Global maps of the magnetic thickness and magnetization of the Earth’s lithosphere
 Foteini Vervelidou^{1}Email author and
 Erwan Thébault^{2}
https://doi.org/10.1186/s4062301503295
© Vervelidou and Thébault. 2015
Received: 30 June 2015
Accepted: 14 September 2015
Published: 26 October 2015
Abstract
We have constructed global maps of the largescale magnetic thickness and magnetization of Earth’s lithosphere. Deriving such largescale maps based on lithospheric magnetic field measurements faces the challenge of the masking effect of the core field. In this study, the maps were obtained through analyses in the spectral domain by means of a new regional spatial power spectrum based on the Revised Spherical Cap Harmonic Analysis (RSCHA) formalism. A series of regional spectral analyses were conducted covering the entire Earth. The RSCHA surface power spectrum for each region was estimated using the NGDC720 spherical harmonic (SH) model of the lithospheric magnetic field, which is based on satellite, aeromagnetic, and marine measurements. These observational regional spectra were fitted to a recently proposed statistical expression of the power spectrum of Earth’s lithospheric magnetic field, whose free parameters include the thickness and magnetization of the magnetic sources. The resulting global magnetic thickness map is compared to other crustal and magnetic thickness maps based upon different geophysical data. We conclude that the largescale magnetic thickness of the lithosphere is on average confined to a layer that does not exceed the Moho.
Keywords
Background
The magnetic field on the Earth’s surface results from the superposition of various sources, both of internal and external to the Earth origin. The surface field is dominated by the field generated within Earth’s outer core (see, e.g., Hulot et al. 2015); however, there are important contributions from the magnetized rocks of Earth’s lithosphere (for reviews on the lithospheric magnetic field see, e.g., Purucker and Whaler 2015; Thébault et al. 2010). The core and the lithospheric magnetic field overlap in the spectral domain. The core field dominates the power spectrum over spherical harmonic (SH) degrees 1 to 15 (Langel and Estes (1982)) approximately, and it is only at larger degrees that the lithospheric field becomes the primary contributor.
As a consequence of the dominant role of the core field, the largescale lithospheric magnetic field corresponding to SH degrees 1 to 15 cannot be recovered through magnetic field measurements alone. To overcome this constraint, studies inferring the thickness and magnetization of the magnetic sources of the lithospheric field have relied on a priori information (e.g., Purucker et al. 2002; Purucker and Whaler 2004). Specifically, these studies assume that over SH degrees 1 to 15 the Moho, a seismic discontinuity that separates the crust from the mantle, is also a magnetic boundary. This approximation is supported by studies that have shown the mantle to be mainly depleted in magnetic minerals (e.g., Wasilewski and Mayhew 1982). However, recent studies, e.g., Ferré et al. 2013; 2014, Friedman et al. 2014; MartinHernandez et al. 2014, propose that there is evidence in some areas for a magnetized upper mantle. This suggests the lower boundary of the magnetic thickness does not necessarily coincide with the Moho. In this study, the term magnetic thickness refers to the part of the lithosphere that contributes to the observable magnetic signal, as opposed to the seismically defined crustal thickness.
New ways of recovering Earth’s largescale magnetic thickness and magnetization, that do not rely on the assumption that Moho is a magnetic boundary, would provide some independent assessment. This in turn could contribute to studies on the composition and the thermal state of Earth’s lithosphere (e.g., Fox Maule et al. 2005), since the minerals are only magnetized down to the Curie isotherm.
