08_mendonÇa e meguid 2008

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Computers & Geosciences 34 (2008) 603–610

Programs to compute magnetization to density ratioand the magnetization inclination from 3-D gravity

and magnetic anomalies$

Carlos A. Mendonc-a�,1, Ahmed M.A. Meguid2

Instituto de Astronomia, Geofısica e Ciencias Atmosfericas, Universidade de Sao Paulo, Rua do Matao, no. 1226,

Cidade Universitaria, CEP: 05508-090 Sao Paulo, Brazil

Received 13 June 2006; received in revised form 24 September 2007; accepted 26 September 2007

Abstract

Vector field formulation based on the Poisson theorem allows an automatic determination of rock physical properties

(magnetization to density ratio—MDR—and the magnetization inclination—MI) from combined processing of gravity

and magnetic geophysical data. The basic assumptions (i.e., Poisson conditions) are: that gravity and magnetic fields share

common sources, and that these sources have a uniform magnetization direction and MDR. In addition, the previously

existing formulation was restricted to profile data, and assumed sufficiently elongated (2-D) sources. For sources that

violate Poisson conditions or have a 3-D geometry, the apparent values of MDR and MI that are generated in this way

have an unclear relationship to the actual properties in the subsurface. We present Fortran programs that estimate MDR

and MI values for 3-D sources through processing of gridded gravity and magnetic data. Tests with simple geophysical

models indicate that magnetization polarity can be successfully recovered by MDR–MI processing, even in cases where

juxtaposed bodies cannot be clearly distinguished on the basis of anomaly data. These results may be useful in crustal

studies, especially in mapping magnetization polarity from marine-based gravity and magnetic data.

r 2007 Elsevier Ltd. All rights reserved.

PACS: PRISM3D; MDRMI

Keywords: Potential fields; Poisson theorem; MDR; MI

1. Introduction

The Poisson theorem can be used to extractinformation about rock properties directly fromprocessing of gravity and magnetic anomaly data.This theorem provides a simple linear relationshipconnecting gravity and magnetic potentials and, byextension, field components that are commonlyderived from geophysical surveys (Blakely, 1995).The assumptions underlying the Poisson theorem

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0098-3004/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cageo.2007.09.013

$Code available from server at http://www.iamg.org/

CGEditor/index.htm�Corresponding author.

E-mail address: [email protected] (C.A. Mendonc-a).1Supported by CNPq (3051005/2002-07) and (140603/2003-4).2Assistant researcher at the Egyptian Petroleum Researches

Institute (EPRI), Ph.D. candidate at IAG-USP.


(or Poisson conditions) are that: (i) causative denseand magnetic sources are common; (ii) magnetiza-tion direction is constant; and (iii) magnetization todensity ratio (MDR) is constant. For geologicalstructures satisfying such constraints, it is possibleto use gravity and magnetic anomalies to estimatethe source MDR, which is related to the slope of thelinear Poisson relationship, and its magnetizationinclination (MI). Poisson-based methods have beenapplied to interpret many geological problems inmarine geophysics (Bott and Ingles, 1972; Cordelland Taylor, 1971) and regional studies of continen-tal crust (Chandler et al., 1981; Hildebrand, 1985;Chandler and Malek, 1991).

Two main approaches have been used to applyPoisson theorem. In the first approach, filtering isapplied to produce the vertical derivative of gravityanomaly and the magnetic anomaly reduced to pole(vertically polarized) (Bott and Ingles, 1972; Chand-ler et al., 1981; Chandler and Malek, 1991). Thesecond approach (Baranov, 1957; Cordell andTaylor, 1971) filters the magnetic anomaly solely,to produce a pseudo-gravity anomaly, whichinvolves magnetic reduction to the pole and itsintegration along the magnetization direction. Ineither case, transformed anomalies should be coin-cident if Poisson conditions are upheld. If thedirection of magnetization is known, MDR valuescan be readily calculated by either approachthrough a linear regression of the transformedgravity and magnetic data. Both approaches canalso be used to estimate an unknown magnetizationdirection by a trying range of trial and errordirections until a maximum correlation is attained.However, such trial and error methods lack theefficiency of a one-step approach and are prone toerror when Poisson conditions are not upheld.

A one-step method that estimates both MDR andMI values from profile data has been developed byusing a vector field formulation for the Poissontheorem (Mendonc-a, 2004). In this approach, thegravity anomaly is processed to furnish its gradient,and the magnetic anomaly processed to give itsparent anomalous vector field. MDR is thenobtained by rationing the amplitude of these fields,and MI by taking their inner product. For 2-Dsources, this approach provides accurate MDR andMI estimations with no prior knowledge of magne-tization direction and no need of iterative proce-dures. Sources violating Poisson conditions lead toapparent MDR and MI values that do not trulyreflect subsurface sources, but they nonetheless can

be used to discern geological contacts and inferspatial variations in rock physical properties (Men-donc-a, 2004). Unfortunately, little is known aboutthe significance of apparent MDR and MI valuesthat are derived in this way from sources that areactually 3-D.

