GEOPHYSICS, VOL. 65, NO. 2 (MARCH-APRIL 2000); P. 540–552, 16 FIGS.

Joint inversion of surface and three-componentborehole magnetic data

Yaoguo Li∗ and Douglas W. Oldenburg‡

ABSTRACT

The inversion of magnetic data is inherently non-unique with respect to the distance between the sourceand observation locations. This manifests itself as an am-biguity in the source depth when surface data are in-verted and as an ambiguity in the distance between thesource and boreholes if borehole data are inverted. Jointinversion of surface and borehole data can help to re-duce this nonuniqueness. To achieve this, we developan algorithm for inverting data sets that have arbitraryobservation locations in boreholes and above the sur-face. The algorithm depends upon weighting functionsthat counteract the geometric decay of magnetic ker-nels with distance from the observer. We apply theseweighting functions to the inversion of three-componentmagnetic data collected in boreholes and then to thejoint inversion of surface and borehole data. Both syn-thetic and field data sets are used to illustrate the newinversion algorithm. When borehole data are inverteddirectly, three-component data are far more useful inconstructing good susceptibility models than are single-component data. However, either can be used effectivelyin a joint inversion with surface data to produce modelsthat are superior to those obtained by inversion of sur-face data alone.

INTRODUCTION

Magnetic data collected on, or above, the earth’s surfacecan provide a good indication of the horizontal location of thecausative body, but they cannot resolve the depth distributionof the source. Inversion of the surface data can easily recoverthe horizontal position of an anomalous body, but it only pro-duces depth information when assumptions are made eitherimplicitly or explicitly about the nature of the anomaly. Forexample, Bhattacharyya (1980) and Zeyen and Pous (1991)

Manuscript received by the Editor July 30, 1997; revised manuscript received July 27, 1999.∗Colorado School of Mines, Dept. of Geophysics, 1500 Illinois St., Golden, Colorado 80401. E-mail: ygli@mines.edu.‡University of British Columbia, Dept. of Earth and Ocean Sciences, 129-2219 Main Mall, Vancouver, British Columbia V6T 1Z4, Canada.E-mail: doug@geop.ubc.ca.c© 2000 Society of Exploration Geophysicists. All rights reserved.

invert for geometrically simple bodies assuming a prior es-timate of values of the parameters. Green (1975), Last andKubik (1983), and Guillen and Menichetti (1984) input intothe inversion the explicit assumption about the body positions.Wang and Hansen (1990) assume polyhedronal causative bod-ies and invert for their vertices. Li and Oldenburg (1996) usea depth weighting function to counteract the depth decay ofthe magnetic kernel so that the recovered susceptibility is dis-tributed with depth. Data measured in boreholes, on the otherhand, have the potential to resolve both the horizontal andvertical locations of the causative body since the data are dis-tributed in three dimensions in the ground and three compo-nents are measured. Considerable work has been done on theuse of three-component borehole data. Most work has assumedgeometrically simple bodies and has focused on determiningtheir positions using geometrical methods (e.g., Levanto, 1959;Bosum et al., 1988; Morris et al., 1995). Silva and Hohmann(1981) invert three-component data to recover a single rectan-gular block assumed to be some distance away from the bore-hole. However, because they often are available in only a fewsparsely located holes, borehole data are mostly sensitive tothe vertical location of the source. Thus, the surface data andborehole data provide complementary information, and it islogical to invert them jointly. Joint inversion of the two datasets is expected to help resolve the ambiguity associated witheither data set and greatly reduce the nonuniqueness of themagnetic inversion. Such nonuniqueness is especially severewhen a 3-D distribution of magnetic susceptibility, instead of asimple body, is sought from the inversion.

As in any geophysical inversion, the problem is highlynonunique even with the complementary information in thetwo data sets. Regularization is therefore necessary. However,straightforward regularization by seeking a flat or smooth sus-ceptibility distribution will not produce the desired result: therecovered model will only have structure concentrating nearthe surface and along the boreholes. The lack of structureaway from the observation locations is caused by the decayof the magnetic kernels, which form the basis functions for

540

Joint 3-D Borehole Magnetic Inversion 541

constructing the model. Since the kernels decay rapidly withthe distance away from the observer, it is easy for the inversionalgorithm to reproduce the data with magnetic susceptibilitythat is physically close to either the surface observation loca-tions or the borehole observation locations. Two consequencesfollow. First, the inversion is unlikely to construct a geologicallymeaningful susceptibility model. Second, the surface data andthe borehole data respond only to model regions that are ad-jacent to their respective locations and do not interact; thus,the complementary information from the two is not used. Toovercome these difficulties, we need to introduce certain priorinformation so the constructed models can resemble the truestructure and so surface and borehole data can be sensitive tothe same susceptibility distribution. For a general algorithm, wecannot assume knowledge that is too specific about the modelthat is sought. However, we can impose a general form of priorinformation. A 3-D weighting function that compensates forthe geometric decay of the kernels is one such form. Whencombined with an objective function that favors a minimumstructure model, it allows one to construct 3-D susceptibilitydistributions from arbitrarily distributed borehole data. Fur-thermore, because susceptibility is away from the borehole, ifpossible, there is a link between the surface and borehole datasets, and the joint inversion is more effective.

