A relatively dry mantle transition zone revealed by geomagnetic diurnal variations

Abstract

The distribution of water within the mantle transition zone (MTZ) has important implications for the material circulation and partial melting of the mantle. Although solubility of hydrogen is very high, leading to speculations that the MTZ plays a key role in the deep-Earth water cycle, the actual water content remains an open question. Electrical conductivity of mantle minerals is very sensitive to water content, so reliable estimates of this physical parameter in the MTZ would provide valuable constraints. Here, we use recently developed joint inversion of geomagnetic diurnal variation for realistic source structure and one-dimensional mantle conductivity profile. Synthetic tests show that the resulting profile is a reasonable proxy for the electrical conductivity distribution of continental mantle over depths where model resolution is best (200 to 600 kilometer), even in the presence of lateral heterogeneity. The inferred water concentration in the MTZ is 0.03 weight %, one to two orders of magnitude below the solubility of wadsleyite and ringwoodite.

The conductivity profile for continental mantle derived from geomagnetic data indicates a relatively dry upper mantle.

INTRODUCTION

Water (or more precisely hydrogen), distributed as point defects in the nominally anhydrous minerals that make up the bulk of the mantle (1), can modify rheological properties (2) and melting relationships of minerals (3), with important implications for the dynamics and geochemical evolution of Earth (4). Although it is broadly agreed that water can be transported deep into the mantle by subducting slabs and returned to the surface by convective upwelling (5), quantitative estimates of water content remain elusive (6). In particular, the water solubility of wadsleyite and ringwoodite is significantly enhanced (7, 8), making the mantle transition zone (MTZ) a potentially key player in the global deep water cycle (9), but estimates of MTZ water content vary from nearly saturated [1 to 2 weight % (wt %)] (10, 11) to nearly dry (≤0.1 wt %) (12, 13). These large differences can be attributed to variations in sensitivity and reliability of the estimation approaches used (14) and to potentially strong mantle heterogeneity implied by geophysical observations (15, 16).

Given the high sensitivity of electrical conductivity to the concentration of hydrogen in mantle minerals (6, 17), reliable estimates of this physical parameter could provide valuable constraints on water content. Measurements of natural variations of geomagnetic fields, due to ionospheric and magnetospheric current systems that induce image currents deep in Earth, can be used to infer mantle electrical conductivity (18–22). These magnetovariational induction methods, which require accurate models of the external source fields, have been most successfully applied to long-period (> 2 day) disturbance storm time (Dst) variations, since their source can be reasonably modeled by a relatively simple magnetospheric ring current (18). However, Dst data at these long periods are primarily sensitive to conductivity of the lower mantle (23). Imaging of depths from 200 to 600 km requires variations with periods of several hours to 1 day (24), i.e., the diurnal variations (DVs) of Earth’s magnetic field, which are generated by dynamo currents flowing in the ionospheric E and lower F region as two asymmetrical horizontal current vortices on the sunlit side of Earth (25). Driven by solar radiation, the DV magnetic fields have a fundamental period of one solar day, along with several harmonics, and exhibit solar-cycle, seasonal, and day-to-day variability (26).

Using DV signals to infer conductivity is challenging, primarily not only due to the complex geometry of near-Earth ionospheric source fields but also due to lateral variations of conductivity in the upper mantle. Previous studies used either carefully selected data for solar quiet (Sq) equinoctial days (27, 28), for which simplified models of source structure are at least tenable, or the mean value of day-to-day asymptotic response, which aims to average the effect of source variability (29). In most cases, single-station data were inverted for local one-dimensional (1D) conductivity profiles (16, 28, 29) or array data for regional models (19, 30). Here, we use a more realistic and complete source model for DV fields and invert geomagnetic observatory data for a globally averaged conductivity-depth profile. Extensive numerical experiments have been performed to investigate the uncertainty and robustness of the model due to 3D deviations from the assumed 1D Earth, to errors in estimated sources, and to inherent nonuniqueness of the inverse problem. The averaged water content of the upper mantle and MTZ is calculated on the basis of the preferred conductivity profile using the available laboratory results (6, 17) and mantle temperature model (31). Our results are most representative of continental mantle in the Northern Hemisphere (Europe, Asia, and North America) since the distribution of observatories is sparse in the Southern Hemisphere and oceans. Throughout this paper, “upper mantle” will refer to the depth range of 200 to 410 km, while upper and lower MTZ will denote depths of 410 to 520 km and 520 to 660 km, respectively.

