Inversion of induced polarization data · Inversion of induced polarization data Douglas W. Oldenburg* and Yaoguo Li* ABSTRACT We develop three methods to invert induced polar ization

GEOPHYSICS, VOL. 59, NO.9 (SEPTEMBER 1994); P. 1327-1341, II FIGS.

Inversion of induced polarization data

Douglas W. Oldenburg* and Yaoguo Li*

ABSTRACT

We develop three methods to invert induced polarization (IP) data. The foundation for our algorithms isan assumption that the ultimate effect of chargeabilityis to alter the effective conductivity when current isapplied. This assumption, which was first put forth bySiegel and has been routinely adopted in the literature,permits the IP responses to be numerically modeled bycarrying out two forward modelings using a DC resistivity algorithm. The intimate connection between DCand IP data means that inversion of IP data is atwo-step process. First, the DC potentials are invertedto recover a background conductivity. The distribution of chargeability can then be found by using anyone of the three following techniques: (I) linearizingthe IP data equation and solving a linear inverseproblem, (2) manipulating the conductivities obtainedafter performing two DC resistivity inversions, and (3)

INTRODUCTION

Induced Polarization (IP) data are routinely collected inmineral exploration surveys and they are also finding theirniche in environmental surveys (Barker, 1990). Excellentreviews on the IP method and case histories can be found inSumner (1976), Bertin and Loeb (1976), Fink et al. (1990),and Ward (1990). The difficulty with interpreting IP data isthe lack of flexible, efficient, and robust inversion algorithms. This is reflected in the remark in Hohmann (1990)regarding the availability of forward algorithms and the lackof general inversion techniques for the IP method. Consequently, much of today's interpretation is still carried out byworking with pseudosections. Yet only in very simplisticcircumstances will the images on the pseudosections emulate geologic structure, and consequently, inferences madeabout the substructure directly from the data are often

solving'a nonlinear inverse problem. Our procedure forperforming the inversion is to divide the earth intorectangularprisms and to assume that the conductivity (J"

and chargeabilityTJ are constant in each cell. To emulatecomplicatedearth structure we allow many cells, usuallyfar more than there are data. The inverse problem, whichhas many solutions, is then solved as a problem inoptimization theory. A model objective function is designed, and a "model" (either the distribution of (J" or TJ)is sought that minimizesthe objective function subject toadequately fitting the data. Generalized subspace methodologies are used to solve both inverse problems, andpositivity constraints are included. The IP inversionprocedures we design are generic and can be applied toI-D, 2-D,or 3-Dearth modelsand with any configurationof current and potential electrodes. We illustrate ourmethods by inverting synthetic DC/IP data taken over a2-D earth structure and by inverting dipole-dipole datataken in Quebec.

incorrect. As an illustration, we present a fairly simpleexample of DC/IP data taken over a 2-D earth structure. Thetrue conductivity and chargeability models and the pseudosection plots obtained by carrying out a pole-pole DC/IPsurvey are shown in Figure 1. Except for the region near thesurface, there is little compelling evidence in the IP pseudosection to indicate a chargeable body at depth.

Methods for inverting IP data do exist but literature on thissubject is sparse. In defining IP data, most authors adopt apresentation given in Siegel (1959) that the ultimate effect ofa chargeable body is to alter its effective conductivity. Assuch, the IP and DC resistivity problems are intimatelylinked, and the inversion of IP data is a two-step process. Inthe first stage, the DC potentials are inverted to recover thebackground conductivity (J" b : The second step accepts (J" b asthe true conductivity of the medium and attempts to find achargeability that satisfies the data. This is usually done by

Presented at the 63rd Annual International Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor AprilS, 1993;revised manuscript received March 1, 1994.*Geophysical Inversion Facility, Department of Geophysics and Astronomy, The University of British Columbia, Vancouver, BC V6T 124,Canada.© 1994 Society of Exploration Geophysicists. All rights reserved.

1327

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1328 Oldenburg and Li

linearizing the equations about ITb to produce a system ofequations that can be solved for the chargeability distribution.

