The geomagnetic/magneto-telluric induction problem with spatial conductivity fluctuations

0.71 MB·

by  R. A. Treumann

ManuscriptpreparedforAnnalesGeophysicæ
withversion3.2oftheLATEXclasscopernicus.cls.
Date:26January2017
The geomagnetic/magneto-telluric induction problem with spatial
conductivity fluctuations
7
1
0 R.A.Treumann∗1andW.Baumjohann2
2 1DepartmentofGeophysicsandEnvironmentalSciences,MunichUniversity,Munich,Germany

n 2SpaceResearchInstitute,AustrianAcademyofSciences,Graz,Austria
a
J ∗VisitingtheInternationalSpaceScienceInstitute,Bern,Switzerland

5
2
Abstract. The electromagnetic induction equation 1 Introduction

]
h (Helmholtz equation) for the electrically conducting
p Earthisgeneralisedtotheinclusionofaspatiallyfluctuating Thepresentnote1dealswithanextensionofgeomagneticin-
- internal conductivity spectrum that is superimposed on a ductiontheorytotheinclusionofaspatiallyfluctuatingcon-
e
c one-dimensional large-scale conductivity reference-profile ductivitydistributioninsidetheEarth.
a which depends solely on the vertical coordinate z>0. This Ingeophysicalprospecting,aswellasingeomagnetismin
p
large-scale profile is assumed to be known. It might consist general,itisofvitalinteresttogatherinformationaboutthe
s
. of a stratification of layers of different conductivity and conductivity structure of Earth’s interior in particular about
s
c thickness but serves just as the background reference for a the distribution of conductivity in Earth’s crust and man-
i distribution of intrusions of differently conducting regions tle. These regions are not directly accessible such that one
s
y whosesizemayspatiallyvaryaccordingtoa(givenoreither is forced to apply indirect methods of determination of the
h unknown) spectrum of fluctuations. The distribution of the propertiesoftheunderneathmaterial.Seismicinvestigations
p fluctuations is allowed to be two-dimensional but can also
[
bethree-dimensional.Weobtaintheconductivity-fluctuation 1This paper has in 2016 been rejected by Ann.Geophys. with
1 generalised Helmholtz equation. It depends on the spectral
thereasonablecommentthattheauthorsarenotfamiliarwiththe
v density of the fluctuations as function of the vertical and progress achieved within the past 3 decades in solving the induc-
2
horizontalcoordinates.Thesearecontainedinanadditional tion problem numerically suggesting several reviews (Alumbaugh
7
term in the Helmhotz equation. In the case of a prescribed etal.,1996;Avdeev,2005;EgbertandKelbert,2012;Kelbertetal,
2
7 fluctuation model the forward problem must be solved. 2014;Meqbeletal,2014;Yangetal,2015).Weareverygratefulfor
0 Application to the inverse geomagnetic/magneto-telluric thesesuggestionasweareindeednotworkinginthisfieldcoming
. problemineitherofitsversionsisdiscussed.Initsanalytical fromplasmaturbulencewithwhatisproposedinthisnotetopossi-
1
blybeapplicabletotheinductionproblemaswell.Sinceourtime
0 approach (Weidelt, 1972, 2005) we show that the inverse
doesnotpermittoinvestmoreeffortsintothisproblemwesimply
7 problem yields the fluctuation spectrum. The relation
provideitherefortheinformationofaninterestedreader.Oneof
1 between the spectral density and the obtained conductivity
: therefereesalsoclaimedthatthefluctuationshadbeenalreadyin-
v profile is given. By chosing different reference profiles the
cluded in theory. In inspecting the above reviews we have indeed
i spectrum could be optimised to fit the data best. Making
X not found any similarity with our approach. Temporal fluctuation
useofthepenetrationdepth,arelationbetweenthespectral spectrahaveofcoursebeentreatedbuttheinclusionofspatialcon-
r
a density of the conductivity spectrum is derived which is ductivity fluctuations to the degree done here is not detectable to
subjecttofurthergeophysicalinterpretation.Thismaybeof us.Thereisnodoubtthatinpraxithenumericalworkisveryvalu-
interestinspaceapplicationsliketheJUICEmission. able.Thereviewerwrote:“Infact,asisshownbydozensofpub-
lishedmanuscripts,thesereasonsincludethepresenceofwaterand
Keywords. Geomagneticinduction,Conductivitystructure,
partialmelt,temperaturevariations,presenceofmetalsandcertain
Magneto-Tellurics,Inversegeomagneticinductionproblem, othervolatiles,suchascarbondioxide,intheEarth’smantleandthe
Earth:Geomagneticeffectsofinternalconductivitystructure crust.Moreover,practical3Dinversionsareabletoconstrainsome
ofthesefeaturesregionallyintheEarth.”Thereisnothingtoaddto
thisstatement.Neverthelesswebelieveitmakessense,tobringthis
Correspondenceto:R.A.Treumann idea/attempttotheattentionofthecommunitywithoutcompletely
([email protected]) reworking.
2 R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem
already provide most important information. However, in- similarthoughdifferentproblemismonitoringthevariations
formation about the conductivity requires additional input ofEarth’sinternalfieldatEarth’ssurfaceinviewoftheirin-
sincethephasevelocitiesofseismicwavesdependonlyindi- formationcontentabouttheconductivitystructureinthelay-
rectlyontheundergroundconductivity.Geomagneticinduc- ersbetweenthedynamoregionandobservationalsites.)By
tionmethodsarethereforehighlydesirable.Thoughbeingin- itsnaturethisisanill-posedproblem.Itsassumptionsarethat
directaswell,theyareboundtotheconductivitydistribution theoriginalinductionfield,avaryingmagneticfieldB(x,t)
via the penetration depth of the external geomagnetic field –mostlyofnaturalorigin–inducesanelectriccurrentflow
into the ground. This is a function of conductivity and fre- j inside the electrically conducting parts/layers of the sub-
quency of the applied natural magnetic induction field. Ac- surfaceinterioroftheEarthwhoseelectromagneticinduction
counting for the induction inside the Earth is important as field penetrations to the surface where it can be detected as
well for separating the internal contribution to the geomag- a modification of the initial induction field. The problem is
netic variation field. An important further application con- quasi-stationary such that any displacement currents can be
cernsthequestionofhowtheinternaldynamo-generatedge- neglected. Moreover, there are no electric charges present.
omagneticfieldpenetratesalmostdiffusivelyacrossthecon- Then, with the (isotropic scalar) conductivity σ(x), on all
ductingmantleandcrust. relevanttimescalesastationaryfunctionoflocationxonly2,
Theoretically, it is quite clear that the most inhomoge- Maxwell’sequationsreduceto
neously electrically conducting parts of Earth are the crust
1
and mantle with the scale of inhomogeneity probably in- ∇2E=µ σ(x)∂ E, ∇× ∇×B=−∂ B (1)
0 t µ σ(x) t
creasingwithincreasingdepthbelowsurface.Moreover,be- 0
lowacertainmantledepthandcertainlyatthetransitionfrom
withtheadditionalconditions
mantletocoreEarth’sinteriorcan,fromanypracticalpoint
ofview,beconsideredasideallyconducting,nomatterwhat ∇·B=0, ∇·E=−E·∇ln∈(x) (2)
thereasonforthegenerationofconductivitywillbe.Thisas-
sumptionisinvalidonlywhendealingwiththegeomagnetic withboundaryconditionsatEarth’ssurfacerequiringthatthe
dynamo.Sincethedynamofieldvariesonalongscaleonly, tangential electric and magnetic fields must be continuous.
