Physics of the Earth and Planetary Interiors, 62 (1990)231—245 231ElsevierSciencePublishersB.V., Amsterdam

Thecomputationof vectormagneticanomalies:a comparisonof techniquesanderrors

Michael E. Purucker

ST Systems Corp., Geology and Geomagnezism Branch, Code 622, Goddard Space Flight Center, Greenbelt, MD 20771 (U.S.A.)

(ReceivedFebruary1, 1990; acceptedFebruary8, 1990)

ABSTRACT

Purucker,M.E., 1990. The computationof vectormagnetic anomalies:a comparisonof techniquesand errors.Phys. EarthPlanet. Inter., 62: 231—245.

The greatercosts and uncertaintiesassociatedwith the experimentaldeterminationof vectormagnetic anomalieshaveresultedin thedevelopmentof techniquesby which total field anomalymeasurementscanbeusedto calculatevectormagneticanomalies.Enhancederrorsare encounteredin all threeof the techniquesconsideredwhen calculating the eastwardanddownwardvectormagneticanomaliesin thevicinity of thegeomagneticequator.Analogouserrorshave been describedfromgeomagneticfield modelsderivedsolely from total field data.Theequivalentsourceinversion technique,although requiringthegreatestcomputerresources,exhibitsthesmallesterrorsandis themostversatile.Errorscan beminimizedby: (1) usingatechniquesuchassingularvaluedecompositionto stabilizetheequivalentsourcesolution; (2) usingboth total field anomalydata and suitably weighted vector anomalydata when available. The space-domainconvolution and frequency-domaintransformationtechniquesrequirecomparablecomputerresources.Morerestrictivethantheequivalentsourcetechnique,theyboth assumea constantgeomagneticfield direction and total field anomalydata collected over a regulargrid at constantaltitude. These techniquesare therefore most appropriatefor aeromagneticsurveys covering small areas.Within thisframework, thespace-domainconvolution is superiorat mostmid to high magneticinclinationsand at themagneticequator.However,thefrequency-domaintransformationis superiorat mostotherlow magneticinclinations.

1. Introduction techniques.These questions are addressedwithcomputersimulations. The simulations consider

Geomagneticfield models derivedsolely from magneticdataacquiredat bothaircraftand satel-scalardata exhibit systematicerrors in the corn- htealtitude.putedvectorcomponents,particularlyin low mag- Vector magneticinformation is particularly im-netic latitudes(Hurwitz andKnapp, 1974;Lowes, portantin low magneticlatitudesbecausethe sig-1975; Stern and Bredekamp,1975; Stern et al., naturesof the equatorialelectrojetand meridional1980; Lowesand Martin, 1987). By analogywith current(Maedaet a!., 1982)can beidentified andthe main geomagneticfield, we might expect that distinguished from crustal sourcesmost easilycrustalfield modelsderivedsolely from total field usingvectormagneticinformation.anomaly measurementswould exhibit similar er-rors in the computedvector anomalies.The pur-pose of this paper is to compare and contrast 2. Theorytechniquesused for computing vector magneticanomaliesandcrustalfield modelswith particular While a general vectorfield requiresthreeor-attention given to the errors associatedwith these thogonal scalarsat each point in space for its

0031-9201/90/$03.50 © 1990 — ElsevierSciencePublishersBy.

232 ME. PURUCKER

characterization,the crustal and main geomag- H~,H~,H, and ~T canbe expressedin termsofnetic fields, in the absenceof external currents, a finite harmonicseriesexpansionand the result-have

ing expansions can be introduced into (5).v x B = 0 (i) Evaluating the derivatives and comparing the

It follows that B canbe expressedas thegradi- terms on both sides of the equation yields aentof a scalarpotentialA. Sincea scalarpotential frequency-domainrelationship between ~T andis determinedby the specificationof a scalarat the vector magneticanomaliesH~,H~,and H~.each point in space,a number of investigators Theserelationshipsare definedby Lourencoand(Lourenco and Morrison, 1973; Bhattacharyya, Morrison (1973) in their eqns.(14)—(16).A previ-1977; Galliher and Mayhew, 1982; Langel et al., ously undescnbedsingularity is present in the1982; Kolesovaand Cherkayeva,1986) have de- algorithmat the geomagneticequatorwhere/ = 1,velopedalgorithmsfor computingvector magnetic m = 0, and n = 0. This algorithmassumesthat theanomaliesfrom the total field anomaly.A brief direction t is fixed. Kolesova and Cherkayevadiscussionof thesealgorithmsfollows. (1986) use an iterative modification of the above