RESULTS

Globally averaged conductivity-depth profile

Our source modeling approach (32) combines geomagnetic observatory data (33) with a data-constrained, physically realistic model of ionospheric source magnetic fields, the thermosphere ionosphere electrodynamics general circulation model (TIEGCM) (34). Initial data processing uses frequency domain principal components analysis (PCA) to extract the dominant modes of spatial variability in DV magnetic signals at frequencies of 1 to 4 cycles per day (cpd) during geomagnetic quiet time (Kp index smaller than 1 in this work). Each of these spatial modes represents the total (external source plus induced internal) magnetic fields sampled at the 127 observatory locations used in the PCA. We estimate the source spatial structure for each spatial mode by fitting to global basis functions derived from statistical analysis of TIEGCM outputs, with corresponding internal magnetic fields computed for a prior Earth conductivity model consisting of a thin heterogeneous sheet (representing the ocean/continent conductivity contrast) over a 1D Earth (35). Once we have estimated the source, we use a conventional 1D electromagnetic (EM) induction inversion (36), based on the thin-sheet model, to improve data fit by refining the 1D mantle conductivity profile. We focus on the first five dominant PCA modes at 1 to 4 cpd with high signal-to-noise ratio. While three magnetic components at all sites are used for the ionospheric source estimate, vertical magnetic components at selected sites (see Materials and Methods) are used for Earth conductivity inversion to minimize effects of complex source structure near electrojets and near-surface 3D conductivity complications (e.g., near coasts) that may be poorly modeled by our global scale thin-sheet model.

This procedure, alternating between inversion for source and for conductivity, is iterated to refine the source estimate and Earth conductivity model. Convergence of the iterative scheme, initialized from two very different 1D conductivity profiles (21, 23), is illustrated in Fig. 1. Results for both starting models after several rounds of source-conductivity refinements are virtually indistinguishable. Note that the conductivity inversion model roughness is penalized by the first derivative operator. The normalized root mean square (nRMS) data misfit of the converged smooth model, computed for the 570 complex data, is 1.54 using error assumptions described in Materials and Methods. The corresponding ionospheric source model (fig. S1) recovers the Sq current system at mid-latitude, auroral electrojet at high latitude, and equatorial electrojet at low latitude, which agrees with the understanding of physics of DV fields (26). The linearized resolution matrix for the preferred model (fig. S2) is concentrated on the diagonal between 200 and 600 km in depth with a typical width of 200 km, indicating that the preferred model is some smooth version of true mantle conductivity. Below 600 km, the strong off-diagonal components show that the model is insensitive to structure below this depth and biased toward the relatively low value of the upper MTZ. This demonstrates that average conductivity in the upper mantle and upper MTZ are well resolved, but the lower MTZ is less well constrained.

Fig. 1. Global conductivity-depth profile based on DV magnetic fields inversion.

Three rounds of source-conductivity refinement are presented, showing convergence starting from models KS0 (21) (warm colors) and PK0 (23) (cool colors). Integers in labels denote iteration, followed by nRMS misfit. The profile labeled as KS3 is our preferred mantle conductivity model. MTZ, mantle transition zone.