atthecorrespondingtimescalesthemantleandcrustcanbe The latter of the last equations is added here for the reason
consideredasbarelyconducting. thatthedielectricconstant(cid:15)=∈(x)·(cid:15)0 maylocallydeviate
Geomagnetic induction theory has found extended appli- fromitsvacuumvalue.Thisisforinstancethecaseinwater
cation for now decades. In the present note we add to it the where one has ∈(H2O)≈80. Moreover the dielectric con-
proposal that the conductivity structure inside Earth – and, stantdependsontemperatureandfrequency.Thiscondition
of course, also on other planets or their satellites like, for isusuallyneglectedinanytreatmentofthegeomagneticin-
instance, Ganymed, if only regular magnetometer measure- ductionproblem.Itmaybecomeimportantwheninterpreting
mentsofthefluctuationsoftheirmagneticfieldswillbeper- inductioneffectsattheoceanicshoreandelsewhereinlayers
formed over a longer period on close-by orbits like those where water is contained. We may assume that ∇∈ is re-
planned for the JUICE mission – can be considered as a strictedonlytotheboundariesofsuchdomains,forinstance
superposition of a main and simple one-dimensional given ocean shores. In this case the above condition just becomes
reference-conductivitymodel,inwhichtheconductivityde- a secondary condition on the solution of one of the above
pendsonlyondepth,andasuperimposedfluctuatingconduc- field equations, where it must necessarily play a role when
tivity of shorter scale which modifies the reference model. consideringinductioneffectsof,forinstance,theequatorial
Weshowhowthecommoninductiontheoryhastobemodi- electrojetoralternativelythegeomagneticSq-variations.
fiedinordertoaccountforthisdistributionandcanpossibly Sincetheelectricandmagneticfieldsarerelatedby∂tB=
provideadditionalinformationabouttheconductivitystruc- −∇×E,itismostconvenienttostayintheelectricTErepre-
tureintheundergrounduptoacertaindepthbeneathsurface. sentation,restrictingtotheelectricfield.Moreover,Lorentz-
gauge implies E=−∂ A, and ∇·A=0 holding outside
t
boundariesofdomainswhere∈changesdiscontinuously.Ig-
2 Briefreviewofbasics noring, as commonly done, such regions, the TE problem
2The condition of stationarity applies to any of the geomag-
The general geomagnetic induction problem is, of course,
netic induction fields considered in the induction problem. How-
subject to the quasi-stationary electrodynamics applied to
ever,whenbeinginterestedinlong-scalevariationslikethesecular
the body of Earth. The idea is to gather information about
variationofthegeomagneticfieldorlong-termchangesintheex-
the conductivity distribution beneath/below Earth’s surface
ternalconditionsofthegeomagneticfield,saysecularaveragesof
downtounaccessibledepthsbymonitoringtheEarth’selec-
geomagnetic disturbances etc., then time scales of plate tectonics
tromagnetic response to variations in the geomagnetic field aswellasvariationsinthegeodynamocomeintoplay.Undersuch
atthesurface.Thisproblemisknownastheinversegeomag- conditionstheconductivitydistributioninsidetheEarthwillexhibit
neticinduction–ormagneto-telluric–inversionproblem.(A long-termvariationsandbecomesafunctionoftime.
R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem 3
can also equivalently be formulated for the vector potential Hence,thefunctionc(x,y,0,s)entersthetemporalconvolu-
A(x,t) tionoftheverticalderivativeofthetangentialelectricfieldin
anygivensurfacepoint(x,y):
∇2A=µ σ(x)∂ A, ∇·A=0 (3)
0 t
(cid:90) ∞ (cid:90) t
ThemagneticfieldfollowsfromB=∇×A. E (0,s)=− dte−st c(0,t−τ)E(cid:48) (0,τ)dτ (11)
T T
In locally homogeneous tangential induction fields E= 0 0
Ee , with e the tangential vector at Earth’s surface, only
x x
one field component remains. This is the simplest imagin- Theresponsefunctionc(x,y,0,s)foldsthefieldanditsver-
able case. If one, in addition, restricts to a mere vertical z- ticalderivative.Itmediatesthetime-delayedresponsetothe
dependence of the conductivity the problem simplifies even tangentialinductionfieldinthisparticularsurfacepoint.
more.TheTEequationthenreducesto Suppliedwiththeboundaryconditionsofcontinuityofthe
tangentialfieldsatEarth’ssurface,theaboveTEequationfor
E(cid:48)(cid:48)(z,t)=µ0σ(z)∂tE(z,t) (4) theelectricfieldandtheresponsefunctionformthebasisfor
theelectromagneticinductionproblem.Herewehavechosen
where (cid:48) indicates differentiation with respect to z. The time
the TE representation. Alternatively an equivalent magnetic
dependencecanquitegenerallybetreatedbyLaplacetrans-
formulation can be given which, however, is formally more
formationofeitherthefieldsorthevectorpotential:
involved.
∞ c+i∞ ThereareseveralapproachestosolvetheTEproblem.The
(cid:90) (cid:90) ds
E(x,s)= dtE(x,t)e−st, E(x,t)= E(x,s)est (5) forward approach assumes a model of the conductivity dis-
2πi tribution in order to reproduce and match the magnetic and
0 c−i∞
electric field components which result from the response of
Thisispreferableasitallowsforapplicationtonon-periodic theEarthtotheapplicationoftheexternalinducingfield.The
perturbations.Theenergydensitiesoftheelectricandmag- complementaryapproachistosolvetheso-calledinversein-
neticfieldsaregivenby ductionproblem.Foralargenumberofmeasurementsofthe
electricfieldcomponentsoreithertheresponsefunctionthe
(cid:15)0(cid:104)E2(cid:105)= (cid:15)0 (cid:90) c+i∞ds(cid:12)(cid:12)E(s)(cid:12)(cid:12)2 (6) system becomes overdetermined. This enables the applica-
2 8π2 tion of well-developed numerical procedures to infer about
c−i∞
theconductivitystructureinoneormoredimensionsandap-
1 1 (cid:90) c+i∞ (cid:12) (cid:12)2
(cid:104)B2(cid:105)= ds(cid:12)∇×E(s)/s(cid:12) (7) plicationofoptimisationmethods(cf.,e.g.,Beusekometal.,
2µ0 8π2µ0 c−i∞ (cid:12) (cid:12) 2010,forapplicationofoptimisationmethodsintheinverse
problem). The third more theoretically oriented method re-
Theintegrandsarethespectralenergydensitiesofthefields
lies on the solution of the theoretical inverse problem. This
averagedoverthelongesttimevariation.
