere of the Maldives Ridge, in agreement with seismic tomography data (Fontaine et al. 2015).
Inversion of TDC data
TDC has less prominent regional bathymetry as compared to GAN (Fig. 3b), and therefore a numerical domain of \(586 \times 586\) km was sufficient. Further, tippers were modeled for periods between 60 and 8264 s, and an overall residual weighted RMS of 0.5 was achieved. The resulting conductivity profile is shown along with its confidence limits by the black solid and dashed lines in Fig. 4c, and the corresponding modeled and observed tippers are shown by the solid line and symbols in Fig. 4d. The model fits the data very well, and only the longest periods of \(Re(T_{zy})\) are not entirely fit within their error limits.
Similar as for GAN, we validate the robustness of our model by additionally inverting for a homogeneous Earth model (HOM). The obtained electrical conductivity of \(\sigma =20\) mS/m roughly corresponds to the average conductivity of the 1D model. The corresponding tippers fit the data with an RMS of 1.1 as compared to an RMS of 0.5 for the 1D model and are shown by the dashed lines in Fig. 4d. As for GAN, the HOM Earth model is not able to fit all of the data, especially for \(T_{zx}\).
Compared to GAN, conductivities are overall higher for TDC with a thinner resistive layer followed by a increase in conductivity that starts at \(\approx 30\) km depth. As for GAN, the conductivity of hydrous olivine (Katsura and Yoshino 2015) is shown by the blue line in Fig. 4c. In comparison with the obtained conductivities and their confidence interval, the presence of melt is therefore not required at depth, although it cannot be excluded either. In addition, the 3D envelope of a recent 3D MT seabottom survey by Baba et al. (2016) is shown by the turquoise lines in Fig. 4c. Baba et al. (2016) concluded that no plumelike structure below the island is necessary to explain the data, but the authors note that their survey layout would not sense smallscale (\(<150\) km) structures directly below the island due to the absence of measurements in the island’s vicinity. Other studies found indications for melt below TDC. For example, Weit et al. (2017) concluded from geochemical considerations that melt fractions of 5% are present at depths of 60–100 km. Also, seismic tomography suggests a crustal lowvelocity anomaly characteristic for a magma feeding system below the island (Ryberg et al. 2017).
Overall, the electrical conductivities obtained by Baba et al. (2016) agree well with this study, with the exception of a more resistive crust and lithosphere, especially at depths of 20–50 km. This confirms the validity of our approach, and the potential to resolve 1D conductivity profiles using tippers and geomagnetic observatory data. Still, we note that results should not be overinterpreted, as the presented approach is only useful if the 1D conductivity assumption is not strongly violated, and if the used model of bathymetry is of good quality and resolution.
Discussion and conclusions
To our knowledge, tippers estimated at geomagnetic observatories have not been used for studying 1D electrical conductivity distributions. However, tippers dominated by 3D effects such as bathymetry also contain information on the vertical conductivity profile (Berdichevsky and Dmitriev 2008; Samrock and Kuvshinov 2013). In consequence, if the 3D effects are known, the 1D conductivity profile can be extracted.
Here, we demonstrated with synthetic tests that tippers can be used to resolve the 1D conductivity structure below islands where 3D effects are constrained by bathymetry. Further, we processed geomagnetic data from two island observatories, Gan in the Indian Ocean and Tristan da Cunha in the South Atlantic, to estimate and invert tippers. The obtained electrical conductivity profile beneath Gan island exhibits no anomalous behavior and is in agreement with other results for old oceanic upper mantle. For TDC, electrical conductivities are higher than for GAN, which may be explained by increased temperatures or increased water content. The presence of melt below the island is not required by the data, but cannot be excluded either. In the same area, an extensive 3D MT seabottom survey (Baba et al. 2016) obtained electrical conductivities that are very similar to our results, confirming the applicability of the proposed method. Additionally, we test the robustness of the obtained 1D profiles by inverting for a homogeneous Earth model and conclude that the additional 1D variability is required by the data.
