Case History

Constrained 3D inversion of magnetic data with structural orientation andborehole lithology: A case study in the Macheng iron deposit, Hebei, China

Shida Sun1, Chao Chen2, and Yiming Liu2

ABSTRACT

We have developed a case study on the use of constrainedinversion of magnetic data for recovering ore bodies quantita-tively in the Macheng iron deposit, China. The inversion is con-strained by the structural orientation and the borehole lithologyin the presence of high magnetic susceptibility and strong rema-nent magnetization. Either the self-demagnetization effectcaused by high susceptibility or strong remanent magnetizationwould lead to an unknown total magnetization direction. Here,we chose inversion of amplitude data that indicate low sensitiv-ity to the direction of magnetization of the sources when con-structing the underground model of effective susceptibility. Toreduce the errors that arise when treating the total-field anomaly

as the projection of an anomalous field vector in the direction ofthe geomagnetic reference field, we develop an equivalentsource technique to calculate the amplitude data from the to-tal-field anomaly. This equivalent source technique is basedon the acquisition of the total-field anomaly, which uses the to-tal-field intensity minus the magnitude of the reference field. Wefirst design a synthetic model from a simplified real case to testthe new approach, involving the amplitude data calculation andthe constrained amplitude inversion. Then, we apply this ap-proach to the real data. The results indicate that the structuralorientation and borehole susceptibility bounds are compatiblewith each other and are able to improve the quality of the re-covered model to obtain the distribution of ore bodies quanti-tatively and effectively.

INTRODUCTION

The Macheng iron deposit is located in the eastern HebeiProvince of the North China Craton (NCC) (Figure 1a), and it be-longs to the eastern Hebei iron metallogenic belt. It is a large, con-cealed, banded iron formation (BIF) deposit and it has more than onebillion tons of iron ore (Wu et al., 2015). Based on the geologic, min-eralogical, and geochemical studies of the Macheng deposit, the Ma-cheng BIF is an Algoma-type iron exhalative formation that formedin an arc-related basin in the Neoarchean (Cui et al., 2014; Wu et al.,2015). The BIF ores, consisting of hematite and magnetite, are hostedin Neoarchean metamorphic rocks that have a significantly lowermagnetic susceptibility compared with the ore body (Wu et al., 2015).

This study focuses on the 3D distribution, especially in the depthdirection, of the Macheng ore bodies. A significant amount ofinformation has been collected to help image the distribution in thisregion; for example, many authors introduced research worksregarding constraints on the metallogenic age and geodynamicevolutionary trend, as well as understandings of the ore-formingmechanism and the genesis of the ores. These aforemetioned studiesmainly focused on the deposit geology, geochronology, and geo-chemistry of high-grade ores and host rocks in the eastern Hebeiiron metallogenic belt (e.g., Nutman et al., 2011; Zhang et al.,2011, 2012b; Cui et al., 2014; Li et al., 2015; Wu et al., 2015;Bai et al., 2016). Meanwhile, geophysical surveys, includingground gravity and magnetic measurements, were conducted over

Manuscript received by the Editor 4 April 2018; revised manuscript received 22 October 2018; published ahead of production 21 November 2018; publishedonline 05 February 2019.

1Huazhong University of Science and Technology, MOE Key Laboratory of Fundamental Physical Quantities Measurement, Hubei Key Laboratory of Gravi-tation and Quantum Physics, Institute of Geophysics, School of Physics, Wuhan, China. E-mail: sdsun@hust.edu.cn.

2China University of Geosciences, Hubei Subsurface Multi-Scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, Wuhan, China. E-mail:chenchao@cug.edu.cn; liuyiming@cug.edu.cn.

© 2019 Society of Exploration Geophysicists. All rights reserved.

B121

GEOPHYSICS, VOL. 84, NO. 2 (MARCH-APRIL 2019); P. B121–B133, 12 FIGS., 1 TABLE.10.1190/GEO2018-0257.1

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this region to determine and delineate the horizontal border of theprospecting targets. Furthermore, there are several boreholesarranged along the key profiles to explore the iron ore and gatherits depth and thickness.The geometry of the ore body and the physical property varia-

tions between the ores and host rocks were analyzed using threedata sets, the first being ground magnetic data. Preprocessed groundmagnetic data provide information about the geometry and mag-netic susceptibility of ore bodies. Three-dimensional inversion, aquantitative interpretation method, can recover the distribution ofthe magnetic susceptibility as a function of the 3D position in thesubsurface (Li et al., 2010), and the corresponding result can de-scribe the formation of source bodies by visualizing the susceptibil-ity anomalies. However, the measured magnetic data containunavoidable noise, and the inversion problem is typically ill posed,both of which mean that inversion of these data would lead to non-unique and unstable solutions. A priori information is introduced tostabilize the solution and constrain the structure of the susceptibilitydistribution. For example, surface geologic mapping demonstratesthat the deposit is located in the lower limb of an overturned syn-clinorium, with a known strike direction and dip angle to the Fe orebodies. Moreover, the BIF iron ores have a gneissic, striated, andbanded structure, with some characterized by a massive structure,based on the photomicrographs of iron ores (Wu et al., 2015). Inaddition, borehole lithology logs, including magnetic susceptibilitymeasurements of rock samples, provide the lithology and corre-sponding magnetic susceptibility bounds as a function of depth.Therefore, the purpose of this study is to incorporate the struc-

tural orientation (mainly the dip angle and strike direction) andborehole susceptibility bounds into a 3D inversion of magnetic datato guide the inversion toward more geologically reasonable results.Two obstacles should be overcome to achieve this. The first is thecompatible incorporation of structural orientation and susceptibilityconstraints, and the other is the inversion of magnetic data in thepresence of high susceptibility and strong remanent magnetization.