We first review the forward modeling for borehole mag-netic data located within the medium. Next, we present ouralgorithm for joint inversion of surface and three-componentborehole data and develop two general weighting methods tocounteract the decay of the magnetic kernels. We then applythe algorithm to data sets from a synthetic example. We also ex-amine the advantages of three-component borehole data oversingle-component data and discuss the effect of borehole ori-entation errors on the actual use of three-component data ininversions. Finally, we apply the joint inversion to a field dataset acquired above an ironstone formation and conclude witha brief discussion.

FORWARD MODELING OF BOREHOLE MAGNETIC DATA

In our inversion, we represent the 3-D magnetic suscepti-bility distribution by a set of rectangular cells and assume aconstant value of susceptibility within each cell. We also as-sume that there is no remanent magnetization and that theself-demagnetization effect is negligible. Under these assump-tions, the magnetization within each cell is uniform and givenby the product of inducing magnetic field H0 and the suscepti-bility κ . The magnetic anomaly produced by such a uniformlymagnetized cell is easily derived. The magnetic potential re-sulting from a distribution of magnetization J is given by

ψm = − 14π

∫�V

J · ∇ 1|r − r′| dv, (1)

where r and r′ are the observation and source positions, respec-tively, ∇ operates on r, and �V is the volume in the cell. Themagnetic field is then defined as Ha = −∇ψm . Since the linearoperator ∇ operates on the observation location r, it followsthat Ha is related to the magnetization by a dyadic Green’sfunction:

Ha = 14π

∫V

∇∇ 1|r − r′| · J dv. (2)

This relation is valid for observation points both inside andoutside of the cell since the singularity at r = r′ is integrable.Assuming that the magnetic permeability is µ0 within theborehole, we obtain the desired expression for the magneticanomaly Ba :

Ba = µ0

4π

∫�V

∇∇ 1|r − r′| · J dv. (3)

Since the magnetization is constant and J = κH0, equation (3)can be written in matrix form as

Ba = µ0

T11 T12 T13

T21 T22 T23

T31 T32 T33

κH0 (4)

≡ µ0κTH0,

where Ti j is given by

Ti j = 14π

∫�V

∂

∂xi

∂

∂x j

1|r − r′| dv, i = 1, 2, 3; j = 1, 2, 3;

(5)

where x1, x2, and x3 represent x, y, and z, respectively. Theexpressions for Ti j can be found in Bhattacharyya (1964) andSharma (1966). (Those expressions are not valid when the ob-servation location is on an edge of a cell. As such, special at-tention is needed to discretize the problem in practice, but thisdoes not pose a serious limitation.) Since T is symmetric and itstrace is equal to −1 and 0 when the observation is inside andoutside the cell respectively, only five independent elementsneed to be calculated. Once T is formed, the magnetic anomalyBa and its projection onto any direction of measurement areeasily obtained.

In a borehole experiment, the three components are mea-sured in the directions of local coordinate axes defined accord-ing to the borehole orientation. Assuming that the boreholedip θ is measured from horizontal surface down and azimuth ϕ

is measured from the north, a commonly used convention hasthe z�-axis pointing downward along borehole and the x�-axispointing perpendicular to the borehole in the direction of theazimuth. The y�-axis completes the right-handed coordinatesystem and is 90◦ clockwise from the azimuth and perpendic-ular to the borehole. Based upon the above definition, the ro-tation matrix that transforms three components of a vector inthe global coordinate system to the components in the localcoordinates is given by

R = cos ϕ sin θ sin ϕ sin θ −cos θ

−sin ϕ cos ϕ 0

cos ϕ cos θ sin ϕ cos θ sin θ

. (6)

If a vector is defined in local coordinates as (�1, �2, �3)T andin global coordinates as (g1, g2, g3)T , then the following tworelations hold:

(�1, �2, �3)T = R(g1, g2, g3)T ,

(7)(g1, g2, g3)T = RT (�1, �2, �3)T .

The rotation matrix R therefore allows measured componentsin local coordinates to be rotated to the global coordinate or

542 Li and Oldenburg

the components of the regional field to be rotated to the localcoordinates for use in regional removal.

Equations (4) and (7) form the basis for forward modelingthree-component magnetic data and for generating the sensi-tivity matrix in the inversion.