We note that our approach relies on the assumption that no additional 3D conductivity anomalies are present. In consequence, any such anomalies may lead to misleading results, or to the unability of the model to fit the data. Clearly, remote and active volcanic islands are anomalous regions, for example due to hotspot mantle plumes. Still, the developed methodology allows us to obtain knowledge of the oceanic lithosphere and upper mantle from available singlesite geomagnetic observatory data that otherwise cannot be used for induction studies. Therefore, as long as we are aware of its limitations, the method will help to better understand the oceanic lithosphere and upper mantle in remote regions where otherwise little or no knowledge is available. Furthermore, when combined with longer period responses due to Sq or magnetospheric sources, this methodology provides a unique opportunity for imaging the electrical structure of the mantle throughout its full depth range. In future, more complete 3D studies with multiple magnetometers or additional measurements of field intensity (Kuvshinov et al. 2016) are also feasible.
Abbreviations
 TDC:

Tristan da Cunha observatory
 GAN:

Gan observatory
 1D:

onedimensional
 3D:

threedimensional
 HOM:

homogeneous Earth model
 RMS:

root mean square (of residuals)
 GEBCO:

General Bathymetry Chart of the Oceans
References
Aubert O, Droxler A (1996) Seismic stratigraphy and depositional signatures of the maldive carbonate system (Indian Ocean). Mar Pet Geol 13(5):503–536. https://doi.org/10.1016/02648172(96)000086
Baba K, Utada H, Goto T, Kasaya T, Shimizu H, Tada N (2010) Electrical conductivity imaging of the philippine sea upper mantle using seafloor magnetotelluric data. Phys Earth Planet Int 183(1–2):44–62. https://doi.org/10.1016/j.pepi.2010.09.010
Baba K, Chen J, Sommer M, Utada H, Geissler WH, Jokat W, Jegen M (2016) Marine magnetotellurics imaged no distinct plume beneath the Tristan da Cunha hotspot in the southern Atlantic ocean. Tectonophysics 716:52–63. https://doi.org/10.1016/j.tecto.2016.09.033
Berdichevsky M, Dmitriev VI (2008) Models and methods of magnetotellurics. Springer, Berlin. https://doi.org/10.1007/9783540778141
Chave AD, Jones AG (2012) Introduction to the magnetotelluric method. In: Chave AD, Jones AG (eds) The magnetotelluric method. Cambridge University Press, Cambridge, pp 1–18. https://doi.org/10.1017/cbo9781139020138.002 (chap 1)
Draper NR, Smith H (1998) Applied regression analysis, 3rd edn. Wiley, New York. https://doi.org/10.1002/9781118625590
Egbert GD, Booker JR (1986) Robust estimation of geomagnetic transfer functions. Geophys J Int 87(1):173–194. https://doi.org/10.1111/j.1365246X.1986.tb04552.x
Farquharson CG, Oldenburg DW (1998) Nonlinear inversion using general measures of data misfit and model structure. Geophys J Int 134:213–227. https://doi.org/10.1046/j.1365246X.1998.00555.x
Farquharson CG, Oldenburg DW (2004) A comparison of automatic techniques for estimating the regularization parameter in nonlinear inverse problems. Geophys J Int 156(3):411–425. https://doi.org/10.1111/j.1365246x.2004.02190.x
Farr TG, Rosen PA, Caro E, Crippen R, Duren R, Hensley S, Kobrick M, Paller M, Rodriguez E, Roth L, Seal D, Shaffer S, Shimada J, Umland J, Werner M, Oskin M, Burbank D, Alsdorf D (2007) The shuttle radar topography mission. Rev Geophys 45(2):RG2004. https://doi.org/10.1029/2005RG000183
Fontaine FR, Barruol G, Tkalčić H, Wlbern I, Rmpker G, Bodin T, Haugmard M (2015) Crustal and uppermost mantle structure variation beneath la Réunion hotspot track. Geophys J Int 203(1):107–126. https://doi.org/10.1093/gji/ggv279
Grayver AV, Kolev TV (2015) Largescale 3D geoelectromagnetic modeling using parallel adaptive highorder finite element method. Geophysics 80(6):E277–E291. https://doi.org/10.1190/geo20150013.1
