Geomagnetic induction in a heterogeneous sphere - Geophysical

Geophys. J. Int. (1998) 135, 650–662

Geomagnetic induction in a heterogeneous sphere:fully three-dimensional test computations and the responseof a realistic distribution of oceans and continents

C. J. Weiss and M. E. EverettDepartment of Geology and Geophysics, T exas A & M University, College Station, TX 77843, USA

Accepted 1998 June 26. Received 1998 June 6; in original form 1997 September 8

SUMMARYLong-period geomagnetic data can resolve large-scale 3-D mantle electrical conductivityheterogeneities which are indicators of physiochemical variations found in the Earth’sdynamic mantle. A prerequisite for mapping such heterogeneity is the ability to modelaccurately electromagnetic induction in a heterogeneous sphere. A previously developedfinite element method solution to the geomagnetic induction problem is validatedagainst an analytic solution for a fully 3-D geometry: an off-axis spherical inclusionembedded in a uniform sphere. Geomagnetic induction is then modelled in a uniformspherical mantle overlain by a realistic distribution of oceanic and continental con-ductances. Our results indicate that the contrast in electrical conductivity between oceansand continents is not primarily responsible for the observed geographic variability oflong-period geomagnetic data. In the absence of persistent high-wavenumber magneto-spheric disturbances, this argues strongly for the existence of large-scale, high-contrastelectrical conductivity heterogeneities in the mid-mantle. Lastly, for several periods thegeomagnetic anomaly associated with a mid-mantle spherical inclusion is calculated.A high-contrast inclusion can be readily detected beneath the outer shell of oceans andcontinents. A comparison between observed and computed c responses suggests thatthe mid-mantle contains more than one order of magnitude of lateral variability inelectrical conductivity, while the upper mantle contains at least two orders of magnitudeof lateral variability in electrical conductivity.

Key words: coast effect, electrical conductivity, electromagnetic induction, finiteelement method.

2–10 volume per cent. Accordingly, the Schultz & LarsenINTRODUCTION

(1987) compilation from the global observatory network (seeLong-period geomagnetic data are capable of resolving Fig. 1 for locations) reveals a wide geographic variation inlarge-scale 3-D mantle electrical conductivity heterogeneities geomagnetic response, consistent with the presence of large(Schultz 1990; Tarits 1994), which are indicators of temper- lateral variability in the Earth’s electrical structure. If a fullyature, mineralogy and pressure variations found in the Earth’s 3-D inversion of geomagnetic induction data were carried out,dynamic mantle (Poirier 1991). Utilizing 1-D forward modelling the resulting electrical conductivity model would complement

and inversion techniques, several independent regional studies the existing seismic velocity (Woodhouse & Dziewonski 1984;

of long-period MT/GDS (magnetotelluric/geomagnetic depth Masters et al. 1996), attenuation (Romanowicz 1995) and mass

sounding) data from a variety of geological provinces have density (Ricard et al. 1993) models and may provide new

insight into complex mantle processes such as subduction ofindicated that lateral variations in mantle conductivity at

200 km depth are as large as two orders of magnitude (Schultz lithospheric plates, upwelling of mantle plumes and the evolution

of convection cells.et al. 1993; Egbert & Booker 1992; Lizzaralde et al. 1995).

Assuming an olivine-dominated mineralogy, Tyburczy (1996) A necessary component of the inversion of surface geo-

magnetic data is a comprehensive forward solver whichshowed that these variations are geologically reasonable and

consistent with variations in mantle temperature of 65–160 K, accurately simulates the low-frequency diffusion of magnetic

fields generated by external and induced currents in thein hydrogen content of a factor of 30 or in melt content of

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Induction in a heterogeneous sphere 651

suggests that the mid-mantle contains more than one order of

magnitude of lateral variability in electrical conductivity, whilethe upper mantle contains at least two orders of magnitude oflateral variability in electrical conductivity.

THE GEOMAGNETIC INDUCTIONPROBLEM

During large magnetic storms, the solar wind interacts withthe Earth’s main field and generates a powerful magneto-

spheric ring current (Rostoker, Freidrich & Dobbs 1997). Atmid-latitudes on the Earth’s surface the magnetic potentialFigure 1. Locations of the 79 magnetic observatories examined indue to this current is nearly azimuthally symmetric and is wellthe Schultz & Larsen (1987) compilation of Schmucker’s c responses.approximated by the first associated Legendre function P0

1,The dark symbols are the locations of a subset of 15 mid-latitude

simply the cosine of the colatitude (Banks & Ainsworthobservatories whose responses are compatible with P01source structure

1992). The recovery phase of a magnetic storm has a charac-and a localized 1-D mantle conductivity profile.

teristic timescale of several days, during which the ring currentexponentially relaxes to its usual intensity (Gonzalez et al.1994). By Faraday’s law of induction, time variations in theheterogeneous Earth. Everett & Schultz (1996) introduced a

finite element method (FEM) approach to the geomagnetic external source induce electric eddy currents in the conductingEarth. The eddies, in turn, generate secondary magnetic fieldsinduction problem and demonstrated favourable comparisons

between the FEM results and previously known, accurate according to Ampere’s law, which diffuse outwards, some to

the Earth’s surface, where they are measured superimposedsolutions for azimuthally symmetric earth models. This paperemploys the FEM forward solver to address a number of on the steady main field. The variations in the period range

T =2–100 days are most sensitive to electrical structure atcurrent issues in global geomagnetic induction research thatmust be resolved before attempting a 3-D inversion. depths of 200–1000 km in the upper and mid-mantle for

conductivities in the range 0.1–1.0 S m−1 (Schultz 1990). SuchWe first complete the validation of the FEM code by perform-

ing a fully 3-D test calculation on the system of eccentrically low frequencies result in an energy transport mechanism withinthe Earth which is a diffusion process rather than the morenested spheres analysed by Martinec (1998). Next, we use the

FEM code to test the hypothesis that the electrical contrast familiar high-frequency wave propagation.