We propose such a methodology in this paper. We choose to work in the spectral domain and rely upon statistical assumptions about the nature of lithospheric field sources, along the lines of studies seeking either to model the large scale lithospheric field or to infer information about magnetic thickness and magnetization. Such studies have been carried out either on a global scale by means of Spherical Harmonic Analysis (SHA) (e.g., Jackson 1990, 1994; Korte et al. 2002; O’Brien et al. 1999; Voorhies 1998; Voorhies et al. 2002; Voorhies 2008) or on local scales using planar geometry (e.g., Bouligand et al. 2009; Li et al. 2013; Maus et al. 1997). However, existing global SH statistical expressions do not represent accurately recent lithospheric field models, and planar geometry is not well suited for largescale studies (e.g., Langel and Hinze 1998, chapter 5.3.2 or Vervelidou 2013, Figure 5.27b). Recently, Thébault and Vervelidou 2015 proposed a statistical SH expression for the power spectrum of Earth’s lithospheric magnetic field that improves the fit to recent lithospheric field models while also being in agreement with global forward models of continental and oceanic magnetization.
In this study, we make use of this expression to construct global maps of Earth’s largescale magnetic thickness and magnetization. In section “Surface spherical cap harmonic power spectrum”, we use the RSCHA methodology (Thébault et al. 2004, 2006, Thébault 2008) and derive an expression for the surface spherical cap harmonic power spectrum, which we show can be directly related to any SH power spectrum. In section “A statistical expression for the Earth’s lithospheric magnetic field”, we present briefly the statistical expression of Thébault and Vervelidou 2015. In section “Synthetic analyses”, we combine this expression with synthetic spherical cap spectra and demonstrate the ability to provide magnetic thickness and magnetization estimates through a misfit analysis. In section “Global models of the Earth’s mean magnetic thickness and magnetization”, we calculate the observational spherical cap spectra based on the satellite, aeromagnetic and marinebased NGDC720 lithospheric field model (Maus 2010), and we present the magnetic thickness and magnetization maps derived for the Earth. We note that a similar approach, based on the statistical expression of Voorhies et al. 2002 and the spatiospectral localization techniques of Dahlen and Simons 2008; Wieczorek and Simons 2007, was followed by Lewis and Simons 2012 for inferring the crustal magnetic thickness of Mars. Finally, we transform our magnetic thickness map into spherical harmonics and compare it to the SH power spectra of other crustal and magnetic thickness models.
Methods
Surface spherical cap harmonic power spectrum
Definition
with V _{1} and V _{2} being the solutions of the boundary values problems 2 and 3.

NeumannNeumann Boundary Value Problem

DirichletNeumann Boundary Value Problem
The second boundary values problem differs from the previous one as it considers Dirichlet instead of Neumann boundary conditions on the lateral boundary of the cap, \(\partial \Omega _{\theta _{0}}\). This problem is obtained by setting \(\delta _{\theta }^{1}=\delta _{\theta }^{2}={\gamma _{r}^{1}}={\gamma _{r}^{2}}=1\) and \(\gamma _{\theta }^{1}=\gamma _{\theta }^{2}={\delta _{r}^{1}}={\delta _{r}^{2}}=0\) in Eqs. 2 and 3. With these boundary conditions, the potential V _{1} has formally the same form as the one given in Eq. 5 but is numerically different. The degrees n _{ k } are now the roots of \(P_{n_{k}}^{m}(\cos \theta _{0})=0\) rather than that of \(\left.{ \frac {dP_{n_{k}}^{m}}{d\theta }}\right \vert _{\theta _{0}}=0\), as is the case for the Neumann boundary condition. The solution V _{2} is identical to Eq. 8.
where, according to the common convention, coefficients \(H_{n_{k}}^{i,m}\) (\(H_{n_{k}}^{e,m}\)) stand for the coefficients \(G_{n_{k}}^{i,m}\) (\(G_{n_{k}}^{e,m}\)) of Eq. 5 with negative order m. By further normalizing the expression (10) by \(S_{\vartheta \Omega _{\rho }}=2\pi \rho ^{2}\left (1\cos \theta _{0}\right)\), the area of the surface 𝜗 Ω _{ ρ }, we obtain the estimation of the mean square magnetic field.