Grayver AV, Kuvshinov AV (2016) Exploring equivalence domain in nonlinear inverse problems using covariance matrix adaption evolution strategy (CMAES) and random sampling. Geophys J Int 205(2):971–987. https://doi.org/10.1093/gji/ggw063
Grayver AV, Schnepf NR, Kuvshinov AV, Sabaka TJ, Manoj C, Olsen N (2016) Satellite tidal magnetic signals constrain oceanic lithosphereasthenosphere boundary. Sci Adv 2(9):e1600798. https://doi.org/10.1126/sciadv.1600798
Hansen N, Ostermeier A (2001) Completely derandomized selfadaptation in evolution strategies. Evol Comput 9(2):159–195. https://doi.org/10.1162/106365601750190398
Katsura T, Yoshino T (2015) The earths heterogenous mantle, chap heterogeneity of electrical conductivity in the Oceanic upper mantle. Springer Geophysics, Cham, pp 173–204. https://doi.org/10.1007/9783319156279_6
Key K (2016) MARE2DEM: a 2D inversion code for controlledsource electromagnetic and magnetotelluric data. Geophys J Int 207(1):571–588. https://doi.org/10.1093/gji/ggw290
Kuvshinov A, Matzka J, Poedjono B, Samrock F, Olsen PS (2016) Probing Earth’s conductivity structure beneath oceans by scalar geomagnetic data: autonomous surface vehicle solution. Earth Planets Space 68(1):189. https://doi.org/10.1186/s4062301605537
Matzka J, Olsen N, Maule CF, Pedersen LW, Berarducci AM, Macmillan S (2009) Geomagnetic observations on Tristan da Cunha, South Atlantic ocean. Ann Geophys Italy 52:97–105. https://doi.org/10.4401/ag4633
Matzka J, Chulliat A, Mandea M, Finlay CC, Qamili E (2010) Geomagnetic observations for main field studies: from ground to space. Space Sci Rev 155:29–64. https://doi.org/10.1007/s1121401096934
Müller RD, Sdrolias M, Gaina C, Roest WR (2008) Age, spreading rates, and spreading asymmetry of the world’s ocean crust. Geochem Geophys Geosyst https://doi.org/10.1029/2007GC001743
NASA, METI, AIST, Japan Spacesystems, US/Japan ASTER Science Team (2001) ASTER global digital elevation model (GDEM) v2. https://doi.org/10.5067/aster/ast14dem.003. NASA EOSDIS Land Processes DAAC
Parkinson WD, Jones FW (1979) The geomagnetic coast effect. Rev Geophys 17(8):1999–2015. https://doi.org/10.1029/rg017i008p01999
Pedersen LW, Merenyi L (2016) The FGE magnetometer and the INTERMAGNET 1 second standard. J Ind Geophys Union 2:30–36
Püthe C, Kuvshinov A (2014) Mapping 3D mantle electrical conductivity from space: a new 3D inversion scheme based on analysis of matrix Qresponses. Geophys J Int 197(2):768–784. https://doi.org/10.1093/gji/ggu027
Rohde JK, van den Bogaard P, Hoernle K, Hauff F, Werner R (2013) Evidence for an age progression along the tristangough volcanic track from new 40ar/39ar ages on phenocryst phases. Tectonophysics 604:60–71. https://doi.org/10.1016/j.tecto.2012.08.026
Ryberg T, Geissler WH, Jokat W, Pandey S (2017) Uppermost mantle and crustal structure at Tristan da Cunha derived from ambient seismic noise. Earth Planet Sci Lett 471:117–124. https://doi.org/10.1016/j.epsl.2017.04.049
Samrock F, Kuvshinov A (2013) Tippers at island observatories: Can we use them to probe electrical conductivity of the Earth’s crust and upper mantle? Geophys Res Lett 40:824–828. https://doi.org/10.1002/grl.50218
Simpson F, Bahr K (2005) Practical magnetotellurics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511614095y
StLouis B (ed) (2012) Technical reference manual. 4.6, INTERMAGNET
Velimsky J, Muslim A, Ziyad I, Jackson A, Kuvshinov A, Samrock F, Arora K, Rao K, Pedersen LW, Finlay C, Riddick J (2014) Geomagnetic observatory Gan. In: Abstracts of XVI IAGA workshop on geomagnetic observatory instruments, data acquisition and processing
Weatherall P, Marks KM, Jakobsson M, Schmitt T, Tani S, Arndt JE, Rovere M, Chayes D, Ferrini V, Wigley R (2015) A new digital bathymetric model of the world’s oceans. Earth Space Sci 2(8):331–345. https://doi.org/10.1002/2015ea000107