Several methods have recently been proposed for forwardbetween oceans and continents is a major contributor to theobserved geographic variability in scalar geomagnetic responses. modelling of geomagnetic induction in spherical geometry.

These include the FEM approach of Everett & Schultz (1996),Finally, we investigate the sensitivity of geomagnetic responses

to a spherical electrical conductivity anomaly embedded in the a semi-analytic method based on the perturbation expansion ofconductivity about a background 1-D model (Zhang & Schultzmantle beneath a realistic distribution of oceans and continents.

References to prior work on these different topics are given 1992), a heterogeneous spherical thin-sheet method (Kuvshinov

& Pankratov 1994), a semi-analytic method based on athroughout the text but an exhaustive review is beyond thescope of this article. generalized spherical harmonic expansion of radially invariant

conductivity in concentric shells (Tarits, Wahr & LognonneThe results presented here encourage and justify the use

of the finite element code towards a fully 3-D inversion for 1995) and a spectral finite element approach for axisymmetricelectrical models (Martinec 1997). The forward modelling inmantle electrical structure, since many questions concerning

the accuracy of the code, the role of the oceans and continents, this paper is based on the FEM approach for two reasons:

first, it is least restrictive on the admissible class of electricaland the sensitivity of geomagnetic responses to mantle structureare now more clearly resolved. The original contribution of this conductivity models, permitting almost arbitrary heterogeneity

to within the resolution of the mesh discretization; second, itpaper is the confirmation of conclusions obtained previously

(e.g. Fainberg, Kuvshinov & Singer 1991a,b; Kuvshinov, is the only approach we are aware of that has been validatedfor multidimensional conductivity models against an analyticPankratov & Singer 1990; Tarits 1994), with a new model that

is fully 3-D and fully tested against a true 3-D analytic solution. solution (Everett & Schultz 1996).

This not only validates our approach but also validates theprevious studies.

FEM FORMULATIONTo summarize the results, we show that the oceans and con-

tinents cannot account for the observed geographic variability A conventional FEM (e.g. Johnson 1995) was used to solvethe geomagnetic induction problem. The method is fullyof geomagnetic responses beyond periods of 2 days, and that

the crust has a negligible effect on the resolution of large-scale described in Everett & Schultz (1996) but for the convenienceof the reader we review here the main details. The linear systemmid-mantle anomalies. The size of the geomagnetic anomaly

on the Earth’s surface is calculated for the case of a mid- of FEM equations is formulated in terms of vector magnetic

and scalar electric potentials, A and W, inside the Earth and amantle spherical inclusion underlying a realistic distributionof oceans and continents, for several different periods and scalar magnetic potential Y outside the Earth—termed the A,

W–Y approach in the electrical engineering literature (Biro &inclusion conductivities. The anomaly is largest when the

excitation frequency corresponds to a skin depth into the Preis 1990). The formulation is based on linear tetrahedralelements. That is, the solution to the governing boundaryinclusion that roughly equals its radius. Finally, we show that

the difference between observed and computed c responses value problem is piecewise linear over each of the elements in

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652 C. J. Weiss and M. E. Everett

the solution domain. To increase the accuracy of the solution, squares interpolant’ scheme, as discussed in the following

if necessary, we adopt a refinement procedure which adds section.more elements to the domain, or alternatively decreases the The linear system of FEM equations is solved usingvolume of some elements at the expense of others. The solution an incomplete LU factorization (ILU) procedure (Everett &domain is defined by the finite element mesh. It is constructed Schultz 1996). The ILU solver can fail to converge for ill-by tetrahedralizing a thick spherical shell and then concen- conditioned systems such as those arising from poorly structuredtrically nesting several of these shells of different radii and meshes or widely heterogeneous conductivity models. In ourthicknesses to form the solid sphere. Shells are geometrically experience, however, the ILU solver converges monotonicallyspaced to ensure high node density near the modelled air– with iteration for well-structured meshes and electrical con-earth interface. This is important not only for accurate calcu- ductivity distributions containing fairly large amounts of laterallation of derivatives (the electric and magnetic fields) at the heterogeneity (see Appendix A).Earth’s surface but also to reflect the greater spatial resolutionof higher-frequency fields which penetrate only the upper

mantle in accordance with the skin effect. 3-D VALIDATIONElectromagnetic induction inside the Earth is governed by

Previously, the FEM code has been validated against solutionsthe usual relations between the electric, E, and magnetic

induction, B, vector fields: Faraday’s law of induction, for two azimuthally symmetric (2-D, or zonal) conductivity

models (Everett & Shultz 1996). The first model is composedV×E= ivB , (1)of a nested spherical inclusion that is offset along but centred

and Ampere’s law, on the polar axis. This model has an analytic solution for

external, zonal source excitation (D’Yakanov 1959; Everett &V×B=sm0E−ivm

0e0E , (2)