Relationship to a spherical harmonic power spectrum
Contrary to the case of the Spherical Harmonic Analysis (SHA), where the whole sphere is under consideration, in the RSCHA only part of the sphere is under consideration. This has direct consequences for the spectral resolution of the RSCHA. It is known that the higher the concentration of a function in the space domain, the poorer its spectral resolution (e.g., Wieczorek and Simons 2005). In the case of SHA, the spectral resolution, i.e. the distance between two consecutive terms of its power spectrum, is equal to one (n _{ i+1}−n _{ i }=1). In RSCHA, the spectral resolution depends on the cap’s halfangle, θ _{0}, and is equal to π/θ _{0}, where θ _{0} is given in radians. The larger the halfangle θ _{0}, the smaller the distance between two consecutive terms of its power spectrum becomes and therefore the better the spectral resolution. For the limit case where θ _{0}=π, we recover the spectral resolution of the SHA.
with ρ the radius of the spherical surface where we estimate the wavelength.
where θ _{0} is given in radians. This expression shows that any regional and global power spectra, experimental or theoretical, despite their differing spectral resolution, can be readily compared with each other.
The implicit assumption
The assumption that the contribution of the Mehler basis functions to Eq. 9 is negligible, puts us in a specific framework. The Mehler basis functions are essential for ensuring the upward/downward continuation of the magnetic field (or conversely for processing data measured at various altitudes). Moreover, they are required when the magnetic field has largescale features and has therefore non zero components on the lateral boundaries of the spherical cap (Thébault et al. 2004). Prior to interpreting Eqs. 10 or 11 as a complete power spectrum, and not as representing the energy of one particular set of basis functions, some investigations are required on the consequences of this assumption.
This equality requires that the radial component of the field at the lower and upper boundary of the spherical cap averages to the same value. Due to the decrease of an internal potential field with increasing altitude, this requirement can only hold true if the radial component of the field averages to zero over every spherical cap surface 𝜗 Ω _{ ρ }, with a≤ρ≤β. This, again, is correct to a good approximation when the magnetic field under study has wavelengths smaller than the cap’s aperture. Possible applications of the power spectrum therefore include studies of lithospheric magnetic fields, whose large scales are currently not considered in magnetic field modelling due to the masking effect of the core field. Such conditions are common in spectral analyses that seek to avoid aliasing and leakage, in particular when Fourier transform is applied to magnetic anomaly data, e.g., for upward/downward continuation purposes (see, e.g., Blakely 1996).
Consequently, Eq. 10 can be interpreted as a spatial power spectrum under the condition that the signal under study averages to zero within the spherical cap. The fulfilment of this condition should be investigated casebycase. In this respect, we note that Hwang and Chen 1997 proposed a similar expression to Eq. 10 for working with sealevel data. However, they did not discuss the implicit assumptions behind neglecting the Mehler basis functions and therefore the applicability of this expression to their context. They applied the expression to a case where the signal does not average out within the cap, thus constructing aliased power spectra.
Once the conditions are fulfilled to neglect the contributions of the Mehler basis functions, it is wise to decide whether the spectral analysis should be performed with Legendre functions computed with the Neumann or with the Dirichlet boundary condition. Since the Neumann boundary condition has the advantage to converge faster (see, e.g., Thébault et al. 2004), it should be preferred.
However, we do not advise to perform regional modelling of magnetic field measurements available at different altitudes using only the Legendre basis functions (or to apply the SCHA as proposed originally by Haines 1985). Both the Legendre and the Mehler basis functions are needed for an accurate representation of the field in the spatial domain. Such a representation becomes possible by solving the DirichletNeumann BVP (Thébault et al. 2006). This BVP is advantageous in terms of the inverse problem because the gradient of its basis functions are orthogonal in the volume of the spherical cap. In this case, the Legendre basis functions are computed with the Dirichlet boundary condition, and they can be used to calculate the regional power spectrum.