method has been put forward by Weidelt (1972, 2005) and
The transformed fields are complex quantities. The field
is based on the reformulation of the Gel(cid:48)fand-Levitan ap-
equationthenreducestotheHelmholtzequationconvention-
proach (Gel(cid:48)fand and Levitan, 1951; Marchenko, 2011) to
allyusedinthegeomagneticinductionproblem
the inverse Sturm-Liouville (stationary Schro¨dinger equa-
(cid:104) (cid:105) tion)problem(cf.,e.g.,Koelink,2008,fortherigorousmath-
∇2E(x,s)=µ σ(x) sE(x,s)−E0 (8)
0 ematicaltheoryofthelatter).AsshownbyParker(1980),the
exactsolutionofthisprobleminthegeomagnetic/magneto-
where E0≡E(x,t=0). With z the vertical coordinate one telluric case is possible only in certain cases. Nevertheless,
maythen,asdoneinTE/magneto-tellurics,definearesponse itprovidesaverylucidunderstandingoftheinversionofthe
functionatEarth’ssurfacez=0by Helmholtz-geomagnetic induction equation and the limita-
tions of any inversions, in particular as the problem itself
E (x,y,0,s) E (x,y,0,s)
c(x,y,0,s)=− T =− T (9) is ill-posed (Jackson, 1972). It does anyway not allow for
E(cid:48) (x,y,0,s) sB (x,y,0,s)
T T anunambiguousreconstructionoftheconductivityprofilein
theunderground.Justfromthispointofviewastatisticalap-
where E (x,y,0), B (x,y,0) are the local orthogonal tan-
T T
proach to the Earth’s crustal and mantle conductivity distri-
gential components of the electric and magnetic fields at
butionisofvitalinterestinpracticalapplicationsaswellas
Earth’s surface obtained in the point (x,y). The response
itprovidesabaseforfurthergeophysicalinterpretation.
function c(x,y,0,s) is a measurable quantity. It is an ana-
lytical function whose general properties have been investi- Inthefollowingweproposeareformulationoftheabove
gated.Thelastequationcanalsobewrittenastheproductof sketchedtheorytotheinclusionofadistributionofconduc-
twoLaplacetransforms tivityintheEarth.Suchanapproachcircumventstheneces-
sityofsolvingaparticulargivenmodelwhichmightnotbe
E (x,y,0,s)=−c(x,y,0,s)E(cid:48) (x,y,0,s) (10) wellsuitedinrealsituations.
T T
4 R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem
3 Spatiallyfluctuatingconductivitydistribution With these conditions in mind we rewrite the basic TE-
Helmholtz equation for the above conductivity profile with
LetusassumethattheconductivityofEarthhasaparticular E→(E +δE)obtaining,afteraveraging
0
structurewhichsonsistsoftwomaincomponents.Thelarge-
scalestructureoftheconductivity,σ(z)isassumedtobever- ∇2E =µ σ (cid:104)F (cid:0)sE −E0(cid:1)+s(cid:10)δFδE(cid:11)(cid:105) (14)
0 0 0 0 0
tical (or radial) with the conductivity organized in shells or
otherwise varies gradually on the vertical scale z>0 only. This equation replaces the original TE equation. It contains
Thismainverticalconductivityprofileisoverlaidwithirreg- theinitialvalueoftheelectricfieldE0≡E(t=0)whichre-
ularlydistributedsmallerregionsoflesserorbetterconduct- sults from the use of the Laplace transform. For harmonic
ing material. Of such regions there may be many. Treating timevariationsoffrequencyω<ωmax itismoreconvenient
theminanyforwardmodelbecomesimpossiblejustforthe to use the temporal Fourier transform. In this case the ini-
reasonthatitwouldbeverydifficulttoadjustforthegeome- tialvaluedropsout,andonehass=iω.Asanewadditional
triesandsatisfythevariousboundaryconditions.Inorderto term,theaboveequationcontainsthecorrelationbetweenthe
circumvent this difficulty the scale of variation of the con- spatial fluctuations of the conductivity and the electric field
ductivitydistributionisassumedtobelessthanthescaleof thatiscausedbythesefluctuations.
the primary induction field, in which case the variation can Subtracting the averaged equation from the original not
be considered smooth and no internal boundary conditions averagedone,thefollowingLaplace-transformedequationis
have to be taken into account. With these assumptions the obtained:
conductivityisrepresentedasthesumoftwoterms ∇2δE=µ σ (cid:104)s(cid:0)F δE+δFδE−(cid:10)δFδE(cid:11)(cid:1)+
0 0 0
σ(x)=σ0(cid:2)F0(z)+δF(z,x(cid:48))(cid:3) (12) +δF(cid:0)sE −E0(cid:1)(cid:105) (15)
0
whereσ isaconstantreferencevalue,F (z)themainver-
0 0 It is understood as an equation that determines the spatial
tical structure function, and δF(z,x(cid:48)) is the fluctuation of
evolution of the fluctuations in the applied electric field in-
the conductivity, in general a three-dimensional function of
cludingitsresponsewhichresultsinthevariationδEofthe
theprimedsmall-scalecoordinatesx(cid:48)thatvariesslowlywith
electric field. Here the fluctuations in the conductivity are
depth on the large scale z. One may assume that the small-
assumed to be given. Since the long-scale variation of the
est scales should be found in the Earth’s crust closer to the
correlation is taken into account in the former equation we
surfacewhilethescaleofthefluctuationsshouldgrowwith
assumethatthetermcontainingthecorrelationontheshort
increasing depth. However, at the present stage no such ex-
scales is of second order and can thus be neglected. More-
plicitassumptionswillbemadeontheparticularformofδF.
over,theaveragetermvariesonthelongscaleonly.Forthis
Theonlyassumptionisthatthedistributionoftheintrusions
reasonitisaconstantonthescaleofthevariationoftheelec-
of better or worse conducting material is such that on the
tricfield.Wedropitinthisapproximationthoughitcouldin
verticalscaleδF averagesoutwhenaveragedoverthemain
principle be retained and added to the last term which pro-
referenceframeofthemainglobalverticalconductivitypro-
videsanotherinhomogeneity.
file.
With these approximations and conventions we have for
This assumption does not imply that the fluctuations in
the spatial electric field fluctuations which are produced in
conductivity are small amplitude; it merely says that their
thepresenceofaspectrumoffluctuationsintheunderground
scalesontheverticalaresubstantiallyshorterthanthescale
conductivity:
of variation of the main stratification of the conductivity.