Lourenco and Morrison (1973) developed a techniquefor the casewheret is variable.frequency-domainmethod for computing vector Bhattachar~ia(1977) utilizes a space-domainmagneticanomaliesfrom the total field anomaly. convolution in order to computevector magneticA Cartesiancoordinatesystemis chosensuchthat anomaliesfrom the total field anomaly.He firstx, y and z= north, east, and down respectively, writes A and ~T in terms of their Fourier trans-Let the direction of the total field anomaly t be forms 4(u, v) and G(u, v) whereu and v are thedefined to be along the direction of the Earth’s angular frequenciesalong the x and y axes. Hemain geomagneticfield suchthat then establishesthe following relationship be-

a a a a tween4l’andG:

+ m~—~+ n~—~ (2) 4(u, v) = —G(u, u)/[i(lu + my)+ ns] (6)where 1, m, and n are the direction cosinesof where~2 = + v2.

the main field. Thenthe anomalyi© thetotal field Using (6) and the equationsestablishingthe(ST) can be expressedin terms of the magnetic relationshipsbetweenA and G and their Fourierpotential A as transformshe writes A as a convolutionintegral

aAy, z)= (3) A(x, y, h)= —1/41T2~T(x,y,0)*I(x, y, h)at

The vector magneticanomaliesH~,H~,and H~ (7)canbe expressedin analogousform as

whereH=-~

ax I(x, y, h)H = —~- exp i(ux+vy) exp(—sh)dudv (8)

Y a~ ffi(/u+mv)+nsH = — (4) Equation (8) can be rewritten as a line integral

z azOperatingon eachside of (3) with first derivatives which in turn can be evaluated using Cauchy’syields residuetheorem.The residuesare evaluatedsep-

3~T aH~ arately for n > 0, n = 0, and n <0. Equation (7)= —~-— can thenbe differentiatedusingeqn.(4) to yield a

a~’r aH~ relationship between ~T and H~,H~,and H~.The vectormagneticanomaliescanthusbe calcu-lated by a digital convolution of the total field

a~T aH~ anomaly with the operators defined by Bhat-= (5) tacharyya(1977) in his eqns.(23)—(30).Unlike the

COMPUTATION OF VECTOR MAGNETIC ANOMALIES 233

frequency-domainmethod,this techniquehas no 3. Computer simulationsknownsingularities.

Galliherand Mayhew(1982) and Langelet al. The computer simulations reported here use(1982) used the equivalent source technique two exactandoneapproximateforwardmodelling(Dampney,1969; Ku, 1977; Mayhew, 1979; Lan- schemesin order to calculate the total field andgel et al., 1984) in order to calculate,in a least- vectormagneticanomaliesexpectedat an arrayofsquaressense,the magneticmomentsof a grid of observationpoints over a magneticsource.Theequivalentsourcedipoles. The dipolesare calcu- grid of total field anomalydata is then usedtolatedas follows: calculatevectormagneticanomalieson that same

= (vD-’v’)s’b (9) grid andcomparedwith the exactor approximateforward modellingresults.

whereThe exact forward modelling algorithmsused

p = vectorwhoseelementsare the magneticmo- herehavebeendevelopedfor usewith rectangularmentsof the dipoles,

S = matrix of the geometricsourcefunction relat- (Bhattacharyya,1964) andpolygonal(Plouff, 1976)ing the j th sourceto the i th position, prisms.They are developedin rectangularcoordi-

natesand henceassumea ‘flat-earth’ geometry.V = matrix whose elementsare eigenvectorsofAn approximatespherical-earthalgorithm(Ku,s’s, 1977; Von Freseet al., 1980) developedfor use

