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3-D Analytic Signal in the Interpretation of Total Magnetic Field Data atLow Magnetic LatitudesIan N. MacLeod*Geosoft Inc.85 Richmond St. W.8th FloorToronto, OntarioM5H-2C9Canada

Keith JonesAshton Mining Limited24 Outram StreetWest PerthWestern Australia, 6005P.O. Box 962, West Perth, W.A.6872

Ting Fan DaiGeosoft Inc.85 Richmond St. W.8th FloorToronto, OntarioM5H-2C9Canada

Abstract

The interpretation of magnetic field data at lowmagnetic latitudes is difficult because the vectornature of the magnetic field increases thecomplexity of anomalies from magnetic rocks.The most obvious approach to this problem is toreduce the data to the magnetic pole, where thepresumably vertical magnetization vector willsimplify observed anomalies. However, RTPrequires special treatment of North-Southfeatures in data observed in low magneticlatitudes due to high amplitude corrections ofthese features. Furthermore, RTP requires theassumption of induced magnetization with theresult that anomalies from remanently andanisotropically magnetized bodies can beseverely distorted.

The amplitude of the 3-D analytic signal of thetotal magnetic field produces maxima overmagnetic contacts regardless of the direction ofmagnetization. The absence of magnetizationdirection in the shape of analytic signalanomalies is a particularly attractivecharacteristic for the interpretation of magneticfield data near the magnetic equator. Althoughthe amplitude of the analytic signal is dependenton magnetization strength and the direction ofgeologic strike with respect to the magnetizationvector, this dependency is easier to deal with inthe interpretation of analytic signal amplitudethan in the original total field data or polereduced magnetic field. It is also straightforward to determine the depth to sources fromthe distance between inflection points of analyticsignal anomalies.

Key wordsanalytic signal, magnetic interpretation, reductionto the pole, magnetic depth

Introduction

When dealing with two-dimensional magneticfield data, an important goal of data processingis to simplify the complex information provided inthe original data. One such simplification is toderive a map on which the amplitude of thedisplayed function is directly and simply relatedto a physical property of the underlying rocks.For example, the gravity field over a body of acertain density contrast exhibits this simplicity.Baranov (1957) proposed converting 2-Dmagnetic data to 'pseudo-gravity' using aprocess that involves reduction to the pole (RTP)and vertical integration. Although Baranov'sprocedure is referred to as pseudo-gravity, theprocess does not imply that the distribution ofmagnetism in the earth is necessarily related tothe density distribution. It is intended only tosimplify the interpretation of magnetic field databy thinking of magnetization in the same way asdensity.

The essential ingredient in Baranov's pseudo-gravity process is the RTP since it presumablysimplifies the vector nature of the magnetic fieldby making the field vertical. However, thisprocess is not straight forward at low magneticlatitudes because of the very large amplitudecorrections that must be applied to North-Southtrending features.

This paper proposes the use of 3-D analyticsignal calculated from the total magnetic field(Nabighian, 1972, Roest, 1992, MacLeod, 1993)as a practical alternative to reduction to the polefrom low magnetic latitudes. In fact, thistechnique has merit at all magnetic latitudes, butthe benefits are most apparent near themagnetic equator.

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Reduction To The Pole

The goal of reduction to the pole is to take anobserved total magnetic field map and produce amagnetic map that would result had an areabeen surveyed at the magnetic pole. Assumingthat all the observed magnetic field of a studyarea is due to induced magnetic effects, polereduction can be calculated in the frequencydomain using the following operator (Grant andDodds, 1972):

( )[ ]L

I i I Dθ

θ=

+ −1

2sin( ) cos( ) cos( )

Equation 1

Where:

θ is the wavenumber directionI is the magnetic inclinationD is the magnetic declination.

From Equation 1, it can be seen that as Iapproaches 0 (the magnetic equator), and (D-θ )approaches π/2 (a North-South feature), theoperator approaches infinity.

This effect, as illustrated by Figure 1, comparesthe magnetic anomalies over an East-West andNorth-South vertically dipping dyke-like body.For the East-West striking dyke, the amplituderemains constant, but the shape changes. Forthe North-South striking dyke, the anomalyshape remains the same at all latitudes, but theamplitude is reduced as the inducing magneticvector is rotated to be along strike at themagnetic equator, at which point the anomalysimply disappears.