Nothing is assumed in the horizontal direction except the ∇(cid:48)2δE=µ σ (cid:104)sF δE+δF(cid:0)sE −E0(cid:1)−s(cid:10)δFδE(cid:11)(cid:105) (16)
0 0 0 0
noted smallness of the scale with respect to the horizontal
scaleoftheappliedinductionfield.Thustheconditionisthat This is an inhomogeneous linear equation for the electric
field fluctuations. After solving it, the result can be used to
(cid:10) (cid:11)
δF =0 (13) obtainanexpressionforthenonlinearterminthefluctuation-
modified Helmholtz equation (14). The prime on the ∇(cid:48)-
holds locally, i.e. on the noted horizontal scale. Monitoring
operator indicates that in this equation it acts on the short
the sub-surface response to the application of the temporal
fluctuationscaleonly.Itdoesnotaffecttheglobalscaleofthe
variation of the induction field along the surface thus pro-
average field E nor the dependence of F (z). The bound-
0 0
vides information about the lateral variation of the conduc-
ary condition is that the total electric field and its vertical
tivity distribution. The angular brackets (cid:104)...(cid:105) indicate aver-
derivative are continuous at z(cid:48)=0, the surface of Earth. It
aging over the vertical scale z of the variation of the main
thus applies to the combination of both the average and the
profile F (z). This also implies that the fluctuating part of
0 fluctuatingfields.Sinceσ=0inz(cid:48)<0,theequationforthe
the induction field and its derivatives that are produced on
fluctuationsaboveEarthreducestotheLaplaceequation
theshortscalesbythespatialfluctuationsoftheconductivity
averageout:(cid:104)δE(cid:105)=(cid:104)δE(cid:48)(cid:105)=0. ∇(cid:48)δE=0, z(cid:48)<0 (17)
R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem 5
As explained above, we may for simplicity drop the last κ-plane
term on the right in (16). Then, on the fluctuation scale all
indexedcoefficientsareconstant,asisalsotheinitialfieldas r
0
it does not participate in the fluctuations yet. Moreover we 0
do not assume any fluctuation model. The final equation is r ψ
0
linearinthefluctuationsandcanthusbesolvedbystandard
methods. We can apply spatial Fourier transforms in x(cid:48),y(cid:48)
and a Laplace transform in z(cid:48), counting z(cid:48)>0 positive in
thedownwarddirection.Thisimpliesthat∇(cid:48)=ik,withk=
(k ,k )theatwo-dimensionalwavevector,yielding
x(cid:48) y(cid:48)
(cid:16) (cid:17) (cid:104)
k2−∇(cid:48)2 δE (z(cid:48))=µ σ sF δE (z(cid:48))+
z k 0 0 0 k
(cid:16) (cid:17) (cid:105)
+ sE −E0 δF (z(cid:48)) (18) Fig.1.Theintegrationcontourinthecomplexκplane.Integration
0 k
isrestrictedtothelowerκ-halfplane.
ThisequationmustnowbeLaplace-transformedinz(cid:48),mul-
tiplying with e−κz(cid:48) and performing the integration over the
interval0≤z(cid:48)<∞.Itsright-handsideprovidesnoproblem. δ(k+k(cid:48))δ(κ+κ(cid:48)). Hence, replacing k(cid:48) →−k, κ(cid:48) →−κ
Onesimplyreplacesallz(cid:48)-dependencieswithκ.Theopera- yieldsforthecorrelationfunction
torontheleftyields,however,twonewterms.Thefinalresult
∞ c+i∞
is (cid:68) (cid:69) 1 (cid:90) (cid:90) dκ
δFδE = d2k ×
µ σ (cid:0)sE −E0(cid:1)δF (κ) (2π)3i k2−κ2−µ0σ0sF0
δE (κ)= 0 0 0 k + (19) −∞ c−i∞
k k2−κ2−µ0σ0sF0 (cid:110) (cid:12) (cid:12)2
× µ σ (sE −E0)(cid:12)δF (κ)(cid:12) − (20)
δE(cid:48)(0)+κδE (0) 0 0 0 (cid:12) k (cid:12)
+ k k
κ2+µ0σ0sF0−k2 −(cid:104)δE(cid:48) (0)−κδE (0)(cid:105)δF (κ)(cid:111)
−k −k k
Theprimeindicatesdifferentiationwithrespecttoz(cid:48)>0,and
theargument0indicatesthatthequantityhastobetakenat In this expression |δFk(κ)|2 is the spectral density of the
z(cid:48)=0.Hencethisexpressionincludestheboundaryvaluesof conductivityfluctuations.Onemaynotethatthefieldissym-
the spatial electric field fluctuations at the surface of Earth. metricink:itholdsthatEk=E−k.Inthisnotationwesup-
Allthesefluctuationsdependonthetransformsofthespatial pressedthedependenceofallquantitiesonthelargescalez.
fluctuationsoftheconductivityδF (κ).
k
Thisexpressionmustbeusedtocalculatetheaveragecor- Intheforwardproblemonewouldchoseadefinitemodel
relation of the field and conductivity fluctuations. This pro- fortheconductivityfluctuationssuchthatitisconsideredas
cedure will yield the explicit form of the ultimate average known. In this version it depends on the large scale coordi-
conductivityfluctuation-generalisedinductionequation(14). nates, however. F0(z), the main spatial profile of the con-
ductivity,dependsonzonly.Clearly,thisexpressionisquite
involved. It is not only complicated by the presence of the
4 Fluctuation-generalisedinductionequation surfaceandinitialvaluesofthefieldbutalsobythedenom-
inator in the integral. For this reason we will farther below
Our aim is to find the fluctuation-averaged induction equa-
introduceasimplification.
tionwhichtellshowthedistributionofelectricalconductiv-
However, before doing this we go one step further in the
ity on different scales affects the electric induction field. It
calculation.Thedenominatorofthecomplexκintegralgen-
willcontainalltheinformationabouttheconductivitydistri-
eratespolesinthecomplexplane.Ifweassumethattheex-
butionintheearthlyhalfspace.Inordertoachievethisgoal
pressioninthecurlybracketsisanalytical,thentheintegral
we must calculate the average correlation function (cid:104)δFδE(cid:105)
canbesolvedbyintegratingoverthepoles.Thiscaneitherbe
between the spatial fluctuations of the conductivity and the doneintheκ-planeorinthek-plane.Definingk2=sµ σ F
0 0 0 0
inducedfluctuationsoftheelectricfield.
and
Calculatingthistermrequiresformationoftheproductof
(cid:113)
theconductivityandfieldfluctuationsandaveragingoverthe κ =± k2−k2≡re∓iψ/2 (21)
± 0
global scales. When representing the fluctuations as the in-
verse Fourier respectively Laplace transformations of argu- thedenominatorcanbefactorised.Undertheassumptionthat
mentsk,k(cid:48) andκ,κ(cid:48).Anumberofexponentialfunctionsof theconductivityspectrumbehavesregularly,theκ-integralis
theseargumentsarise.Thespatialaveragesoverthesefunc- formallysolved.Bothk andsarethemselvescomplexvari-
tions lead to products of Dirac-delta functions of the kind ables.Letus,forsimplicity,assumethats→iω,withω the
6 R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem
(cid:41)
frequencyoftheappliedinductionelectricfield.Itisassumed (cid:104) (cid:105)
− δE(cid:48)(0)−κ δE (0) δF (κ ) (26)
that a broad range of frequencies may be applied by taking k + k k +
their Fourier transform instead of the Laplace transform. In
this case the intial value of the field E0 drops out from the where κ is to be expressed through k. Solution of the k-
+
aboveequation.Wehavek2≡iωµ σ F =iρ2andcomplex
0 0 0 0 0 integral is inhibited if no model is assumed for the conduc-
k=ρexp(iχ)withρthemodulusandχthephase.Then tivityfluctuations.Noteagainthatκ ,k ,F ,δF alldepend
+ 0 0 k
(cid:112) on the large scale z. Hence, the new additional term in the
κ =± ρ2exp(2iχ)−iρ (z) (22)
± 0
Helmholtz equation caused by the spatial conductivity fluc-
Thefrequencyωisreal.Thepolesofκinthecomplex(r,ψ)- tuationsturnsouttobearathercomplicatedfunctionofthe
planelieat globaldepth-coordinatez.