D = matrix whosediagonalcontainsthe eigenval- with MAGSAT data, was also used for forwarduesof S’S, modelling. Thetechniqueutilizes Gauss—Legendre

b = vectorof observations.The matrix D is modified using singularvalue quadratureintegrationandlinearinterpolationbe-

tweenadjacentboundarypoints in order to calcu-decompositionso that small eigenvaluesand their late equivalentsourcedipolesgiven sourcevolumeinversesare set to 0. E~,the percentageof the shapesand magnetizations.The magnetic field

trace of D that is retained,is selectedso as to calculatedfrom eachequivalentsourcedipole isreducethe standarddeviation to typical mid-lati- then summedat the observationpoint. For effec-tude values. An of between95 and 99% is tive resolution, the distance from the magneticfound to satisfy this requirement.An E~of the body to the observationpoint must be greater97% is usedin this analysisunlessotherwisemdi-

than the distancebetweennodesor dipoles(Ku,cated.

The elementsof the geometricsourcefunction 1977). In the casediscussedhere the observationmatrix sare: planeis 430 km abovethe Earth’ssurfacewhereas

the equivalentsourcedipolesare 8—55 km apart.a F.s,1= (10) 3.1. Frequency-domaintransformation

These S13 relate the F = (AT, H~,H~,H~)magneticfield at i(r, 9, ~) to the magnitudeof The accuracyof the frequency-domaintrans-the dipolesource(p) at j( r’, 9’, ~‘) formation can be evaluated by comparing the

Total field or vector magneticanomaliescan vector magneticanomaliescomputedusing thisthen be calculatedfrom these dipoles. This al- transformationto the valuescalculatedemployinggorithm will work with either fixed or variable t anexact forward modellingscheme(Plouff, 1976).and can be used with ungriddedmeasurements The vector magneticanomaliesare calculatedastaken at multiple elevationsabovethe magnetic follows: The total field anomaly is first trans-source.Becauseof theseadvantages,it has been formed using a two-dimensionaldiscrete Fourierthe methodof choicefor satellitemagneticsurveys, transform (FF1’) with an array size of 64 x 64.The equivalentsourcemethod,however, requires This size was chosenin order to minimize edgegreater computer resources than either the effects and to produce smooth contours. Array.frequency-domaintransformationor the space-do- sizes as large as 90 x 90 and as small as 8 x 8main convolutionmethod. producequalitatively similar results.The discrete

234 ME. PURUCKER

DELTA T H H

FREQUENCY DOMAIN

TRAN SF0 R MAT ION

::: :1: ~CONTOUR INTERVAL: I UNIT

MODEL DATA SYNTHETIC DATA

Fig. 1. Comparisonof ‘model’ and ‘synthetic’magneticfield datacalculatedusinga frequency-domaintransformation.Themagneticsourceis outlined by dashedlines. Negativevaluesareindicatedwith hashmarks. ‘Model’ data, shownon the left, are calculatedusinganexactalgorithm(Plouff, 1976)developedfor useon a ‘flat earth’. Frequency-domaintransformationsarethen appliedto thetotal field anomalyin order to computethecorrespondingvectormagneticanomalies.Theselatteranomaliesaretermed ‘synthetic’fields and areshownon theright.

The simple magneticsourceis a prism with horizontal dimensionsof 12.8X 12.8 units and a vertical depth of 10 units. Themagneticsourceis centredon a 64 X 64 grid andthetransformationis doneusingthat samegrid. Themagneticsourceis magnetizedby induction.All magneticfields arecalculatedat a distanceof 6.4 unitsabove thesource.

Fourier transformis then multiplied by an oper- anomaliescalculatedusing the frequency-domainator and finally back-transformedto the space transformationof the total field anomalyarere-domain.A depth-limitedmagneticsourcelocated ferredto as ‘syntheticdata’ (Fig. 1).in the centreof the arrayensuresthat the mean The differencesbetweenthe model and syn-valueof the field is effectively 0. This meanvalue theticdatasetscanbequantifiedand expressedasis evaluatedusingeqns.(18)—(20) of Lourencoand a percentageerror ~ whereMorrison (1973).