Further computational experiments—included varying data selection criteria (quiet versus active conditions, subsets of observatories or spatial modes used), modifications in the regularization, and constrained inversion tests—are summarized here with additional details in figs. S3 and S10. Inverted conductivity profiles obtained with all choices of datasets are consistent, always exhibiting a nearly homogeneous structure between 250 and 500 km in depth, with deviations from the converged model within roughly a factor of 2 at all depths. We test for possible discontinuities in electrical conductivity associated with mineral phase transitions (37, 38) by turning off the regularization smoothness constraints at depths of 410 and 520 km. Even with the regularization turned off, there is no obvious increase in conductivity at the top of the MTZ. At the 520-km discontinuity, turning off the smoothness constraint results in a jump of nearly one order of magnitude. However, this model shows a gradually decreasing conductivity from 300 to 520 km, required to maintain the well-constrained total conductance over the 300 to 600 km depth range. Although both models are compatible with observations, the smooth inversion result should be preferred, given that conductivity generally has positive correlation with temperature that increases with depth in the deep mantle (6). However, a jump in conductivity at the wadsleyite-ringwoodite transition, as suggested by laboratory results (6, 17), is consistent with the data, as shown by explicit constrained inversion tests discussed later (fig. S10).

DISCUSSION

The effect of mantle heterogeneity on inversion

An example of data fitting is provided in Fig. 2 (see also fig. S4). Misfits for horizontal and vertical magnetic field components have similar characteristics: Most observatories with large misfits are within the equatorial and auroral electrojet zones, near coasts, or on islands. As noted above, these were omitted from the 1D global conductivity inversion. Beyond these anomalous stations, some regional patterns in signed misfit for the vertical components correlate well with tectonic blocks, as exemplified by the opposite sign of relative error for groups of observatories on the Indian Shield and North China Craton. Since the amplitude of the vertical magnetic field is generally reduced for a more conductive Earth (39), sites with negative misfits suggest a more conductive underlying mantle and vice versa. The misfit plot thus suggests a more conductive mantle beneath the North China Craton than the Indian Shield, which is consistent with the high- and low-conductivity anomalies beneath eastern China and western Yangtze Craton adjoining the Indian Shield, respectively (22). The coherent spatial pattern in the signed misfit implies that there are strong lateral variations in the upper and mid-mantle conductivity. Since the vertical magnetic field used for conductivity inversion is susceptible to distortion by both deep and shallow lateral Earth conductivity gradients, the reliability of a 1D interpretation needs to be carefully examined. We thus tested our full source/conductivity inversion procedure on synthetic data generated with hypothetical 3D conductivity models.

Fig. 2. Difference between predicted and observed data.

The amplitude of the vertical magnetic field components for the first mode at 1 cpd, normalized by the amplitude of fields, is plotted. Warm colors indicate that the predicted amplitude is smaller than the observed. Note that all stations are used for the external source estimate, but only those marked with green outlines are used for the global 1D Earth conductivity inversion.

These 3D conductivity models, derived by modifying the global averaged profile in (21) (KS0 in Fig. 1), include four with checkerboard layers at varying mantle depths and one by scaling an averaged shear wave velocity model (fig. S5D) (40), were combined with the source field estimates to generate realistic synthetic datasets (see Materials and Methods). The simulation results show that the scattered fields induced by heterogeneity, which can exceed 10% of the primary fields, delineate lateral conductivity boundaries (fig. S6). The iterative two-step procedure, estimation of source structure followed by 1D inversion for mantle conductivity, was then applied to each synthetic dataset, starting from a completely different 1D conductivity profile [from (23), PK0 in Fig. 1]. Results are used to assess the recovery of both source (fig. S7) and average conductivity profiles in the presence of 3D conductivity variations.

