Geophysical Research Letters
Supporting Information for
Electrical Resistivity Image of the South Atlantic Continental Margin Derived from
On- and Offshore Magnetotelluric Data
G. Kapinos 1,2 , U. Weckmann 1 , M. Jegen-Kulcsar 3 , N. Meqbel 1 , A. Neska 4 , T. T.
Katjiuongua 5 , S. Hoelz 3 , O. Ritter 1,6
1 - GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany.
2
– now at: Federal Institute for Geosciences and Natural Resources (BGR), Stilleweg 2, 30655 Hannover,
Germany.
3
– GEOMAR - Helmholtz Centre for Ocean Research, Wischhofstr. 1-3, 24148 Kiel, Germany.
4 – Institute of Geophysics, Polish Academy of Sciences, Ul. Ksiecia Janusza 64, 01-452 Warszawa,
Poland.
5 – Geological Survey of Namibia, Ministry of Mines & Energy, 6 Aviation Road, Windhoek, Namibia
6 -
Freie Universität Berlin, Institute of Geophysics, Malteserstr. 74-100, 12249 Berlin, Germany.
Corresponding author: O. Ritter
Text S1
Figures S1 to S9
Supplement material includes information on methodology used (magntotellurics), dimensionality and directionality analysis (geo-electric strike), technical details on the 2D inversion approach, and 3D modelling of the ocean bathymetry.
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Text S1.
Methodology
Magnetotellurics (MT) is an electromagnetic (EM) geophysical deep sounding method applied to study electrical properties of Earth’s interior. The method is based on measurements of time variations of orthogonal components of electric and magnetic fields at surface of the Earth. The sounding depth depends of the frequency content of the observed EM fields and the electrical resistivity of the subsurface (skin effect); it is typically in a range between a few hundred meters to tens of kilometers.
The sources for the magnetotelluric method are naturally occurring electromagnetic field variations. Electric currents are excited over a wide frequency range in the earth’s magnetosphere by solar activity and global lighting discharges. While these sources are available everywhere on earth and at all times, their signal strength varies considerably.
Electromagnetic signals produced by man-made installations are superimposed on the natural EM fields and act as noise. The so-called remote-reference technique is an effective way to improve magnetotelluric data quality by referencing the electromagnetic fields recorded at a local site to simultaneously recorded, undisturbed fields at a remote site.
The information on the electrical resistivity structure of the subsurface is contained in the complex, frequency-dependent impedance tensor Z , expressed in Cartesian coordinate system [e.g., Berdichevski 1960, 1964; Tikhonov and Berdichevski, 1966]:
(1)
With E constituting the electric field and B the magnetic field. A similar relation exists for the geomagnetic transfer function T between the vertical and the two horizontal magnetic field components [e.g. Schmucker 1970]:
(2)
For a 1D Earth (horizontally layered or uniform), the diagonal components of the impedance tensor Z are zero and Zxy = −Zyx. In a 2D environment, the diagonal elements are only zero if the coordinate system of the observations matches or is rotated to the geo-electrical strike direction.
Typically, the elements of the impedance tensor are presented as apparent resistivity ( ) and phase ( ) curves:
(3)
ω is the angular frequency and all quantities are frequency dependent and i,j = x,y . The sounding depth depends on frequency and resistivity of the subsurface in a highly nonl inear manner. Solving the underlying mathematical equations (Maxwell’s equations) and construction of electrical resistivity models requires numerical approximations.
Dimensionality and Directionality Analysis
The MT data in northern Namibia were acquired along a transect roughly perpendicular to the expected strike of the main tectonic structures of the Kaoko Belt and the NNW-SSW striking ocean-land boundary (shoreline). While we expect that these features also dominate the geo-electric strike (e.g. the predominant current flow in the subsurface), this must be tested prior to subsequent 2D modelling. Prior to a directionality analysis the magnetic declination of 10.5° was considered.
If the electric and magnetic fields in a 2D system are aligned (or mathematically rotated) to the geo-electrical strike, the main-diagonal elements of the impedance tensor become zero (small) and the off-diagonal elements decouple into the tangential electric
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mode (TE or E-polarization) and the tangential magnetic mode (TM or B-Polarization). In the TE- mode the induced electric currents flow parallel to the strike and in the TM- mode current flow is perpendicular to the strike [e.g. Cantwell, 1960; Vozoff, 1987;
McNeill and Labson, 1987].