This paper presents two Fortran programs thatuse gridded gravity and magnetic data to estimateMDR and MI values for 3-D sources. ProgramPRISM3D computes responses over simple 3-Dmodels and gives some guidance for interpretingfield-acquired data sets. Program MDRMI can beused to process field-acquired gravity and magneticdata sets to estimate MDR and MI values for thecausative sources. MATLAB scripts to visualizeprogram outputs are also included.

2. Poisson relationship in vectorial form

Poisson theorem (Blakely, 1995) relates themagnetic potential, Vm, to gravity potential, U, as

Vm ¼ �pqU

qm, (1)

under the assumption that Poisson conditions areheld. By Eq. (1), magnetization direction is given byunit vector m,

p ¼ cDMDr

, (2)

where DM is the intensity of the magnetizationcontrast DM ¼ DMm (bold types denoting vectors),Dr is the source density contrast, c ¼ m0=4pG,where m0 ¼ 4p10�7 H=m is vacuum magnetic per-meability and G ¼ 6:67� 10�11 m3 kg�1 s�2 is thegravitational constant.

Gravity anomaly, gz, is obtained from gravitypotential U as gz ¼ z � rU , with r denoting thegradient operator, and z a unit vector aligned to thevertical (positive downward). Total field magneticanomaly, Tt

m, is obtained from magnetic potentialVm as Tt

m ¼ �t � rV m, where t is the unit vectoralong the local magnetic field.

For 2-D sources (Mendonc-a, 2004), the MDR,r � DM=Dr, is obtained by

r ¼1

c

jTmj

jrgzj(3)

and MI, a, such that

sinðaÞ ¼Tm � rgz

jTmjjrgzj. (4)

ARTICLE IN PRESSC.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603–610604


For 3-D sources, the vector quantities related inEqs. (3) and (4) are such that

Tm ¼ Txmex þ Ty

mey þ Tzmez (5)

and

rgz ¼qgz

qxex þ

qgz

qyey þ

qgz

qzez. (6)

For real 3-D sources, substitution of fields (5) and(6) in Eqs. (3) and (4) yields apparent values forquantities r and a (MDR and IM, respectively),which eventually may reflect the true values for theunderlying sources.

3. MDR–MI from 3-D models

The basic building block of our formulation is avertical prism with a horizontal top and bottom. Bycompounding blocks we can readily compute thefield components and derivatives that reflect acomplex, multi-source data set. Vector fields Tm

and rgz can be evaluated simply by addingmagnetic field components and gravity derivativesfrom adjacent prisms. Once obtained, these fieldsenter Eqs. (3) and (4) to give MDR and MI.

To evaluate magnetic fields from an isolatedprism, we use formula adapted from Plouff (1976),

Ttm ¼

m04p

DM ðmN þ nMÞ1

2log

R� x

Rþ x

� ��

þ ðlN þ nLÞ1

2log

R� y

Rþ y

� �

þ ðlM þmLÞ1

2log

R� z

Rþ z

� �

þ lL tanyz

xR

� �þmM tan

xz

yR

� �

þ nN tanxy

zR

� ��, ð7Þ

in which ðL;M ;NÞ and ðl;m; nÞ are cosine directors,respectively, for unit vectors t (geomagnetic fielddirection) and m (magnetization direction) andR ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2 þ y2 þ z2Þ

p. For a local field with inclina-

tion I and declination D, L ¼ cosðIÞ cosðDÞ,M ¼ cosðIÞ sinðDÞ, and N ¼ sinðIÞ. For a magneti-zation direction with inclination i and declination d,l ¼ cosðiÞ cosðdÞ, m ¼ cosðiÞ sinðdÞ, and n ¼ sinðiÞ.Using Eq. (7), the field components Tx

m, Tym and Tz

m

can be evaluated by assigning cosine directionsðL;M ;NÞ, respectively, equal to ð1; 0; 0Þ, ð0; 1; 0Þand ð0; 0; 1Þ.

For gravity field evaluation, we start from Banerjeeand Das Gupta (1977) formula,

gz ¼1

2GDr x log

R� y

Rþ y

� �þ y log

R� x

Rþ x

� ��

þ 2z tanxy

zR

� ��ð8Þ

obtaining gravity derivatives as

qgz

qx¼

1

2GDr log

R� y

Rþ y

� �,

qgz

qy¼

1

2GDr log

R� x

Rþ x

� �,

qgz

qz¼ GDr tan

xy

zR

� �. (9)

To our knowledge, the expressions in (9) were notpublished previously and they were consequentlytested by comparing their results with numericallyevaluated derivatives that were derived through afinite difference scheme with Eq. (8).