For the conductivity inversion step, we tested both the global inversion approach outlined above (all reliable stations inverted simultaneously) and single-station inversions for local profiles. Performance of the inversion step is illustrated in Fig. 3, where we plot ratios of recovered/actual average conductivity for the two best resolved mantle layers: upper mantle and upper MTZ. The global inversions recover both layer averages to within better than a factor of 2. By contrast, the single-station inversions exhibit much larger divergence, with estimates for one or both layers frequently off by an order of magnitude or more, although in most cases, the data misfit is still acceptable. The problem is that the 1D inversion can falsely interpret the scattered fields associated with shallow lateral conductivity gradients in terms of deep structure (fig. S8). However, these localized 3D effects largely cancel out with simultaneous inversion of a global array of stations, allowing recovery of the average background model. There is some suggestion from our synthetic data experiments that the global 1D inversion is systematically biased to slightly lower conductivity in the presence of 3D structure, so it is possible that the true globally averaged profile is more conductive than the preferred model derived from real data. On the basis of inversion experiments (figs. S3 and S10) and the 3D synthetic data tests, we conclude that an upper bound on our estimates of upper mantle and upper MTZ conductivity is about a factor of at most 3 higher, while the lower bound is about a factor of 2 lower. Synthetic models with a lower MTZ conductivity increased by factor of 3 exhibit a steady rise of nRMS at most periods and modes (fig. S10C), so we suggest a similar uncertainty for this layer.

Fig. 3. Summary of results from global and local synthetic inversion tests.

Ratio of estimated σ_inv to actual σ_true layer-averaged conductivity is plotted, with horizontal axis for the upper MTZ layer (410 to 520 km) and the vertical axis for the upper mantle (200 to 410 km). For this comparison, σ_true is taken to be the geometric mean of mantle conductivity beneath all stations included in the inversion. Different colors correspond to different synthetic models with 3D structure (fig. S5) noted in legend. Symbols with black outlines denote global inversion results. Those without represent single-station inversion results for 40 randomly selected stations; note that some deviate from true underlying conductivity by more than two orders of magnitude and are not shown. The gray box denotes deviations from σ_true by less than a factor of 2.

Upper mantle water content

Average water content in the upper mantle and upper and lower MTZ layers can be estimated by comparing our conductivity model with profiles derived from high-pressure, high-temperature experiments for electrical conductivity of hydrous mantle minerals. This study uses laboratory results conducted by two independent groups, which are reviewed by Karato and co-workers (6) (K11) and Yoshino and co-workers (17) (Y13), respectively. Here, we assume a simplified pyrolytic model for mantle composition: 60% olivine + 20% pyroxene + 20% garnet for the upper mantle, 60% wadsleyite + 40% garnet for the upper MTZ, and 60% ringwoodite + 40% garnet for the lower MTZ. Figure 4A plots laboratory-based profiles under ambient pressure and temperature with some fixed water content that best explain the preferred conductivity model. The approach for calculating bulk conductivity, which incorporates water partitioning among coexisting minerals, is described in Materials and Methods.

Fig. 4. Water content in the upper mantle and MTZ.

(A) Conductivity-depth profile (black line), compared to laboratory-based profiles derived from mineral physics experiments on nominally anhydrous minerals with hydrogen (6, 17, 59, 60) using adiabatic mantle temperature profile (31). The pink dot-dashed line is the dry peridotite model SEO3 with quartz-fayalite-magnetite buffer (42). (B) Layer averaged bulk conductivity of minerals aggregate as a function of bulk water content and the averaged conductivity derived from preferred inversion profile (black line) with uncertainty (gray). From top to bottom: the upper mantle, upper MTZ, and lower MTZ. MORB, mid-ocean ridge basalt; OIB, ocean island basalt. (C) Conductance-depth profile of plausible inverted models derived from various data and inversion schemes. The black dotted line denotes the upper bound of conductance derived from preferred conductivity profile, accounting for estimated uncertainty. The thick purple lines denote calculated conductance based on laboratory studies (6, 17, 59, 60) with fixed water contents in upper mantle and MTZ: Solid line corresponds to best match to preferred conductivity profile, and dotted lines represent upper and lower bounds. For the lower bounds, both the upper mantle and lower MTZ are “dry” (0.001 wt % water). Olivine, ol; wadsleyite, wad; ringwoodite, ring; garnet, gt; and pyroxene, py; UM, upper mantle.