To determine a common (regional) geo-electrical strike we used the decomposition scheme of Becken & Burkhardt [2004], which infers the existence of a regional 2D structure and geo-electric strike direction from an ellipse parametrization of the impedance tensor columns. The ellipticities of the impedance tensor columns are rotationally variant; they vanish in case of 2D conditions if the data are rotated to the regional strike. To determine the strike direction, the sum of squared ellipticities weighted with their variances are minimized by rotating the coordinate system. The minimization is applied for a range of periods (and sites) simultaneously. The results for a period range between 10 and 1,000 s for sites along the profile are presented in Fig. 3 of the main text.
A similar strike direction is also derived from the induction vectors (IVs) which are calculated from vertical magnetic transfer functions (eq. 3). In the Wiese convention
[Wiese, 1962], they tend to point away from regions of higher electrical conductivity (low resistivity) and in a 2D environment they are aligned perpendicular to strike. The induction vectors of all sites are shown for two representative periods in Fig. S1. At long periods (large induction volume) the IVs are dominated by the contrast between land and the highly conductive Atlantic Ocean, particularly at sites in the western part of the
Kaoko Belt. At shorter periods (<1 s) the IVs reflect more local features. They show, for example, a reversal over the N-S trending Purros Mylonite Zone, thereby indicating that this shear zone is an elongated structure with lower resistivity than the surrounding areas.
Another illustrative presentation of the dimensionality and directionality of the underground provides the phase tensor, which is defined as the relationship between the real and imaginary parts of the impedance tensor and which is unaffected by galvanic distortion. Fig. S2 shows a pseudo-section of the phase tensor at frequencies between
1 kHz and 0.1 mHz for all off- and onshore sites used for the 2D inversion. The major or minor axes of the ellipses correspond to the regional strike direction and the colors of the ellipses (skew angle

) is a measure of the complexity of the subsurface (3D structures).
Overall, we observe that the structural grain of the Kaoko Belt, particularly the NS trending shear zones, and the first order anomaly of the Atlantic Ocean dominate the MT and vertical magnetic data. A regional strike, which is required to identify the TE- and
TM- modes for 2D inversion, can be established at approximately 6 degrees. The analysis also shows, however, that the data are influenced by 3D (off-profile) features, which cannot be fully explained with 2D modelling.
2D Inversion of the data
With 2D inversion one can transform the data (apparent resistivity, phases and magnetic transfer functions) into a resistivity depth section. The challenge is doing this along a profile across the entire passive continental margin, with the inherent complexities of the on and offshore data. Modeling of marine data, for example, requires inclusion of the bathymetry which in turn can only be done numerically in 3D. However a joint 3D inversion becomes unfeasible, when considering that it would have to include detailed 3D bathymetry, OBMT stations aligned along a profile with sparse site spacing, land data which are sampled with denser spacing and better areal coverage, and the expected highly complex geological imprint of this margin. Thus we address the problem in a hybrid approach. 3D forward modeling is used to estimate the effect of bathymetry
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(see next section of the supplement) but our focus is on 2D inversion along the amphibious transect (see Fig. 1 in main text). With 2D inversion we can use a much finer mesh design, test inversion parameters and the influence of individual data sets more rigorously. A 2D approach seems also natural because the data are acquired along a profile.
We used the Non-Linear Conjugate Gradients (NLCG) algorithm of Rodi & Mackie
[2001] to explain data in the period range between 0.001 and 10,000 s collected at 82 on- and offshore sites projected onto the profile shown in Fig. 1. The model space was discretized by using a Finite Difference (FD) mesh with 271 x 59 cells in the horizontal and vertical directions, respectively. The model space is much larger than the profile to satisfy numerical boundary conditions. The mesh is equidistant in the central part, where the MT sites are located, with cell sizes of 4 km offshore and 2 km on land. The column widths increase approximately exponentially towards the periphery in accordance with exponentially decaying EM fields. In the vertical direction, row thickness increases slowly in order to image appearance of the seafloor with as much detail as possible; just below the seafloor bottom the smallest cell thicknesses are 10 m.