Schultz 1995). The second model is a zonal, inhomogeneouswhere the underlying e−ivt time dependence is implicit, while thin sheet buried at a depth of 400 km within a uniformlythe magnetic permeability m0 and electric permittivity e0 of free conducting sphere. The response of this model has been foundspace are constant. The second term on the right-hand side of by an integral equation method (Kuvshinov & Pankratoveq. (2) is the contribution due to Maxwell’s displacement 1994). While good agreement was obtained in both cases withcurrent and is neglected from further consideration since the corresponding FEM solutions, the 2-D models do notperiods T =2–100 days and likely mantle conductivities in the contain the full 3-D physics of electromagnetic induction.range s~0.0001–10 S m−1 result in ve0/s~10−17–10−10. For the purely zonal geometries of the azimuthally symmetricSource terms are absent from eqs (1) and (2) since at periods problems, galvanic effects are absent since the source and itsless than several years the geomagnetic variations are due induced currents are everywhere perpendicular to conductivityalmost entirely to external magnetospheric and ionospheric gradients, resulting in no charge build-up within the model.sources. In the FEM formulation these sources enter as In fully 3-D models, induced currents are forced alongDirichlet conditions on the outermost shell of the finite element conductivity gradients, which results in charge accumulations.mesh. The steady main field due to the internal dynamo is

Our 3-D validation is based on a recent analytic solutionneglected since it varies over much longer periods than those

by Martinec (1998) for multiple off-axis nested sphericalwe examine in this study.

inclusions. Galvanic effects are present and accounted for inInside the Earth the induction B is the curl of the vector

both the analytic and the FEM solutions. We compare resultspotential, B=V×A. The electric field E inside the Earth is

for the model shown in Fig. 2 containing a single off-axisequal to ivA−VW. Together with this reformulation, eqs (1)

inclusion.and (2) combine to yield the following governing equation for

The solution for multiple nested spherical inclusions pre-potentials inside the Earth:

sented by Martinec (1998) is formulated in terms of theV×V×A− ivm

0sA+m

0sVW=0 . (3) magnetic field H, which is expressed as a truncated expansion

of the product of vector spherical harmonics Ylmn

of order mOutside the Earth the magnetic induction field is simply theand degree n and of the spherical Bessel functions j

land y

lnegative gradient of the magnetic scalar potential, B=−VY.(see Abramowitz & Stegun 1964; Varshalovich, Moskalev &The divergence-free constraint on the induction, VΩB=0,Khersonskii 1988). These are eigenfunctions of the vectorimplies that Y satisfies Laplace’s equation and establishes theHelmholtz equation, V2H+k2H=0, the governing equationgoverning equation outside the Earth,for low-frequency magnetic fields inside source-free regions of

V2Y=0 . (4) the Earth, where k2=−ivm0s. We can compare FEM results to

Martinec’s solution since the vector Helmholtz and the A, W–YTo ensure uniqueness of the potentials (Biro & Preiscoupled-potential systems are equivalent representations of1990), the Coulomb gauge (VΩA=0) is enforced along withthe geomagnetic induction problem. The FEM and analyticdivergence-free current density (VΩJ=0) and zero normalsolutions are shown in Fig. 3, and good agreement is obtainedcurrent density at the Earth’s surface. At the air–Earth interface(see discussion below). Note especially that the H

wcomponent(radius r=RE ) we require continuity of the induction vector,

is generated entirely by galvanic interactions and is presentV×A|r=RE

=−VY|r=RE

. Further details regarding the internalonly in fully 3-D calculations.and external boundary conditions are discussed in Everett &

When appraising the goodness of fit between the analyticSchultz (1996). Once the potentials are computed, they mustand FEM solutions it is important to recognize possiblebe numerically differentiated to obtain the electric E andsources of numerical error. Some of the error in the solutioninduction B vectors. The numerical differentiation of the

electromagnetic field is carried out using a ‘moving least- can be attributed to the fineness of the mesh discretization.

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THE COASTLINE EFFECT

Nearly 70 per cent of the Earth’s surface is covered by theoceans, a highly conductive medium (s~3.2 S m−1) that can

significantly affect the secondary magnetic fields generatedinside the Earth by induced electric currents. The coastlineeffect is the change in the surface ratio Z/H of the vertical Z

and horizontal H induction field components near the con-tinental boundaries due to the high conductivity constrast,up to three orders of magnitude, between continental crust

and sea water. In flat-earth simulations with uniform hori-zontal source currents and layered conductivity structure,the impedance Z/H is zero owing to the absence of a vertical

magnetic component. Near a coastline the vertical componentZ has non-zero magnitude and the horizontal component His generally directed perpendicular to the coastline, giving riseFigure 2. From Martinec (1998), geometry of the 3-D nested-sphere

configuration used for the test calculation. Shown is a vertical slice of to the coast effect. A textbook numerical simulation of thethe 3-D body through the centres of both the inclusion and host coast effect in Australia is illustrated in Lilley & Corkerysphere. An inclusion of radius r=3500 km is centred at Oi , a distance (1993).d= 2700 km from the centre O of the host sphere with radius 6371 km. In a spherical earth Z/H is a function dependent uponThe inclusion centre is located at polar angle hi=40° and azimuth

geographic latitude and therefore it is more convenient to35°. The electrical conductivities of the inclusion and host are 10 S m−1

consider Schmucker’s c response (Schultz & Larsen 1987),and 1 S m−1, respectively.

c=RE tan hB

r(h)

2Bh(h)

, (5)

This effect is most pronounced in the Hw

component in Fig. 3

but is present in all three components. Furthermore, we where RE is the Earth’s radius and h is the geomagneticcolatitude. This combination is independent of h for the specialattribute the error in the region r~0 km to the inclusion–host

interface, which lies at r~−800 km (see Fig. 2). Lastly, since case of a spherically symmetric earth and an external P01

source. For a layered spherical earth with an outermostthe analytic solution is given in terms of the magnetic field H,the FEM solutions for the potential A must be differentiated heterogeneous shell of ocean–continent conductivities, the c

response at locations far from a coastline approaches thein order to compare results. Several authors have worked on

the problem of accurate differentiation of FEM electromagnetic value for a laterally uniform sphere composed of the local 1-Dconductivity profile. Near the coasts, the c response can havepotentials and an excellent review of the various methodologies

is found in Omeragic & Silvester (1996). It is well established a more complex structure, but in general, for long periods, it

smoothly varies between the values found far inland and thosethat direct differentiation of the FEM basis functions (resultingin piecewise constant derivatives) generally yields unacceptable far seawards.

