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Exploring effects in tippers at island geomagnetic observatories due to realistic depth and timevarying oceanic electrical conductivity
Earth, Planets and Space volumeÂ 73, ArticleÂ number:Â 3 (2021)
Abstract
Vertical magnetic transfer functions (tippers) estimated at island observatories can constrain the onedimensional (1D) conductivity distribution of the oceanic lithosphere and upper mantle. This is feasible due to the bathymetrydependent ocean induction effect (OIE), which originates from lateral conductivity contrasts between ocean and land and leads to nonzero tippers even for 1D conductivity distributions below the ocean. Proper analysis of island tippers requires accurate threedimensional (3D) modeling of the OIE, for which so far was performed assuming constant sea water electric conductivity with depth. In this study, we exploreâ€”using rigorous 3D electromagnetic modelingâ€”to what extent realistic, depthdependent, oceanic conductivity affects island tippers. The modeling is performed for 11 island observatories around the world in the period range \(10^{1}\) to \(10^{4}\) s. We also investigate the effect of seasonal variations of the oceanic conductivity and to which extent this could explain the observed systematic seasonal variation of tippers. Our model studies suggest that for most of the considered island observatories the effect from depthvarying oceanic conductivity is tangible and exceeds the error floor of 0.025, which usually is assigned to tippers during their inversion. The effect varies significantly with location, depending on regional bathymetry. Contrarily, the effects from seasonally varying oceanic conductivity were found to be too small to be worth consideration.
Introduction
One of the geophysical methods to probe the physical parameters of the Earthâ€™s mantle is Geomagnetic Depth Sounding (GDS; Banks 1969; Weidelt 1972). GDS exploits magnetic field variations of magnetospheric and/or ionospheric origin, and allows to constrain electrical conductivity at depth. The main data source for GDS is magnetic field measurements performed at the global net of geomagnetic observatories. Longperiod (> 3 h) variations are routinely used in GDS to constrain electrical conductivity of the Earthâ€™s mantle (approximately from 400 km down to 1500 km) either in terms of local onedimensional (1D; Olsen 1998; Utada etÂ al. 2003; Munch etÂ al. 2018; Chen etÂ al. 2020; Zhang etÂ al. 2020, among others) or threedimensional (3D; Kelbert etÂ al. 2009; Semenov and Kuvshinov 2012; Koyama etÂ al. 2014; Sun etÂ al. 2015; Li etÂ al. 2020, among others) conductivity distributions.
Recent studies (Samrock and Kuvshinov 2013; Morschhauser etÂ al. 2019) have also shown that vertical transfer functions (tippers) estimated from shortperiod (< 3 h) variations at island observatories can be used to constrain conductivity distribution beneath oceans, where our knowledge is still very limited. This becomes feasible due to the ocean induction effect (OIE; cf. Parkinson and Jones 1979), which originates from lateral conductivity contrasts between ocean and land. The OIE leads to nonzero tippers even for 1D conductivity distribution beneath the ocean (cf. Samrock and Kuvshinov 2013); therefore, the inversion of island tippers can provide information on the electrical properties of the crust and upper mantle in remote oceanic regions. However, the interpretation of island tippers requires accurate 3D electromagnetic (EM) modeling of the OIE that takes into account the bathymetry around the observatory. Hitherto, during such modeling, the researchers assumed constant oceanic electric conductivity with depth (cf. Samrock and Kuvshinov 2013; Morschhauser etÂ al. 2019). In many regions of the world, however, oceanic conductivity varies significantly within the uppermost few hundred meters of the water column (cf. Tyler etÂ al. 2017). In this study, we explore the extent to which realistic depthvarying oceanic conductivity affects island tippers. Our analysis is performed for 11 island observatories located in the Pacific, Atlantic, and Indian Oceans (cf. Fig. 1). In addition, we investigate the effect of seasonal variations of oceanic conductivity and to which extent this could explain the systematic seasonal variations in tippers investigated by ArayaÂ Vargas and Ritter (2016) and attributed to the variability of external magnetic source fields.
Methods
Tippers
In nonpolar regions, the source of the magnetic field variations with periods shorter than 3 h is well approximated by a vertically incidenting plane wave. The planewave assumption allows one to relate the vertical component \(B_z\) with the horizontal component \({\mathbf {B}}_H = (B_x~ B_y)\) at the location rÂ =Â (x,y,z)Â via the socalled tipper \({\mathbf {T}} = (T_{zx}~ T_{zy})\) (e.g., Berdichevsky and Dmitriev 2008):