For the upper mantle (200 to 410 km), water content estimates derived from different experiments are similar: The averaged conductivity can be explained by olivine with 0.015 wt % water if the lower bound from (6) is used. The estimated water content agrees remarkably well with petrological evidence from mid-ocean ridge basalts (MORB) (Fig. 4B) (41). In the upper MTZ, on the basis of the K11 laboratory results, pyrolytic mantle with 0.01 to 0.15 wt % water can explain our conductivity profile in the upper MTZ within estimated uncertainties. A broader range, with up to 0.3 wt %, is obtained using the Y13 results. Both estimates are in fact consistent with 0.075 wt % water estimated for ocean island basalt (OIB) source mantle (41) and are more than an order of magnitude below the saturation level (7). In the lower MTZ layer, K11 results suggests a water content of 0.001 to 0.02 wt %, while the Y13 results overlap within our uncertainties, as long as the water content is lower than 0.3 wt %.

The water content estimate is also influenced by the mantle temperature profile (see fig. S9 for the conductivity of olivine, wadsleyite, and ringwoodite versus temperature and water content). The K11 results show high sensitivity to water content and relatively weak dependence on temperature (∼170 K results in a variation in water content of a factor of 2), while Y13 results show stronger temperature sensitivity and can barely match our preferred MTZ conductivity estimate. As shown in Fig. 4B, the broad range of estimated water content obtained with Y13 is mainly caused by the insensitivity of their estimates of MTZ minerals’ conductivity to hydrogen. Here, we chose the K11 laboratory results for further investigation. Note that variations in mantle compositions, partition coefficients of water between minerals, and bulk conductivity calculation method do not bring significantly different estimate result: Only slight hydration is needed to explain the observed electrical conductivity profile. Constrained inversions (fig. S10A) provide firmer constraints on MTZ water content. We generated laboratory-based MTZ conductivity profiles with homogeneous water content of 0.03, 0.04, and 0.06 wt % and then ran the inversion with the resulting MTZ conductivity profile fixed, and only shallower and deeper layers free to vary. While it is possible to fit the data with these more conductive MTZ profiles (nRMS misfits for the two cases are 1.55 and 1.57), the resulting models have compensating low-conductivity layers between 200 and 400 km in depth. Especially for the case of 0.06 wt % water, these layers are more resistive than dry peridotite (42) at ambient temperature and thus are not physically plausible (43).

Figure 4C summarizes the well-constrained conductance distribution of plausible conductivity profiles over the 300 to 670 km depth range, which reveals that the upper mantle and MTZ are both nearly dry, with 0.015 and 0.03 wt % water, respectively. Considering the possibility of low-conductivity bias caused by 3D structure, the upper bound, on average, MTZ water content of 0.06 wt %, as revealed by the constrained inversion, can be loosened to 0.15 wt % with lower values preferred. Certainly, some parts of the MTZ may contain substantially more water, for example, due to accumulation from water-rich subducting slabs (44). The geomagnetic observatories used in this study are biased toward Europe, Asia, and North America, and our estimated averages (conductivity and water content) are most likely similarly biased. Figure S10B compares our preferred model with conductivity profile derived from the joint inversion of satellite detected magnetospheric and oceanic tidal signal (38). These two models exhibit quite similar feature at 200 to 600 km in depth, and the differences between them are within our model uncertainty, although the joint inversion model, which inclines to structure beneath oceans with better global coverage, is more conductive by a factor of ~2.

First-principle calculation for mineral elasticity indicates that a variant of global seismic model AK135 can be best fit by relatively dry (<0.1 wt %) upper mantle and MTZ with 50% olivine or a slightly less well with 1 wt % water in the MTZ and a pyrolytic mantle (45). Our study supports the first (drier) option and suggests that the current MTZ and upper mantle may not be wet enough to generate dehydration melting (3) above the 410-km discontinuity. Although the lower MTZ is more poorly resolved by the DV data, considering the water partitioning between wadsleyite and ringwoodite (46), the lower MTZ is more likely to be unsaturated, as suggested by a recent study of mantle water capacity across the 660-km discontinuity (47). This mantle conductivity model (Fig. 4C) constrains the water stored in nominally anhydrous minerals in the upper mantle and MTZ at present to 0.06 to 0.3 and 0.09 to 0.4 the mass of the surface oceans, respectively, which would suggest a small role for the mantle in Earth’s water circulation (48, 49). On the basis of our study, the water content of the MTZ is not significantly higher than that of the upper mantle, consistent with a recent modeling study of whole mantle circulation over geological time scales (50).