Much attention has been paid to the coastline effect and itsresults. Our preliminary analyses confirm this fact and we

therefore adopt a ‘moving least-squares interpolant’ scheme consequences on the determination of upper- and mid-mantleelectrical properties (Rikitake 1961; Bullard & Parker 1970;(Tabbara, Blacker & Belytschko 1994) since it is easy to

implement, computationally efficient and yields very good Parkinson & Jones 1979; Fainberg & Singer 1980; Roberts

1984; Takeda 1993; Chen & Dosso 1997). Several factorsresults.

Figure 3. Magnetic field components (A m−1 ) computed for v=3×10−7 rad s−1 (T =242 days) along a line from the centre of the host sphere

to a point on its surface at polar angle h=13° and azimuth 0° (see Fig. 2). The analytic solution (Martinec 1998) is shown in solid lines while the

FEM solution is represented by symbols. The large symbols correspond to FEM solutions on a finely discretized mesh (94 208 tetrahedra) while

the small symbols correspond to those derived from a coarsely discretized mesh (19 456 tetrahedra). As the Hw

component is absent in any

azimuthally symmetric calculation, its presence here is due entirely to 3-D induction effects.

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contribute to the magnitude of the coastline effect: frequency

v (or period T ), outline and bathymetry of the ocean basins,

and the presence of subduction zones or crustal sutures which

may serve as conductive pathways through the resisitive crustto deeper conductors (Gough 1989; Wannamaker et al. 1989;

Kurtz, DeLaurier & Gupta 1990; Park et al. 1991). A global

tour of regional observations is presented in Parkinson &

Jones (1979). They note that while the short-period (T<1 day)coast effect is almost ubiquitous, there are anomalous regions

such as the western Mediterranean and northern Europe where

it is absent, as well as northern Australia and Japan where itappears to be augmented by the effects of complex crustal Figure 5. Centroids of the tetrahedra in the outermost layer (upperstructure. As for the long-period coast effect, early studies such 70 km of the model earth) of the finite element mesh. In accordanceas Roberts (1984) predict a substantial effect at periods with the ocean–continent model shown in Fig. 4, the heavy symbols

are centroids of tetrahedra assigned continental-crust electrical con-extending to T =100 days while later works by Kuvshinovductivity sc , and the light symbols are centroids assigned oceanic-et al. (1990), Takeda (1993) and Tarits (1994) indicate acrust conductivity so . The assignments are made by applying thesignificant decrease in its magnitude at periods between 1 andpoint-in-polygon algorithm (see Appendix B).15 days. The latter three studies show that at periods T >15

days the coast effect accounts for less than 9 per cent of the

geomagnetic variations. Our computations of c responses overFig. 6. Ocean model I is a high-contrast end-member modela realistic distribution of oceans and continents support thiswith so=1.0 and sc=0.01 S m−1 while ocean model II is aresult.low-contrast model with so=0.24 and sc=0.05 S m−1. TheIn the ocean–continent conductivity model examined here,latter is in closer agreement with realistic estimates of thecontinental outlines are each approximated by a simple sphericalresistivity–thickness product between continental and oceanicpolygon, that is, a small number of vertices connected byregions. Ocean–continent model II takes into account theminor great-circle arcs. Fig. 4 illustrates the Earth’s actualresistive lithospheric mantle. The use of simple ocean–continentcontinental outlines and the simplified ‘continents’ used in themodels is consistent with the low resolving power of long-model. The ocean–continent model is a heterogeneous outerperiod global induction data to structure within the uppershell of constant thickness overlying a uniform mantle. In the~100 km of the Earth (Schultz 1990). This simplification isFEM, this outer shell consists of packed tetrahedra, as before,further justified by noting that the attenuation of magneticwhich are assigned an electrical conductivity of either so or sc fields at periods T >2 days is less than 1 per cent over 20 kmcorresponding to ‘oceanic’ or ‘continental’ conductivity. Toof crustal material with conductivity s<0.01 S m−1. The FEMdetermine which conductivity value to assign, we have developedcalculation for the ocean–continent models is done here toa point-in-polygon algorithm (see Appendix B) to classify therule out any possibility of long-period distortion of geo-tetrahedra on the basis of the location of their centroidsmagnetic responses by near-surface structure. Takeda (1993)relative to the continental outlines. The shell is a binary con-analysed a similar model composed of a uniform s=0.1 S m−1ductivity function so that conductivity jumps are present, asmantle overlain by a smoothly varying heterogeneous thinwith the spherical-inclusion test. A map showing the classifi-sheet whose conductivity was determined by conductancecation of tetrahedra in the shell is shown in Fig. 5.estimates to 10 km depth of rock (s=0.01 S m−1) and sea

Two ocean–continent models were constructed. A schematicwater (s=4.0 S m−1) in the presence of realistic bathymetry.

representation of the cross-section of both models is shown inThe geomagnetic fields associated with models I and II were

computed using the FEM code. From these fields Schmucker’s

Figure 6. Schematic representation of a cross-section of the ocean–

Figure 4. Map of the continental outlines and their corresponding continent model showing the uniform crustal thickness d=70 km. The

mantle conductivity was fixed at 0.1 S m−1, while the oceanic andspherical polygon approximations (shaded). The highlighted magnetic

observatories from Fig. 1 are shown by the filled circles. Also shown continental crust conductivities were varied according to two models:

high-contrast model I, sc=0.01 S m−1, so=1.0 S m−1; low-contrastis the epicentre (square) of the mantle inclusion and the profile A–B

described in Fig. 10. model II, sc=0.05 S m−1, so=0.24 S m−1.