where \(\omega = 2\pi /P\) is the angular frequency of magnetic field variations with period P. The x and ydirections are defined in this paper as the directions to geographic North and East, respectively, and z is directed vertically downwards. As a consequence of the planewave excitation, \(B_z\) (and, thus, \({\mathbf {T}}\)) are nonzero only above non1D conductivity structures. In fact, one can interpret \({\mathbf {T}}\) as a measure of the tipping of the magnetic field out of the horizontal plane above twodimensional or/and 3D conductivity structures.
Global oceanic conductivity model
The global oceanic electric conductivity model used here is that by Petereit etÂ al. (2019), which is based on the Coriolis Oceanographic data set for ReAnalysis (CORA5.0, Cabanes etÂ al. 2013) provided by the Copernicus Marine Environment Monitoring Service. The model consists of monthly 3D oceanic electrical conductivity distributions from 1990 to 2016. The conductivity is computed on a lateral grid of \(0.5^\circ \times 0.5^\circ\) resolution and at 152 vertical levels between the sea surface and 2000 m. More details on the oceanographic data set and the calculation of the global oceanic conductivity model can be found in Petereit etÂ al. (2019). From 2000 m down to the ocean bottom, the sea water conductivity was set to 3.2 S/m (cf. Tyler etÂ al. 2017). As an example, Fig. 2 presents global maps of the depthaveraged oceanic conductivity for a number of depth intervals for December 2015, illustrating the fact that oceanic conductivity indeed varies laterally and with depth.
Constructing island 3D conductivity models
The (Cartesian) 3D conductivity models were constructed separately for each observatory. The models include a nonuniform oceanic bathymetry and a landmass with a uniform conductivity of 0.01 S/m. We decided to use the same land conductivity for all observatories as a mean representative value, since we did not succeed to find better data to set the land conductivity specific for each location. The landmass is defined here as the upper crust from the maximum elevation down to the maximum depth of the ocean in the considered region. Note that landmass conductivity of 0.01 S/m was also adopted in the previous studies (Morschhauser etÂ al. 2019; Samrock and Kuvshinov 2013) which addressed modeling of island tippers.
The landmass and the ocean comprise the 3D part of the model, which is underlain by a crust and mantle with the 1D conductivity distribution (cf. Fig. Â 3) from Grayver etÂ al. (2017). Their 1D model was obtained by joint inversion of satellitedetected tidal and magnetospheric signals and is believed to represent the globally averaged 1D conductivity profile beneath the oceans. More details on our 3D models are as follows.
First, we note that as far as we exploit the integral equation (IE) based solver (to be discussed in the next section) to compute magnetic fields, the modeling domain is confined to the 3D part of the model. The lateral size of the 3D modeling domain was taken as \(356 \times 356\) km\(^2\) square with the island observatory in its center. The island was placed in the center of the modeling region (laterally) to avoid potential edge effects in the results. The vertical range of the domain is from the maximum topography down to the maximum depth of the ocean in the region of interest. The 3D part of the models is constructed using bathymetry and topography data from the General Bathymetry Chart of the Oceans (GEBCO 2019), which is a global map compiled from a variety of sources with 15 arc sec horizontal resolution (corresponding toÂ 0.46 km at the equator). Bathymetry/topography is converted to Cartesian coordinates by the use of the Transverse Mercator map projection and then linearly interpolated to a uniform grid with 1 km horizontal resolution. Note that we performed a comprehensive model study to justify the chosen lateral size of the modeling domain and cell sizes.
The uppermost left panel in Figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15 shows the bathymetry/topography in the vicinity of the corresponding island observatory. Vertically, the 3D modeling domain was discretized in 100 m layers from the maximum topography down to 1000 m depth; from 1000 m down to the maximum bathymetry in the region, the domain was discretized in 500 mthick layers (see Fig. 4). Oceanic conductivity within each vertical layer is assumed to be constant and is set to the 3D average value of the 2015 December (or June) oceanic conductivity model for the layerâ€™s volume. The models from 2015 year are chosen, since most of experimental tippers are derived from the data for this year (or adjacent years). Sedimentary layers were not incorporated into the models, since sediment thickness is negligible in the oceanic regions considered in this study (Straume etÂ al. 2019).
Outside the 3D volume, at depths between the surface and maximum depth of the ocean, conductivity is assumed to be laterally uniform (1D) and is set to the same oceanic conductivity profile as inside the volume, underlain by the 1D model for crust and mantle. The air conductivity is set to \(10^{8}\) S/m. The uppermost right panel in Figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15 depicts the vertical profile of oceanic conductivity (in purple), calculated by horizontally averaging oceanic conductivity (Fig.Â 2) in a \(5^\circ \times 5^\circ\) region centered at the considered island observatory. The panel demonstrates that the oceanic conductivity varies significantly with depth down to approximately 1 km depth. Below this depth, the values are close to 3.2 S/m.