Vector and total field anomaliescalculated I Fm — Iusing an exact forward modellingschemeare re- = v’ V F X 100 (11)ferredto as ‘model data’ (Fig. 1). The vectorfield I m I

COMPUTATION OF VECTOR MAGNETIC ANOMALIES 235

~ g

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236 ME PURUCKER

DELTA T H H

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SPACE DOMAIN 0Q CONVOLUTION 0

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MODEL DATA SYNTHETIC DATA

Fig. 3. Comparison of ‘model’ and ‘synthetic’ magneticfield data calculatedusing a space-domain convolution. The space-domain

convolution is applied to the total field anomaly in order to compute the corresponding vector magnetic anomalies. Thespace-domain convolution is calculated using a 64x 64 window.

The total field anomaly is calculated at a distance of 3.4 units above the source. The vector magnetic anomalies are calculated at adistance of 6.4 units above the source. Other information is as given in Fig. 1.

where curs in low-inclinations, particularly in theFm = the ‘model data’ and andH2 components.F~= the ‘syntheticdata’.

The sumsare takenover that portion of the 3.2. Space-domainconvolutiondata set which is not contaminatedby edgeef-fects. The accuracyof the space-domainconvolution

The results,presentedin Figs. 1 and2, indicate can be evaluatedby comparingthe vector mag-a maximumpercentageerror in excessof 90% for netic anomaliescomputedusing this approachtothe H~and H, componentsnear the magnetic the valuescalculatedusinganexactforward mod-equator.The closestcorrespondencebetweenthe elling scheme(Plouff, 1976).The vectormagneticmodel and synthetic data sets occurs at mid to anomaliesare calculatedas follows. A space-do-high inclinations.The poorestcorrespondenceoc- main convolution is applied to the total field

COMPUTATIONOF VECTOR MAGNETIC ANOMALIES 237

U

z .P !I~ 0< j I

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~ flUCL ~ F- ( ~ . . . ‘~(I) <Z Z = ~ N

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238 ME. PURUCKER

anomaly using a window of (64 x 64). This size with singular value decomposition(Langel et al.,was chosenin order to comparethe resultswith 1984) in order to estimatethe valuesof magneticthat of the frequency domain transformation. momentslocatedon a grid at the Earth’ssurfaceArray sizes as small as 21 x 21 produce qualita- from a much larger set of ungridded magnetictively similar results. field valuescollectedat multiple altitudes. These

The results(Figs. 3 and4), presentedin away dipolescanthenbe used to calculatethe expectedanalogousto the frequency-domaintransforma- magneticfield at anyaltitude.tion, indicatea maximum percentageerror in ex- Thedipolesin the equivalentsourcerepresenta-cessof 60% in the H,, componentnear5°incina- tion areassumedto carry a magnetizationonly inlion. The closest correspondencebetween the the direction of the ambient field. The inversionmodel and synthetic data sets occurs at mid to producesboth positive and negativevalues. Ashigh inclinations.The poorestcorrespondenceoc- discussedby Harrisonet al. (1986), only positivecursin low inclinations,particularlyin the H,, and valuesshould result from induction, which is as-H2 components.The improvementof the space- sumedto be the primarysourceof magnetization.domain convolutionbetween5° inclination and Thepresenceof bothpositiveandnegativevaluesthe magneticequatoris a consequenceof using illustratesthat the zero level is non-unique.Sincedifferent convolutionoperatorsfor the two cases. the dipolesare only an intermediatestepandnotIf the convolution operatorused for n > 0 had a final product, we havenot applied an annihila-beenusedat the magneticequator,the percentage tor (e.g. Parker and Huestis, 1974) to produceerrors on H,, andH2 would havedoubled. exclusivelypositivevalues.