Early algorithms for inverting DC and IP data generallyparameterized the earth model into a relatively small numberof blocks and kept the same parameterization for invertingthe DC and IP data (Pelton et al., 1978; Sasaki, 1982; Rijo,1984). Overdetermined systems of equations were solvedand algorithm convergence was judged on the basis of datamisfit alone. There are practical difficulties that arise withthis approach. The electrical conductivity structure of theearth is complicated and rarely does a representation by afew blocks adequately represent the true distribution of thisphysical property. Also, anomalous regions of high or lowconductivity do not necessarily correspond to regions ofhigh chargeability. If only a few blocks are used, keeping thesame parameterization may preclude the possibility of finding a meaningful solution. In addition, the restriction of usingonly a few model cells does not allow insight about thenonuniqueness that is inherent in the inverse problem. Thedifficulties with respect to parameterization can be overcomeby discretizing the earth into a large number of cells. Theinverse problem is then solved as an optimization problemwhere an objective function of the model is minimizedsubject to adequately fitting the data. An archetypal exampleapplied to a geophysical inverse problem was presented byConstable et al. (1987). LaBrecque (1991) has applied thismethodology to carry out a 2-D inversion of IP data in a

cross-borehole tomography experiment, and Beard andHohmann (1992) have provided an approximate inversion ofIP data that is valid when resistivity contrasts are small.

In this paper, we use Siegel's (1959) formulation anddevelop three methods by which to invert IP data. In the firstmethod, we assume that the chargeability is small andlinearize the equations. The technique is similar to thatpresented in LaBrecque (1991), but we use a more generalmodel objective function, we incorporate a subspace methodology to bypass the large computations normally requiredto invert the full matrix system, and we work with chargeabilities directly in the inverse problem rather than with theirlogarithms. The second method makes use of formal mappings connecting DC/IP voltages with a conductive andchargeable earth. Two DC resistivity inversions of differentdata are carried out, and the chargeability is obtained bymanipulating the recovered conductivities. The third methodmakes no assumptions about the size of TJ and solves anonlinear inverse problem to recover the chargeabilities.The resultant algorithm is essentially the same as that usedto invert the DC resistivity data. All of our techniques areapplicable to any dimension of earth structure and to anyconfiguration of electrodes. Our paper begins by defining theforward mapping for the IP data. Next, we introduce the 2-Dsynthetic example and invert pole-pole DC potentials torecover the background conductivity. This is presented insome detail because the subspace method used to invert theDC data also plays the dominant role in inverting IP data.

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FIG. 1. A synthetic 2-D conductivity model is shown in (a). Surface electrodes are spaced at 10 m over the interval x = (-100,100) m. Current is input at each electrode site in turn, and potentials are observed at the remaining 20 sites. The pole-polepotentials <P<T are converted to apparent conductivity ITa = I1(211'r<P<T)' where r denotes the distance between current andpotential electrodes and are plotted in (b). The apparent conductivity value is plotted midway between the current and potentialelectrodes at a (pseudo) depth of z = 0.86r. The grey scales indicate conductivity in mS/m. The synthetic chargeability modelis given in (c) and the apparent chargeability, plotted in the pseudosection format is shown in (d). The grey scale indicateschargeability in percent.


Inversion of Induced Polarization Data 1329

Three methods for inverting IP data are explained andillustrated with the synthetic example. We address the issuesof positivity, of how well the background conductivity needsto be known, and practical considerations pertaining to thedata to be inverted. We also examine the merits of using TJ orIn TJ as the variable in the inversion and show how minimizing different objective functions can be useful in hypothesistesting and in exploring nonuniqueness. A field data set ofdipole-dipole data is inverted, and the paper concludes witha discussion about the relative merits of the inversionalgorithms.

and the boundary conditions are a<l>cr/an = 0 at the earth'ssurface and <l>cr ~ 0 as Ir - rsI~ 00, where r s denotes thelocation of the current electrode.