Weit A, Trumbull RB, Keiding JK, Geissler WH, Gibson SA, Veksler IV (2017) The magmatic system beneath the Tristan da Cunha island: insights from thermobarometry, melting models and geophysics. Tectonophysics 716:64–76. https://doi.org/10.1016/j.tecto.2016.08.010
Wessel P, Smith WHF (1998) New improved version of the generic mapping tools released. EOS Trans Am Geophys Union 79:579. https://doi.org/10.1029/98EO00426
Wiese H (1965) Geomagnetische Induktionspfeile in der ČSSR, Hervorgerufen durch Grossräumige Elektrische Leitfähigkeitsstrukturen. Stud Geophys Geod 9:415–419. https://doi.org/10.1007/BF02607678
Author's contributions
AM processed and analyzed the magnetic field data, AG performed the inversions and synthetic tests, AK created the concept of the study, and FS and JM obtained and provided information on the GAN and TDC magnetic field data, respectively. AM drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The reviews by Paul A. Bedrosian and one anonymous reviewer greatly helped to improve this manuscript. We thank ETH Zürich, the Gan Meteorological Office, and GFZ Potsdam for supporting the operation of the observatories GAN and TDC, and INTERMAGNET for promoting high standards of magnetic observatory practice (www.intermagnet.org). We gratefully acknowledge the work of Ahmed Muslim who locally operates the GAN observatory and Robin Repetto who locally operates the Tristan da Cunha observatory. Without their help and enthusiasm, this work wouldn’t have been possible. Also, we acknowledge the work of the GEBCO group and are grateful for using their data (’The GEBCO_2014 Grid, version 20150318, www.gebco.net’). As well, we are grateful for using the ASTER GDEM data (ASTER GDEM is a product of METI and NASA). Finally, we thank Wessel and Smith (1998) for sharing their Generic Mapping Tools (GMT).
Availability of data and materials
The results presented in this paper rely on 1 Hz data collected at TDC and GAN geomagnetic observatories. These data were digitally filtered to produce one minute means that are available from the INTERMAGNET data repository.
Competing interests
The authors declare that they have no competing interests.
Funding
This work was supported by a Grant from the Swiss National Supercomputing Centre (CSCS) under Project ID767. The research stay of AM at ETH Zürich was funded by DFG (“Deutsche Forschungsgemeinschaft”) under Grant No. MO2598/11.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Author information
Appendices
Appendix A: Data processing
The GAN geomagnetic observatory was installed in April 2011 in the area of the GAN International Airport, and accepted as an INTERMAGNET observatory in October 2013. The observatory is operated by ETH Zürich and Gan Meteorological Office (Velimsky et al. 2014). Vector magnetic field variations are recorded by a DTU Fluxgate magnetometer (Pedersen and Merenyi 2016) at a sampling rate of 1 Hz. These data are digitally filtered to produce one minute means and subsequently calibrated for publication at INTERMAGNET. In this study, we use unpublished 1 Hz data which we calibrate and rotate to geographic coordinates using the available observatory baselines.
The Tristan da Cunha geomagnetic observatory is operated by GFZ Potsdam and was established in October 2009. Since 2012, it is part of the INTERMAGNET network of observatories. Similar to GAN, 1 Hz vector magnetic field data are recorded by a DTU fluxgate magnetometer (Pedersen and Merenyi 2016), and calibrated and digitally filtered minute means are available on INTERMAGNET. For more details, the reader is referred to Matzka et al. (2009, 2010). Similar to GAN, we use unpublished 1 Hz data which we calibrate and rotate to geographic coordinates using the available observatory baselines.