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c response was computed. Figs 7 and 8 show as a function ofSENSITIVITY STUDY

period T the percentage change in c response from a uniformmantle due to the addition of the near-surface structure To examine the sensitivity of c response values to 3-D

mantle heterogeneity we evaluate three models of a uniformcontained in ocean models I and II, respectively. Note the

dramatic reduction in anomalous c response with increasing mantle overlain by the heterogeneous ocean–continent shell(model II), containing a single, spherical conductivity inclusion.period T . We also computed the difference max Dc between

the maximum and minimum c response values due to each A schematic cross-section of the models is shown in Fig. 10,

where ocean model II is used for the shell conductivities,ocean–continent model. The legend in Fig. 9 shows max Dc forT =10 days for each model, in addition to the observed value and the inclusion conductivities si are 1, 4 and 10 S m−1. A

spherical inclusion, like that used in the 3-D Martinec com-taken from the 15 geomagnetic observatories highlighted in

Figs 1 and 4. In the more realistic case (model II), the predicted parison, is placed beneath the Philippine Plate, at the locationindicated in Fig. 4. Centred 5000 km from the Earth’s centre,max Dc value due to ocean/continent heterogeneity is less than

7 per cent of the total observed (shaded bar), suggesting that the r=1000 km radius inclusion spans the upper and lower

mantle, is centrally located between the observatory locationseddy currents induced in the oceans are not the primaryagent for the observed geographic variability in long-period in Australia, Japan and the western Pacific, and lies beneath

both oceanic and continental regions. For the period rangec response. For the less realistic, high-contrast model I the

predicted max Dc value rises to 32 per cent for the real T =2–120 days the conductivities si of the inclusion were suchthat the dimensionless size parameter L =|r√vm

0si rangescomponent. This places an approximate upper bound on the

ocean–continent contribution. from electromagnetically ‘dim’ (L =0.9) to electromagnetically

Figure 7. Percentage change in Schmucker’s c response of high-contrast ocean model I from that of a uniform mantle of electrical conductivity

0.1 S m−1. The effects of the conducting ocean clearly decrease with increasing period from T =0.5–10 days, such that by T =10 days the variation

is less than ~10 per cent in the real component and ~3 per cent in the imaginary.

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Figure 8. Percentage change in Schmucker’s c response of the more realistic, low-contrast ocean model II from that of a uniform mantle. In

agreement with previous work (e.g. Takeda 1993), the effects of the conducting ocean decrease with increasing period from T =0.5–10 days, such

that by T =2 days the variation is less than ~5 per cent in the real component and ~1 per cent in the imaginary.

‘bright’ (L =21.4) values. Admittedly, the inclusion is really The results shown in Fig. 11 were further analysed to revealthe size of the region containing the anomalous c response.big, but the response of the dimmest model should approximate

that of a brighter, smaller inclusion. The latter is defined as the relative difference between themodulus of the fully 3-D c response and that of the backgroundThe computed c responses for the 1 and 10 S m−1 inclusion

models are shown in Fig. 11 as functions of epicentral distance 1-D c response of the uniform mantle model, expressed as a

percentage. The anomalous c response decays with epicentralfrom the inclusion centre. For clarity of illustration, the FEMresults were post-processed by computing the mean and distance, as shown. The size D10 of an anomalous region is

defined as the epicentral distance beyond which the anomalystandard deviation for 12° epicentral bins. The size of an errorbar is therefore a rough measure of the scatter in the FEM has decayed to less than 10 per cent. Beyond this distance the

effects of the inclusion become increasingly difficult to discern.solution. The scatter in the numerical computations is expected

because of geometric mismatches between the spherical Shown in Fig. 12 is a plot of D10 as a function of period T .The values used in this figure are obtained from the resultsinclusions and the tetrahedra of the FEM mesh. Test compu-

tations (not shown) on the inclusion model with a uniformly presented in Fig. 11, along with the additional calculation for

the 4 S m−1 inclusion. The peak in a D10 (T ) curve indicatesconducting outer shell (1 S m−1 ) yield similar response curvesto those shown in Fig. 11. Thus the ocean–continent distri- the period at which an inclusion generates its largest anomalous

region over the Earth’s surface. The 10 S m−1 inclusionbution has a negligible effect on the total c response. This is

consistent with the results of the previous section. In all cases, generates a maximum anomalous region of D10~50° at aperiod T ~100 days, while the effects of the 1 and 4 S m−1the c response at large epicentral distances approaches the

value for the uniform mantle. inclusions extend to D10~32° and D10~45° at periods T ~45

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days and T ~60 days, respectively. These periods corre-

spond to skin depths into the inclusion that roughly equal itsradius, in other words, L ~√2. It is not surprising that themost conductive inclusion generates the largest geomagnetic

anomaly on the Earth’s surface.Fig. 13 shows the maximum size of the geomagnetic anomalies

generated by the three inclusion models considered. Our

hypothetical mantle inclusions underlying the Philippine Plate

noticeably affect the Earth’s long-period geomagnetic responseseveral thousand kilometres away, as distant as the Indian

subcontinent and the Bering Strait. If our hypothetical sub-

Philippine Plate anomaly were real, its geomagnetic effects

would be recorded by at least four and as many as nine of the15 observatories highlighted in Fig. 1.