Bhattacharyya(1977) illustrated his technique The grid of equivalentsourcedipoles used inwith an examplewhich crossesthe geomagnetic the inversion extendsover the entire map. Theequator.Errorsof the typediscussedin this paper dipoles are not confined to the region of thecanprobablybeseenin the H,, vectoranomalyat magneticbodybut includea larger region,encom-thegeomagneticequator(fig. 9 of Bhattacharyya, passing the entire region shown on Fig. 5. Re-1977) but are not apparent in H~(fig. 10 of stricting the dipolesto the magneticbody impliesBhattacharyya, 1977) although they may be an a-priori knowledgeof the location of the mag-maskedby the lack of significant anomaliesnear netic body. In the generalcase,the locationof thethe geomagneticequator. Bhattacharyyaclaims magneticbody is unknown.that hisapproachis superiorto the frequency-do- The resolution of the equivalentsource tech-main transformationin low latitudesaround the nique is such that the distancebetweenthe ob-geomagneticequator.As shownin Figs. 2 and4, if servationplaneandthe equivalentsourcedipolesthe percentageerroris usedto evaluateand com- must belarger than the distancebetweenindivid-pare the two techniques,the space-domaincon- ual equivalentsource dipoles(Ku, 1977). In ourvolution is superiorat mostmid to high magnetic case the observationplane is 430 km above theinclinations and at the magneticequator. How- Earth’s surfacewhereasthe equivalentsourcedi-ever, the frequency-domaintransformationis su- polesare 110 km apartand are locatedwithin theperior at most other low magnetic inclinations, upper40 km of the Earth’scrust.Improvementscould be madein the space-domain Singularvalue decompositionis used to stabi-convolution techniqueby evaluatingequation(8) lize the equivalentsource solution. Instability isto higher order using Cauchy’s residuetheorem. particularly acutein the vicinity of the magneticAs notedearlier, whenthis is doneat the geomag- equator,wherea straight inversionproduces‘ring-netic equator,thepercentageerror decreasedby a ing’ in the resultantdipoles, with large negativefactor of 2 for H,, andH2. andpositivevaluesalternating.

Vector and total field anomaliescalculatedat3.3. Equivalentsourceinversion satellitealtitude using the approximatespherical-

Equivalent sourceinversionusesa least-squares earthalgorithm are referred to as ‘model data’.algorithm (Mayhew et al., 1984), in conjunction Thesefields are shown in the upper two rows of

COMPUTATION OF VECTOR MAGNETIC ANOMALIES 239

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240 ME. PURUCKER

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COMPUTATION OF VECTOR MAGNETIC ANOMALIES 241

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242 ME. PURUCKER

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COMPUTATION OF VECTOR MAGNETIC ANOMALIES 243

Fig. 5A, B. The ‘model data’ are then inverted to experimentaldata (Fig. 7). The noise has an ap-an equivalentsourcedistribution which in turn is proximate r.m.s. of 1 nT and zero mean. Thisusedto calculatethevectorandtotal field anoma- noise level is comparableto that encounteredinlies at satellite altitude again. Theselatter fields the MAGSAT data (Langel et al., 1982). Forare referredto as ‘synthetic data’ and are shown comparison,the absolutemaximum of the totalin the middle two rows of Fig. 5A, B. These field anomalymodelis 6 nT. The model datawith‘synthetic’ fields are calculatedusing total field added noise are then inverted to an equivalentanomaly data alone, shown in the left two col- source distribution as above. For the inductionumns of Fig. 5A, B, and using vector and total caseusing total field anomalydata aloneat lowfield anomaly data, shown in the two righthand magnetic latitudes, the results indicate that thecolumnsof Fig. 5A, B. The differencebetweenthe nearly orthogonalH,, and H2 vectorcomponents‘model data’ and the ‘synthetic data’ is then are not recoveredat all in the presenceof noisecalculatedand is shown in the lower two rows of with minimal (E~= 99.99999) singular value de-Fig. 5A, B. Theseexperimentsare performedfor compositionapplied to the least-squaressolution.both low-latitude (Fig. 5A) and mid-latitude(Fig. The ‘ringing’ or ‘bulls-eye’ patternpresentin theSB) magneticdistributions.The illustrated results field value solutions is analogousto the patternare for sourcesmagnetizedby induction, presentin the underlyingdipole solutions.Singu-