As shown in Eqs. (7)–(9), field components andderivatives are evaluated at the origin of thecoordinate system. To be evaluated at a genericposition ðx0; y0; z0Þ, terms ðx; y; zÞ are substitutedby ðx0 � xi; y0 � zi; z0 � ziÞ, i ¼ 1; 2, in whichterms ðxi; yi; ziÞ denote the coordinates for a prismvertices.

4. MDR–MI from data processing

Current potential field data provide only a singlecomponent for underlying vector anomalousfields. In magnetic surveys, the measured quantitycommonly is regarded as being the total fieldanomaly, Tt

m, which means the field componentalong the direction t, of the local geomagneticfield (Blakely, 1995). Gravity surveys carried outwith common gravimeters provide the verticalcomponent, gz, of the anomalous gravity field.Therefore, a processing scheme is required toevaluate magnetic field components and gravityderivatives in Eqs. (5) and (6), and by extension inEqs. (3) and (4). It can be done by exploiting well-known properties of potential fields, which allow tocompute field components and derivatives byapplying a suitable set of linear transforms (Gunn,1975; Blakely, 1995).

In the wavenumber domain, a potential fieldCðx; yÞ measured at a constant height can be

ARTICLE IN PRESSC.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603–610 605


transformed into a field Dðx; yÞ by making

FfDðx; yÞg ¼ F ðkx; kyÞFfCðx; yÞg, (10)

in which F denotes the Fourier transform, F ðkx; kyÞ

is a mathematical (or filter) expression for thedesired transformation, written in terms of wave-numbers kx and ky corresponding to directions x

and y of a Cartesian system. For operationsinvolving a component change from a measuredtotal field anomaly, filter F ðkx; kyÞ assumes ageneral form of

F ðkx; kyÞ ¼Aðkx; kyÞ

iLkx þ iMky þNjkj, (11)

with Aðkx; kyÞ being equal to ikx, iky, and jkj �ðk2

x þ k2yÞ

1=2 to, respectively, compute x, y, and z

components of Tm. To compute derivatives ofgravity anomaly, term F ðkx; kyÞ is made equal toikx, iky, and jkj, respectively, to provide derivativesalong x, y, and z directions. A flow chart tocompute MDR and MI is presented in Fig. 1.

5. Program description

We describe here two main programs developedunder Fortran 90 environment. The first one,PRISM3D, implements Eqs. (7)–(9) to calculatemagnetic field components, gravity derivatives,and MDR–MI parameters from a set of juxtaposedprisms. It works as a forward model for MDRand MI quantities in the sense that from a knownprism model, one can evaluate MDR–MI values

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Fig. 1. Flow diagram for MDR–MI processing of gravity anomaly gz and total field magnetic anomaly Ttm. T t

m, for t ¼ x; y; z; t, is tcomponent of the anomalous vector field Tm; FfCg denotes the Fourier transform of C, h is the continuation height (positive downward)

for gravity and magnetic fields, kx and ky are wavenumber for the x and y directions, i ¼ffiffiffiffiffiffiffi�1p

, and jkj ¼ ðk2x þ k2

yÞ1=2. Geomagnetic field

direction, t, is defined by director cosines ðL;M;NÞ, and magnetization direction, m, by director cosines ðl;m; nÞ. Unit vectors et, for

t ¼ x; y; z, define the axis of the coordinate system.

C.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603–610606


expected from fields observable at the groundsurface. The input of a particular geophysical modelis done by file modpar.txt, by entering basicinformation on grid size, number of reading points,observation height, local geomagnetic field, andnumber and size of prisms. For each prism, 11parameters are required; two to define prismposition, six to define size and depth, four toassign physical properties and one to establish the

rotation angle with respect to northward pointingx-axis. PRISM3D was validated in comparisonwith the gravity and magnetic anomalies evaluatedwith programs gbox and mbox developed byBlakely (1995, p. 377 and 379). Besides a jointevaluation of gravity and magnetic fields,PRISM3D evaluates fields from rotated prisms,which is not implemented in published versions ofgbox and mbox.

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Fig. 2. Magnetic anomaly, MDR and MI evaluated from PRISM3D program for models with two adjacent prisms, 2.5 km tick and depth

to the top at 1 km. In all cases, the magnetization of the southern prism is 0:25A=m, with inclination of 40� and declination of 10�,

coincident with the local geomagnetic field. The northern prism, otherwise, assumes variable magnetization intensity of 0.25, 0.50, and

0:50A=m, respectively, in Cases (a), (b), and (c), as well as variable direction (reversed, normal, reversed) in Cases (a), (b), and (c). Both

prisms have a density contrast of 0:1 g=cm3. MDR values for Cases (a), (b), and (c) are then 2.5, 5.0, and 5:0mA:m2=kg3.