Prior to the estimation of tippers, obvious spikes were manually removed from the selected timeseries data. Then, for each discrete frequency \(\omega _i\), data were split into overlapping and tapered windows of two period lengths before the data in each window were transformed to spectral domain. Subsequently, tippers were obtained using robust linear regression based on the Huber norm (e.g., Farquharson and Oldenburg 1998; Püthe and Kuvshinov 2014). The quality of the obtained transfer functions was evaluated by estimating confidence intervals \(\delta _i\) for each frequency \(\omega _i\) from linear propagation of data noise (Draper and Smith 1998), and data noise was estimated from the residuals. However, it should be noted that this approach often underestimates real uncertainties (Egbert and Booker 1986).
Appendix B: Numerical model
The presence of the ocean requires to solve Maxwell’s equations for a given 3D conductivity distribution. To this end, we solved
for the complexvalued electric field vector \(\mathbf {E}\). Here, \(\varOmega \subseteq \mathbb {R}^3\) is the modeling domain, \(\mu _0\) the magnetic permeability of free space, and \(\sigma\) denotes the realvalued 3D electric conductivity. Further, we applied the inhomogeneous Dirichlet boundary conditions, and \(\mathbf {E}_0\) resulted from the solution of Maxwell’s equations on the boundaries. We discretized Eq. (2) using the finiteelement method on a nonconforming hexahedral mesh. Finally, the corresponding magnetic fields were calculated by virtue of Faraday’s law. For more details on the underlying numerical scheme and solver, the reader is referred to Grayver and Kolev (2015).
The corresponding numerical mesh was adapted to the regional bathymetry that was obtained from the General Bathymetry Chart of the Oceans (GEBCO). This dataset provides a global bathymetry map at a resolution of \(30^{\prime \prime }\) that has been compiled from a variety of sources (Weatherall et al. 2015), including the land topography dataset SRTM30 at the same resolution (Farr et al. 2007). The mesh was refined at the seafloor and along the coastlines, as well as close to the observatory. For Tristan da Cunha, additional ASTER topography data with a resolution of 30 m have been included due to strong topographic gradients (NASA et al. 2001). For GAN, the minimal cell size of the locally refined mesh is \(62.5 \times 62.5 \times 15.6\) m (\(x\times y\times z)\).
In order to invert the observed tippers \(\mathbf {d} \in \mathbb {R}^{N}\) for the model parameters \(\mathbf {m} \in \mathbb {R}^{M}\), we minimized the regularized functional
where \(w_i\) are the data weights, \(f_i\) represents the forward operator obtained from solving Eq. 2, \(\beta\) is a regularization parameter, \(\mathbf {l}_i\) is a regularization operator for the ith model parameter penalizing model parameter jumps, and scalar p controls the norm of the regularization term. The data weights \(w_i\) were defined from the estimated confidence intervals \(\delta _i\) and an absolute error floor \(e_i=0.025\) such that \(w_i=1/\max (e_i,\delta _i)\). Please note that the level of error floor is quite arbitrary, but the resulting RMS of the residuals should in principle approach one under the assumption of Gaussian noise. The unknown model parameters were given by \(\mathbf {m} = [\lambda (\sigma _1) \cdots \lambda (\sigma _M)]\), where \(\lambda (\cdot )\) represents a logbased transformation used to bound parameters (Key 2016). Here, \(\sigma _i\) denote the 1D conductivity profile of the subsurface that was divided into \(M=11\) “layers” of increasing thickness (from 0.5 km at the top to 30 km at the bottom) and with homogeneous conductivities.
The minimization problem (3) was solved using a stochastic optimization method called Covariance Matrix Adaptation Evolution Strategy (CMAES, Hansen and Ostermeier 2001). This method is capable to reach global minima and is independent of the starting model. It permits usage of other than \(L_2\)norm metrics, and supplies information on model uncertainty. Here, we used \(p=1.5\) (Grayver and Kuvshinov 2016), and the regularization parameter \(\beta\) was chosen using the discrepancy principle (Farquharson and Oldenburg 2004).
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Morschhauser, A., Grayver, A., Kuvshinov, A. et al. Tippers at island geomagnetic observatories constrain electrical conductivity of oceanic lithosphere and upper mantle. Earth Planets Space 71, 17 (2019). https://doi.org/10.1186/s4062301909910
Received:
Accepted:
Published:
Keywords
 Electrical conductivity
 Tippers
 Ocean effect
 Tristan da Cunha
 Maldives
 Geomagnetic observatories
 Oceanic upper mantle
 Inversion