GENERAL METHODOLOGY OF INVERSION

First, we define the inverse problem and outline the assump-tions necessary for obtaining its solution. Given a set of Nmagnetic anomaly data, d = (d1, . . . , dN )T , we seek to con-struct a 3-D distribution of magnetic susceptibility beneath thesurface. Each data point can be located above the earth’s sur-face or in a borehole. Let the source region be divided into a setof M rectangular cells by an orthogonal 3-D mesh and assumea constant magnetic susceptibility κ within each cell. Underthe assumptions made in the preceding section, the magneticanomaly is related to the susceptibility distribution by a linearrelationship

d = Gκ, (8)

where κ = (κ1, . . . , κM)T is the susceptibility in the cells andG is an N ×M matrix sensitivity whose elements Gi j quantifythe contribution of a unit susceptibility in the jth cell to theith datum. Given a set of observed data, the objective is toconstruct a susceptibility distribution κ that reproduces thedata.

Because of the inherent ambiguity associated with the po-tential field data and because of the finite number of inaccuratedata points available, the inverse problem of constructing thesusceptibility distribution is highly nonunique. To select oneparticular model out of the many that can reproduce the data,we seek to construct a minimum structure model by employ-ing the well-established Tikhonov regularization technique, inwhich the inverse problem is solved as an optimization:

min. φd + µφm

s.t. φd = φ∗d (9)

κ ≥ 0,

where φd is the data misfit function, φm is the model objectivefunction, and µ is the regularization parameter. The first con-straint ensures that the observed data are reproduced, whilethe second constraint ensures that the constructed model isphysically plausible.

Since the magnetic data to be inverted always have errorsassociated with them, we seek to misfit the data to a targettolerance determined according to the estimated errors. Thisensures that the inversion result is minimally affected by noisein the data and that the inversion reproduces the data justaccurately enough to extract as much information as possible.We have used a misfit measure defined by

φd =N∑i=1

(dobsi − dpre

i

εi

)2

(10)= ∥∥Wd(dobs − dpre)

∥∥2,

where εi is the standard deviation of the ith datum and Wd =diag{1/ε1, . . . , 1/εN }. Because of the lack of knowledge aboutthe actual noise statistics, we usually assume that errors are

independent and Gaussian. The misfit φd is then a χ 2 distri-bution, and its expected value is equal to the number of data,N .

An objective function used by Li and Oldenburg (1996) pe-nalizes both the deviation of the recovered susceptibility modelfrom a reference model and the structural complexity in threespatial directions. A key feature of that objective function is adepth weighting function that has the form w(z) = (z+ z0)−3/2.The objective function is defined as

φm(κ) = αs

∫V

ws{w(z)[κ(r) − κ0]}2 dv

+ αx

∫V

wx

{∂w(z)[κ(r) − κ0]

∂x

}2

dv

(11)

+ αy

∫V

wy

{∂w(z)[κ(r) − κ0]

∂y

}2

dv

+ αz

∫V

wz

{∂w(z)[κ(r) − κ0]

∂z

}2

dv,

where κ is the sought susceptibility model; κ0 is the referencemodel; ws , wx , wy , and wz are spatially dependent weightingfunctions; and αs , αx , αy , and αz are coefficients that affect therelative importance of different components in the objectivefunction. They are chosen so that the ratios αx/αs, αy/αs, αz/αsare much greater than unity, and the recovered model becomessmoother as these ratios increase. The depth weighting func-tion w(z) is used to overcome the tendency of concentrating thestructures near the surface and to distribute the recovered sus-ceptibility anomaly with depth. The parameter z0 of the depthweighting is chosen such that w2(z) approximates the decay ofkernels. Empirical results have shown that minimizing equa-tion (11), subject to fitting the data, produces a model thathas the susceptibility anomalies at approximately the correctdepth.

A general data set containing both surface and boreholedata can be inverted using the same methodology. The depthweighting function, however, is no longer applicable becausethe data are sparsely distributed in 3-D space and they areoften inside the volume in which the susceptibility model isto be constructed. Nevertheless, the philosophy of the depthweighting—that is, to weight preferentially the cells of lowsensitivity at depth so that all cells have approximately equallikelihood of deviating from the reference model—can still beadopted. For a data set containing general observation loca-tions, a 3-D weighting function is needed since the decay ofkernels is not dominated by any particular direction. The ob-jective function of the susceptibility defined by equation (11)becomes

φm(κ) = αs

∫V

ws{w(r)[κ(r) − κ0]}2 dv

+ αx

∫V

wx

{∂w(r)[κ(r) − κ0]

∂x

}2

dv

(12)

+ αy

∫V

wy

{∂w(r)[κ(r) − κ0]

∂y

}2

dv

+ αz

∫V

wz

{∂w(r)[κ(r) − κ0]

∂z

}2

dv,

Joint 3-D Borehole Magnetic Inversion 543

where w(r) is the 3-D weighting function, and it preferentiallyweights the cells of low sensitivity irrespective of their loca-tions.