C.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603–610 607


Our second program, MDRMI, implements thealgorithm presented in Fig. 1, by calling a set ofsubroutines. Subroutine tmag evaluates the mag-netic fields components Tx

m, Tym and Tz

m fromprocessing the total field anomaly Tt

m. It has asame structure of program signal, which evaluatesx, y, and z derivatives for gravity anomaly gz.Auxiliary programs fmdr and finc compute Eqs. (3)and (4) to estimate MDR and MI values. Sub-routine fourn from Press et al. (1990) was used tocompute the discrete Fourier transform for a griddata set in both tmag and signal routines. Sub-routine signal to compute jrgzj was kindly providedby Richard Blakely (personal communication).Functions dircos (to compute direction cosines frominclination and declination of an unitary vector) andkvalue (to evaluate wavenumber coordinates forsubroutine fourn) are found in Blakely (1995).

Grid gravity and magnetic data entering inMDRMI is read by program read_matlab, in aformat able to be plotted by using load and pcolorfunctions from MATLAB. Output from programsPRISM3D and MDRMI can be displayed byMATLAB script files prism3d.m and mdrmi.m.

6. Numerical experiments

We apply program PRISM3D to obtain MDRand MI estimates over a model composed by twojuxtaposed prismatic bodies. The southern body ofthe model has induced magnetization only whereasthe northern one has a variable magnetization asillustrated by Cases (a), (b), and (c) in Fig. 2. InCase (a), magnetization intensity is constant in allbodies but the northern one is reversely magnetized.In Case (b), magnetization direction is the same for

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Fig. 3. MDR and MI evaluated from the model in Case (c) of Fig. 2 (left column) and obtained by applying program MDRMI on

corrupted noise data for both gravity and magnetic anomalies. Additive uniformly distributed noise with amplitude equal to 1% of the

peak-to-peak anomalies was applied.

C.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603–610608


the two bodies but magnetization intensity is twiceas much in the northern body. Finally, in Case (c),the northern body has a doubled magnetizationintensity and a reversed magnetization directionrelative to the southern body. MDR for the south-ern body is 2:5mAm2=kg3, whereas for the north-ern body it is equal to 2.5, 5.0, and 2:5mAm2=kg3,respectively. Magnetization polarity for northernbody is reversed, normal, and reversed, respectively.An induced magnetization with inclination of 40�

was assumed, normal (positive) polarity and re-versed (negative) polarity are relative to thisdirection.

Fig. 2 demonstrates that even when the twosources cannot be clearly discriminated from themagnetic anomaly data, they are readily discernablein the MDR and MI data. In all cases magnetizationpolarity is well recovered from MI mapping andgood MDR estimates are obtained over the centralportions of the prisms. These results suggests apotential application of MDR–MI mapping incrustal studies, especially with regard to mappingthe magnetization polarity of the ocean floor inmarine geophysics.

Fig. 3 illustrates the capacity of the method inrecovering suitable MDR and MI values fromgravity and magnetic anomalies in the presence ofnoise. The left column of the figure shows MDRand MI values evaluated from the model by usingforward program PRISM3D program; its rightcolumn shows corresponding values obtained withprocessing program MDRMI applied to noise-corrupted data. To simulate noise in data, gravityand magnetic anomalies were disturbed with ad-ditive, uniformly distributed noise, with amplitudeequal to 1% of the peak-to-peak anomalies.

7. Concluding remarks

Results from MDR–MI analysis can be degradedin two major ways: (1) the computation of gravityanomaly gradient involves the application of a noiseenhancing derivative filter, and (2) the evaluation ofmagnetic z component for fields measured at lowmagnetic latitudes. The latter operation exhibits thesame sort of instability previously recognized inreducing to the pole, a total field magnetic anomalymeasured close to the magnetic equator. In thistransformation, the denominator of filter expressionin Eq. (11) tends to zero at a particular direction inthe kx–ky plane, thus unboundedly enhancing dataspectral content along this direction. For field in low

magnetic latitudes, specialized procedures (Silva,1986; Mendonc-a and Silva, 1993) to stabilize theoperation of component change must be applied.

Noisy data should be low-pass filtered prior to theapplication of MDRMI or upward continued to asuitable height. Upward continuation attenuates thedata’s short wavelength content, thus indirectlydiminishing noise effects. Having managed noiseamplification in gravity and magnetic data proces-sing, we believe programs presented here can beused to process real data sets in continental andmarine studies.

Acknowledgements

Thanks are given to Val Chandler for significantimprovement in text clarity and Richard Blakely forproviding subroutine signal.for.

Appendix A. Supplementary data

Supplementary data associated with this articlecan be found in the online version, at doi:10.1016/j.cageo.2007.09.013

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