The combination of structural orientation and susceptibility con-straints in a constrained inversion of potential field data has beenwidely researched. Li and Oldenburg (2000) develop a model ob-jective function based on their inversion framework presented by Liand Oldenburg (1996), which allows the strike direction and dipangle to be incorporated into geophysical inversions on a regionalbasis to recover linear structures. Lelièvre and Oldenburg (2009a)use the structural orientation to summarize available geologic infor-mation, such as the strike and dip of a geologic body, and they ex-pand the function so that different orientations can be specified ateach mesh cell center on a local scale. Sun and Li (2010) introduce ajoint inversion approach that constrained the inversion using dipinformation and density bound constraints, in which the dip infor-mation was estimated from the joint inversion of surface and bore-hole gravity measurements with density bound constraints. Wu(2017) uses the estimated orientations and fault slips, extracted froma migrated seismic image, to construct convenient structure, stratig-raphy, and fault-guided regularizations for geophysical inversion torecover models that conform to stratigraphic features. Moreover, theregularization proposed by Wu (2017) can integrate well-log prop-erties into the inversion.For the second obstacle, the high susceptibility (≥ 0.1SI) may

cause strong self-demagnetization effect (e.g., Guo et al., 2001;Krahenbuhl and Li, 2017) and large magnetic anomalies. The ob-vious self-demagnetization effect and strong remanent magnetiza-tion obscure the direction of total magnetization of the source, butthis direction is crucial for inversion of the total-field anomaly. Liet al. (2010) develop a comprehensive set of methods to tackle thisproblem. If a single magnetization direction is a valid assumptionfor the whole study region, the direction can be estimated by thecrosscorrelation method and can be used in inversion. The cross-correlation method obtains the particular magnetization directionby calculating the maximum crosscorrelation coefficient betweentwo types of magnetic total-field transforms. One of these transfor-mations is related to the reduced-to-pole (RTP) anomaly, which is

sensitive to the assumed magnetization direc-tions; the other transform is related to some mag-netic quantities that are weakly sensitive to thedirections of the source magnetization, such astotal gradient (Nabighian, 1972; Roest et al.,1992), amplitude data Ta (Stavrev and Gerovska,2000), and normalized source strength (NSS)(Wilson, 1985). Several combinations of thesetwo types of anomalies have been published,e.g., the vertical derivative of the RTP anomaly(RTPz) and the total gradient of the RTP anomaly(RTPxyz) (Dannemiller and Li, 2006), the RTPanomaly and Ta (Gerovska et al., 2009),and the RTP anomaly and NSS (Rao et al.,2016; Li et al., 2017). Considering that thetwo types of anomalies should have the same de-cay rates with the distance to the source (Dan-nemiller and Li, 2006), RTPz may be a betterchoice than RTP when combined with NSS. Ifthe foregoing assumption is not valid, then itis better to directly invert the above, direction-in-sensitive magnetic quantities. Many of these datahave been used to recover an effective suscep-tibility model, such as the total gradient (Shearer,

Figure 1. (a) Tectonic subdivisions of the NCC and (b) simplified geologic map of theMacheng BIF iron deposit (modified from Cui et al., 2014). The survey area is outlinedin red.

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2005), amplitude data (Shearer, 2005; Li et al., 2010; Liu et al.,2013, 2015; Li and Li, 2014; Krahenbuhl and Li, 2017), andNSS (Beiki et al., 2012; Clark, 2012; Pilkington and Beiki,2013). Meanwhile, the magnetization vector inversion is anothereffective approach for solving the problems caused by the unknowndirection of total magnetization (e.g., Lelièvre and Oldenburg,2009b; Li and Sun, 2016; Liu et al., 2018; Sun and Li, 2018).Moreover, the measured total-field anomaly is generally consid-

ered to be a good approximation of a component of the anomalousfield vector in the direction of the reference field, under the condi-tions that the magnitude of the reference field is much larger thanthe anomalous field and that the direction of the reference field overthe study region is approximately constant (Blakely, 1996). How-ever, the maximum value of the measured total-field magneticanomaly (Figure 2) over the study region exceeds 104 nT, invali-dating the first condition. Yuan et al. (2015) analyze the differencebetween the measured total-field anomaly and the projection of theanomalous field vector in detail. However, the errors between thereal amplitude data and the data transformed from the measuredtotal-field anomaly were not included.In this study, first, the geologic setting of the study area is outlined

and the ground magnetic data and borehole magnetic susceptibilitybounds are introduced. Then, we introduce an equivalent source tech-nique to convert the total-field anomaly to amplitude data. Moreover,we analyze the errors of amplitude calculation and test the efficacy ofthe constrained amplitude inversion approach through a synthetic ex-ample in the presence of high susceptibility and strong remanentmagnetization. Following this, we use the same approach for the realcase, and we compare the recovered models, produced using generaland constrained inversions, respectively. The last part contains a sum-mary of this study and our conclusions.