In this paper, we solve the Neumann boundary condition for the Legendre basis functions, making sure the signal under study is close to zero on average. We do not, however, interpret the model in the physical space. For this, the complete solution, based both on the Legendre and the Mehler basis functions would be necessary.
A statistical expression for the Earth’s lithospheric magnetic field
where lat stands here for the latitude of the center of the cap.
where F _{ n }(ε), C _{ n } and F _{ latitude } are given by Eqs. 16, 17, and 18, respectively.
By a misfit analysis between the regional power spectrum, as described by Eqs. 10 and 11 and the statistical expression of Eq. 19 it is possible to carry out regional analyses and estimate for any spherical cap the mean magnetization and the mean magnetic thickness. In the following, all regional spectra, observational or statistical, have been downscaled by the factor \(\frac {\pi }{\theta _{0}}\) (according to Eq. 13).
Results
Synthetic analyses
where R _{ n } is the regional power spectrum of the synthetic data and E{R _{ n }} its expected value as given by Eq. 19.
The coestimation of the three unknown parameters leads to some uncertainty as they are not easy to separate (see also Bouligand et al. 2009; Maus et al. 1997). Therefore, we set the susceptibility power law to 1.36 for the whole globe (in accordance to the preferred estimated mean global value of Thébault and Vervelidou 2015) and compute the value of χ ^{2} for a variety of values for the parameters ε and M. The magnetic thickness ε is allowed to vary between 0 and 80 km and the magnetization M between 0 A/m and 2 A/m. This investigation provides us with the 2D function χ ^{2}(ε,M), whose minimum corresponds to the maximum likelihood values for the mean magnetic thickness and the mean magnetization.
The mean value of the 2000 regional mean magnetic thickness estimations is 39.3 km, with a standard deviation due to regional variability of 4.3 km. The mean value for the mean magnetization is 0.94 A/m with a standard deviation of 0.2 A/m. The expected values lie within the uncertainty of the estimated values, despite a small bias in the estimated maximum likelihood.
From this figure we also see that the maximum of the power spectrum moves to higher degrees as the magnetic thickness reduces (see also Thébault and Vervelidou 2015). Moreover, the maximum of the spectrum becomes smoother and spreads over a larger bandwidth. We estimate the 10 km magnetic thickness to be the lower limit of our resolution. Observational spectra of higher harmonic degree would be necessary to detect thinner magnetic layers as discussed by Thébault and Vervelidou 2015.
Global models of the Earth’s mean magnetic thickness and magnetization
Data
We follow the same methodology as in section “Synthetic analyses” to infer the largescale mean magnetic thickness and magnetization for Earth. To achieve this, we need global magnetic field measurements to a high spatial resolution. Many aeromagnetic, marine, and satellite magnetic measurements have been collected worldwide and merged together into the World Digital Magnetic Anomaly Map (WDMAM) (see Khorhonen et al. 2007 for its first version and Dyment et al. 2015a, 2015b for the recently published version 2.0, which can be downloaded at www.wdmam.org). The WDMAM results from vast international efforts to represent globally lithospheric magnetic field structures at a wide range of spatial scales. Three different grids were built as candidates for the first edition of the WDMAM (Hemant et al. 2007; Hamoudi et al. 2007; Maus et al. 2007). Differences between these candidate grids, which rely on different processing and merging techniques, show that the WDMAM, although selfconsistent, is imperfect. An attempt to improve, this grid was proposed by Maus et al. 2009 who constructed the EMAG2 grid using updated aeromagnetic and marine compilations and adding a priori information in the oceanic domain. We choose not to use the WDMAM or the EMAG2 scalar grid to compute the regional power spectra, as the computation of regional models from scalar data only proved to be unstable near the equatorial regions. This is a result of the Backus effect and of the fact that the radial component of Earth’s main magnetic field cancels out at the magnetic equator. Therefore, we use the NGDC720 model (Maus 2010), built upon the EMAG2 grid, for which this problem was addressed by a global regularization. The NGDC720 model is a hybrid construction that uses the MF6 lithospheric magnetic field model (Maus et al. 2008) derived from CHAMP satellite (Reigber et al. 2002) measurements for SH degrees 16 to 130 and then the EMAG2 grid up to SH degree 720. This model therefore offers a maximum horizontal spatial resolution of about 50 km. We refer to Thébault and Vervelidou 2015 for a discussion on the limitations of the EMAG2 grid and the NGDC720 model for carrying out spectral analyses.