(cid:113) Thisexpressioninitsimplicitformistobemultipliedwith
r = 4 ρ4cos22χ+(ρ2sin2χ−ρ2)2 (23)
0 iωandinsertedintothefluctuation-averagedinductionequa-
(cid:18)ρ2sin2χ−ρ2(cid:19) tion (14). While the integral containing the spectral density
ψ =tan−1 0 (24)
ρ2cos2χ of the conductivity fluctuations adds to the linear term in
thatequation,theintegralcontainingtheboundaryvaluesap-
In particular r=ρ ,∞ and 1ψ=−1π,0 for ρ=0,∞, re-
0 2 4 pears as an inhomogeneity. The TE induction equation thus
spectively. They are conjugate to the positive real axis in
becomesaninhomogeneousHelmholtzequation.
this plane. When the complex horizontal wavenumber k
Becausewe assumedthat theglobal modelis knownand
changes,theymovefromρ →∞ontwoarcsectionsfrom
0 is strictly one-dimensional, depending solely on the global
ψ=∓1π→ψ=0. Figure 1 shows the integration contour
4 vertical conductivity profile F (z), the Helmholtz equation
inthecomplexκ-plane. 0
remainsaone-dimensionalequationalsowhenconductivity
After factorisation, the solution of the κ-integration for-
fluctuationsareincluded.Thesefluctuationsareatleasttwo-
mallyyieldsthesumoftheresiduaintheκ-poles
dimensional, depending on the vertical and one horizontal
(cid:68) (cid:69) 1 (cid:90)∞(cid:90)2π kdkdφ (cid:40)k2 coordinate. Their large-scale variation in the horizontal di-
δFδE = 0 × (25) rection is intrinsic to the k-dependence of the new terms. It
8π2 +(cid:112)k2−k02 F0 becomes explicit only after solution of the equation and re-
0 0
transformationfromk toconfigurationspace.TheTEprob-
(cid:12) (cid:12)2 (cid:104) (cid:105) (cid:27)
×(cid:12)δF (κ )(cid:12) E − δE(cid:48)(0)−κ δE (0) δF (κ ) lemcouldbetreatedasaforwardproblemwhenanappropri-
(cid:12) k + (cid:12) 0 k + k k +
atemodelforthefluctuationsoftheconductivityisassumed.
Of the two residua that with κ=κ is selected by the StillthisisdifficultbecausethecoefficientsintheHelmholtz
+
requirement that the Laplace-transform of δF (κ) exists. equationarenotconstants.Moreover,itsinhomogeneityre-
k
We rewrote the two-dimensional volume element of the k- quiresconstructionoftheGreen’sfunction.Subsequentlythe
integral in terms of k and the azimuthal angle φ. One may solutionbecomesanintegraloftheGreen’sfunctionandthe
note that the root in the denominator is κ whose explicit inhomogeneity.
+
formintermsofρandχisgivenabove. The integration variable k is complex. When the residua
Theimportantterminthisexpressionisthetermcontain- areentireor analyticalfunctionspossessingsimple polesin
ingthemainelectricfieldE (z).Thecorrelationtermaddsa thecomplexk-plane,thepresenceoftherootinthedenomi-
0
linearcontributiontotheoriginalaverageinductionequation, natoroftheintegrandintroducestwoalgebraicbranchpoints
withthenonlinearitybeingabsorbedintothespectraldensity at
oftheconductivityfluctuationswhichappearsasaspacede-
pendentfactor.Theappearanceoftheboundaryvaluesinthe k =(cid:112)µ ωσ F e±iπ/4=ρ e±iπ/4 (27)
±b 0 0 0 0
last bracket complicates the problem. They represent an in-
homogeneityinthefluctuation-averagedinductionequation.
which depend on z and frequency ω, and one additional
Furtherreductionofthenonlinearcorrelationtermcanbe
branch point at k=∞. The latter provides no further prob-
doneassumingthatthefluctuationsdependonlyonzandx,a
lembecauseallvariablesvanishsufficientlyfastfork→∞.
caseofpracticalimportancebecauselocallyitwillalwaysbe
Of the two finite branch points only that in the positive k-
possibletoidentifyamaindirectionofstrikeofthehorizon-
plane is selected for reasons of convergence of the inverse
talvariationoftheconductivity.Inthiscasethek-integration
Fouriertransform.Thecontourofintegrationinthecomplex
isone-dimensionalwithφ-integrationdisappearing.Theen-
k-planeisshowninFig.2.Thus,thecutistobemadefrom
tireproblemisthentwo-dimensional,andtheaverageofthe
the upper branch point k=k ≡+k along a line at angle
+b 0
conductivityandfieldfluctuationssimplifiesto
+π/4toinfinitywhenintegratingwithrespecttok overthe
∞ (cid:40) upperk half-plane.Inprinciple,thecontourisclosedalong
(cid:68) (cid:69) 1 (cid:90) dk k2(cid:12) (cid:12)2
δFδE = 0(cid:12)δF (κ )(cid:12) E − theimaginaryaxisfrom+i∞downtotheorigin.Itcan,how-
4π (cid:112)k2−k02 F0(cid:12) k + (cid:12) 0 ever,extendedtotheentireupperk-halfplanebyanalytical
0
R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem 7
The ultimate fluctuation-generalised Helmholtz equation
reads
k-plane
nary k-axis branch-cut E0(cid:48)(cid:48)(z)−k02E0(z)(cid:34)1+(cid:16)(cid:112)2kF020k(cid:88)R(cid:54)=eksb(cid:12)(cid:12)(cid:17)δFk(cid:112)((cid:112)k2k−2−k02k02)(cid:12)(cid:12)2(cid:35) (29)
magi = k02k(cid:88)(cid:54)=kbδFk k2−k02 (cid:104)δE(cid:48)(0)−κ δE (0)(cid:105)
i branch-point 2 (cid:112)k2−k2 k + k
Res 0
wheretheresiduaarecontributedbythepolesofthefluctua-
0
tionspectruminsidetheintegrationcontourintheupperhalf
real k-axis
ofthebranch-cutcomplexk-planeshowninFig.2.Itmustbe
stressed that without any assumption about the fluctuations
nothingisknownabouttheexistenceand/orformofthepoles
inthespectrumofconductivityfluctuations.Forthisreason
Fig.2.Theintegrationcontourinthecomplexkplane.Integrationis
the representation given in the above generalised equation
restrictedtotheupperrightk-halfplane.Therelevantbranchpoint
isimplicit.However,theargumentisthatanyphysicallyrea-
islocatedatk =ρ exp(+iπ/4).Thebranch-cutextendsfromk
b 0 b
sonabledistributionofconductivityfluctuationsintheunder-
toinfinitywhereforanyreasonabledistributionthespectrumvan-
ground,i.e.intrusionsofhighorlowconductivitystructures
ishes.
will necessarily give rise to a number of poles or, in more
complicated cases, branch points integration around which
continuation.Thevalueoftheintegralbecomes contributeresidua.Onemightassumethatanyphysicalcon-
ductivity distribution will not give rise to essential and thus
∞ (cid:34) ∞ (cid:35)
(cid:90) dk(cid:110) (cid:111) (cid:90) dk (cid:110) (cid:111) non-integrablesingularities.Thusapplicationoftheimplicit
... = P −2πRes(k ) ...