When surface and borehole data are inverted jointly, thereare two dominant decays in sensitivities: the decay with depthfor the surface data and the decay with the distance away fromthe boreholes. Thus, the weighting function for a joint inver-sion should be formed such that it counteracts both decays andplaces the susceptible material at about the right location bothin depth and at distance from boreholes.

For numerical solution, equation (12) is discretized using afinite-difference approximation, and a matrix representation isgenerated:

φm(κ) = (κ − κ0

)TZTWTWZ(κ − κ0

), (13)

where the diagonal matrix Z contains the discretized depthweighting function. We define a weighted model and the cor-responding weighted sensitivities as

κZ = Zκ(14)

GZ = GZ−1.

Thus, the data equation and the model objective function, re-spectively, acquire the usual form of

d = GZκZ (15)

and

φm(κ) = (κZ − κZ0)TWTW(κZ − κZ0). (16)

The minimization of equation (9) is carried out in the newvariables κZ .

The weighting function is but one of many competing factorsin the inversion algorithm that determines the final model. Itexpresses the algorithm’s preference to place susceptibilitiesin the regions far from the observations by having reducedweights there. However, it does so under the condition thatthe observations be reproduced satisfactorily. Therefore, thealgorithm does not exclude models that have significant sus-ceptibilities near the surface or the boreholes. One can alwaysreproduce the data resulting from distant causative bodies bysusceptibilities near the observations; however, it is much moredifficult—and sometimes impossible—to reproduce the dataresulting from nearby causative bodies using susceptibilitiesat distant locations. If the anomaly is originally caused by abody near the surface or a borehole, the inversion will pro-duce a model that has the susceptibility at those locations. Thisis confirmed by numerical tests, but for brevity we have notreproduced the results here.

GENERALIZED DEPTH WEIGHTING FUNCTION

We have explored two approaches to generating a 3-Dweighting function. The first is based on the sensitivity ma-trix, and it depends on the overall sensitivity of the entire dataset to a particular cell in the model. The second approach isa generalization of the method used in surface data inversion.These approaches are general and applicable to surface, bore-hole, or joint data sets. In particular, an inversion of surfacedata using these two approaches will produce results similar toan inversion using the depth weighting w(z). In the following,we present the methods and then illustrate them with examplesarising in a synthetic problem.

Sensitivity-based weighting function

Assuming that the problem has been discretized, we definean rms sensitivity, Sj , as the measure of the sensitivity of theentire data set to the jth cell:

Sj =(

N∑i=1

G2i j

)1/2

, j = 1, . . . ,M, (17)

where N is the number of data, M is the number of cells in thediscretized model, and Gi j are the elements of the sensitivitymatrix. The rms sensitivity Sj is small if a cell is far away from alldata points, and it is large when a cell is close to one or severaldata points. When only surface data are present, the functionclosely approximates the decay of the magnetic kernel withdepth. When replacing w(z) with

√Sj in equation (11), the

inversion produces susceptibility models that have about theright depth for the recovered anomaly. Thus, it is reasonable toextend the use of

√Sj as the weighting to the cases where we

have borehole data. We choose a general form for the weightingfunction as

w j =(

N∑i=1

G2i j

)β/4

, 0.5 < β < 1.5, (18)

where larger β means stronger weighting. The value for β

should usually be close to 1.0. [A similar form of weightingfunction corresponding to β = 1.0 is used by Wang (1995) inseismic refraction tomography.] The weights, w j , are used as adiscrete representation of a continuous function w(r) in equa-tion (12).

This weighting function captures the general decay of themagnetic kernel with distance, but it has a minor drawback.Since the magnetic kernels are direction dependent, the re-sulting weighting function is also variable with the direction inwhich a cell is located relative to a borehole. This variation isclearly visible when only single-component data are available;however, it is completely eliminated when three-componentdata are used.

Distance-based weighting function

Our second weighting function is defined by the distancebetween the cells and observation locations, and it representsa direct generalization of the depth weighting function used inthe surface inversion. Let

w j =

N∑i=1

[∫�V j

dv

(Ri j + R0)3

]2

β/4

, j = 1, . . . ,M,

(19)where �Vj is the volume of jth cell and Ri j is the distancebetween the ith observation and any point within �Vj . Theparameter R0 is a constant and is usually chosen to be one-halfof the cell width used in the inversion. The parameter β is be-tween 0.5 ≤ β < 1.5 as in equation (18) and determines thestrength of the weighting to be applied. This weighting functionessentially combines the distance characteristics of the origi-nal depth weighting for surface data with the generality of theweighting function based on rms sensitivity. It is independent ofthe sensitivity calculation and is not affected by the directionalvariation.