Accurate models of external source magnetic fields in the DV band (10⁴- to 10⁵-s periods) enable more reliable imaging of electrical conductivity at depths of 200 to 600 km. Here, we have focused on estimating global averages using a quasi-1D approach supplemented by 3D synthetic data tests. Our results suggest only slight hydration of the upper mantle and MTZ, and estimated water content in the MTZ is approximately 0.03 wt %. However, the geomagnetic observatories we have used represent an incomplete sample of Earth. In particular, there are almost no sites in ocean basins, and our result is probably more representative of continental mantle in the Northern Hemisphere. The iterated source/conductivity inversion approach can be extended to consider 3D conductivity inversions (51) and to incorporate satellite magnetic data to improve global coverage (32). Another useful extension would be to integrate data with periods longer than 1 day using improved source models for Dst, extended to allow for low-frequency magnetospheric source complications (51). Our determination of water content can also be refined by incorporating equilibrium mineralogy for the upper mantle and the MTZ instead of a simplified pyrolytic model (52). These extensions will better constrain Earth’s conductivity distribution more and further enhance understanding of global water circulation.

MATERIALS AND METHODS

Data processing

Data processing and source modeling are described in detail in our previous study (32). Here, we provide a brief summary. Time series for our analysis are from 127 (of 182 available) geomagnetic observatories for years 1997 to 2018 (33), selected to provide more uniform global coverage, with reduced bias toward Europe. The magnetospheric component of the CHAOS-6 model (53) was subtracted, before high-pass filtering of time series (0.5-cpd cutoff), and then transformed to the frequency domain using a series of overlapping 8-day time segments. PCA (54) was then used to estimate the K dominant modes of coherent spatial variability in a set of J nonoverlapping bands U_jk, k = 1, …, K, j = 1, …, J. The full PCA model (with J = 12 bands, K = 20 modes) explains ∼95% of the total variance in the observatory array, including data from all latitudes and all geomagnetic conditions. Here, we focus on the strongest and best determined modal components: modes k = 1, ..., 5 for frequency bands centered at 1 to 4 cpd.

Source estimates

The spatial modes U_jk represent the total (external source plus induced internal) magnetic fields sampled at the observatories. We seek to separate these components with an iterative approach, starting with an assumed model for Earth conductivity, and estimating source structure as in (32). Ionospheric source spatial structure for modes k in band j are taken to be a linear combination of spatially continuous basis functions

$${\mathrm{\Psi }}_{\mathit{jk}}={\displaystyle {\sum }_i^I}{c}_{\mathit{ijk}}{\mathrm{\Phi }}_{\mathit{ji}}^{\text{ext}}$$ (1)

where the functions ${\mathrm{\Phi }}_{\mathit{ji}}^{\text{ext}}$ represent external potential functions (27), derived by applying a frequency domain PCA scheme, similar to that used for the data, to year-long runs of TIEGCM (34) outputs. Coefficients c_ijk are obtained by fitting the source model defined by Eq. 1 to the estimated spatial modes U_jk using weighted-regularized least squares, with damping parameter determined by leave-one-out cross-validation (32). For the fitting, internal components of the magnetic fields are derived using the thin-sheet modeling code described below to compute induced components associated with each source basis function ${\mathrm{\Phi }}_{\mathit{ji}}^{\text{ext}}$. In this work, we fit all three magnetic field components at all 127 observatories using I = 50 source basis functions.