100

0

0

100

50

-50

0

-100

0

0

-100

100

0100

0100

0100

0100

0

-600 6000 -600 6000

I = 90°

I = 70°

I = 45°

I = 20°

I = 0°

South-to-North profile over West-to-East profile overan East-West dyke a North-South dyke

Figure 1. The shape of total magnetic fieldprofiles over a vertically dipping dyke depends onthe direction of the dyke. On the left, profiles areshown for an East-West striking dyke, and on theright profiles are shown for a North-South strikingdyke. Reduction to the pole involves correctingthe shape of East-West features and correctingthe amplitude of North-South features to producethe same profile as would be observed at aninclination of 90°.

The very large amplitude correction required forNorth-South features at low latitudes alsoamplifies the North-South components of noiseand magnetic effects from bodies magnetized indirections different from the inducing field.Numerous authors have addressed the noiseproblem in the published literature.

The methods either modify the amplitudecorrection in the magnetic North-South directionusing frequency domain techniques (Hansenand Pawlowski, 1989, Mendonca and Silva,1993), or calculate an equivalent source in thespace domain (Silva, 1986). We have found thatthe simplest and most effective technique is thatdeveloped by Fraser Grant and Jack Dodds inthe development of MAGMAP FFT processing

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system in 1972. They introduced a secondinclination (I') that is used to control theamplitude of the filter near the equator as shownin Equation 2.

In practice, (I') is set to an inclination greaterthan the true inclination of the magnetic field (orless than the true inclination in the Southernhemisphere). By properly attenuating thecomplex amplitude while not altering thecomplex phase from Equation 1 to Equation 2,anomaly shapes will be properly reduced to thepole, but by setting (|I'| > |I|), unreasonably largeamplitude corrections are avoided. Controllingthe RTP operator then becomes a matter ofchoosing the smallest I' that still givesacceptable results. This will depend on thequality of the data and the amount of non-induced magnetization present in the area understudy.

The application of Grant and Dodds RTPprocessing on a simple model magnetized byinduction and remanence, is illustrated in Figure2. Note that when the body is magnetized onlyby induction (Figure 2.a), a reasonable RTP canbe produced using Equation 1, the result ofwhich is shown in Figure 2.c. However, addingremanence to the same body produces the totalfield anomaly shown in Figure 2.b, and the RTPprocess produces the very poor result shown inFigure 2.d. When remanent effects such asthese are recognized, the amplitude inclination I'in Equation 2, can be increased to diminish theproblem, as shown in Figure 2.f. However, thiscorrection is at the expense of the RTP for theinduced prism shown, which is shown in Figure2.e. Adding noise to the data will introduce

similar problems, although noise rejection filterscan be designed to address the specificcharacteristics of the noise, which is often line-to-line levelling error.

Given these difficulties with RTP, and ouroriginal objective of simplifying the magneticfield, it would seem preferable to produce aresult that simply provides a measure of theamount of magnetization, regardless of direction.

3-D Analytic Signal

Nabighian (1972, 1984) developed the notion of2-D analytic signal, or energy envelope, ofmagnetic anomalies. Roest, et al (1992),showed that the amplitude (absolute value) ofthe 3-D analytic signal at location (x,y) can beeasily derived from the three orthogonalgradients of the total magnetic field using theexpression:

A x ydTdx

dTdy

dTdz

( , ) = � � +�

��

�� +

�

��

��

2 2 2

Equation 3

Where:

),( yxA is the amplitude of the analyticsignal at (x,y)

T is the observed magnetic field at(x,y).