544 Li and Oldenburg

Examples

As an example, we have calculated the generalized weight-ing functions for data sets associated with the model in Fig-ure 1. The susceptibility model consists of two prisms buried ina uniform half-space. Assuming an inducing field direction ofI = 65◦ and D= 25◦, the total field anomaly on the surface isas shown in Figure 2a. Figure 2b shows the three componentsof the anomaly field measured in three boreholes, which areplaced around the region occupied by the two prisms. The po-sitions of the holes are indicated by the circled dots in Figure 1.

Figure 3 shows the sensitivity-based weighting functions de-fined in equation (18) corresponding to the data shown in Fig-ure 2. The quantity plotted in the figures is the square of theweighting function, w2(r), since it directly reflects the decayrate of the sensitivities. In all three cases, we have used a valueof 1.0 for β. Figure 3a is for the surface data. We can see thatthe weighting function decreases with depth and the rate of de-crease is quite uniform for different horizontal locations. Theeffect of this weighting is to put the recovered susceptibilitypreferentially away from the surface as long as the data arereproduced. Figure 3b shows the weight for the borehole data.It is almost uniform with the depth and decreases horizontallyaway from the boreholes. This weighting function favors mod-els that have high susceptibilities in the central region or in theregion outward from the boreholes, if no data require suscep-tible cells adjacent to the boreholes. Figure 3c shows the cor-responding weight for the joint data set containing both thesurface and borehole data. This weighting function reflects thestructure of the two preceding weighting functions; it has highvalues near the surface and near the boreholes.

Figure 4 shows the distance-based weighting functions, de-fined in equation (19), for the three data sets. They are calcu-lated with β = 1.0 and R0 = 12.5 m. These weighting functionsare similar to those shown in Figure 3, and they exhibit the same

FIG. 1. Two sections of a susceptibility model, consisting of twoprisms buried in a uniform background. The collar positions ofthree vertical boreholes are indicated on the plan section.

characteristic decay as the distance between a cell and obser-vation locations increases. Differences between the weightingfunctions in Figures 3 and 4 cause differences in details of thereconstructed models, but the general characteristics are thesame. In general, either one can be used in the inversion of adata set, but the distance-based weighting might be preferablewhen only single-component borehole data are available.

SYNTHETIC EXAMPLE

We now apply the method to a synthetic example. The truesusceptibility model consists of two prisms buried in a nonsus-ceptible half-space as shown in Figure 1. Both prisms have awidth of 125 m in each horizontal direction. The first prismextends from a depth of 50 m to 175 m, and the second prismextends from a depth of 100 m to 275 m. The horizontal sep-aration between the two prisms is 75 m. The inducing field isassumed to be at I = 65◦ and D= 25◦. The total field anomalywas calculated on the surface at an interval of 25 m along east–west lines spaced 100 m apart, resulting in a total of 175 data.Three-component borehole data are calculated in three holesshown in Figure 1. There are 16 observation locations spaced25 m vertically in each hole, and the total number of data inthree holes is 144. Gaussian noise having a standard deviationof 2 nT has been added to all data, and the resulting simulatedobservations are shown in Figure 2. The surface data mainlyshow the shallow prism and have little indication of the second

FIG. 2. The top panel is the total-field anomaly on the surfaceproduced by the model in Figure 1 under an inducing field inthe direction I = 65◦ and D= 25◦. The bottom panels are thethree-component anomalies in three vertical boreholes. Differ-ent components are identified by the labels beside the curves.

Joint 3-D Borehole Magnetic Inversion 545

FIG. 3. The general weighting function based on the sensitivities of different data sets. The squares of the weighting functionsare shown here, and the value is multiplied by 1000 for the purpose of display. (a) One cross-section and one plan section of theweighting function for the surface data shown in Figure 2a. (b) The borehole data shown in Figure 2b. The weighting is highest nearthe boreholes. (c) The joint data set, consisting of surface and borehole data.

FIG. 4. The general weighting function based on distances. The squares of the weighting functions are shown here and the value ismultiplied by 1000 for the purpose of display. (a) The weighting function for the surface data shown in Figure 2a. (b) The boreholedata shown in Figure 2b. The weighting is highest near the boreholes. (c) The joint data set consisting of surface and borehole data.

546 Li and Oldenburg

prism. The borehole data are more difficult to interpret directly,but the difference in the peak positions for different boreholesmay indicate the presence of two source bodies.

We adopt a discretization that divides the model into cubiccells of 25 m on a side. This yields 24 cells in each horizontaldirection and 16 cells from the surface to a depth of 400 m. Thesurface, the borehole data, and the combined surface and bore-hole data are all inverted using the same mesh. The weightingfunction is calculated from the sensitivity matrix of the respec-tive data set with β = 1.0 (see Figure 3).

We first invert the surface data. Figure 5 displays one plansection and one cross-section of the recovered susceptibilitymodel. This model clearly shows the presence of the shallowprism at the correct location, but it does not give a clear indica-tion of a separate, deeper prism. There is only a broad zone oflow susceptibility extending from the high-susceptibility zone.The vertical extent of the recovered anomaly is poorly defined.