κ+ (cid:112)k2−k02 b form of the above induction equation is for a large class of
−∞ −∞ fluctuationsjustified.Thisclassencompassesallfluctuations
k(cid:88)(cid:54)=kb 1 (cid:110) (cid:111) thatsatisfytheconditionthattheirscalesareshorterthanthe
=2πi ... (28) verticalscaleofthereferencezero-ordermodelF (z).
(cid:112) 0
k2−k2
Res 0 The right-hand side of the above euqation contains any
possible residua which are contributed by the correlation of
ItisthesumoftheHilbert-principalvalueP oftheintegral
thesurfacefluctuationelectricfieldandtheconductivityfluc-
takenalongtherealk-axis,plusthenegativeresiduuminthe
tuations.Inaforwardproblemsolutiononewoulddetermine
positivebranchpoint.Thissumequalsthesumofallresidua
this term from the boundary conditions. Here, for simplic-
intheupperpositivequadrantofthek-planecontributedby
ity,weassumethatitcanbeputtozerobecausenocorrela-
thepolesoftheintegrandinthecurlybraces.Allresiduacon-
tributed by poles on the real axis must be multiplied by 1. tionisexpectedbetweenthefluctuationsinconductivityand
2 field at the surface, a condition which can, in principle, be
Hence,thek-integralisthesumofallresiduaontherightof
relaxed when solving the external problem at z(cid:48)<0 for the
thelastexpression.Iftherearenoadditionalpoles,i.e.ifthe
fluctuations. We restrict to the demonstration of the modifi-
spectral density of the conductivity fluctuations is an entire
cationoftheinductionproblembyinclusionofconductivity
function,thentheintegralvanisheswithvanishingright-hand
fluctuations. We therefore don’t do this here, relegating the
side.Theproblemdegenerates.Thefluctuationsinthatcase
inclusionoftheboundaryeffectstoamoreextendedinvesti-
do not visibly contribute to the induction field. The above
gation3.UnderthissimplifyingassumptiontheTEinduction
integral, split into imaginary and real parts, then provides a
equationreducestozeroontheright.
so-called“dispersionrelation”forthefluctuations.Itrelates
The conductivity fluctuations just generate an additional
therealandimaginarypartsofktothefluctuationspectrum
contribution to the linear term in the Helmholtz equation.
and the induction electric field. On the other hand, if the k-
Formallythistermisamodificationoftheconductivitypro-
integral in this case is interpreted as a principal-value inte-
file while being of second order in the applied frequency.
gral, then the relevant residuum is taken to be the residuum
atthebranchpoint:+2πi|δF (0)|2.
k=kb 3The assumption is that the surface values of the fluctuation-
The physically important case is when the fluctuation
inducedelectricfieldvariationisofshortscalesuchthatthehori-
spectrum of the conductivity possesses poles and thus gen-
zontalvariationoftheelectricfieldisentirelyattributedtoit.Inthis
erates non-vanishing residua. This case is realistic because
casethevalueoftheconductivity-fluctuation-causedfieldisgiven
one must expect that the presence of conductivity fluctua- by its internal value, and the external field in the non-conducting
tions causes a nonnegligible effect in the induced electric airjustbehavespassivelyanddecaysexponentiallywithincreasing
field. altitudeabovethesurfaceasdeterminedbyLaplace’sequation.
8 R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem
This was expected. Hence, it seems that little has been σ (x,z).Intheaboveapproachtheverticalprofileofthecon-
0
gained by assuming the presence of a fluctuating conduc- ductivity is assumed to be known. Thus, in any given point
tivity. Moreover, due to the dependence of all quantities on x on the surface the inversion provides information about
the large-scale coordinate z the spatial variation of the co- the average spectral density of the conductivity fluctuation
efficient of E (z) is complex. Indeed, a forward analytical as function of the global coordinates z and x. These coor-
0
solutionofthebasicfluctuation-averagedTEequationseems dinates are prescribed by the zero-order model profile. The
improbableevenformostsimplefluctuationmodels.Numer- spectraldensitythusinformsabouttheaveragedeviationof
icalsolutionsareonewayout. the real conductivity structure from the model. In principle,
Letusassumethattherearempolesintheupperhalfk- this could be used in an optimisation of the vertical model
planewhichleadto1≤n≤mresiduawithk thevalueof structure by variational methods. In the horizontal direction
n
k at the n-th pole of the expression under the sum in the therestrictionisthatthedistanceofmeasuringpointsislim-
aboveequation.Then,byFourier-transformingtheinduction ited by the scale of the horizontal variation of the external
equation back into real configuration space with horizontal inductionfield.Thisisaconsequenceoftheassumptionofa
coordinatexweobtainitstwo-dimensionalversion homogeneousfield.
(cid:34) In application of the inverse problem it is more conve-
k2(z)
E(cid:48)(cid:48)(z)δ(x−x )−k2(z)E (z) δ(x−x )+ 0 × nient to return to the one-dimensional Eq. (29) than using
0 0 0 0 0 4πF0(z) its x-transformed two-dimensional version (30). The two-
dimensionalityissecondaryonlysincek appearssimplyas
(cid:88)m eikn(x−x0) (cid:12)(cid:12) (cid:18)(cid:113) (cid:19)(cid:12)(cid:12)2(cid:35) aparameter.Withtheright-handsidesetntozero,theinduc-
(cid:112)k2−k2(z)(cid:12)(cid:12)δFkn kn2−k02(z) (cid:12)(cid:12) =0 (30) tionequationassumesitscanonicalform
n=1 n 0
kn(cid:54)=+k0
E(cid:48)(cid:48)(z,ω)=iωµ σ Σ(z,ω)E (z,ω) (32)
Thisholdsunderthestrictconditionthat 0 0 0 0
whereweunderstandE (z,ω)→E (z,ω)/E (0,ω).More-
k2(cid:54)=k2=iρ2 (31) 0 0 0
n 0 0 over, σ Σ(z) is the fluctuation modified equivalent conduc-
0
The field on and along the Earth’s surface refers to x as tivityand
0
an arbitrary reference point. The horizontal variation of the  (cid:12) (cid:16)(cid:112) (cid:17)(cid:12)2
contribution of the fluctuating conductivity is thus given as Σ(z)=F 1+ k02 (cid:88)m (cid:12)(cid:12)δFkn kn2−k02 (cid:12)(cid:12)  (33)
anusmupbeerrpsoksit.ion of m harmonic modes with complex wave 0 2F0 n=1 (cid:112)kn2−k02 
n kn(cid:54)=+k0
Theparameterρ containsthetimevariationoftheinduc-
0 where all quantities ρ ,F ,k are functions of z. This de-
tion field. It corresponds to the inverse penetration depth of 0 0 n
pendence complicates the calculation but is, for the inverse
theelectromagneticfieldoffrequencyωintotheconducting
problem, not of primary importance. It simply implies that
Earth, which thus enters in a rather complicated way. This
the conductivity structure is not as simple as the originally
complication reflects the complex reflection and absorption
assumedstratificationmodelthatiscontainedinthefunction
processoftheinductionfieldinthevariousdifferentlycon-
F (z).Withthisnormalisationthesurfaceresponsefunction
ducting conductivity intrusions. The statistical formulation 0
atz=0is
has the advantage that it avoids the taking into account of
manyinternalconductivityboundariesbelowEarth’ssurface 1
c(ω)=− (34)
in a complicated and still unrealistic structure model of the E(cid:48)(0,ω)
0
conductivity. Only the average response of the conductivity
afunctionwhichbehavesanalyticalifonlyΣ(z,ω)isanalyt-
structureenterstheapproach.
ical,sincethisisoneoftheconditionswhichmustgenerally
holdforapplicationofourtheory.