GEOLOGIC SETTING, MAGNETIC DATA, ANDBOREHOLE INFORMATION

Geologic setting

The NCC (Figure 1a) is the largest and oldest Precambrian cratonin China, which is bounded by the Central Asian orogenic belt tothe north and the Paleozoic to Triassic Qinling-Dabie-Sulu orogenicbelt to the south (e.g., Li et al., 1993, 2007; Sengör et al., 1993;Meng and Zhang, 2000; Xiao et al., 2003). The craton is composedof eastern and western blocks, separated by a central (trans-north)orogen with an Archean to Paleoproterozoic metamorphosed base-ment (e.g., Zhao et al., 2001; Wilde et al., 2002; Santosh et al.,2006, 2007; Santosh, 2010; Kusky, 2011).The eastern Hebei region, located on the eastern block of the

NCC, contains Archean metamorphic rocks and granitic gneisses,covered by Proterozoic to Cenozoic platform rocks and intruded byMesozoic granites (e.g., Zhai and Liu, 2003; Wilde et al., 2008;Wan et al., 2011; Zhai and Santosh, 2011). These Archean rocksexhibit low- to high-grade regional metamorphism, which alsoaffected the BIF deposits, e.g., the Shirengou, Shuichang, Mengjia-gou, Sijiaying, and Macheng BIF-related iron deposits. Thesedeposits constitute the eastern Hebei iron metallogenic belt.The Macheng deposit is located on the lower limb of the

Macheng compound synclinorium (Figure 1b). The Macheng BIFis mainly hosted in the Neoarchean metamorphic rocks of theBaimiaozi Formation, Luanxian Group. The Baimiaozi Formationis composed of plagioclase gneiss, biotite leptynite, plagioclase

amphibolite, migmatite, and migmatitic granite (Wu et al., 2015).Based on the necessary conditions for the deposition of BIFs (Tren-dall, 2002), as well as the deposit geology and geochemical results,Wu et al. (2015) suggest that the Macheng BIF is an Algoma-typeformation, likely formed in a back-arc basin setting, which is con-sistent with previous studies of other BIFs in the eastern Hebei ironmetallogenic belt (e.g., Zhang et al., 2011, 2012a, 2012b; Xianget al., 2012; Cui et al., 2014). During the formation of this deposit,the high-temperature hydrothermal fluids in the back-arc basin set-ting were discharged on the seafloor and interacted with seawater.The activities of these fluids controlled the formation of theMacheng BIF, producing gneissic, striated, and banded features.The Fe orebodies are generally north–northwest-striking and dip

to the southwest or northwest, with angles ranging from 20° to 56°,which are mainly controlled by the strike direction of the Machengcompound syncline (Figure 1b). The Macheng Fe ores extend ap-proximately 6 km along-strike and occur between 46 and 1404 mdepth, forming a monoclinal structure consisting of magnetitequartzite and tremolite magnetite quartzite with minor martitequartzite. Compared with the average susceptibility and remanentmagnetization of the main host rocks listed in Table 1, i.e., plagio-clase gneiss, 0.009 SI and 0.764 A∕m, respectively, the parametersof the magnetite quartzite are 1.105 SI and 156.372 A∕m, respec-tively. The strong contrast between the host rocks and ore bodiesprovides a basis for the use of magnetic data to recover the distri-bution of ore bodies with relatively higher susceptibility.

Magnetic data

Ground magnetic data were collected over the deposit with 100 mline spacing and 20 m point spacing. Diurnal variations and thelocal inducing field, with a strength of 54,000 nT, inclination of

Figure 2. The observed ground magnetic total-field anomaly overthe Macheng iron deposit in Hebei, China. The inducing field is inthe direction of I ¼ 57.9° and D ¼ −7.1°. The gray dots indicatethe locations of boreholes.

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57.9°, and declination of −7.1°, were removed. Using a minimum-curvature algorithm with 50 m gridding intervals, the gridded total-field anomaly was produced (Figure 2). The range of the anomalyvalue is between −2795 and 11,401 nT. The elevation within thearea ranges from 3 to 42 m, with an average value of 15 m. Weignore the effect of the topography by considering the local topog-raphy as flat.

Borehole information

More than 90 boreholes (the gray dots in Figure 2) are located in thestudy area. The depths of these boreholes range from 298 to 1883 m.The core sample characteristics, especially of the high-grade ores, notonly help to better understand the ore-forming mechanism of the de-posit and genesis of the ores (Wu et al., 2015), but they also providemeaningful physical property bound constraints for 3D inversion.Based on the drill core lithology logs and magnetic parameter

measurements, we can extract the relations between the depthand susceptibility bounds at the locations of individual boreholes.The statistical results of the magnetic parameters of the ore and rocksamples are listed in Table 1. The average susceptibility and rema-nent magnetization of ore bodies such as magnetite quartzite aremuch larger than the host rocks, such as gneiss and migmatite.Therefore, we divide the drill core rocks into two clusters basedon the large susceptibility differences, i.e., the ore body, with high

susceptibility, and the host rock, with low susceptibility. The dis-tribution of the two clusters is shown in Figure 3, and the boundsof susceptibility of the ore body and host rocks are set as (0.2 SI,1.5 SI) and (0 SI, 0.09 SI), respectively, based on the parameterslisted in Table 1.Considering the self-demagnetization effect caused by the high