The results indicatea maximumpercentageer- lar valuedecompositionsusingan E~of less thanror of 25% in the H,, componentnearthe geomag- 99.9%arenecessarybeforethis featureis removednetic equator.Theclosestcorrespondencebetween from the solutions.the model and synthetic datasetsoccurs in mid-latitudes (Fig. SB) where both vector and totalfield anomalydataareused in the inversion.The 4. Discussionpoorest correspondencebetweenthe model andsynthetic data sets occurs in low latitudes (Fig. Enhancederrors are encounteredin all threeof5A) wheretotal field anomalydataaloneare used the discussedtechniqueswhen calculating thein theinversion, eastwardand downwardanomaliesin the vicinity

Histogramsof the percentageerrors are shown of thegeomagneticequator.Analogouserrorshavein Fig. 6. Thesehistogramsillustrate that while beendescribedfrom geomagneticfield modelsde-mid-latitudemodelsare recoveredmoreaccurately rived solelyfrom total field data. Theerrors in thethan low-latitudemodels,the presenceor absence geomagneticfield models have been related byof vector anomaly data in the inversion is the someto the proof of Backus(1970), that the scalarprimaryfactor in determininghow well the model potential A is not uniquely determinedby thewill berecovered, measurementof the magnitudeof the total field

If the observationplaneis changedfrom satel- outsidethe magneticsource.lite altitude (430km) to aircraftaltitude(1 km) we The equivalentsourceinversion technique,al-may use the exact algorithms of Bhattacharyya thoughrequiringthe greatestcomputerresources,(1964),developedfor useon a ‘flat earth’.A closer generatesthe smallest errors and is the mostcorrespondenceexistsbetweenthe model andsyn- versatile. Errors can be minimized by using athetic data(Fig. 6) when the dataare acquiredat techniquesuch as singularvaluedecompositiontoaircraft altitude. The ‘aircraft’ data are also stabilize the equivalent source solution. At lownoteworthy becauseof the absenceof any de- latitudesin the presenceof noise,vector compo-gradationin the total field anomaly data set at nentsoblique to the total field anomaly are notlow latitudes when the synthetic total field recoveredat all from the total field anomalydataanomalydata are reconstructedfrom the model aloneif singular valuedecompositionis not used.total field andvectoranomalydata. An E~of 99.9% or less will allow first-order

Normally distributedpseudo-randomnoisewas recovery of the vector componentsand will alsoadded to the model data in order to simulate suppress‘ringing’ or ‘bulls-eye’ patternsin the

244 ME PURUCKER

vectorsolutions.Errorscanalso be minimizedby Referencesusing both total field anomalydataand suitablyweightedvectoranomalydatawhenavailable.Be- Backus,G.E., 1970. Non-uniquenessof theexternalgeomag-

cause vector data are noisier than total field netic field determinedby surfaceintensitymeasurements.J.

anomalydatafor a givenuncertaintyin altitude, it Geophys. Res., 75: 6339-6341.

may be necessaryto use differentweights for the Bhattachar~ia, B.K., 1964. Magnetic anomalies due to prism-shaped bodies with arbitrary polarization. Geophysics, 29:differentdatatypes.Whenwe invert observational 517531

data using the equivalent source technique,we Bhattacharyya, B.K., 1977. Reduction and treatment of mag.

assign to eachobservationa weight which corre- netic anomalies of crustal origin in satellite data. J. Geo-

spondsto the standarderrorof all observationsof phys. Res., 82: 3379—3390.Dampney, C.N.G., 1969. The equivalent source technique.

its type (i.e. total field or vector)in the immediate Geophysics, 45: 39—53.

vicinity. The dipoles are then calculated as fol- Gaffiher, S.C. and Mayhew, M.A., 1982. On thepossibility of

lows: detecting large-scale crustal remanent magnetization withMAGSATvector magnetic anomaly data. Geophys. Res.

p = (vD1V’)S’wb (12) Lett., 9: 325—328.where: Harrison,C.G.A., Cane,H.M. and Hayling, K.L., 1986. Inter-

V = matrix whose elementsare eigenvectorsof pretationof satelliteelevationmagneticanomalies.J. Geo-

s’ws, phys. Res.,91: 3633—3650Hurwitz, L. and Knapp, D.G., 1974. Inherent vector dis-D = matrix whosediagonalcontainsthe eigenval- crepanciesin geomagneticmain field models basedon

uesof S’WS, scalarF.J.Geophys.Res., 79: 3009—3013.

