Computers & Geosciences 34

Programs to compute magnand the magnetization incli

and magnetic a

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

r 2007 Elsevier Ltd. All rights reserved.

connecting gravity and magnetic potentials and, byextension, eld components that are commonly

ARTICLE IN PRESS

Corresponding author.E-mail address: mendonca@iag.usp.br (C.A. Mendonc-a).

1Supported by CNPq (3051005/2002-07) and (140603/2003-4).

derived from geophysical surveys (Blakely, 1995).The assumptions underlying the Poisson theorem

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

doi:10.1016/j.cageo.2007.09.013

2Assistant researcher at the Egyptian Petroleum Researches

Institute (EPRI), Ph.D. candidate at IAG-USP.PACS: PRISM3D; MDRMI

Keywords: Potential elds; 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 relationship

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

CGEditor/index.htmInstituto de Astronomia, Geofsica 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 eld formulation based on the Poisson theorem allows an automatic determination of rock physical properties

(magnetization to density ratioMDRand the magnetization inclinationMI) from combined processing of gravity

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

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

existing formulation was restricted to prole data, and assumed sufciently 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 MDRMI 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.(2008) 603610

etization to density rationation from 3-D gravitynomalies$

www.elsevier.com/locate/cageo

11 3 1 2

ARTICLE IN PRESSC.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603610604(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 apply

Poisson theorem. In the rst approach, ltering 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) lters 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 theefciency of a one-step approach and are prone toerror when Poisson conditions are not upheld.A one-step method that estimates both MDR and

MI values from prole data has been developed byusing a vector eld 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 eld. MDR is thenobtained by rationing the amplitude of these elds,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 truly

reect subsurface sources, but they nonetheless canmeability and G 6:67 10 m kg s is thegravitational constant.Gravity anomaly, gz, is obtained from gravity

potential U as gz z rU , with r denoting thegradient operator, and z a unit vector aligned to thevertical (positive downward). Total eld magneticanomaly, Ttm, is obtained from magnetic potentialVm as T

tm t rVm, where t is the unit vector

along the local magnetic eld.For 2-D sources (Mendonc-a, 2004), the MDR,

r DM=Dr, is obtained by

r 1c

jTmjjrgzj (3)

and MI, a, such that

sina Tm rgz

jT jjrgzj . (4)be used to discern geological contacts and inferspatial variations in rock physical properties (Men-donc-a, 2004). Unfortunately, little is known aboutthe signicance of apparent MDR and MI valuesthat are derived in this way from sources that areactually 3-D.This paper presents two Fortran programs that

use 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 interpretingeld-acquired data sets. Program MDRMI can beused to process eld-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 pqUqm

, (1)

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

p cDMDr

, (2)

where DM is the intensity of the magnetizationcontrast DM DMm (bold types denoting vectors),Dr is the source density contrast, c m0=4pG,where m0 4p107 H=m is vacuum magnetic per-m

evaluated derivatives that were derived through a

ARTICLE IN PRESSC.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603610 605 nN tan xyzR

, 7

in which L;M ;N and l;m; n are cosine directors,respectively, for unit vectors t (geomagnetic elddirection) and m (magnetization direction) andR

x2 y2 z2

p. For a local eld with inclina-

tion I and declination D, L cosI cosD,M cosI sinD, and N sinI. For a magneti-zation direction with inclination i and declination d,l cosi cosd, m cosi sind, and n sini.Using Eq. (7), the eld components Txm, T

ym and T

zm

can be evaluated by assigning cosine directionsL;M ;N, respectively, equal to 1; 0; 0, 0; 1; 0For 3-D sources, the vector quantities related inEqs. (3) and (4) are such that

Tm Txmex Tymey Tzmez (5)and

rgz qgz

qxex

qgz

qyey

qgz

qzez. (6)

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

3. MDRMI 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 theeld components and derivatives that reect acomplex, multi-source data set. Vector elds Tmand rgz can be evaluated simply by addingmagnetic eld components and gravity derivativesfrom adjacent prisms. Once obtained, these eldsenter Eqs. (3) and (4) to give MDR and MI.To evaluate magnetic elds from an isolated

prism, we use formula adapted from Plouff (1976),

Ttm m04p

DM mN nM 12log

R xR x

lN nL 12log

R yR y

lM mL 12log

R zR z

lL tan yzxR

mM tan xz

yR

and 0; 0; 1.nite difference scheme with Eq. (8).As shown in Eqs. (7)(9), eld components and

derivatives are evaluated at the origin of thecoordinate system. To be evaluated at a genericposition x0; y0; z0, terms x; y; z are substitutedby x0 xi; y0 zi; z0 zi, i 1; 2, in whichterms xi; yi; zi denote the coordinates for a prismvertices.