DISCUSSION

Our model of geomagnetic induction uses a simple P01

morphology for the external source currents. Campbell (1996)

has noted that solar-derived charged particles in the magneto-

sphere during storm times form very complicated spatial

patterns at 3–5 Earth radii. These patterns are created byparticles that spiral around geomagnetic field lines, bounce

back and forth along field lines between northern and southern

hemisphere mirror points, collide with one another, etc. (Chen,

Schulz & Lyons 1997). However, there is a net westward driftto the ensemble of charged particles that is associated with the

Earth’s rotation. Banks & Ainsworth (1992) show conclusively

that the long-period magnetic effect on the Earth’s surface of

the net westward drift is equivalent to that of an equatorial,filamentary ‘ring current’. A simple ring current generates aP01

scalar magnetic potential. At periods less than 2 days, aFigure 9. Schmucker’s c response (Schultz & Larsen 1987) from themore complicated scalar potential is generated by the presence15 observatories highlighted in Fig. 1 plotted on the same axis toof complex magnetospheric current systems (Banks &emphasize geographic variability. At the period T =10 days (frequency

0.1 c.p.d.) the difference between the maximum and minimum values Ainsworth 1992). We note that the FEM code can handleof the observed c response, max Dc, is shown by the shaded bar and arbitrary Pm

l(cos h) spherical harmonic terms for the external

compared in the legend to that of ocean models I and II. The small source.variation in the c response from models I and II compared to that Geomagnetic response functions are useful for global con-observed indicates that the electrical contrast between oceans and

ductivity studies since they depend primarily on the internalcontinents is not the main agent for the geographic variability.

conductivity structure and little on the strength of the external

source currents (Egbert & Booker 1986). For example,

Schmucker’s c response is a suitable choice of response function

when the spherical electrical conductivity distribution is1-D, or weakly 3-D, beneath an observatory site. However,

at geographical locations where a 1-D representation of the

underlying electrical conductivity cannot be justified, a tensor

response function is preferable. Schultz & Zhang (1994)have accordingly defined a new second-rank tensor response

function, denoted f(v). Although the finite element model

readily calculates f responses, we have used Schmucker’s c

response in this study mainly for historical reasons and becauseworkers in geomagnetic induction are familiar with it.

Fig. 9 shows the observed geographic variability in cFigure 10. Schematic representation of a spherical inclusion embedded response. We have already shown that this variability is likelyin a uniform mantle overlain by the ocean–continent crustal model. because of upper- and mid-mantle electrical conductivityThe inclusion of radius r=1000 km is centred R=5000 km from the

heterogeneities, at least in the absence of persistent, high-Earth’s centre at latitude 25° and longitude 135°, beneath the Philippine

wavenumber external source currents. Our modelling resultsPlate (profile A–B on Fig. 4). Using the more realistic ocean model II

provide a clue to the amount of heterogeneity in mantlefor the calculations, two models were run with the inclusion con-electrical structure that is required to generate the observedductivity si set to 1 and 10 S m−1. This corresponds to a range in sizegeographic variability. Fig. 10 shows the range of c values thatparameter L =|r√vm

0si from 0.9 to 21.4 for the modelled period range

T =2–120 days. would be found in the vicinity of a large spherical mantle

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Figure 11. Real (top) and imaginary (bottom) components of Schmucker’s c response as a function of epicentral distance from the spherical

inclusion described in Fig. 10. The filled circles correspond to si=10 S m−1 and the open circles are those for si=1 S m−1. As the conductivity of

the inclusion is increased, the magnitude of its effect on the c response also increases. At large epicentral distances the c response tends to the

value for the uniform mantle.

inclusion. At periods T =10 days an inspection of Fig. 10 where a fully 3-D inversion of long-period geomagnetic data

should be carried out, especially considering the imminentreveals that the 1 S m−1 inclusion within the 0.1 S m−1 back-ground mantle (solid symbols) generates 80 km (real ) and arrival of the f response and the continuing development of

more powerful computers. We have demonstrated that the300 km (imaginary) c response variations over the Earth’s surface.

The observed amount at T =10 days (frequency 0.1 c.p.d.) FEM forward solver for geomagnetic induction in a hetero-geneous sphere can be used as the basis for such a non-linearis, from Fig. 9, considerably more: 560 km (real) and 415 km

(imaginary). The 10 S m−1 inclusion, however, generates roughly inversion. The large amount of lateral heterogeneity in mid-

mantle electrical structure, as suggested by the large observedthe same amount of geographic variability as that observed,particularly in the imaginary component of the c response. geographic variability in c response, makes the inverse problem

attractive. The sparsity of the existing 15-site c response dataThese comparisons argue that the Earth’s mid-mantle hetero-

geneity contains electrical conductivity variations greater than set and the high computational cost of the FEM forward solvercurrently requires special techniques, including local para-a factor of 10, but perhaps less than a factor of 100, over

continental length scales. At shorter periods, a similar analysis metrization in electrical conductivity (e.g. Weiss & Everett 1996)and efficient model search strategies for the solution to the inverseof Figs 9 and 10 reveals that the spherical-inclusion models

generate approximately an order of magnitude less c response problem. We will present the results of a non-linear inversion

and its geodynamical implications in a future publication.variability than is observed. This result supports a stronglyheterogeneous upper mantle with lateral electrical conductivityvariations greater than the two orders of magnitude already

ACKNOWLEDGMENTSdetected (Lizzaralde et al. 1995).