susceptibility and strong remanent magnetization, amplitude inver-sion is a better choice than inversion of the total-field anomaly(Krahenbuhl and Li, 2017). Existing methods for amplitude dataconversion from the total-field anomaly, such as the FFT-basedand equivalent-source methods, assume that the total-field anomalyapproximates a component of the anomalous vector. However, thisassumption is invalid when the value of the total-field anomaly is aslarge as the maximum of the data shown in Figure 2, with amagnitude of 13,000 nT. In this situation, these existing methodsmay corrupt the results. Therefore, without any assumptions, anequivalent source technique that constructs the equivalent sourcesby fitting the total-field anomaly is introduced in the next section.

EQUIVALENT SOURCE TECHNIQUE FORAMPLITUDE CALCULATION

Total-field magnetic anomaly and the amplitude data

The total-field magnetic anomaly is actually obtained by

ΔT ¼ jTj − jT0j; (1)

where ΔT is the total-field anomaly and jTj and jT0j are the mag-nitudes of the total field T and regional field T0 at the same point,respectively. The total field T is the vector sum of the regional fieldT0 and the anomalous field vector Ta. The three components of Ta

are Hax, Hay, and Za in the x-, y-, and z-directions, respectively.The amplitude of the Ta is defined as

Ta ¼ jTaj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH2

ax þH2ay þ Z2

a

q: (2)

In general, the total-field anomaly is approximately equal to thecomponent of Ta in the direction of the reference field if jT0j ≫ Ta;it is a potential and satisfies Laplace’s equation if the direction ofthe reference field is approximately constant (Blakely, 1996). Underthese two conditions, the three components of Ta can be derivedfrom the total-field anomaly through FFT-based processing (Peder-sen, 1978) or the equivalent-source technique (Dampney, 1969).However, these existing techniques are no longer applicable ifthe above conditions are invalid. Therefore, a new method is neededto calculate the amplitude data from the total-field anomaly.

Figure 3. The distribution of ore bodies and host rocks in thelocation of boreholes at depth as a volume-rendered image.

Table 1. Statistics of magnetic property measurement on ore and rock samples of the Macheng iron-ore deposit, Hebei, China.

Ores and rocks Sample numberSusceptibility

κ (4π × 10−6 SI)Induced magnetization

Mi ¼ κT0∕μ0 (10−3 A∕m)Remanent magnetization

Mr (10−3 A∕m)

Magnetite quartzite 24 87,920 47,477 156,372

Plagioclase gneiss 3 686 370 764

Migmatite 19 122 65 161

Migmatitic granite 12 344 185 493

Diabase 12 2078 1122 1294

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Method

Let H0x, H0y, and Z0 represent the components of T0 in thex-, y-, and z-directions, respectively. Then, the three componentsof T in the same three directions can be represented asðH0x þHaxÞ; ðH0y þHayÞ, and (Z0 þ Za). Therefore, equation 1is updated to

ΔT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH0x þHaxÞ2 þ ðH0y þHayÞ2 þ ðZ0 þ ZaÞ2

q

−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH2

0x þH20y þ Z2

0

q; (3)

where the three components of T0 can be calculated from theInternational Geomagnetic Reference Field model, whereas thethree components of Ta are usually unknown. The formulationof equation 3 is similar to equation 2 with the same unknown quan-tities and square root calculation. Furthermore, the process of con-structing the equivalent source by inverting the total-field anomalythrough equation 3 is similar to the inversion of amplitude data.In this study, a 3D equivalent source was used, similar to 3D

inversion. The advantages of a 3D equivalent source have been dis-cussed by Li and Li (2014). The model region is divided into rec-tangular cells within constant and isotropic susceptibility. A modelobjective function is necessary beside the total-field anomaly misfitfunction for stabilizing the solution. The sum objective function ap-plies the depth-weighting function, the form of which is given in Liand Oldenburg (1996), and the logarithmic barrier method (Li andOldenburg, 2003) for the susceptibility positive constraint. Due tothe nonlinear relationship between the total-field anomaly and thesusceptibility model (see equation 3), we use the Gauss-Newtonmethod to solve this nonlinear optimization problem. The wholeprocess is the same as the inversion of amplitude data except forthe Jacobian matrix of sensitivities in each iteration. Here, this ma-trix is introduced, and more details can be found in Shearer (2005)and Li et al. (2010).Assuming the susceptibility model ismðnÞ at the nth iteration, the

elements of the Jacobian matrix of sensitivity Jij are equal to thevalues differentiating the ith observation total-field anomaly withrespect to the susceptibility in the jth cell, which can be describedas