W= diagonalweight matrix. Kolesova,V.1. and Cherkayeva,Ye.A., 1986. Computationof

and with other symbols as defined in (9). The the componentsof the anomalousmagneticfield from the

additionof weightsto the problemhas the further data of absolute surveys for a variable direction of thevectorof the main field. Geomagn.Aeron.,26: 884—885.

advantagethat errors can then be calculatedon Ku, C.C.,1977. A direct computationof gravity and magnetic

the dipoles, andby propagation,on the field val- anomaliescausedby 2- and 3-dimensional bodiesof arbi-ues. trary shapeand arbitrary magneticpolarization by equiv-

The space-domainconvolutionand frequency- alent-point method and a simplified cubic spline. Geo-

domain transformation techniquesrequire corn- physics,42: 610—622.Langel, RA. and Estes,R.H., 1985. The near-earthmagnetic

parablecomputerresources.More restrictivethan field at 1980determinedfrom MAGSAT data.J. Geophys.

the equivalentsourcetechnique,theybothassume Res.,90: 2495—2509.

a constantgeomagneticfield direction and total Langel, R.A., Slud, E.V. and Smith, PJ., 1984. Reductionoffield anomalydatacollectedovera regulargrid at satellite magneticanomalydata. J. Geophys.,54: 207-212.

constantaltitude. Thesetechniquesare therefore Langel,R.A., Schnetzler,C.C.,Phillips, J.D. andHomer, Ri.,1982. Initial vector magneticanomaly map from MAG-most appropriatefor aeromagneticsurveyscover- SAT Geophys.Res Lett., 9: 273—276.

ing small areas. Within this framework, the Lourenco, J.S. and Morrison, H.F., 1973. Vector magnetic

space-domainconvolutionis superiorat mostmid anomaliesderivedfrom measurementsof a single compo-to high magneticinclinations andat the magnetic nent of the field. Geophysics, 38: 359—368.

Lowes, F.J., 1975. Vectorerrorsin sphericalharmonicanalysisequator.However, the frequency-domaintransfor- of scalardata.Geophys.J. R. Astron.Soc., 42: 637—651.

rnation is superior at most other low magnetic Lowes, F.J. and Martin, J.E., 1987. Optimum useof satelliteinclinations, intensity and vector data in modelling the main geomag-

netic field. Phys. Earth Planet. Inter., 48: 183—192.

Maeda, H., Iyemori, T, Araki, T. and Kamei, T., 1982. NewAcknowledgments evidenceof a meridional current system in the equatorial

ionosphere. Geophys. Res. Lett, 9: 337—340.I thank Robert Langel for encouragementand Mayhew,M.A., 1979. Inversion of satellitemagneticanomaly

for his thoughtfulreviews during the early stages data. J. Geophys., 45: 119—128.Mayhew,M.A, Estes,RH. and Myers, D.M., 1984. Remanent

of this project. Reviewsby PatTaylor, Jim Heirt- magnetizationand three-dimensionaldensity model of thezler, andMichael Mayhewhelpedclarify the writ- Kentuckyanomalyregion. Final report of NASA contract

ing. NAS5-27488,90 pp.

COMPUTATION OF VECTOR MAGNETIC ANOMALIES 245

Parker,R.L. andHuestis,S.P.,1974. The inversionof magnetic Stern,D.P., Langel, R.A. andMead,G.D., 1980. Backuseffectanomaliesin thepresenceof topography.J. Geophys.Res., observedby MAGSAT. Geophys.Res. Lett., 7: 941—944.79: 1587—1593. Von Frese, R.B., Hinze, W.J., Braile, L.W. and Luca, AJ.,

Plouff, D., 1976. Gravity and magnetic fields of polygonal 1980. Sphericalearthgravity and magneticanomalymodel-prisms and application to magnetic terrain corrections. ing by Gauss-Legendre quadrature integration. Final reportGeophysics,41: 727—741. of NASA ContractNAS5-25030,116 pp.

Stern,D.P. and Bredekamp,J.H.,1975. Error enhancementingeomagneticmodelsderivedfrom scalardata. J. Geophys.Res., 80: 1776—1782.