This paper has addressed some important issues in current We thank the National Science Foundation for financialsupport. Illustrations in this paper were generated by plotxy,global geomagnetic induction research, including the role of

oceans and continents, the sensitivity of surface responses to kindly provided by R. Parker, and the StarBase HewlettPackard graphics library. We also thank Hazel Everett formantle heterogeneities and the fully 3-D validation of the

forward solver. These issues are now resolved to the point introducing us to the theorem used as the foundation of our

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Martinec, Z., 1998. Geomagnetic induction in multiple eccentrically APPENDIX A: CONVERGENCEnested spheres, Geophys. J. Int., 132, 96–110. PROPERTIES OF THE ILU SOLVER

Masters, G., Johnson, S., Laske, G. & Bolton, H., 1996. A shear-velocity

model of the mantle, Phil. T rans. R. Soc. L ond., A, 354, 1385–1411. The ILU solver performs well for the models we have examinedOmeragic, D. & Silvester, P.P., 1996. Numerical differentiation in so far but may be ill-suited as a general matrix solver. For

magnetic field postprocessing, Int. J. Numer. Mod., 9, 99–113. finite element analyses, the ILU solver is attractive since itPark, S.K., Biasis, G.P., Mackie, R.L. & Madden, T.R., 1991. exploits the inherent sparsity in the system of equations, a

Magnetotelluric evidence for crustal suture zones bounding thefeature which results in massive savings of computer storage.

southern great valley, California, J. geophys. Res., 96, 353–376.For example, only 0.5 per cent of the matrix elements are non-

Parkinson, W.D. & Jones, F.W., 1979. The geomagnetic coast effect,zero for a moderately sized mesh containing 19 465 tetrahedra.Rev. Geophys. Space Phys., 17, 1999–2015.The algorithm for the ILU is fully described in Everett &Poirier, J.P., 1991. Introduction to the Physics of the Earth’s Interior,Schultz (1996) and is a straightforward adaptation of theCambridge University Press, Cambridge.discussion presented in Duff, Erisman & Reid (1986).Ricard, Y., Richards, M., Lithgow-Bertelloni, C. & Le Stunff, Y., 1993.

A geodynamic model of mantle density heterogeneity, J. geophys. We note that the ILU is guaranteed to converge if and onlyRes., 98, 21 895–21 909. if the spectral radius r of the iteration matrix (I−LU

~A) is less

Rikitake, T., 1961. Sq and Ocean, J. geophys. Res., 66, 3245–3254. than unity. We have not computed the spectral radius explicitlyRoberts, R.G., 1984. The long period electromagnetic response of the in order to prove convergence; however, the ILU solver

Earth, Geophys. J. R. astr. Soc., 78, 547–572. converges monotonically for all reasonable Earth conductivityRomanowicz, B., 1995. A global tomographic model and shear

models, as we now demonstrate.attenuation in the upper mantle, J. geophys. Res., 100, 12 375–12 394.

To examine the convergence properties of the ILU andRostoker, G., Friedrich, E. & Dobbs, M., 1997. Physics of magneticshow its suitability for geomagnetic induction studies we havestorms, in Magnetic Storms, eds Tsurutani, B.T., Gonzalez, W.D.,performed a suite of tests on various conductivity modelsKamide, Y. & Arballo, J.K., AGU Geophys. Monogr., 98, 149–160.based on previous regional geomagnetic studies. These studiesSchultz, A., 1990. On the vertical gradient and associated heterogeneity

(Egbert & Booker 1992; Schultz et al. 1993; Lizzaralde et al.in mantle electrical conductivity, Phys. Earth planet. Inter., 64, 68–86.

Schultz, A. & Larsen, J.C., 1987. On the electrical conductivity of 1995) reveal a wide range of geographic variability, up to twothe mid-mantle—I: calculation of equivalent scalar magnetotelluric orders of magnitude at ~100 km depth and decreasing toresponse functions, Geophys. J. R. astr. Soc., 88, 733–761. approximately one order of magnitude below ~300 km depth.

Schultz, A. & Zhang, T.S., 1994. Regularised spherical harmonic On the basis of the compilation of regional profiles shown inanalysis and the 3-D electromagnetic response of the Earth, Geophys.

Fig. A1, 3-D electrical conductivity models were generatedJ. Int., 116, 141–156.

which preserve the gross depth dependence in the Earth’sSchultz, A., Kurtz, R.D., Chave, A.D. & Jones, A.G., 1993. Conductivity

conductivity but contain varying amounts of randomizeddiscontinuities in the upper mantle beneath a stable craton, Geophys.lateral heterogeneity. Convergence of the ILU is shown inRes. L ett., 20, 2941–2944.Fig. A2 along with a detailed description of the conductivityTabbara, M., Blacker, T. & Belytschko, T., 1994. Finite elementmodels.derivative recovery by moving least squares interpolants, Comput.

Meth. appl. mech. Eng., 117, 211–223.

Takeda, M., 1993. Electric currents in the ocean induced by model

Dst field and their effects on the estimation of mantle conductivity, APPENDIX B: POINT-IN-POLYGONGeophys. J. Int., 114, 289–292. ALGORITHM

Tarits, P., 1994. Electromagnetic studies on global geodynamicA spherical polygon is a set of points on a sphere connectedprocesses, Surv. Geophys., 15, 209–238.