4. MDRMI from data processing

Current potential eld data provide only a singlecomponent for underlying vector anomalouselds. In magnetic surveys, the measured quantitycommonly is regarded as being the total eldanomaly, Ttm, which means the eld componentalong the direction t, of the local geomagneticeld (Blakely, 1995). Gravity surveys carried outwith common gravimeters provide the verticalcomponent, gz, of the anomalous gravity eld.Therefore, a processing scheme is required toevaluate magnetic eld 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 elds, which allow tocompute eld components and derivatives byapplying a suitable set of linear transforms (Gunn,1975; Blakely, 1995).In the wavenumber domain, a potential eldFor gravity eld evaluation, we start from Banerjeeand Das Gupta (1977) formula,

gz 12GDr x log

R yR y

y log R x

R x

2z tan xyzR

8

obtaining gravity derivatives as

qgz

qx 12GDr log

R yR y

,

qgz

qy 12GDr log

R xR x

,

qgz

qz GDr tan xy

zR

. (9)

To our knowledge, the expressions in (9) were notpublished previously and they were consequentlytested by comparing their results with numericallyCx; y measured at a constant height can be

transformed into a eld Dx; y by makingFfDx; yg F kx; kyFfCx; yg, (10)in whichF denotes the Fourier transform, F kx; kyis a mathematical (or lter) expression for thedesired transformation, written in terms of wave-numbers kx and ky corresponding to directions xand y of a Cartesian system. For operationsinvolving a component change from a measuredtotal eld anomaly, lter F kx; ky assumes ageneral form of

F kx; ky Akx; ky

iLkx iMky Njkj, (11)

with Akx; ky being equal to ikx, iky, and jkj k2x k2y1=2 to, respectively, compute x, y, and z

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

5. Program description

We describe here two main programs developedunder Fortran 90 environment. The rst one,PRISM3D, implements Eqs. (7)(9) to calculatemagnetic eld components, gravity derivatives,and MDRMI 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 MDRMI values

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aly gz

urier

x and

tizatio

C.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603610606Fig. 1. Flow diagram for MDRMI processing of gravity anom

component of the anomalous vector eld Tm;FfCg denotes the Fofor gravity and magnetic elds, kx and ky are wavenumber for the

direction, t, is dened by director cosines L;M;N, and magne

t x; y; z, dene the axis of the coordinate system.and total eld magnetic anomaly Ttm. Ttm, for t x; y; z; t, is t

transform of C, h is the continuation height (positive downward)

y directions, i 1

p, and jkj k2x k2y1=2. Geomagnetic eld

n direction, m, by director cosines l;m; n. Unit vectors et, for

expected from elds observable at the groundsurface. The input of a particular geophysical modelis done by le modpar.txt, by entering basicinformation on grid size, number of reading points,observation height, local geomagnetic eld, andnumber and size of prisms. For each prism, 11parameters are required; two to dene prismposition, six to dene 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 elds,PRISM3D evaluates elds from rotated prisms,which is not implemented in published versions ofgbox and mbox.

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D pr

rn pr

other

le dire

C.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603610 607Fig. 2. Magnetic anomaly, MDR and MI evaluated from PRISM3

to the top at 1 km. In all cases, the magnetization of the southe

coincident with the local geomagnetic eld. The northern prism,

0:50A=m, respectively, in Cases (a), (b), and (c), as well as variab3prisms have a density contrast of 0:1 g=cm . MDR values for Cases (aogram for models with two adjacent prisms, 2.5 km tick and depth

ism is 0:25A=m, with inclination of 40 and declination of 10,wise, assumes variable magnetization intensity of 0.25, 0.50, and

ction (reversed, normal, reversed) in Cases (a), (b), and (c). Both2 3), (b), and (c) are then 2.5, 5.0, and 5:0mA:m =kg .

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

ym and T

zm from

processing the total eld anomaly Ttm. 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 les 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

ARTICLE IN PRESS

Fig.

dditiv

C.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603610608Fig. 3. MDR and MI evaluated from the model in Case (c) of

corrupted noise data for both gravity and magnetic anomalies. Apeak-to-peak anomalies was applied.2 (left column) and obtained by applying program MDRMI on

e uniformly distributed noise with amplitude equal to 1% of the

j.cageo.2007.09.013

ARTICLE IN PRESSC.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603610 609the 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 two

sources 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 MDRMI mapping incrustal studies, especially with regard to mappingthe magnetization polarity of the ocean oor inmarine geophysics.Fig. 3 illustrates the capacity of the method in

recovering suitable MDR and MI values fromgravity and magnetic anomalies in the presence ofnoise. The left column of the gure 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 MDRMI analysis can be degradedin two major ways: (1) the computation of gravityanomaly gradient involves the application of a noiseenhancing derivative lter, and (2) the evaluation ofmagnetic z component for elds measured at lowmagnetic latitudes. The latter operation exhibits thesame sort of instability previously recognized inreducing to the pole, a total eld magnetic anomalymeasured close to the magnetic equator. In thistransformation, the denominator of lter expressionin Eq. (11) tends to zero at a particular direction inthe kxky plane, thus unboundedly enhancing data

spectral content along this direction. For eld in lowReferences

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application of MDRMI or upward continued to asuitable height. Upward continuation attenuates thedatas short wavelength content, thus indirectlydiminishing noise effects. Having managed noiseamplication 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 signicantimprovement in text clarity and Richard Blakely forproviding subroutine signal.for.

Appendix A. Supplementary data

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ARTICLE IN PRESSC.A. Mendonc-a, A.M.A. Meguid / Computers & Geosciences 34 (2008) 603610610

Programs to compute magnetization to density ratio and the magnetization inclination from 3-D gravity and magnetic anomaliesIntroductionPoisson relationship in vectorial formMDR-MI from 3-D modelsMDR-MI from data processingProgram descriptionNumerical experimentsConcluding remarksAcknowledgementsSupplementary dataReferences