5 Inverseproblem Inthefollowingwearenotprimarilyinterestedinabrute
forcenumericalsolutionofthegeomagneticinversionprob-
Solvingtheaboveequationinaforwardcalculationisdiffi- lem.Thereaderisreferredtoabroadliteratureonthissub-
cult even when a reasonable conductivity fluctuation model ject(cf.,e.g.,thereviewsbyBackusandGilbert,1967;Bai-
isassumedthoughitcan,ofcourse,beattackedbynumerical ley, 1970; Johnson and Smylie, 1971; Parker, 1971, 1980,
methodsandwiththehelpofsupercomputers. 1994; Gubbins, 2004, and references therein). Rather we
Anotherwayoutistheapplicationoftheinversegeomag- seek to relate it to the analytical formulation of the one-
neticinductionproblem.Theinverseinductionprobleminei- dimensional inverse problem as given by Weidelt (1972),
therversionseekstogatherinformationaboutthecoefficient whofirsttransformedthegeomagneticinductionintothesta-
oftheelectricfieldinthesecondtermoftheaboveequation. tionarySchro¨dingerequationformsuchthatitbecomestreat-
Ittheirordinaryversiontheinverseproblemsaimonadirect ablebythestandardexactGel(cid:48)fand-Levitaninversionproce-
reconstruction of the one- or two-dimensional conductivity dure.
R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem 9
InviewofapplicationoftheGel(cid:48)fandandLevitan(1951) data, the difficulty is shifted from the solution of the inver-
theory in its adapted Weidelt (1972) version we adopt the sionproblemtotheinterpretationoftheobtainedresult,i.e.
Weidelttransformations to the interpretation of V(ζ,ξ) in terms of a model of the
(cid:90) z subsurfacestructureoftheconductivitydistribution.Itisthe
(cid:112) (cid:112)
ξ:= k (0)= iµ ωσ , ζ(z,ξ):= dy Σ(y,ξ) (35) great achievement of Weidelt (1972) of having discovered
0 0 0
0 theabovetransformationswhichhavebroughttheinduction
forthevariables,andforthefunctions equation into its general form Eq. (38) making it treatable
bythegeneralinversiontheory(Gel(cid:48)fandandLevitan,1951;
(cid:104) (cid:105)1
u(ζ,ξ):= Σ(ζ,ξ) 4 (36) Marchenko,2011)ashadbeendevelopedforthewaveequa-
tion and the inverse scattering problem. Since this theory is
f(ζ,ξ):= u(ζ,ξ)E (ζ,ξ) (37) completeandhasbeenshowntoyieldauniquesolution,itis
0
theappropriatewayofsolvingtheinductionproblem.Parker
Intheseexpressionsthenormalisedfrequencyξ occursasa
(1980)hasdiscusseditsprosandcontras,itsadvantagesand
parameteronly.
deficiencies.
With (cid:48) now indicating differentiation with respect to ζ Weidelt(1972),infollowingGel(cid:48)fandandLevitan(1951),
the conductivity-fluctuation-generalised induction equation
hasshownthatthesolutionoftheinverseTEinductionprob-
transforms into the standard Sturm-Liouville (or stationary lemreducestothesolutionofthelinearGel(cid:48)fand-Levitanin-
Schro¨dingerequation)form
tegralequationforsomefunctionG(ζ,η)(seeAppendix2)
(cid:104) (cid:105)
f(cid:48)(cid:48)(ζ,ξ)= ξ2+V(ζ,ξ) f(ζ,ξ) (38) +ζ
(cid:90) (cid:110) (cid:111)
G(ζ,η)=K(ζ+η)+ dτG(ζ,τ) K(η+τ)+K(η−τ) (40)
This has been demonstrated by Weidelt (1972) by straight-
−ζ
forwardthoughsomewhattediousalgebra(seeAppendix1).
The“equivalentpotential”functionV(ζ,ξ)isdefinedsolely underthecondition|η|<ζ.Itrequiresknowledgeoftheker-
and uniquely through the conductivity Σ(ζ,ξ) respectively nelfunctionK(ζ),whichitselfisgivenbythemeasurements
u(ζ,ξ)as c(ξ)atEarth’ssurfacewhichistheinverseLaplacetransform
V(ζ,ξ)≡u(cid:48)(cid:48)(ζ,ξ)(cid:14)u(ζ,ξ) (39) 1 (cid:90) (cid:15)+∞ (cid:104) (cid:105)
K(ζ)= dξ 1−ξc(ξ) eξζ (41)
4πi
Itisthisfunctionwhichisconsideredunknownintheinver- (cid:15)−i∞
sion problem. It is to be determined by inverting the induc- HavingdeterminedK(ζ)fromtheobservationscontainedin
tion equation for any given frequency ξ respectively ω or – c(ξ),thesolutionoftheaboveGel(cid:48)fand-Levitanequationcan
more generally – any applied frequeny spectrum of the ex- beobtainedbystandardnumericalmethodsiteratingthein-
ternal geomagnetic induction field. On may, however, note tegral equation, i.e. expanding its solution into a Neumann
that ξ is just a parameter reserved for later use, not the pri- series.ThisprocedureyieldsG(ζ,η).Thesolutionofthein-
maryimportantvariableintheinversionproblem.Therele- verseTEproblemsubsequentlyfollowsfrom
vantvariablethatenterstheinversionisζ.
(cid:90) +ζ
Inverting the induction equation will yield the function
u(ζ)=1+ dτ G(ζ,τ) (42)
u(ζ,ξ), from which one also obtains V(ζ,ξ) on the way of
−ζ
differentiation.Thisfinallyprovidesthewanteddependence
It directly produces the vertical profile of the conductivity
of Σ(ζ,ξ) on the depth ζ for given frequency ξ. Once this
Σ(ζ,ξ).Straightforwarddifferentiationwithrespecttoζ ul-
hasbeenachieved,oneencoutersthedifficultproblemofin-
timately yields the wanted ”equivalent potential” function
terpreting the result in terms of the spectral density of the
V(ζ,ξ).
conductivity fluctuations. The advantage up to this point is
Once the inverse TE problem has been solved, u(ζ,ξ) is
that by relegating all the complicated distribution of the in-
locally given. This implies that V(ζ,ξ) and by it also the
trusions of conducting material to the fluctuation spectrum
vertical profile Σ(ζ,ξ) are available at some site x−x at
of conductivity reduces the inverse geomagnetic induction 0
Earth’s surface. Pointwise construction of the vertical con-
problemtoastrictlyone-dimensionalproblem.Onlythede-
ductivity profile over an area of prospection on Earth’s sur-
pendence on depth ζ enters. Any lateral dependence is re-
face then, for given frequency ξ, allows obtaining the con-
constructedpointwisealongaprofileobtainedattheEarth’s
ductivitystructurepointwisebelowthatareaofprospection.
surface.