[ ]DIiI −⋅⋅− )cos()cos()sin( 2θ

( ) [ ] [ ] IIIIifDIIDII

L =<−⋅+⋅−⋅+

= '|),||'(|,)(cos)(cos)(sin)(cos)'(cos)'(sin 222222 θθ

θ

Equation 2

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e f

c dRTP

I = -10I' = -10D = -20

RTPI = -10I' = -45D = -20

a bTotal Field

Induced Magnetization Only Induced plus 90° Remanence

Figure 2. RTP applied to synthetic data for a depth limited prism 200x200 units square, 100 unitsthick, 50 units deep. a) shows the magnetic field that results from induced magnetization only in a field withan inclination of -10°, and declination 20° West, and b) shows the same prism in the same inducing field, butwith vertical remanence. c) and d) are the RTP results using Equation 1, applied to a) and b) respectively. c)shows the desired results, but d) shows a severely distorted anomaly after RTP. Applying Equation 2, withan amplitude inclination (I') set to 45° produces the results shown in e) and f). A more acceptable anomaly isproduced for the remanently magnetized prism f), but at the expense of distortion in the induced prism e).

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An important comment at this point is the easewith which the analytic signal can be calculatedwith commonly available computer software.The x and y derivatives can be calculateddirectly form a total magnetic field grid using asimple 3x3 filter, and the vertical gradient isroutinely calculated using FFT techniques.

The analytic signal anomaly over a 2-D magneticcontact located at (x=0) and at depth h isdescribed by the expression (after Nabighian,1972):

( ) 2122

1)(xh

xA+

= α

Equation 4Where:

α is the amplitude factor

α = −2 1 2 2M d I Asin ( cos ( ) sin ( ))

Equation 5

and

h is the depth to the top of the contact

M is the strength of magnetization

d is the dip of the contact

I is the inclination of the magnetizationvector

A is the direction of the magnetizationvector

The analytic signal described by Equation 4 is asimple bell shaped function in which alldirectional terms are contained in the amplitudefactor α, which is a constant. Therefore, only theamplitude of the analytic signal is affected by thevector components of the model. The shape ofthe analytic signal is dependant only on depth.

Similarly, it can be shown that the analytic signalanomaly over a 2-D magnetic sheet (or dyke) isdescribed by the equation:

( )A xh x

( ) =+

α1

2 2

Equation 6

and over a cylinder:

( )A x

h x( ) =

+α

22 2

32

Equation 7

Figure 3 shows the results of analytic signalprocessing over the same simple prism modelused for reduction to the pole in Figure 2. Theuse of a 3-D perspective presentation is toclearly show the how the amplitude of theanalytic signal peaks over the edges of themodel. Note that the amplitude of the peaks isproportional to the magnetization at that edge asdefined by Equation 5. In this case, the prismwidth is four times the depth. For widths lessthan the depth, the peaks of the analytic signalwill merge.

MacLeod, et al (1993), showed how the analyticsignal could be calculated from the verticalintegral of the magnetic field in order to producea result that was more similar to the analyticsignal of pseudo-gravity. This had the effect ofemphasizing deeper, or more regionalinformation in the magnetic field and diminishingsurface noise.

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Total Magnetic Field Analytic Signal

I = 90°induced only

I = -10°induced only

I = -10°induced plus

90° remanence

Figure 3. The analytic signal of the total magnetic field produces similar results regardless of thedirection of magnetization. In this case, the analytic signal is shown for the same model used in Figure 2,under the same magnetic conditions. Note that the analytic signal peaks over the edges of the model, andthe amplitude of the peak is proportional to the magnetization at that edge.

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Depth from Analytic Signal

The analytic signal shape can be used todetermine the depth to the magnetic sources.Atchuta et al (1981) and Roest et al (1992) usedthe anomaly width at half the amplitude to derivethe depths. Atchuta also used othercharacteristics of the analytic signal (which hereferred to as 'complex gradient') to help resolvethe effect of overlapping edges. The followingrelationships can be derived directly fromEquations 4, 6 and 7:

For a contact:

x h h122 3 346= = .

Equation 8

For a thin sheet (dyke):

x h122=

Equation 9

For a horizontal cylinder:

x h h12

3 122 4 1 1533= − =( ) .

Equation 10

Where:

x12

is the width of the anomaly at half theamplitude

h is the depth to the top of the contact.

However, the measured amplitude of ananomaly can be in error due to overlappinganomalies, and even where the amplitude canbe well determined for a single source, theresulting depth will be significantly in error if thecorrect model is not used.