Furthermore, for each observatory, we construct a 3D model with a constant oceanic conductivity of 3.2 S/m in all vertical layers. Outside the 3D volume, the oceanic conductivity is set to 3.2 S/m, underlain by the 1D model by Grayver etÂ al. (2017) for crust and mantle.
Modeling tippers
We compute magnetic fields (and then tippers) using the 3D EM forward modeling solver PGIEM2G which is based on a volume integral equation method with contracting kernel (Pankratov etÂ al. 1995; Singer 1995), and which was presented and validated (against finiteelement solver by Grayver and Kolev 2015) in Kruglyakov and Kuvshinov (2018).
We compute tippers at 25 periods spanning the range from \(10^{1}\) to \(10^{4}\) s. Note that, nowadays, most observatories provide magnetic field data in the form of minute means. This allows researchers to estimate tippers at periods starting from approximately 300 s. However, there is a growing investment from the scientific community in providing magnetic observatory data in the form of second means, which would enable estimating tippers for periods down to a few seconds. Tippers for even smaller periods can also be estimated at observatory locations by performing a shortterm (a few days) measurements with the use of induction coils; note that the standard observatory instrument to measure magnetic field variations is a fluxgate magnetometer. Bearing in mind the above considerations, we modeled tippers at periods shorter than 300 s to obtain an idea what is the smallest period at which ocean induction effect becomes noticeable.
Estimating tippers from the data
We estimated tippers at nine island geomagnetic observatories from the International RealTime Magnetic Observatory Network (INTERMAGNET) using definite minutemean data that are timeseries of three components of magnetic field. In addition, we estimated tippers at two nonINTERMAGNET island observatories: St. Helena (SHE) and SantaMaria/Azores (SMA), also using minutemean data. Prior to the tipper estimation, obvious spikes were removed from the corresponding timeseries. Table 1 summarizes information about the observatories, including the time interval used for the tipper estimation. This interval varies from observatory to observatory and was selectedâ€”based on a trialanderror approachâ€”to obtain smooth (with respect to period) tippers with low uncertainties.
For each period, data were split into overlapping tapered windows of two period length. Data in these windows were Fourier transformed, giving the spectrum of the corresponding component from 300 to 9600 s. Tippers and their uncertainties were then estimated using a robust, sectionaveraging (Olsen 1998) linear regression scheme based on the Huber norm (e.g., Aster etÂ al. 2005; PĂĽthe and Kuvshinov 2014).
Results
Effect of depthvarying oceanic conductivity
Modeling results for all 11 island observatories are presented in Figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15. In the figures, panel (a) shows the bathymetry/topography map with the corresponding observatory located in the center. The dashed line indicates the Westâ€“East running bathymetry/topography profile, as shown in panel (b). Panel (c) shows the regional depthvarying oceanic conductivity down to 2000 m (purple solid line) and the constant reference oceanic conductivity (3.2 S/m, dashed orange line). Panels (dâ€“g) present the \(\text {Re} \, T_{zx}\), \(\text {Re}\,T_{zy}\), \(\text {Im}\,T_{zx}\), and \(\text {Im}\,T_{zy}\) tipper components. Red and orange curves correspond to tippers calculated in the models with depthvarying and depthconstant oceanic electrical conductivity, respectively. The blue curves represent tippers and their uncertainties estimated from the data. One can see that both observed and modeled tippers fulfill the property (Marcuello etÂ al. 2005) that is often used as a plausibility check for the tippers, namely, at periods where \(\text {Re} \, T_{zx}\) or \(\text {Re} \, T_{zy}\) reaches a maximum (or a minimum) value, \(\text {Im} \,T_{zx}\) or \(\text {Im}\,T_{zy}\) changes the sign. Finally, panels (h) and (i) show the effect of depthvarying oceanic conductivity on island tippers. This effect is assessed by the following difference:
where \(i \in [x,y]\), and superscripts â€śvâ€ť and â€ścâ€ť correspond to the tippers calculated in the models with depthvarying and depthconstant oceanic conductivity, respectively. Here, we consider the effect as non negligible if it exceeds a value of 0.025, which is conventionally used as an error floor in tipper inversions (e.g., Morschhauser etÂ al. 2019; Yang etÂ al. 2015; Tietze and Ritter 2013; Rao etÂ al. 2014; Bedrosian and Feucht 2014) and which is shown in the panels as horizontal dashed line.
Three observations can be made from panels (dâ€“g) that are independent of modeling being done with depthvarying or depthconstant oceanic conductivity:

1.
The manifestation of the OIE in the modeled tippers varies from observatory to observatory. Given that depthvarying oceanic conductivity profiles (panels (c) in all figures) present similar values for all observatories, except for SMA, this variability of the OIE is expected to be due to the different bathymetry distributions around the islands.

2.
Modeled and experimental tippers agree rather well for most observatories, for the full spectrum from 300 to 9600 s, for both components and for both the real and imaginary parts. Any remaining discrepancy can most probably be attributed to regional deviations of the crustal and mantle conductivity structure from the global 1D conductivity structure used for modeling, and, partly, to limitations in bathymetry data resolution.

3.
The modeled OIE in tippers can be traced to periods as short as 0.2 s.
As for the effect from depthvarying oceanic conductivity, it varies from observatory to observatory, from component to component, and shows different behaviour with respect to period. This variability, like the variability of OIE itself, is expected to be from the different bathymetry distributions around the islands. The largest effectâ€”reaching 0.15â€”is observed at Apia observatory in \(\text {Re}\, T_{zx}\) component.
Effect of timevarying oceanic conductivity
Figure 16 presents global maps of differences between December and June oceanic conductivity in the same depth intervals as in Fig. 2. As expected, the difference varies laterally, it is the largest at shallower depths (reaching 20 percent of the mean value of oceanic conductivity) and decreases with depth.
Finally, we modeled the effect of timevarying oceanic conductivity on island tippers. The effect is assessed by analyzing the difference:
where \(i \in [x,y]\), and superscripts â€śDâ€ť and â€śJâ€ť correspond to December and June results. For the observed (i.e., estimated from the data) tippers, December and June results stand for tippers estimated from observatory data of corresponding 2015 months. As for modeled tippers, these results mean tippers calculated in 3D models with depthvarying oceanic conductivity models for 2015 December and June months.
In all considered observatories, modeled \(\Delta T_{zx}\) and \(\Delta T_{zy}\) are very small at all periods in the \(10^{1}\) to \(10^{4}\) s period range. FiguresÂ 17 and Â 18 present observed and modeled differences as filled circles on a global map for six representative periods from 300 to 9600 s. Observed and modeled differences are colored in Fig.Â 17 by light and dark red, and in Fig.Â 18â€”by light and dark blue. It is clearly seen that the effect due to timevarying oceanic conductivity is negligible; indeed, the filled circles depicted the modeled difference look as â€śdotsâ€ť. It is interesting to note that the temporal variability of the experimental tippers increases with period and, overall, it is larger in \(T_{zy}.\)
Conclusions
In this study, we performed the first ever analysis of the effects of realistic depth and timevarying oceanic electrical conductivity on island tippers. The analysis is based on 3D EM modeling, which was carried out for 11 island observatories located in the Pacific, Atlantic, and Indian Oceans. The conductivity models specific for each observatory were constructed using bathymetry/topography data with the highest spatial resolution available (GEBCO 2019) and a 3D, timedependent and physicsbased global model of oceanic conductivity (Petereit etÂ al. 2019). The Cartesian EM forward solver by Kruglyakov and Kuvshinov (2018) was used for tippersâ€™ modeling. Modelings were performed in wide period range (\(10^{1} \text {to} 10^{4}\) seconds) and demonstrated that ocean induction effect in tippers can be traced to periods as short as 0.2 s.
The effect due to depthvarying oceanic conductivity was assessed by comparing the tippers obtained from the depthvarying and depthconstant oceanic conductivity models. Our model studies show that this effect is tangible in all observatories except for TDC. It exceeds the error floor of 0.025, which is usually assigned to tippers during their 2D or 3D inversion, and reaches values of aboutÂ 0.1 at API, GAN, and HON observatories. The appearance of the effect with respect to period and its strength varies from observatory to observatory. Since depthvarying conductivity profiles are rather similar for all locations, such variability of the effect is most probably due to different bathymetry distributions around the islands.
On the contrary, the modeled effects from timevarying oceanic conductivity appeared to be too small to explain the observed seasonal variations in tippers.
It is worth noting that introducing a more complex ocean conductivity did not lead to an improvement in the agreement between modeled and experimental responses at many observatories. The most possible reason for this is a deviation of local subsurface conductivity structure from the global 1D section used in the paper. Obviously, to improve the agreement between modeling and experimental tippers, one needs to invert tippers for each location, and thus obtain local conductivity profile beneath each island observatory. Such inversions are out of the scope of this paper, but will be the subject of future study. It is planned to invert island tippers jointly with the longer period responses (cf. Munch etÂ al. 2020) to constrain conductivity throughout the entire depth range. The reliability of the inversions in particular depends on how accurate we represent in the model the oceanic conductivity. In this context, we believe that it is worth effort to use depthvarying oceanic conductivity during inversions provided that these data are considered to be trustworthy.
Availability of data and materials
The results presented in this paper rely on 1 Hz data collected at geomagnetic observatories. These data were digitally filtered to produce 1 min means that are available from the INTERMAGNET data repository.
Abbreviations
 EM:

Electromagnetic
 1D:

Onedimensional
 2D:

Twodimensional
 3D:

Threedimensional
 OIE:

Ocean induction effect
 GEBCO:

General Bathymetry Chart of the Oceans
 MT:

Magnetotellurics
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Acknowledgements
The authors acknowledge national institutes around the world that operate geomagnetic observatories, and INTERMAGNET (www.intermagnet.org) which promotes high standards of observatory practice. We also acknowledge the work of GEBCO group for providing publicly available global bathymetry data.Â We thank the two anonymous reviewers whose comments greatly improved this manuscript.
Funding
RR was supported by CNPq, Process 133345/20181. MK was supported by grant 200500001 from the Russian Foundation for Basic Research. AK was partially supported by the European Space Agency through the Swarm DISC project. KP was supported by FAPERJ (Jovem Cientista do Nosso Estado, Process 202.748/2019). Â EM was supported by the state assignment of the Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences. JP was supported by the German Research Foundationâ€™s priority program 1788 â€śDynamic Earthâ€ť.
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RR estimated tippers from observatory data, prepared 3D conductivity models, performed 3D modeling, and analyzed the results. MK provided the 3D EM modeling code PGIEM2G and assisted RR with the modeling. EM and MK provided the codes for converting global bathymetry/topography and ocean conductivity data into 3D conductivity models. AK created the concept of the study, and AK and KP supervised the RRâ€™s work. JP prepared global oceanic conductivity data. JM obtained and provided SHE and SMA magnetic field data. RR drafted the manuscript. All authors read and approved the final manuscript.
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Rigaud, R., Kruglyakov, M., Kuvshinov, A. et al. Exploring effects in tippers at island geomagnetic observatories due to realistic depth and timevarying oceanic electrical conductivity. Earth Planets Space 73, 3 (2021). https://doi.org/10.1186/s40623020013393
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DOI: https://doi.org/10.1186/s40623020013393