Tarits, P., Wahr, J. & Lognonne, P., 1995. A solution to the problem by great-circle arcs. We have developed a point-in-polygonof internal and external electromagnetic induction in a spherical algorithm (PIPA) to determine whether an arbitrary point onheterogeneous Earth (abstract), EOS 1995 Fall Meet. Suppl., 76, the surface of a sphere is contained within a given sphericalF165. polygon. The PIPA is based on a theorem in computational

Tyburczy, J.A., 1996. Lateral variations of electrical conductivity in geometry:the 200 to 400 km depth range: interpretations based on mineral

any great-circle arc drawn between a point p∞ locatedphysics results (abstract), EOS 1996 Fall Meet. Suppl., 77, F168.

inside a spherical polygon and a point p located outsideVarshalovich, D.A., Moskalev, A.N. & Khersonskii, V.K., 1988.will intersect the polygon an odd number of times; if p∞ isQuantum T heory of Angular Momentum, p. 216, World Scientific,located outside the spherical polygon, any great-circle arcSingapore.between p∞ and p will intersect the polygon zero or anWannamaker, P.E., Booker, A.G., Jones, A.G., Chave, A.D., Filloux, J.H.,

Waff, H.S., Law, L.K. & Young, C.T., 1989. Conductivity cross even number of times.

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Figure A2. Convergence of the ILU solver as a function of model

heterogeneity. Each model is composed of FEM tetrahedra whoseFigure A1. A collection of conductivity (s) profiles (symbols) fromconductivity values are drawn from a uniformly distributed parentthree different tectonic settings: Canadian Shield (Schultz et al. 1993),population. The parent population is centred on the five-layer profileNorth American Basin and Range (Egbert & Booker 1992) and NEshown in Fig. A1 and spans a depth-dependent envelope of conduc-Pacific Plate (Lizzaralde et al. 1995). On the basis of the regionaltivity values. The width of the envelope used for each model isobservations, a five-layer global conductivity profile was constructedindicated by the italicized numbers and follows the same naming(heavy line). Symmetric about this profile and shown by the inter-convention as used in Fig. A1. Shown in the inset is the number ofmediate shaded region labelled 1.0 is a depth-dependent envelopetetrahedra which lie in each of the five layers.of conductivity values which roughly bound the observed regional

variations. Increasing the width of the 1.0 envelope by 40 per cent

(in log conductivity!) yields the lightly shaded region labelled 1.4 while

decreasing by 50 per cent yields the more darkly shaded 0.5 envelope.

By this labelling convention, the 0.0 envelope (not labelled in the

figure) is coincident with the heavy black line.

The regions ‘inside’ and ‘outside’ must be defined since a

collection of vertices and edges describes two complementaryspherical polygons, each of which is the ‘outside’ region ofthe other. The above theorem lacks full mathematical rigour

since arcs that coincide with polygon edges or which intersectpolygon vertices are not properly treated. However, in practice,these special cases do not pose a problem. A graphical illus-

tration of the theorem is shown in Fig. B1. Bevis & Chatelain(1987) developed a similar algorithm for sorting earthquakeepicentres into geographical regions. Figure B1. Graphical illustration of the central theorem to the point-

Consider the spherical polygon with vertices ai shown in in-polygon algorithm. Shown is the spherical polygon used to representFig. B1. A test point p∞ and the outside reference point p are North America with vertices ai connected by great-circle arcs. A great-

circle arc between point p outside the polygon and the test point p∞also indicated. The epicentral angle between any two verticesintersects the polygon an odd number of times only if p∞ lies withindefining a polygon edge is strictly less than 180°. This ensuresthe polygon.that each edge of the spherical polygon is a minor great-circle

arc. Similarly, we only consider the minor arc between p andp∞. If p is antipodal to p∞ no unique minor arc exists andanother point p must be chosen.

(4) Arc pp∞ intersects arc ai−1ai when either of the followingThe primary steps in the PIPA are outlined as follows.conditions on minor arc length |Ω| is satisfied:

(1) Set iteration counter k=0.(2) Loop over polygon edges, i. (i) |ps|+|sp∞|=|pp∞|, |ai−1s|+|sai |=|ai−1ai | ,(3) Compute the intersection of two great circles: one

passing through points p–p∞, and the other coincident with oredges ai−1ai. The intersection consists of two antipodal points,s and s∞. (ii) |ps∞|+|s∞p∞|=|pp∞|, |ai−1s∞|+|s∞ai |=|ai−1ai | .

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(5) If condition (i) or (ii) is satisfied, update intersection one of the following factors: the finite numerical precision

of the computer, or a pathological case when arc pp∞ iscounter k=k+1 and proceed to the next edge, i+1.(6) Apply the central theorem when all edges have been coincident with a polygon edge or intersects a polygon vertex.

In practice, conditions (i) and (ii) of the stated outline arevisited: if k is odd then p∞ is located inside the polygon, if k is

zero or even then p∞ lies outside. enforced within a tolerance e to compensate for the propa-gation of truncation error within the computer; typicallye~10−6 for double-precision arithmetic. The pathologicalNon-convex polygon features such as ‘bays’ or ‘peninsulas’

pose no additional computational burden for the PIPA. cases are easily identified and avoided by choosing differentoutside points p.However, when errors do appear, they are the result of

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Geomagnetic induction in a heterogeneous sphere - Geophysical