Thin-sheet algorithm

A thin-sheet global EM solver (35) is used for both the source estimation step described above and the quasi-1D conductivity inversion, to be described below. The global conductivity model in the thin-sheet algorithm is a radially symmetric Earth overlain by an inhomogeneous surface conductance layer representing the ocean-continent conductivity contrast. In our study, the surface conductance layer is derived by the procedure described in (55) with prescribed seawater conductivity (56) and sediment thickness (57). Model resolution for the thin-sheet model is 1° × 1°. The underlying 1D model is represented with 50-km layers, except for the topmost layer of 10 km.

3D finite difference algorithm

For synthetic data experiments, we used a recently developed 3D finite difference (FD) solver for Maxwell’s equations in spherical coordinates based on a secondary field formulation (39)

$$\nabla \times \nabla \times {\mathit{E}}_s+i\mathrm{\omega }\mathrm{\mu }({\mathrm{\sigma }}_s+{\mathrm{\sigma }}_p){\mathit{E}}_s=-i\mathrm{\omega }\mathrm{\mu }{\mathrm{\sigma }}_s{\mathit{E}}_p$$ (2)

In our application, the primary electric field E_p is the thin-sheet solution obtained for the background layered Earth model σ_p, and σ_s is the added 3D heterogeneity. The linear equations resulting from numerical discretization are solved iteratively with a biconjugate gradient stabilized method with incomplete LU preconditioner. Surface magnetic fields required for our study are easily computed from the sum E_s + E_p. For generating synthetic datasets, FD horizontal grid resolution was 2° × 2°, and the vertical grid was logarithmic, starting from 50 m at the surface with an increase factor of 1.1.

Quasi-1D conductivity inversion

For the conductivity inversion step, we take the external source and inhomogeneous surface conductance layer as known and invert for the 1D layered Earth conductivity. The penalty functional for the regularized inversion is thus

$$P(\mathit{m})={(\mathit{f}(\mathit{m})-\mathit{d})}^{\mathrm{T}}{\mathit{W}}_d(\mathit{f}(\mathit{m})-\mathit{d})+\mathrm{\lambda }{\mathit{m}}^T{\mathit{L}}^T\mathit{Lm}$$ (3)

where m is the model parameter (1D mantle conductivity), d is the data vector (vertical component of DV magnetic fields), f is the data functional (thin-sheet magnetic field solution, forced by corresponding source estimates and evaluated at observatory locations), W_d is the inverse of data error variance, λ is the regularization parameter, and L defines the regularization. For the latter, we used a first derivative operator, but in some experiments, the smoothness constraint was turned off at one or more depths. The penalty functional is minimized using an Occam algorithm (36) to obtain the smoothest conductivity model achieving a specified misfit. The Jacobian matrix needed for the Occam scheme was computed by the perturbation approach.

Sites used in the conductivity inversion were selected on the basis of two criteria: (i) geomagnetic dipole latitude between 3° and 60°, where the DV source model is simplest, and (ii) quality of fit in the initial estimates of source structure. Specifically, misfits for all three magnetic field components had to be below the 0.8 quantile among corresponding components at all sites. The relative error floor of data is set as 5% of amplitude computed site by site.

Workflow

Overall workflow consists of iterating the external source estimate and earth conductivity inversion steps successively (51, 58). The procedure is initialized by computing the source estimate for an Earth conductivity model (e.g., 1D models KS0 and PK0 displayed in Fig. 1), followed by conductivity inversion with the source from the first step. These steps are repeated to convergence, typically requiring just two to three iterations.

Synthetic tests

The 3D complications in the real Earth may compromise our estimates of both source structure and Earth conductivity. We combined hypothetical 3D conductivity models with realistic sources to generate synthetic data to investigate these issues.

The 3D conductivity models tested include a series of checkerboard models, with alternating blocks of conductivity 10^1/2 or 10^−1/2 times background. Models with this heterogeneity embedded at three distinct depths are considered, along with one that incorporates checkerboards at all depths (Table 1 and fig. S5). A fifth model (“Vs model” in Fig. 3) is derived from a shear wave velocity model (40) by scaling dV_s to dlogσ, so that the total range of variations is one order of magnitude (as with the checkerboard models). In all cases, the global 1D profile KS0 (21) is used as the background model (see Fig. 1).