The inversion algorithm aims to minimize an objective function, which consists of data misfit and regularization terms. The best compromise between both terms is controlled by the regularization parameter tau which is determined iteratively by repeating inversions with different values of tau. Fig. S3 shows expected result of this trade-off between model smoothness and data fit, the optimum value is found at the kink in the L-curve for a value of tau = 5.
In a 2D world, electrical conductivity varies in the vertical plane but remains constant along an infinitely extending geological strike. The data are then referred to the
TE- and TM- modes, with induced electrical currents flowing parallel or perpendicular to the strike, respectively. The TE- mode data are generally more sensitive to deep conductive structures (embedded in a resistive host) but are more affected by off-profile
(3D) structures. Since the TM- mode data are more immune to 3D effects, practitioners often resort to TM-mode only inversions to avoid artificial structures which may be introduced by 2D inversion to “explain” 3D features. To test robustness and reliability of the inversion models and to identify inversion artefacts, we assess the results of the individual mode inversions (TE-, TM- mode, and vertical magnetic fields) and examine their influence on the joint inversion model.
For the offshore data it is particularly important to control the effect of the coast and the seafloor topography. Anomalous magnetic fields are induced due to the high conductivity contrast between seafloor topography and saltwater. Particularly at the continental margin, where the seafloor shallows towards the shore, strong electrical currents are induced into the highly conductive ocean, i.e. right above the measurement points. These currents cause secondary anomalous magnetic fields, which eventually act like internal sources [see Key, 2003; Constable et al., 2009; Key and Constable,
2011; Kapinos, 2011; Wheelock, 2011). In the offshore TE- mode data, this phenomenon is expressed as cusps in the apparent resistivity curves and by phases leaving the quadrant. For the TM- mode apparent resistivities we obtain a static-shift-like effect and such an effect is evident in our offshore data as the TM-mode apparent resistivity curves are consistently smaller then TE- mode data. The more resistive the underground is, the more severe is the effect.
2D inversion can reproduce the anomalous currents of the TE- mode quite well but shifts in the TM- mode leads to biased models with underestimated resistivities.
Consequently, Fig. S4 shows a TM- mode inversion model of the offshore data set with a much less resistive underground than the TE- mode inversion model.
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Fig. S5 shows exemplary inversion results of the onshore data for the TM- and TEmodes, respectively. The TE- mode inversion produces a model with a high conductivity structure embedded in a resistive host. This high conductivity structure appears in the east because there the TE-mode long period apparent resistivity curves drop to very low values. High RMS errors when compared to the model derived from the inversion for the
TM- mode (TE =2.83; TM=1.77) and relatively high skew values indicate that this conductor may be an artificial structure, possibly caused by off-profile structures.
Our preferred model presented in Fig. 4 of the main text is obtained using the following strategy: As a first step, the error floors of the onshore data were set to 20% for the apparent resistivity and 1.45° for the phases of the TM- mode data (TE- mode ignored). For joint inversion of the on- and offshore data in a second step we set an error floor to 100% for the apparent resistivity in the TE- and 1,000% in the TM- mode but tight error floors of 0.58° for both phases. Thus the inversion focuses to fit the undistorted phases and mitigates the influence of the biased TM-mode marine data. The high error floor for the apparent resistivity in the TE- mode is necessary to down-weight the effects of off-profile features which particularly affect the onshore data. For the vertical magnetic transfer functions (VTF) which are free of static shift effects a relatively small absolute error floor of 0.02 was used for all discussed inversion runs.
Using separate inversion runs for the onshore and offshore data sets was also motivated because they show completely different convergence behaviors. We found for example that inversion of the offshore data often stopped after a few tens of iterations, while inversion of the on-shore data typically required hundreds of iterations before reaching convergence. It is a principal problem of non-linear inversion algorithms that they often converge to a local minimum without any certainty that the solution is (close to) a global minimum [e.g., Asster, 2005]. Our experience with direct joint inversions of the entire on- and offshore data set suggested that the inversion often stopped at local minima. Better results in terms of data misfit and coherent model structures could be obtained by running our successive inversion scheme.
The error floor has a significant impact on the inversion models and the data fit. In order to find a compromise between a physical (geological) reasonable model and mathematical solution (minimization of the objective function) we tested this extensively by trial and error, i.e. by calculating numerous inversion models testing a range of error floors for the on- and offshore data, TE- and TM- modes, magnetic transfer functions.