We have taken the approach of using thedistance between the inflection points of theanalytic signal anomaly to determine the depth.The inflection points occur higher up the flanksof the anomaly and therefore should be lesssusceptible to interference from neighboringanomalies. Taking the second derivative ofEquations 4, 6 and 7 with respect to x producesthe following results:

For a contact:

d A xdx

x h

h x

2

2

2 2

2 252

2( )

( )=

−

+α

Equation 11

x h hi = =2 1 414.

Equation 12

For a thin sheet (dyke):

d A xdx

x hh x

2

2

2 2

2 2 3

2 3( ) ( )( )

=−

+α

Equation 13

hhxi 155.132 ==

Equation 14

For a horizontal cylinder:

d A xdx

x h

h x

2

2

2 2

2 272

6 4( ) ( )

( )=

−

+α

Equation 15

x hi =

Equation 16

Where:

A is the analytic signal calculated from (3)

α is the amplitude factor from (5)

h is the depth

xi is the width of the anomaly betweeninflection points

Although we have not mathematically solved fora vertical cylinder, model tests show that resultsare very close the those predicted by Equation16.

The summaries of the two approachs fordeterimining depth from analytic signalanomalies are shown in Table 1. Note that usingthe width at half the amplitude and assuming a

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contact model, the true depth for dyke/sill bodieswill be 1.73 times the modelled depth, and thetrue depth for a cylindrical body will be 2.25%times the modelled depth. In contrast, using thedistance between inflection points the true depthfor a dyke/sill body will be 1.24 times themodelled depth, and the true depth of acylindrical source will be 1.41 times the modelleddepth.

Table 1

Analytic signal anomaly widths as a functionof depth (h).

Sourcegeometry

Distance betweeninflection points

Width at halfthe amplitude

contact 1.414h 3.46h

dyke/sill 1.155h 2h

cylinder 1h 1.533h

The depth of a dyke-like source calculated fromthe inflection point widths as a function of thebody width to depth ratio is shown in Figure 4.These curves were compiled by measuring thewidth of the second horizontal derivativecalculated from synthetic model curves. Thesecond derivative was calculated by applyingsimple three-point convolution filter (-0.5, 1, -0.5)to the synthetic curves.

Note that when the body width is less than thedepth, the apparent depth will increase with thewidth to a maximum of almost 2.5 times the truedepth. When the width is greater than or equalto the depth, the method is able to distinguishthe two edges of the body as separate edges.Results are shown for both an assumed dykeand sill, and the limiting cases agree with thetheoretical prediction summarized in Table 1.

The effect of varying thickness of a step as afunction of depth is shown in Figure 5. As wewould expect, as the bottom of a stepapproaches the top, the model becomes morelike a sill, and the sill model results approach thetrue depth. It is interesting to note that when thethickness of a step is the same as the depth tothe top, correct results are obtained using thestep model, and as the thickness increased fromthat point, the apparent depth only increases to amaximum of 2% greater than the true depth, andthen gradually returns to the depth predected bytheory as the thickness increases.

From these results, we would conclude that it isbest to use assume a contact model whenworking with more regional data. In the worstcase, that of a very thin dyke or sill, the contact

model will under estimate the depth by 18%.Both the dyke and the sill model will overestimate the depth of dyke-like bodies whosewidth approaches its depth. Vertical cylindermodels can be rejected based on the fact thatthey produce a single isolated solution, andhorizontal cylinders are rare.

apparentdepthtrue

depth

widthdepth

dyke

dyke

contact

contact

depthwidth

analytic signalx i

Figure 4. This figure shows therelationship between the apparent depthcalculated from the width of the anomaly at theinflection points and the true depth as a functionof the width for a dyke-like source. Results areshown assuming a dyke source (heavy line) and acontact source (light line). Note that where thewidth is less than the depth, the apparent depthincreases to a maximum depth=width. Where thewidth is greater than the depth the anomaly isresolved into two separate edges, which produceseparate depths as shown for width/depth valuesgreater than 1.