Jij ¼∂ΔTi

∂mj

¼ ðH0xi þHðnÞaxiÞGxij þ ðH0yi þHðnÞ

ayiÞGyij þ ðZ0i þ ZðnÞai ÞGzijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðH0xi þHðnÞaxiÞ2 þ ðH0yi þHðnÞ

ayiÞ2 þ ðZ0i þ ZðnÞai Þ2

q ; (4)

where ΔTi is the ith observation total-field anomaly; mj is the sus-ceptibility in the jth cell;HðnÞ

axi,HðnÞayi, and Z

ðnÞai are the three predicted

components of Ta at the ith observation location by forward calcu-lation of mðnÞ at the nth iteration; and Gmijðm ¼ x; y; zÞ is theelement of the sensitivity matrix quantifying the correspondingmagnetic vector component produced at the ith observation locationby a unit susceptibility in the jth cell.The method we used for determining the optimal regularization

parameter is generalized cross-validation (GCV) (e.g., Li andOldenburg, 2003; Farquharson and Oldenburg, 2004). The conver-gence conditions of the whole process are set where the rate ofchange of the sum objective function is smaller than 0.5% and

the maximum of the model perturbation is smaller than0.001 SI. If these conditions are both satisfied, the process is termi-nated and the latest recovered susceptibility model is output as thefinal equivalent source. This equivalent source can be used to re-produce the total-field anomaly and calculate the three componentsof the anomalous vector and then the amplitude data in equation 2.

SYNTHETIC EXAMPLE

We set the synthetic model by simplifying the real case, and di-vided the subsurface into 31 × 31 × 15 cubes with each cube being50 m along each edge. The synthetic model consisted of two paralleldipping causative bodies with a total magnetization directionðIm;DmÞ ¼ ð45°; 75°Þ and effective susceptibility of 1.5 SI(Figure 4a). The inducing field had a strength of 57,000 nT, an in-clination of 65°, and a declination of −25°. Figure 4b shows thehorizontal positions and lithologic variations with depth in the nineboreholes.

Figure 4. (a) The 3D distribution of a synthetic model consisting oftwo parallel dipping causative bodies with total magnetization di-rection (Im;DmÞ ¼ ð45°; 75°) and effective susceptibility of 1.5 SI.(b) The horizontal positions and the corresponding lithologic var-iations of nine assumed boreholes in the depth direction.

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Error analysis of amplitude data calculation

Here, we set ΔT1 to represent the projection of the anomalousfield vector in the direction of the inducing field and set ΔT2 torepresent the total-field anomaly that is calculated from equation 1.Figure 5a and 5b shows ΔT1 and ΔT2 at a uniform height of 50 mwithout noise contamination, respectively. The difference betweenΔT1 and ΔT2 is shown in Figure 5c. The maximum value of thisdifference is close to 500 nT, which is too large for precise magneticmeasurements, the typical accuracy of which is 1–2 nT.The true amplitude data Ta are displayed in Figure 6a. Regarding

ΔT1 and ΔT2 as a component of Ta in the direction of the inducing

field, we converted them to their corresponding amplitude datathrough FFT-based processing. The conversion was executed byrunning a MATLAB package (MMATrans) designed by Gerovskaand Araúzo-Bravo (2006). Figure 6b and 6d displays the amplitudedata Ta1 and Ta2, which were transformed from ΔT1 (Figure 5a)and ΔT2 (Figure 5b), respectively. As expected, the difference be-tween Ta2 and Ta (Figure 6e) is larger than the difference betweenTa1 and Ta (Figure 6c). The distribution and range of values in theresult, shown in Figure 6e, are similar to the result shown in Fig-ure 5c. Therefore, in the presence of large total-field magneticanomalies, the conversion from total-field anomaly to the three

Figure 5. The difference betweenΔT1 andΔT2. (a) The termΔT1 represents the projection of the anomalous field vector in the direction of theinducing field and (b) ΔT2 represents the total-field magnetic anomaly based on its measured and preprocessed procedure. (c) The differenceusing ΔT2 minus ΔT1. The value of this difference is constant nonnegative.

Figure 6. The differences between the true amplitude data and the amplitude data that are transformed from ΔT1 and ΔT2 through FFT-basedprocessing. (a) The true amplitude data Ta that are calculated directly from the model. (b) The amplitude data (Ta1) converted from ΔT1 and(c) the corresponding difference between Ta1 and Ta. (d) The amplitude data (Ta2) converted from ΔT2 and (e) the corresponding differencebetween Ta2 and Ta.

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components of Ta in the three orthonormal directions usingFFT-based methods and the subsequent calculation of amplitudedata introduce obvious errors, almost 400 nT in this example.We then applied the equivalent source technique to obtain 3D

equivalent sources from inverting anomalies and to predict ampli-tude data through forward calculation. Here, we use the data misfitfunction ϕd to evaluate the difference between the inverted data dand the predicted data dpre through

ϕd ¼ kWðd − dpreÞk22 ¼Xni¼1

½Wiiðdi − dprei Þ�2; (5)

where n is the number of inverted data and W is a diagonal dataweighting matrix and its elements (i.e., Wii) are reciprocals of thestandard deviations of noise in the inverted data d. In all of the be-low synthetic tests, the number of the inverted data was 961 and thestandard deviations of noise were all assumed to be 1 nT. The firsttest was to invert ΔT1 to construct equivalent sources based on thelinear relationship between the data and sources. The final data mis-fit of ΔT1 is 0.502. The predicted amplitude data(Ta3) from forward modeling of the sources areshown in Figure 7a. Compared with the con-verted result using FFT-based method (Fig-ure 6c), the difference between Ta3 and Ta issmaller (see Figure 7b). The second test wasthe application of the equivalent source tech-nique introduced in this paper, by inverting ΔT2.The final data misfit of ΔT2 is 0.498. Thepredicted amplitude data (Ta4) are displayed inFigure 7c, and their difference against Ta isshown in Figure 7d. The difference shows similardistribution characteristics with the one shown inFigure 7b. The third test was to perform the proc-ess of the first test, but using ΔT2 instead; i.e.,ΔT2 is treated as ΔT1 to be inverted to constructequivalent sources. We use the same regulariza-tion parameter as the one in the above tests, andthe final data misfit is 264.108. The large differ-ence between ΔT2 and ΔT1 is shown in Fig-ure 5c. In accordance with the expectation, thedifference between this predicted amplitude data(Ta5) (Figure 7e) and Ta is at the same level withthe difference between ΔT2 and ΔT1 (seeFigure 7f).Therefore, we conclude that if the value of the