The important point to be made is that though V(ζ,ξ) is
substantially more complicated than in the non-fluctuation 6 Reconstructionofthefluctuations?
models,thereisnoprincipaldifferencebetweenthosemod-
els and the present one what concerns the inversion theory. For practical purposes the above result suffices. It provides
Once one has succeeded in determining V(ζ,ξ) from the the vertical conductivity structure within a certain surface
10 R.A.Treumann,W.Baumjohann:GeomagneticInductionProblem
region. Its advantage is that it yields the corrections to a Inthisexpressionthedependenceondepthζ andfrequency
(cid:112)
simple analytically solvable geophysically reasonable verti- ξ is explicitly contained in k ∼ ξF (ζ). It is also im-
0 0
cal conductivity-reference model that had been initially as- plicittok andthusnotdirectlyacessiblewithoutfurtheras-
n
sumed. sumptions.Onerealises,however,thatthespectraldensityof
This is what one would like to know from the practical conductivityfluctuations|δF |2couldinprinciplebedeter-
kn
point of view. One might, however, be interested in the re- minedifonlythedependenceofk ondepthandfrequency
n
constructionofthedistributionofconductivityitself,i.e.in- wouldbeknown.
feraboutthenatureoftheconductivityfluctuations.Thisis, Determination is possible of the horizontal wavelengths
unfortunately,notcompletelypossibletoachieveindetailfor λ =2π/kr of the contributing conductivity fluctuations at
n n
the reason that the information contained in the conductiv- thesurfaceofEarthatz=0foreachappliedrealfrequency
ity profile, though of practical use, is just statistical. Infer- ξ aswellastheirsurfaceamplitudes.Thisrequiresthemea-
ence about the nature of the fluctuations in real space re- surement of c(ξ) as function of x−x in an area on the
0
quires not only the separation of the conductivity spectrum surface along the direction of dominant variation. (A sim-
butalsoitsre-transformationfromwavelength/wavenumber ilar procedure would be possible also for two-dimensional
space into real space. Below we show that this requires ad- dependencies.)
ditionalknowledgewhichmightbeavailableonlyinimplicit It is reasonable to assume that the real parts kr of the
n
form. wavenumbers k do not (or only weakly) depend on depth
n
Applied to our case which involves fluctuations, the ζ.Iftheyareindependentonz thenthevariationoftheam-
conductivity profile differs from the assumed large-scale plitudes of the fluctuation spectrum is included in the de-
reference-modelprofileσ0F0(z).Sincetherelationbetween pendenceofk0(z)whichisgivenby F0(z).Inthisapprox-
these two is linear, one can subtract the model profile. The imation, determination of kr(z=0) fixes the real part of
n
deviation from it is the wanted contribution that is caused the wavenumbers of the fluctuations over a certain range of
solelybythespectrumofconductivityfluctuationsthatissu- depths.
perimposedonthereferenceprofile: Theimaginarypartki(ξ,ζ=0)forgivennandfrequency
n
ξ on the Earth’s surface can be determined from the be-
∆Σ=Σ−σ0F0=ξ22 (cid:88)m (cid:12)(cid:12)(cid:12)δFkn(cid:112)(cid:16)(cid:112)k2k−n2−k2k02(cid:17)(cid:12)(cid:12)(cid:12)2 (43) h∆aΣvi(oxu−roxf0,ζ=0)∼exp[−kni(x−x0)] (46)
n=1 n 0
kn(cid:54)=+k0 along the profile x−x . For ki >0 the amplitude ∆Σ in-
0 n n
with left-hand side now known from the solution of the creasesaslongasx<x0.Itreachesmaximumatx=x0and
inverse problem. Here k02(ζ,ξ)=ξ2F0(ζ) contains the de- decaysforx>x0.Hence,x0(n,ξ)isidentifiedasthewave-
pendence of the fluctuations on ζ respectively z. One may lengthandfrequencydependentlocationofthemaximumre-
remember that ξ is complex. Similarly, the poles k (ζ,ξ), sponsein∆Σatthesurface.Ideallythisbehaviourisexpo-
n
where the residua have been taken, are as well functions of nential. It should be treated by considering the logarithmic
ζ,ξ.Sofartheyareunknown.Wouldthefluctuationspectrum profile. Its slope provides the imaginary part kni(ζ=0,ξ),
beknown,thiswouldbethefinalresult.Thisis,however,not howeveronlyatthesurfaceforanygivenfrequencyξ.
thecase. Incontrasttothewavelengthandrealpartknr,theimagi-
Determinationoftheconductivityfluctuationsrequiresthe narypartkni(ζ,ξ)is,however,afunctionofdepthζ andfre-
inversionofthesumontherightandthedeterminationofthe quencyξbecauseitcontainsthespatialdamping/penetration
variousk (ζ)foraspectrumoffrequenciesξ. effectoftheconductinglayers.Thedeterminationofitsde-
n
There is only a finite number of poles and therefore only pendenceonζposesamajorproblemwhichcannotbesolved
a finite number of wave numbers k . We also know that all easily.Forthisreasontheexplicitdeterminationofthecon-
n
k =kr+iki havepositiveimaginarypartski >0only.Re- ductivity fluctuations in real space is practically impossible
n n n n
transformation from k-space into horizontal coordinates x from the measurement of the response function at Earth’s
thenyields surface. One has to stay with the information contained in
∆Σ(ζ,ξ).
∆Σ(ζ,x)=4ξπ2 (cid:88)m e(cid:112)ikkn2(x−−xk02)(cid:12)(cid:12)(cid:12)(cid:12)δFkn(cid:18)(cid:113)kn2−k02(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)2 (44) preTchiseerley.isTnhoisoinbavbioiluitsywinahyiboiftshtohwepkrniec(ζis)ecfouunlcdtiobneailnrfeecrorend-
n=1 n 0 struction of the conductivity fluctuations, i.e. it inhibits the
in the one-dimensional case under consideration. The sum precise knowledge about the physical nature of the fluctu-
thus consists of a limited number n≤m of harmonic func- ations. The solution of the inverse problem only provides
tions.Foreachharmonicwecanwrite the contribution of the fluctuation spectrum to the conduc-
tivity, it does not say for whatever geophysical reason the
ξ2 1 (cid:12)(cid:12) (cid:18)(cid:113) (cid:19)(cid:12)(cid:12)2 conductivity structure has the partucular spatial distribu-
∆Σn(ζ,ξ)=4π(cid:112)k2−k2(cid:12)(cid:12)δFkn kn2−k02 (cid:12)(cid:12) (45) tionobtainedfromthesolutionofthefluctuation-generalised
n 0