We next invert the three-component borehole data. Figure 6displays the recovered model at the same sections. The modelnow clearly shows two regions of high susceptibility at locationscorresponding to the true prisms, and the recovered depthsagree well with the true depths. There is no excessive struc-ture away from the high susceptibility zones in the center. Themain drawback is that the amplitude of the shallow prism israther small, and this makes the model look more like a sin-gle body. Despite this, the model in Figure 6 is an encouragingresult, considering that there are only three widely separatedboreholes and that the inversion has no explicit informationregarding where to place the magnetic materials. This illus-trates that three-component borehole data can be inverted toconstruct a 3-D susceptibility distribution and that the generalprior information encapsulated in the weighting function is ef-

FIG. 5. The susceptibility model recovered from inverting thesurface total-field anomaly in Figure 2a using the weight-ing function shown in Figure 3a. The solid lines indicatethe outlines of the prisms. The shallow prism is shown as ahigh-susceptibility zone with a poorly defined depth extent.The deep prism is shown only as a broad zone of lower suscep-tibility.

fective in helping extract the information in the borehole data.Last, we jointly invert the surface and borehole data to re-

cover a susceptibility model that simultaneously reproducesboth data sets. The model recovered from inverting 319 obser-vations is shown in Figure 7. This model combines the merits

FIG. 6. The susceptibility model recovered from inverting thethree-component borehole data in Figure 2b. The definition ofthe deeper prism is reasonably good since it is close to one ofthe boreholes.

FIG. 7. The susceptibility model recovered from the joint inver-sion of total-field surface data and three-component boreholedata. The sensitivity-based weighting function is used in this in-version. This model clearly defines both prisms and illustratesthe improvement achieved by joint inversion of surface andborehole data.

Joint 3-D Borehole Magnetic Inversion 547

of the models in Figures 5 and 6: the two target prisms arewell defined in both horizontal and vertical locations, and theiramplitudes are comparable to those of the true model. Of thethree susceptibility models generated from the inversion of dif-ferent data sets, this model provides the best representation ofthe true model in Figure 1 and illustrates the advantages of thejoint inversion of surface and borehole data.

As a final comment, the good results of joint inversion werenot produced by the particular placement of the boreholes usedhere. Additional tests have shown that consistent results areobtained when the three holes are shifted to a wide range ofpositions. This gives us confidence for the future success of thetechnique.

PRACTICAL CONSIDERATIONS

Successful application of our inversion algorithm requiresthat the data comply with the basic assumptions used in the for-mulation. In particular, we assume that the borehole anomalydata are the fields produced by the susceptible targets aloneand that these data can be reliably reduced from field obser-vations. These assumptions are not always met in practice. Inparticular, violations arise if the borehole intersects magneticmaterial or if the orientation of the borehole is not accuratelyknown.

When a borehole intersects magnetic materials, its cavitywill produce large responses because of the effective magneticcharges on the wall. The cavity effect is highly dependent uponthe geometry of the hole and the position of the magnetic sen-sor within the borehole. Attempts have been made in the lit-erature to correct for the borehole effect using solutions forthe field produced by a cylindrical borehole. However, uncer-tainty in borehole geometry at depth complicates such correc-tions. It is therefore necessary to winnow measurements thatare within magnetic bodies. Fortunately, winnowing these datais not a serious limitation of the algorithm. First, these dataare relatively easy to identify by their high amplitudes and er-ratic spatial variation caused by local structure of magneticsource and cavity effects. Second, the usefulness of invert-ing borehole data is mainly in delineating off-hole targets—especially when the drillholes have missed them. Thus, datawinnowing is usually not an issue, or it will not affect the finalobjective.

The orientation (dip and azimuth) of a borehole at depth isnot precisely known. This leads to two different types of errors.First, the location and orientation of the three-component mea-surements are erroneous; thus, the calculated sensitivity matrixis inaccurate. However, our numerical experiments indicatethat this is not a major concern. Given that modern boreholeprobes can accurately measure the borehole dip, we have sim-ulated errors in azimuth by adding uniformly distributed noiseup to 10◦ in the synthetic borehole data shown in Figure 2.Although the inversion converges more slowly as the azimutherror increases, the final recovered model does not degradesignificantly. The second type of error is produced when theresidual field is extracted by subtracting the background fieldfrom measurements. The three components of the magneticfield are measured in a local coordinate system that is relatedto the borehole orientation. The projected components of thebackground field will be erroneous if the borehole orientationis not precisely known. Since the background field is several or-

ders of magnitude greater than the anomalous field, a small er-ror in borehole orientation will produce an unacceptably largeerror in the calculated residual data. For example, assuming aborehole with a dip of 45◦ and an azimuth of 0◦, and assumingan inducing field that has a strength of 50 000 nT and a directionaligned with the borehole, an error of 1◦ in dip and 2◦ in azimuthof the borehole orientation will produce errors exceeding 800and 1200 nT in xl - and yl -components, respectively. Errors ofsuch a magnitude render the three-component data ineffective.Successful use of these data would require an intermediate stepof recovering the true borehole orientations—a topic beyondthe scope of this paper.