total-field anomaly is large (exceed 104 nT inthis example) there might be a large differencebetween the total-field anomaly and its generalapproximation, i.e., one component of theanomalous field vector in the direction of theinducing field. In this situation, the equivalentsource technique, introduced in this paper, out-performs the existing methods in obtaining thedistribution of sources and reproducing or pre-dicting data, such as the amplitude data.

Constrained inversion of magnetic data

We used the 3D inversion algorithm developedby Li and Oldenburg (1996) to invert ΔT1, we

applied the equivalent source technique that we introduced to invertΔT2, and we introduced the inversion approach developed byShearer (2005) and Li et al. (2010) to invert the amplitude data.The structural orientation, such as the dip angle and strike direction,was incorporated into the inversion (Li and Oldenburg, 2000). Wechose the logarithmic barrier method to solve bound constrainedinverse problems (e.g., Li and Oldenburg, 2003), and we used GCVto determine the optimal regularization parameter automatically(e.g., Li and Oldenburg, 2003; Farquharson and Oldenburg, 2004).We first inverted the noise-contaminated ΔT1 and ΔT2 displayed

in Figure 8a and 8b, respectively. We assumed that the directions ofmagnetization of the sources are the real directions. Here, we usegeneral inversion to describe the inversion without combining apriori information to distinguish it from the constrained inversion.The determined optimal regularization parameters for these twosolutions were the same. The general inversion of ΔT1 and ΔT2

with the susceptibility positivity constraint are shown in one crosssection at 750 m north and one plan section at 225 m depth. Thesusceptibility distribution shown in Figure 8c, recovered from the

Figure 7. The differences between the true amplitude data and the amplitude data thatare forward calculated from equivalent sources. (a) The amplitude data (Ta3) are pre-dicted from forward modeling of the source, which is obtained by inverting ΔT1 basedon the linear relationship between ΔT1 and the source. (c) The amplitude data (Ta4) arepredicted by inverting ΔT2 using the new equivalent source technique and subsequentforward modeling. (e) The amplitude data (Ta5) are predicted by using the same pro-cedure for ΔT1, whereas the input data are ΔT2. (b, d, and f) The correspondingdifferences between the true amplitude data and Ta3, Ta4, and Ta5, respectively.

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inversion of ΔT1, is similar to the case shown in Figure 8d, recov-ered from the inversion of ΔT2. The location, spatial extent, and dipof the anomalous bodies are both well-recovered, but the parallellayered characteristic is not reflected. This is thought to be due tothe lack of depth resolution in the magnetic field, making it difficultto identify two closely parallel bodies in the depth direction.The inversion constrained only by borehole susceptibility bounds

is not displayed here because the number of boreholes is inadequateto noticeably improve the structure of the recovered model. Theresults from inverting ΔT1 or ΔT2, only incorporating structuralorientation, are similar to the general inversion results, which arenot displayed here either. Figure 8e and 8f shows the constrained

inversions of ΔT1 and ΔT2 when integrating the structural orien-tation and the borehole susceptibility bounds, respectively. The ori-entation angles (ϕ, ξ, η) were (0°, 135°, 0°), respectively, and thelength scales (Lx, Ly, Lz) were (5000, 1, 100,000), respectively.The minimum, maximum, and original susceptibility (kmin, kmax,kori) of the host rocks and ore bodies in the borehole were (0,0.005, 0.001 SI) and (1.4, 1.51, 1.401 SI), respectively; the samesettings at positions without boreholes were (0, 10, 0.001 SI). Thesetwo constrained inversion results are able to construct multilayerparallel dipping structures and improve the spatial resolution inthe horizontal and depth directions. Therefore, the structural orien-tation and borehole susceptibility bounds are compatible with each

other and can be combined into the inversionjointly to raise the quality of the recoveredmodels.We then performed the amplitude inversion

approach with and without incorporating con-straints, respectively. Figure 9a displays the am-plitude data that have been contaminated by 5 nTGaussian noise from Figure 6a. The inversion ofTa is usually performed by assuming that themagnetization direction of the whole model is thesame as the direction of the inducing field. Theeffective susceptibility distribution, recoveredfrom general inversion of Ta, is displayed in Fig-ure 9b. The result reflects the overlying shallowersource body but can hardly obtain informationabout the deeper one. Figure 9d shows the con-strained inversion of Ta using the same con-straints and weights as introduced above. Theconstraints do have a positive effect on improv-ing the quality of the inverted image, includingthe horizontal positions and the resolution in thedepth direction. However, the effective suscep-tibility distribution of the constrained resultshows fewer smooth features compared with re-sults shown in Figure 8e or 8f, especially in thedip direction. Therefore, under the situation thatstrong remanent magnetization is present and thegeometries of sources are complex, like in thisexample, the general choice, that the magnetiza-tion direction of the source is the same as thedirection of the inducing field, might not be op-timal for recovering geologically meaningfulmodels.Based on the low sensitivity of Ta to the direc-