The preferred combined on-offshore model explains the data within an overall root mean square misfit (RMS) of 4.11, while the inversion of the “onshore-only” data (TM- mode and VTF) yields a total RMS of 1,69. The high overall RMS value is caused by the offshore data, probably due to the very small error floor (0.58°) assigned to the TE- mode phases.
To test if the offshore MT data is sensitive to structure and in particular the resistivities below the sea water saturated shallow sediments, we ran constrained 2D inversions. These studies are aimed to sustain our conclusion of atypically thickened and moderately resistive crust along the Walvis Ridge (see Results section of the main text).
For the constrained inversion study we used the offshore section of our preferred model as the starting point (Fig 4, main text), which fitted the TE mode and VTF data to a final RMS of 2.84. We added half spaces at varying depths and with varying resistivity and re-ran the 2D inversions. The average oceanic crust is approximately 7 km thick and can be highly to moderately resistive (Key, 2012), while the oceanic lithosphere is commonly resistive > 10,000
 m. Therefore, resistive bodies of either 800
 m or
1,500
 m were included below depth levels (possible thickness of the crust) of 8.2 km,
11.8 km, 17.0 km, 20.8 km and 42.5 km reaching down to 200 km depth. The resulting
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half spaces below the upper boundary of the inserted bodies were locked and the inversion was restarted keeping all other inversion settings unchanged. It took between
20 and 80 iterations until the data fit (total RMS) could not be improved any further.
Tests with a resistive half space instead of a resistive body with limited maximum depth extent of 200 km resulted in extremely poor data fit, indicating that lower resistivities are required by the TE mode MT data somewhere at great depth (not shown).
Two end-member constrained inversion results are shown in Fig. S6 together with the data fit in form of total RMS values. When inserting a more resistive subsurface at the typical depth of oceanic lithosphere (corresponding to the 8.2 km case in Fig. S6a) the TE-mode data cannot be fitted, particularly between periods of 100 s and 1,000 s and at stations between profile kilometers 120 and 280. Increasing the depth towards the top of the resistive lithospheric gradually to a maximum depth of 42.5 km (see
Fig 6b) shows similarly gradually improving data fits. However, only towards the endmember models with a top of the lithospheric body at 20.8 km and below yielded a data fit comparable to the unconstrained inversion (Fig. S6c).
A similar inversion study was performed using a combination of the TM and VTF data. As mentioned before, the TM mode apparent resistivities seem to be prone to bias effects and therefore we focused the inversion on the phases of the TM mode. We used the offshore section of our preferred model as the starting point, which fitted the TE mode and VTF data to a final RMS of 1.9. Resistive half spaces of either 800
 m or
1500
 m were then introduced at the same depth levels and locked as this was done for the previous study using the TE mode data. Interestingly, the TM mode data appear to be insensitive to higher conductivities below 200 km.
Figs. S6d and e show inversion results of both models with the top of the resistive half space at 8.2 km and 42.5 km depth together with the data fit in form of total RMS values in Fig. 6f. When inserting a more resistive subsurface at 8.2 km depth the data cannot be fitted, particularly at long periods and at stations between profile kilometer 250 and 300. To compensate for the higher resistivities at relatively shallow depth, the inversion inserts extremely low resistivity material of 0.1
 m down to depths of approximately 5 km depth (but without reaching a reasonable data fit). Increasing the depth towards the top of the resistive lithosphere improves data fit, but only the endmember model with the deepest top of the lithosphere at 42.5 km yielded a similar fit as the unconstrained inversion (Fig. S6c).
Additional inversion tests for both modes were run using so-called tear zones.
Along their boundaries, regularization is disrupted so that sharp resistivity changes are not penalized by the smooth inversion approach. These zones were included into the preferred “offshore-only” model in 8.2 km and 17.0 km depth. For the TE mode data, tear zone inversion did not alter structures nor the RMS, while in the TM mode, the inversion model resulted in a sharper conductivity contrast outside the region with station coverage and slightly but insignificantly improved total RMS values. Expected higher resistivities could not be fostered in both studies.