It is possible, however, to use three-component boreholedata even in the presence of significant orientation error. Fieldvalues in three orthogonal directions accurately define the am-plitude of the total field. The amplitude difference between thetotal field and the background field then gives the total fieldanomaly that is, to a high degree of approximation, the pro-jection of the anomalous field in the direction of the inducingfield. Thus, we always have good single-component data avail-able for use in inversions. We expect that the single-componentdata themselves may not contain enough information to con-struct a good susceptibility model; however, they can providethe required complementary information in a joint inversionwith surface data. We illustrate this with the synthetic datain Figure 2. Three-component observations are first simulatedby adding the corresponding component of the backgroundfield to each component of the borehole anomaly in Figure 2.A total field anomaly is then obtained by taking the differ-ence of total and background field amplitudes. The resultantdata are used first in an individual inversion and then in ajoint inversion with the surface data. In both inversions, wehave used the distance-based weighting function shown in Fig-ure 4. The model recovered by inverting borehole total-fieldanomaly data is shown in Figure 8. The single-component dataclearly cannot resolve the two separate prisms, and the max-imum value in the recovered susceptibility lies between theprisms. In contrast to the model in Figure 5, which is obtainedusing three-component data, this is a poor result. When thejoint inversion of surface and borehole total-field anomaly iscarried out, we obtain the model shown in Figure 9. It clearlyimages both prisms and represents an improved result com-pared to any of the individual inversions. This model is similarto that shown in Figure 7, which is obtained by the joint in-version using three-component borehole data. The success ofthis inversion demonstrates that single-component boreholedata contain much information that complements surface dataand joint inversion using single-component data, such as com-puted total field anomaly and can produce an improved sus-ceptibility model compared to an inversion using surface dataonly.

FIELD EXAMPLE

We test the new inversion algorithm on a set of magnetic datacollected in Australia. The target of the surface magnetic sur-vey was an ironstone formation hosted in unaltered siltstoneand graywacke. The magnetic anomaly is produced entirelyby the ironstones. The total-field magnetic data collected onthe surface are shown in Figure 10. Three-component bore-hole magnetic data were also collected in seven holes. These

548 Li and Oldenburg

holes are inclined to the north, and five intersect the ironstonebody. The borehole positions are shown in Figure 10 in relationto surface anomaly by their projections on three orthogonalplanes. Figure 11 shows the measured three-component datain three of the holes.

The inversions were first carried out using a mesh whose cen-tral region consists of cubic cells of 25 m on a side. Invertingthe surface total field anomaly with a sensitivity-based weight-ing function produces the model shown in Figures 12 and 13.The model is basically a magnetic body of variable susceptibil-ity dipping west. Its center agrees with the known position ofironstone at shallow depth, but the recovered susceptibility isshifted to the north at greater depths. The recovered suscep-tibility anomaly is also spread out over a larger volume thanthat of the known ironstone formation.

Borehole data were collected at an interval of 1 m, but weuse only a subset in the inversion. We select one observationper cell and choose the location as close to the cell center aspossible. We also restrict the data to lie outside the magneticbody. The selected three-component data are then reducedby subtracting the respective component of the backgroundfield. Unfortunately, the joint inversion of surface data withthese three-component borehole data failed to converge to asatisfactory data misfit, and the resultant susceptibility modelhad spurious structures. The lack of success was judged to becaused by the uncertainties in the azimuth of the boreholes. Asdiscussed in the preceding section, the erroneous orientationdata can result in large errors in each component.

The total field anomaly computed from three-componentdata is much more accurate, and we have performed a joint in-version using surface data and total-field anomaly in the bore-

FIG. 8. The susceptibility model recovered from the inversionof total-field borehole data computed from three orthogonalcomponents. This model does not define two prisms but showsonly a broad, single susceptibility anomaly.

holes. Eighty-four total-field anomaly data in seven holes areused, and the inversion constructs a model that reasonably re-produces both the surface and borehole data. Figure 14 com-pares the observed and predicted total-field anomaly in threeselected boreholes; there is good fit between the observationsand the predicted data from the joint inversion. This is in sharpcontrast to the large discrepancy between the observations andthe data predicted by the susceptibility model recovered fromsurface data alone. This contrast demonstrates that the inclu-sion of the borehole data has had a significant effect on the dis-tribution of the recovered susceptibility. The model recoveredfrom the joint inversion is shown in Figures 15 and 16. Thereis marked improvement in the correspondence between therecovered anomaly and the ironstone formation in both hor-izontal and vertical locations. The enclosure of the recoveredsusceptibility by the boundary of the ironstone in the cross-section is especially encouraging. On a detailed level, there isa discrepancy between the geologic information supplied to usand our inversion results. Geologic interpretation indicates asingle body, but the recovered susceptibility suggests the pres-ence of two bodies located at different depths and horizontallyoffset. There is already some hint of this offset in the inversionof the surface data, and the borehole data seem to accentuateit. There may therefore be two disjoint bodies, but resolutionof this discrepancy is impossible without more detailed drillinginformation.