tion of the source magnetization and the inducingfield, we can infer that all Ta show good consis-tency with the one produced by the same source,in which the inclination and declination of thesource magnetization and inducing field are(90°, 0°). We compared Ta, which is produced bythe susceptibility model (Figure 4a), in which thesource magnetization direction and inducingfield direction are (90°, 0°), with Ta shown inFigure 9a. These two values are close to eachother and show great consistency on the spatialdistribution features. Finally, we inverted Ta

displayed in Figure 9a again using the source

Figure 8. Magnetic anomalies and their corresponding inversion results. (a and b) TheΔT1 and ΔT2. Both of them have been contaminated by 5 nT Gaussian noise. (c and d)Susceptibility models recovered from general inversion ofΔT1 andΔT2, respectively. (eand f) Susceptibility models recovered from constrained inversion of ΔT1 and ΔT2,respectively.

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magnetization direction (90°, 0°) and inducing field direction (90°,0°). Using the same regularization parameters and weights as intro-duced above, the corresponding general and constrained inversionswere performed. The recovered effective susceptibility anomalousbodies of the general inversion (Figure 9c) and constrained inver-sion (Figure 9e) have more accurate horizontal positions than theresults shown in Figure 9b and 9d. Moreover, the latter one shownin Figure 9e delineates a clearer parallel dipping characteristic thanthe result shown in Figure 9d, especially for the deeper anomalousbody. Hence, we suggest that the inversion of Ta, by assuming thesource magnetization direction and inducingfield direction to be (90°, 0°), is a better choicein this example. In addition, because the ampli-tude data are insensitive to, but not totally inde-pendent on, the magnetization direction, theremight be an unacceptably large difference be-tween the true amplitude data and the amplitudedata calculated by assuming that the source is lo-cated at the pole. Therefore, it should be notedthat the above suggestion can be used in situa-tions similar to the synthetic example, but notall situations.

INVERSION RESULTS ANDDISCUSSIONS

As previously mentioned, we prefer to use aconstrained inversion approach that combinesground magnetic data and a priori information,to recover the effective susceptibility distributionfor providing the formation of ore bodies quan-titatively. The chosen a priori information is thestructure orientations and borehole informationin the study region. We choose the amplitudedata to be inverted in the presence of strong re-manent magnetization and probable self-demag-netization effect. We use the equivalent sourcetechnique, introduced in this paper, to calculatethe amplitude data from the measured total-fieldanomaly with the maximum value exceeding104 nT. Moreover, through the synthetic exam-ple, we not only obtain reference values ofweight functions, such as length scales, but wealso present recommendations regarding the op-timal directions of the source magnetization andinducing field for inversion of amplitude data.

Preparation for amplitude data and apriori information

We first performed the conversion from themeasured total-field anomaly (Figure 2) to mag-nitude data using the proposed equivalent sourcetechnique. The numbers of the total-fieldanomaly and the predicted magnitude data are65 × 91 ¼ 5915, and the standard deviations ofthe noise in the total-field anomaly are all 1 nT.The 3D equivalent source is the same as themodel in the 3D inversion, which contains 65 ×91 × 34 ¼ 201;110 rectangular cells, with each

cell being 50 m along each edge. The assumed magnetization di-rection of the equivalent source, ðIm;DmÞ ¼ ð57.9°;−7.1°Þ, is thesame as the direction of the inducing field. The regularizationparameter is set to be relatively smaller than the one in general3D inversions to obtain a smaller data misfit; hence, a data misfitof 723 is achieved. The amplitude data calculated from the recov-ered equivalent source through forward calculation are displayed inFigure 10.The structural orientation of this case has been introduced in the

“Geologic setting” section. For convenience, we set the orientation

Figure 9. The amplitude data and their corresponding inversion results. (a) The ampli-tude data Ta that have been contaminated by 5 nT Gaussian noise. (b and c) Effectivesusceptibility models recovered from general inversion of Ta. (d and e) Effective sus-ceptibility models recovered from constrained inversion of Ta. The directions of sourcemagnetization in (b and d) are assumed as the direction of the inducing field. Thedirections of the source magnetization and the inducing field in (c and e) are set as(90°, 0°).

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angles (ϕ, ξ, η) to be (0°, 135°, 0°), respectively, and the corre-sponding length scales (Lx, Ly, Lz) were (500, 1, 10,000). Asshown in Figure 3, the horizontal positions of the boreholes andtheir corresponding lithology in the depth direction are known.Combined with the statistical results of the magnetic parameterslisted in Table 1, the minimum, maximum, and original effective

susceptibility (kmin, kmax, kori) of the host rocks and ore bodiesin boreholes were set to be (0, 0.09, 0.001 SI) and (0.2, 1.5,0.201 SI), respectively; the (kmin, kmax, kori) of the positions outsideof boreholes were (0, 10, 0.001 SI).Except for the regularization parameter, the above constraints and

their corresponding weights, the general and constrained inversionsused the same inversion parameters. The magnetization directionsof the sources were set at an inclination of 90° and a declination of0°; and the inducing field had an inclination of 90°, a declination of0°, and a strength of 54,000 nT. The effective susceptibility refer-ence model was set as a zero-value model. The logarithmic barriermethod was chosen to solve the bound constrained inverse problem.