Table 1. 3D checkerboard model series.

These global 3D conductivity models are used for synthetic tests presented in Fig. 3.

Notation in Fig. 3	Target depth (km)	Block size
Lithosphere	10 to 250	20° × 20°
Upper mantle	250 to 410	40° × 40°
MTZ	410 to 660	40° × 40°
Mixed	Merge three models above	Merge three models above

Ionospheric sources derived from realistic data (fig. S1) are used to force either the thin-sheet model or the 3D FD algorithm to generate synthetic magnetic fields. These are sampled at the actual observatory locations, and 5% random errors are added. We used synthetic data from both 1D and 3D conductivity models to test robustness and accuracy of the source estimation scheme when the initial conductivity model is incorrect and when different subsets of data components are used in the fitting. In our synthetic source estimate tests, profile KS0 (21) and the lithosphere checkerboard model are used to generate data in 1D and 3D scenarios, but a different 1D profile PK0 (23) is used to estimate the ionospheric source. Two variants on the fitting approach were tried: using only horizontal field components (H_x and H_y) and using all three components (H_x, H_y, and H_z). See fig. S7 for the relative difference between the actual and estimated source coefficients c_i11 for the first mode at 1 cpd that defines the source through Eq. 1.

Synthetic data generated from all five 3D conductivity models were then used to test the full source/conductivity estimation scheme for both global and single-station cases. For each synthetic 3D dataset, the ionospheric source was first estimated, starting from the initial conductivity PK0 (23) followed by inversion for 1D conductivity, both using the set of all reliable stations (as defined by real data), and for the local inversion tests from 10 randomly selected stations. For the global inversions, the whole process was repeated, with one additional source/conductivity inversion step. The single-station inversions were only run on the first step and only for the four checkerboard models, for a total of 4 × 10 = 40 local inversions. Results are summarized in Fig. 3 as ratios of inverted to actual average conductivity in the best resolved broad layers, i.e., the upper mantle (200 to 410 km) and the upper MTZ (410 to 520 km).

Calculation of laboratory-based conductivity profile

The laboratory measurements of mineral conductivity can be described by a function of activation enthalpy, temperature, water content, and other factors (e.g., pressure and iron content) for different conduction mechanism. The electrical conductivity of hydrous minerals for fixed water contents have been calculated under a revised adiabatic mantle temperature model (31) by using laboratory results summarized in K11 (6) and Y13 (17, 59, 60). Moderate partition coefficients of H₂O between olivine and other phases are used (48, 61): $D_{{\mathrm{H}}_2\mathrm{O}}^{\text{gt}/\text{ol}}=1$, $D_{{\mathrm{H}}_2\mathrm{O}}^{\text{py}/\text{ol}}=5$, $D_{{\mathrm{H}}_2\mathrm{O}}^{\text{wad}/\text{gt}}=20$, $D_{{\mathrm{H}}_2\mathrm{O}}^{\text{ring}/\mathit{gt}}=10$. The influence of oxygen fugacity has been omitted as there are only limited constraints from laboratory experiments. The bulk conductivity of mineral aggregate is calculated by Hashin-Shtrikman lower bound (62). While the calculated bulk conductivity using Hashin-Shtrikman upper bound is only slightly higher, our current scheme gives a more conservative estimate of water content, and the realistic case might be even drier. Note that the Y13 results does not include experiments for hydrous garnet, so laboratory results for dry garnet (60) conducted by the same group are used instead. This replacement would overestimate the bulk water content, but the influence should be marginal since garnet has relatively small volume fraction in the upper mantle and very limited water storage capacity due to the water partitioning among coexisting MTZ minerals.

Supplementary Materials

This PDF file includes:

Table S1

Figs. S1 to S10

References