For this field example, the following summary statementscan be made. The inversion of surface data delineates the gen-eral location and structure of the anomaly. Three-componentborehole data are ineffective because of uncertainties inthe borehole azimuth, but joint inversion of surface and

FIG. 9. The susceptibility model recovered from the joint in-version of surface and borehole total-field data. This modelis similar to that shown in Figure 7. Both prisms are clearlyimaged.

Joint 3-D Borehole Magnetic Inversion 549

FIG. 10. The grayscale contours show the surface total-fieldmagnetic anomaly above an ironstone formation in Australia.The contour interval is 10 nT. The direction of the inducingfield is I = −51◦ and D= 5◦. The difference between the mag-netic declination and the alignment of the positive and neg-ative peaks indicates that the source body is dipping. Sevenboreholes were drilled to intersect the source of the anomaly.Their locations are shown by projection onto three orthogonalplanes. The data collected in holes labelled as A, B, and C areshown in Figure 11.

FIG. 11. Three-component borehole data collected in threeselected drillholes as indicated in Figure 10. The data havebeen rotated to northing, easting, and vertical components,and a regional field with a magnitude of 50 670 nT has beenremoved.

FIG. 12. Susceptibility recovered from inverting surface total-field anomaly data is shown in four horizontal slices. The solid whiteline indicates the boundary of ironstone inferred from drillholes.

550 Li and Oldenburg

FIG. 13. One cross-section at N = 9960 m of the susceptibilitymodel recovered from surface data. The anomaly agrees withthe dipping structure of the ironstone formation. The white lineindicates the boundary of ironstone on a composite section.

FIG. 14. Comparison of the observed and predicted total-field data in three selected holes. The observed data are shown as triangles,and the predicted data are shown as crosses. The left column shows the borehole data predicted by the susceptibility model fromsurface-data inversion (Figures 12 and 13). The predicted data are poor representations of the observations. The column on theright shows the predicted data from the joint inversion. There is a good match between the observation and prediction.

totalfield borehole data has improved the localization of theanomaly.

DISCUSSION

The 3-D magnetic inversion algorithm developed earlier bythe authors has been extended to include borehole magneticdata. The principal difficulty is to design a weighting functionthat distributes the susceptibility away from the observationlocations. We have developed two approaches to form such aweighting function, and they are applicable to data sets havingarbitrary observation locations ranging from the earth’s surfaceto boreholes. The first approach is based upon sensitivity, andthe second is based upon distance between the observationlocation and model cells. The two approaches generate similarweighting functions; when used in an inversion, they producesimilar susceptibility models.

The newly formulated weighting functions are shown,through numerical examples, to be effective in inverting 3-Ddata sets, and we have developed a practical algorithm thatallows the construction of susceptibility distributions from

Joint 3-D Borehole Magnetic Inversion 551

FIG. 15. Joint inversion of surface data and total-field borehole data. This result shows marked improvement over the result fromsurface-data inversion.

borehole data. The weighting functions also act as a linkagebetween the surface and borehole data and therefore enablethe joint inversion to use the complementary information in thetwo data sets. Joint inversion of surface and borehole data fromsynthetic models has produced superior susceptibility modelsthat better define the deeper susceptibility anomalies to whichthe surface data are insensitive. Three-component boreholedata are superior to single-component data, but the successfuluse of three component data requires that the borehole ori-entation be accurately known. If the orientation errors arelarge, then the total-field anomaly computed from the threecomponents can still supply information that can improve theinversion compared to that obtained by inverting surface dataalone.

ACKNOWLEDGMENTS

We thank Colin Barnett and Newmont Gold Company forsupplying the field data and discussing the interpretation ofthe inversion results. This work was supported by an NSERCIOR grant and an industry consortium, Joint and Coopera-tive Inversion of Geophysical and Geological Data. Partici-pating companies are Placer Dome, BHP Minerals, NorandaExploration, Cominco Exploration, Falconbridge, INCO Ex-ploration & Technical Services, Hudson Bay Exploration andDevelopment, Kennecott Exploration Company, NewmontGold Company, Western Mining Corporation, and CRA Ex-ploration Pty.

FIG. 16. One cross-section of the susceptibility model fromjoint inversion of surface and borehole total-field data. Thecross-section is at N = 9960 m. The black line indicates the com-posite boundary of the formation, which is well represented bythe recovered anomaly.

552 Li and Oldenburg

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