General inversion

Then, we performed the general inversion of amplitude data witha positive constraint and the same length scales as one in all threedirections. The recovered effective susceptibility distribution is dis-played in Figure 11. The horizontal position of six cross sections isshown as the white lines in Figure 10. The range of ore bodies in-ferred from the borehole core lithology logs and banded character-istics of Fe ores are marked as the white line in each cross section.The general inversion result can reflect the range of horizontal po-sitions and depths of the main orebodies. However, the effectivesusceptibility distribution is hardly able to provide additional struc-tural and geometric information about the sources, such as relativelysmaller range ore bodies (Figure 11a and 11d), dip information(Figure 11b and 11c), and multilayered parallel ore bodies(Figure 11f).

Constrained inversion

Finally, we integrate the constraints into the inversion of ampli-tude data. The recovered effective susceptibility is shown in thesame cross sections as Figure 12. Compared with the result of

Figure 10. The amplitude data calculated from the total-fieldanomaly from the Macheng deposit displayed in Figure 2 usingthe proposed equivalent source technique. The gray dots indicatethe locations of boreholes. The white lines show the six east–westcross sections.

Figure 11. General inversion result. (a-f) Cross sections from left (west) to right (east) through the recovered effective susceptibility model ofwhich location have been denoted by the white lines in Figure 10. The solid white lines in each section represent the range of ore bodiesinferred from the borehole core samples and the banded characteristics of Fe ores.

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the general inversion, the constrained inversion constructs a morecompact effective susceptibility model with higher susceptibilityvalues and spatial resolution. As with the foregoing results shownin the synthetic example, the combined structural orientation andborehole susceptibility bounds can help the inversion of Ta to re-cover a model with multilayer dipping characteristics (Figure 12cand 12f). Moreover, the constrained inversion result is able to delin-eate relatively small orebodies, like Figure 12a and 12d, which areimportant in the interpretation of magnetic data to describe the ironformation quantitatively.

CONCLUSION

The combined structural orientation (dip angle and strike direc-tion) and borehole susceptibility bounds are integrated into the in-version of magnetic data to quantitatively delineate the Fe oreformation in the Macheng deposit. The ore bodies have multilay-ered dipping and banded characteristics from geologic and geo-chemical information. Meanwhile, the ore bodies are known tohave high susceptibility and strong remanent magnetization frommagnetic parameter measurements of rock samples.To reduce the effect of self-demagnetization caused by high sus-

ceptibility and strong remanent magnetization, we chose the ampli-tude inversion to recover the effective susceptibility distribution. Inthe presence of a large total-field anomaly value, we introduced anequivalent source technique to calculate the amplitude data from thetotal-field anomaly. This technique can effectively reduce the errorscaused by treating the total-field anomaly as one component of theanomalous field vector in the direction of the inducing field. Weapplied the equivalent source technique to a synthetic model, con-sisting of two parallel dipping dikes with strong remanent magneti-zation, to analyze the errors in the calculation of amplitude data.Compared with the calculated results from the FFT-based techniquein the wavenumber domain or the existing equivalent source tech-nique, the result from this introduced equivalent source technique

shows a smaller data misfit with the true amplitude data. We thenperformed the general and constrained inversions of amplitude data.The inversion results not only indicate that the structural orientationand borehole susceptibility bounds are compatible with each otherand improve the quality of the recovered distribution, but they alsoprovide a suggestion for choices about the directions of the sourcemagnetization and the inducing field in the inversion of magneticamplitude data when significant remanence is present.Using the optimal parameters and suggested directions, we car-

ried out the inversion of magnetic amplitude data in the study regionconstrained by the dip information and borehole susceptibilitybounds. The inverted effective susceptibility presented a distribu-tion of ore bodies with relatively higher magnetic and multilayereddipping characteristics. Moreover, the constrained inversion resultdelineated small ore bodies that are important in improving theunderstanding of ore body formation through quantitative interpre-tation. Therefore, the approach outlined in this study, and its results,demonstrate the efficacy of combining structural orientation andborehole susceptibility bounds into an inversion of magnetic datafor ore delineation.

ACKNOWLEDGMENTS

Wewould like to thank H. Wang and Z. Guo for assistance on thepreprocessing of magnetic data, as well as J. Du, Q. Liang, Y. Wu,X. Niu, and Q. Zhang for their helpful suggestions. We also thankthe associate editor and three anonymous reviewers for theirconstructive comments. This work was supported by the NationalProgram on Key Basic Research Project (973 Program) of China(grant no. 2012CB416800) and National Natural Science Founda-tion of China (grant nos. 41804067, 41604060, 41574070).

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Figure 12. Constrained inversion result. (a-f) Cross sections from left (west) to right (east) through the recovered effective susceptibility modelof which location have been denoted by the white lines in Figure 10. The solid white lines in each section represent the range of ore bodiesinferred from the borehole core samples and the banded characteristics of Fe ores.

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