This constrained inversion studies suggest that the offshore MT data can only be explained if moderately resistive material reaches down to at least 20 km of the subsurface. When focusing on the TM mode phases, ignoring the TM mode resistivities, a comparable model to the TE mode inversion can be obtained.
3D modeling the bathymetry
In marine MT, bathymetry plays a far more important role than topography in onshore studies. To assess the results of the 2D inversion in this respect, we performed
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3D forward modeling using ModEM [Egbert&Kelbert, 2012; Meqbel, 2009] to estimate the effect on the data (apparent resistivity and phase curves) along our profile offshore.
Fig. S7 shows the complex bathymetry of the Walvis Ridge together with the locations of six exemplary sites. The 3D mesh design is described in the main text of the paper and shown in Fig. S8. The modeling results are summarized in Fig. S9 and presented together with the data (symbols). The responses (apparent resistivities and phases) shown as dashed lines of the 3D model which consists only of a homogeneous half space (1,000
 m) and the ocean (0.3
 m) clearly shows the effect of the bathymetry and a strong period dependency. Obviously though, these curves do not resemble the offshore data (blue and red dots in Fig. S9). After adding a layer of higher conductivity (1
 m) close to the seafloor the general shape of the apparent resistivity and phase curves is much more similar to the measured and modeled offshore data.
Since the calculated transfer functions of this simple 3D model (solid lines) can reproduce the trend of the data and the 2D inversion results, we conclude that the bathymetry can be described in 2D. Hence 2D inversion of the offshore data along the
Walvis Ridge is justified and meaningful.
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Figure S1. Real induction vectors (Wiese convention) in northern Namibia. a) At long periods (724 s) sites in the western part of the Kaoko Belt are dominated by the transition from the resistive land to the highly conductive saline water of the Atlantic
Ocean. b) At a period of 1 s the induction vectors reflect the distribution of more local crustal structures of the Kaoko Belt.
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Figure S2. Phase tensor ellipse map to illustrate the dimensionality along the 2D profile
(from West to East). Ellipses which appear as circles are consistent with a 1D subsurface. Red, blue and black colors mark high beta (skew) values (beta > 3 or beta
<-3) which are indicative for 3D structures.
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Figure S3. : Regularization parameter tau. The best compromise between model smoothness (low roughness) and data fit (low RMS error) is found at the kink of the Lcurve for tau = 5.
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Figure S4.
2D inversion model of the offshore-only data. Top: TE- mode (RMS = 2.83); bottom: TM- mode (RMS = 1.64).
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Figure S5 : 2D Inversion of the onshore-only data Top: TE- mode (RMS = 2.83); bottom:
TM- mode (RMS = 1.77)
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Figure S6: Offshore constrained inversion study: a) Dashed black lines in 8.2 km,
11.8 km, 17.0 km and 20.8 km indicate depth levels below which resistivities of either
800
 m (Fig. S6a) or 1500
 m (Fig. S6b)) were inserted into the preferred TE+VTF offshore model and locked (black dots). The image shows the inversion result after additional 24 iterations until the inversion was terminated because the final RMS was reached. b) Inversion result of a resistive oceanic lithosphere of 1500
 m below 8.2 km depth. c) Total RMS values of the constrained TE mode + VTF inversions are listed for the inclusion of a resistive lithospheric body at varying depths. d) Dashed black lines in
8.2 km, 11.8 km, 17.0 km and 20.8 km indicate depth levels below which resistivities of either 800
 m (Fig. S6d) or 1500
 m (Fig. S6e)) were inserted into the preferred
TM+VTF offshore model and locked (black dots). The image shows the inversion result after additional 50 iterations until the inversion was terminated because the final RMS was reached. f) Total RMS values of the constrained TM+VTF inversions .
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Figure S7. : Colors indicate the bathymetry of the Walvis Ridge. Black dots indicate MT station locations. Red asterisks and numbers mark stations for which measured transfer functions and model responses are presented below.
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Figure S8. : Simple 3D mesh to model of the bathymetry of the Walvis Ridge (see also
Fig. S6). left) Map view, right) vertical slice along the red line in a). See text for model variations.
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Figure S9. : Observed MT data (symbols) at 6 exemplary sites are shown together with calculated model responses of a 3D model consisting of a homogeneous half space with ocean bathymetry (dashed line) and an additional conductive layer representing sea water saturated sediments (solid line).
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