A quality index for the global maps
where χ is given by Eq. 20 and N are the number of the terms of the regional spectrum.
The regional spectrum over Antarctica, a region where EMAG2 has a sparse aeromagnetic coverage (see Fig. 7 of Maus et al. 2009), presents an abrupt drop of energy after degree 130. This pattern is similar to the cutoff illustrated in Fig. 1. We attribute the drop in power to the lack of nearsurface data. The immediate consequence of this socalled spectral gap is that the large degrees are artificially low. This drop off causes an underestimation of the mean magnetization. At the same time, the comparatively large energy at low degrees leads to an overestimation of the magnetic thickness. Consequently, estimations from such regions should not be considered.
Another example of a poor fit is around the North Pole. Figure 8 shows the spectrum of a cap located over central Greenland. We see again a drop of energy for degrees larger than 130. However, Greenland has good aeromagnetic coverage (see again Fig. 7 of Maus et al. 2009), suggesting this feature could be genuine. This would indicate the statistical expression for the power spectrum is locally not in agreement with the statistical properties of the magnetic sources. Alternatively it could indicate a poor spectral content of the respective aeromagnetic grids. Interestingly, this spectrum is also characterised by a drop of energy between degrees 50 and 100. This, again, could be a genuine feature. But, it could also be related to how the satellite data were corrected for the external fields along the satellite orbits or how the underlying MF6 model was regularized (see, e.g., Lesur and Maus 2006; Maus et al. 2008; Thébault et al. 2012). In this respect, we note that over the same cap satellitebased magnetic field models more recent than MF6 (Maus et al. 2008) e.g., MF7, GRIMM (Lesur et al. 2013), CM5 (Sabaka et al. 2015) are more energetic over the same bandwidth, although still less energetic than predicted by the statistical expression.
Another continental example of mismatch is found in Russia, over the broader area around the Kursk magnetic anomaly. For this major magnetic anomaly (see, e.g., Taylor et al. 2003), the underlying hypotheses of the statistical expression may not hold.
A final example of a spectrum with low QI is for a cap located over the Eastern Pacific Ocean. This spectrum is in clear disagreement with the form given by the statistical expression. Further caps, mainly located over oceanic regions, exhibit this behaviour. Testing different values for the power law γ does not improve the fit. This mismatch could result from the extrapolation scheme used to fill in oceanic areas void of direct marine measurements (see Maus et al. 2009).
The above examples illustrate the spectral variability of the NGDC720 model, which depends on a multitude of factors. Lack of aeromagnetic and marine data, incomplete data processing, and/or genuine geophysical features are all possible sources of incompatibility between the observational spectra and the statistical expression. Clearer information about the underlying datasets would help to discern the origin of the mismatch on a casebycase basis. In any case, for the purposes of our study, we find that the QI index suitably accounts for various possible sources of incompatibility, and we use it as an indicator for the degree of reliability of the magnetic thickness and magnetization estimates.
The maps
In Fig. 12, we see that the spectrum carries energy from SH degree 0 to 36. The maximum SH degree that we can recover is dictated by the spatial distribution of the spherical caps over Earth’s surface. The more caps, the denser the spatial sampling of the magnetic thickness and therefore the higher the recovered maximum degree. For our distribution of 2000 caps, the maximum SH degree is equal to 36. Whether the thickness map has energy or not up to that degree depends on the cap’s halfangle, since it is this parameter that defines the actual spectral content of our map. For the extreme case of θ _{0}=π, the magnetic thickness would be a constant for the whole Earth and so only SH degree 0 would carry energy. For θ _{0}=15, which is our case, the thickness map carries energy up to degree 36. Even more interestingly, it also carries energy over SH degrees 1–16, which correspond to wavelengths that are absent from the known lithospheric magnetic field due to the masking by the core field.