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apparentdepthtrue

depth

thicknessdepth

dyke/sill

contact

thickness

depth

analytic signalx i

Figure 5. This figure shows therelationship between the apparent depthcalculated from the width of the analytic signalanomaly between inflection points and the truedepth as a function of the width for a step/contactsource. Depths are calculated assuming a dykesource (heavy line) and a contact source (lightline). Note that where thickness is equal to orgreater than the depth, the contact modelproduces accurate results. As the thicknessapproaches 0, the step becomes a sill, and thedyke/sill model results are correct.

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Sangihe Island, Indonesia

An airborne magnetic and radiometric surveywas conducted over the southern half of SangiheIsland, which forms part of a volcanic island arcextending over 400 km from the Northeasternarm of Sulawesi to Mindanao in the Philippines.The helicopter survey was carried out on north-south lines with a nominal 200 metre lineseparation, and a cesium total magnetic fieldsensor was towed at a nominal terrain clearanceof 60 metres.

The geology of Sangihe Island is characterizedby Miocene to recently active calc-alkaline stratavolcanoes, which become progressively youngernorthwards. Gold mineralisation occurs within asequence of andesitic flows, sills and dykes,lithic and crystal tuffs and andesite porphyryintrusives. The main purpose of the survey wasto help map areas of alteration within thevolcanic pile. These are characterized by areduced magnetic response, associated with thedestruction of magnetite by mineralizing fluidswithin the andesitic sequence. Areas ofalteration are also characterized by potassiumanomalies in the radiometric data. The magneticdata has been used to map lithology and majorstructural features which are important infocusing mineralisation. Because of therelatively young age of the rocks, remanentmagnetization was an issue in the interpretationof the total magnetic field.

Figure 6. Total magnetic field contoursover an apparently remanently magnetized bodyin Indonesia at a magnetic inclination of –10°.The source of the anomaly is interpreted to beroughly 1500 metres in length and lies along theaxis defined by the East-North-East trendinganomaly.

The original magnetic anomaly shown in Figure6 is asymmetric with a strong low on the Northand a high on the South. Our interpretation of

this feature is a roughly 1500 metre longmagnetic unit that strikes in an East to North-East direction. Strong remanent magnetizationis suspected because induced magnetizedbodies at this latitude would produce an almostsymmetric low. In order to improve theinterpretability of the data, standard reduction tothe pole was performed using Equation 2, withthe result shown in Figure 7.

Figure 7. The anomaly shown in Figure 6,has been reduced to the pole using Equation 2with an amplitude inclination (I') of 45°, which wasnecessary to reduce strong North-Southstretching of the anomaly. The result is anasymetric anomaly that might be interpreted as ashallow North-dipping body. However, thisinterpretation does not account for the North-South attenuation that was required. The fact thatsuch attenuation was required confirms remanentmagnetization in the source.

It was necessary to use an amplitude inclinationof 45° in order to reduce North-South stretchingof the anomaly, which further supports thesuspicion of remance. The pole reducedanomaly could be interpreted as a shallow Northdipping body, which would might explain thestrong low, but this interpretation does notaccount for the under corrected amplitude in theRTP.

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Figure 8. The analytic signal of the originaltotal field data clearly shows anomalous highswhich outlines the body interpreted from Figure 6.A simple interpretation of the outline of the bodyand magnetic lineations/faults are shown forreference. The interpretation was based on thecurvature plot shown in Figure 9.

Figure 8 shows the analytic signal calculatedfrom the original total field data together with asimplified interpretation of the outline of the bodyand interpreted cross-cutting lineations/faults.Note how the analytic signal anomaly peaksclearly outline the North and South edges of theinterpreted unit. Figure 9. shows the positivecontours of the analytic signal after applying asimple 3x3 Laplacian filter, which has thefollowing coefficients:

0 -0.25 0

-0.25 1 -0.25

0 -0.25 0

The Laplacian filter produces a curvature map inwhich inflection points in the original data arelocated at the zero contour. The contour map inFigure 9. takes advantage of this by plotting onlypositive contours. The same effect can beachieved on a colour image by using a solid andcontrasting colour for the negative parts of theLaplacian grid. From the Laplacian, depths canbe estimated by measuring the widths of theanomalies and applying the factors in Table 1. Anumber of measured depths are shown in Figure9, to illustrate the measurement of anomalywidths. Automated methods similar to thosedescribed by Roest et al (1992) can also beapplied to produce depth maps.