Discussion
The most prominent feature of the derived magnetic thickness map (Fig. 9) is the dichotomy between the oceans and the continents. The oceans are characterised by a smaller magnetic thickness than the continents. This is in agreement with our current understanding about crustal thickness (see, e.g., the CRUST1.0 model, Pasyanos et al. 2014). This dichotomy is visible in the map as its spectral content starts from very low SH degrees.
In Fig. 12, we show the power spectra of two other models: (1) the seismic crustal model CRUST1.0 (Pasyanos et al. 2014, http://igppweb.ucsd.edu/˜gabi/crust1.html) (magenta line) and a recent version (pers. comm. M. Purucker) of the magnetic thickness map based on the methodology of Purucker et al. 2002, which was applied over Antarctica by FoxMaule et al. 2005 and on a global scale by FoxMaule et al. 2009 (cyan line). The second model is based on the 3SMAC seismic model (Nataf and Ricard 1996) for the SH degrees up to 15 and on the lithospheric magnetic field model MF7 (see Maus et al. 2008 for the latest published version of the MF series) for SH degrees larger than 15. We see that our model follows very closely the spectra of the other two models from SH degree 11 up to degree 18. It is, however, somewhat more energetic than the other models from SH degree 1 up to SH degree 10. This probably results from the overestimation of the magnetic thickness over large areas such as Antarctica, Africa, and the Arctic (see section “A quality index for the global maps” for an explanation). A greater data coverage will allow more realistic estimates for the magnetic thickness, and we believe it will lead to a decrease in the energy of our power spectrum over these wavelengths.
From SH degree 19 and onwards, the spectrum of our model follows closely the spectrum of Fox Maule et al. 2009, while both of them are a little more energetic than the spectrum of CRUST1.0. This could indicate the depth of Moho is actually shallower than the depth of the Curie isotherm over this waveband, and that parts of the upper mantle are magnetized. However, both the spectrum of the hybrid model of Fox Maule et al. 2009 and of the crustal thickness model CRUST1.0 lie within the errors of the spectrum of our model, indicating that these two models are statistically equally energetic to ours.
Comparing our magnetic thickness and magnetization maps, we observe differences in their main features. There are three particularly notable examples. Australia’s magnetic thickness pattern is almost reversed with respect to its magnetization pattern. The low magnetization feature over China and parts of Indonesia does not correspond to a particular feature on the magnetic thickness map. The same is true for the low magnetization feature that covers almost all of South America. Interestingly, the low magnetization pattern over South America has also been observed by Purucker and Whaler 2004 (see also Purucker and Whaler 2015). Due to their methodology (they transform the MF3 lithospheric field model (Maus et al. 2006) to a minimum magnetization map (see Whaler and Langel 1996), while considering the magnetic thickness to be constant), their magnetization map is expected to resemble more closely our VIM map rather than our magnetization map. Indeed, comparing their magnetization map (see Fig. 10, Purucker and Whaler 2015) with our VIM map (Fig. 11), we find common features that include the signatures of the Bangui and Kursk magnetic anomalies, but also, e.g., an anomaly over central Australia and a dichotomy in the signals of North and South America. Figure 11 b also shows an apparent dichotomy between West and East Antarctica. This feature could have important implications for inferring the thermal state of the crust in Antarctica and is in close agreement with previous works (see, for instance Fig. 1 in Fox Maule et al. 2005).
Conclusions
In this study, we present the surface spherical cap harmonic power spectrum, based on the RSCHA modelling method (Thébault et al. 2006). It can be directly interpreted in terms of wavelengths, in the same way as the SHA power spectrum. Such a regional spectrum can have various applications that go beyond magnetic field studies, as long as the signal under study averages to zero over the spherical cap surface. It can also be applied on planets other than Earth (see for example Lewis and Simons 2012 for an application of a different regional spectrum (Dahlen and Simons 2008; Wieczorek and Simons 2007) for Mars).
Here, we use this regional power spectrum to calculate global maps of the large wavelengths of the mean magnetic thickness and mean magnetization of Earth. We do so by combining spherical cap observational spectra with a recently proposed statistical expression for the power spectrum of Earth’s lithospheric magnetic field (Thébault and Vervelidou 2015). This expression depends on the mean magnetic thickness, the mean magnetization and a power law of the susceptibility power spectrum. The regional observational spectra are obtained by inversion of the NGDC720 lithospheric field model (Maus 2010) over a series of spherical caps, covering the whole Earth.
The NGDC720 model relies on information from satellite, aeromagnetic, and marine magnetic measurements. The resulting regional power spectra therefore cover a broad range of wavelengths. The first outcome of our study is that by exploiting simultaneously satellite and near surface information, separate estimates of the mean magnetization and mean magnetic thickness are provided.
However, our estimates are not equally reliable globally. For this reason, we accompany them with a quality index, called QI (see Fig. 6). This variability stems mainly from incomplete, and inhomogeneous in terms of quality and resolution, nearsurface data coverage. For regions, however, with dense coverage of good quality nearsurface data, a low QI can act as an indicator that the regional properties of the magnetic sources are different from the assumptions underlying the statistical expression of Eq. 15, e.g., the assumption of purely induced magnetization. Consequently, a second outcome of our study, as illustrated through Figs. 7 and 8, is that the RSCHA, and its power spectrum offer a way to study and evaluate regionally the spectral content of data compilations and models over a large bandwidth. This can be very useful for regional magnetic studies. The WDMAM project (see Dyment et al. 2015a, 2015b for its recently published version 2.0, which can be downloaded at www.wdmam.org) and regional efforts to enrich the available nearsurface data sets (e.g., Gaina et al. 2011; Golynsky et al. 2013), provide currently a favourable framework for such studies.
Maps of magnetic thickness and magnetization can be valuable in understanding further the temperature, composition, and structure of Earth’s lithosphere. Efforts to produce such maps are hindered because of the masking effect of the core field on the large scales of the lithospheric magnetic field. A third outcome of our study, as illustrated by Fig. 12, is the feasibility of estimating the large scales of the magnetic thickness and magnetization using magnetic data. Keeping in mind the uncertainties involved, we conclude from Fig. 12 that the large scales of the magnetic thickness are on average confined to a layer that does not exceed the Moho. Updates to our maps (Figs. 9 and 10), and consequently to the SH power spectrum shown in Fig. 12, are especially welcome, as the worldwide compilation of magnetic data gets richer, both in the spatial and the spectral domain. Both the SWARM satellite mission (see, e.g., Olsen et al. 2013) and the continuous updating of the WDMAM move in this direction.
Declarations
Acknowledgements
The authors would like to thank three anonymous reviewers, the handling editor Prof. Toshi Yamazaki and the editor in chief, Prof. Yasuo Ogawa, for their constructive comments that helped ameliorate the manuscript, V. Lesur for useful discussions and M. Brown for an early review of the manuscript. FV was partly funded by Région Îlede France and partly by the DFG SPP1488 PlanetMag. ET was partly funded by the Centre National des Etudes Spatiales (CNES) within the context of the project of the “Travaux préparatoires et exploitation de la mission Swarm” and partly by INSU through the “Programme National de Planétologie”. Figures 6, 9, 10 and 11 were drawn with the Generic Mapping Tools (Wessel and Smith 2001).
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.
Authors’ Affiliations
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