Figure 9. Applying a simple Laplacian filterto the analytic signal produces curvature mapfrom which inflection points are mapped by thezero contour. A simplified interpretation basedon this data shows the outline of the source bodyand interpreted lineation/faults. Depthscalculated from the curvature anomaly widths arealso shown.

Conclusions

Reduction to the pole attempts to simplify themagnetic field by rotating the magnetizing vectorto be vertical. However, this technique presentssignificant problems when the direction ofmagnetization is not in the same direction as theearth's field, which is particularly important whenthe data has been collected at low magneticlatitudes. This problem can be partly overcomeby modifying the amplitude correction of the polereduction operator, but this must be done at theexpense of distortion in anomalies frominductively magnetized rocks. In contrast, theanalytic signal of the total magnetic field reducesmagnetic data to anomalies whose maximamark the edges of magnetized bodies, andwhose shape can be used to determine thedepths of those edges.

Since amplitude of analytic signal anomaliescombines all vector components of the field intoa simple constant, a good way to think of analyticsignal is as a map of magnetization in theground. With this in mind, stronger anomaliescan be expected where the magnetization vectorintersects a magnetic contrasts at an acuteangle. As a result, inductively magnetized rocksat the equator will have stronger analytic signalat their North and South edges. The samethinking can be applied to remanently oranisotropically magnetized bodies which havemagnetization vectors in different directions.

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Once an analytic signal grid is available, depthscan be calculated by applying a Laplacian filter tothe grid and measuring the distance betweeninflection points on anomalies of interest. Thisdistance is directly proportional to the depth tothe top of the contrast. Although a depthcalculated in this way will depend on theassumed source model, we have shown that thedifference between source models will produceresults generally accurate to within 20%.However, this method does not solve thecommon ambiguity between depth and thicknessfor dike-like bodies where the width is less thanor near the depth.

Also, because an analytic signal anomaly simplymarks a magnetic contrast, one does not knowthe sense of the contrast from the analytic signalalone. Because of this, the original total fieldmap should be used help resolve these aspectsof an interpretation.

Acknowlegements

The authors would like to thank Ashton MiningLimited for providing the geophysical data fromSangihe Island and Geosoft Inc. for their suportof this work. Thanks are also due to SergioVierra of WMC Brazil for proving the practicalapplication of these techniques in his own worknear the magnetic equator in South America.

References

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Baranov, V., 1957, A new method forinterpretation of aeromagnetic maps:Pseudo-gravemetric anomalies:Geophysics, 22, 359-383.

Bhattacharyya, B., K., 1966, Continuousspectrum of the total magnetic field anomaly

due to a rectangular prismatic body:Geophysics, 31, 99-121

Grant, F.S., Dodds, J., 1972, MAGMAP FFTprocessing system development notes,Paterson Grant and Watson Limited.

Hansen, R.O., and Pawlowski, R.S., 1989,Reduction to the pole at low latititudes byWeiner filtering: Geophysics, 54, 1607-1613.

Jost, H., De Oliveira, A.M., Stratigraphy of thegreenstone belts, Crixas region, Goias,central Brazil: J. South American Sciense, 4,201-214.

Macleod, I. N., Vierra, S., Chaves, A. C., 1993,Analytic signal and reduction-to-the-pole inthe interpretation of total magnetic field dataat low magnetic latitudes, Proceedings of thethird international congress of the Braziliansociety of geophysicists.

Mendonca, C.A., and Silva, B.C., 1993, A stabletruncated series approximation of thereduction-to-the-pole operator: Geophysics,58, 1084-1090.

Nabighian, M.N., 1972, The analytic signal oftwo-dimensional magnetic bodies withpolygonal cross-section: Its properties anduse for automated anomaly interpretation:Geophysics, 37 507-517.

Nabighian, M.N., 1974, Additional comments onthe analytic signal of two-dimensionalmagnetic bodies with polygonal cross-section, Geophysics, 39, 85-92.

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Roest, W.E., Verhoef, J., Pilkington, M., 1992,Magnetic interpretation using 3-D analyticsignal: Geophysics, 57, 116-125.