If the ground is chargeable, then the potential <1>'1]' recorded after the constant current is applied, will differ from<l>cr' According to Siegel's formulation, the effect of thechargeability of the ground is modeled by using the DCresistivity forward mapping ?:F de but with the conductivityreplaced by 0" = 0"(1 - TJ). Thus

(3)

FORWARD MODEL or

or

The IP datum, which we refer to as apparent chargeability, isdefined by

(6)

(5)

Equation (6) shows that the apparent chargeability can becomputed by carrying out two DC resistivity forward modelings with conductivities 0" and 0"(1 - TJ).

Equation (6) defines the forward mapping for the IP data.The data can be inverted by: (1) linearizing this equation andsolving a linear inverse problem, (2) introducing a formal DCinverse operator ?:F;;'} and obtaining TJ by manipulating theconductivity models obtained after applying ?:F icl to <1>'1] andto <l>cr' or (3) solving a nonlinear inverse problem thatinvolves linearization but iterates until data predicted fromequation (6) are in agreement with the observations. Irrespective of the method, the inversion of the IP data requiresthat the potentials <l>cr be inverted to recover the backgroundconductivity. We address this issue first.

Complete understanding of the microscopic phenomenonthat result in the macroscopic IP response has not beenachieved. Here we adopt a macroscopic representation ofthe physical property governing the IP response that was putforth in Siegel (1959). Basically, he introduces a macroscopicphysical parameter called chargeability to represent all of themicroscopic phenomena. As such, our earth model is described by the two quantities: conductivity O"(x, y, z) andchargeability TJ(x, Y, z). Both are positive, but whileconductivity varies over many orders of magnitude, chargeability is confined to the region [0, 1). We note that Siegel'smodel refers only to the volumetrically distributed polarization and does not apply to highly conductive targets in whichsurface polarization dominates the IP effect. Fortunately,this is not an important limitation for the majority of practical situations.

A typical IP experiment involves inputting a current I tothe ground and measuring the potential away from thesource. In a time-domain system, the current has a dutycycle that alternates the direction of the current and hasoff-times between the current pulses at which the IP voltagesare measured. A typical time-domain signature is shown inFigure 2. In that figure, <l>cr is the potential that is measuredin the absence of charge ability effects. This is the "instantaneous" potential measured when the current is turned on.In mathematical terms

(1)INVERSION OF DC POTENTIALS

FIG. 2. Definition of the three potentials associated with theIP survey.

where the forward mapping operator ?:F de is defined by theequations Equation (1) is a nonlinear relationship between the ob

served potentials <l>cr and the conductivity 0". The goal of theinverse problem is to find the function 0" which gave rise tothose observations. There have been many attacks on thisproblem. For the example problem in this paper where thestructure is presumed to be 2-D, we use the subspaceinversion method given in Oldenburg et al. (1993). Thismethodology will be used for inverting both DC data and IPdata, and we therefore outline essential details of the approach.

Let the data be denoted generically by the symbol d andthe model by m. To carry out forward modeling to generatetheoretical responses, and also to attack the inverse problem, we divide our model domain into M rectangular cellsand assume that the conductivity is constant within eachcell. Our inverse problem is solved by finding the vectorm = {m I , m 2, ••• , m M} which adequately reproduces theobservations do = (dOl' d o2,"" dON)'

(2)

Time

T


(7)

(9)

Oldenburg and Li

The solution of equation (11) requires that y T Oy~Wm +I1JT.vV be inverted and the numerical efficiency of theinversion is therefore realized since this is a q x q matrix. Ateach iteration in the inversion, we desire a model perturbation that minimizes Iji m and alters Iji d so that it achieves aspecific target value Iji~. To prevent the buildup of unnecessary roughness, the target misfit begins at an initial value(usually a fraction of the misfit generated by the startingmodel) and decreases with successive iterations towards afinal value selected by the interpreter. Convergence isreached when the data misfit reaches this final target and nofurther reduction in the model norm is obtained with successive iterations.

In the IP survey carried out to produce Figure 1, surfaceelectrodes are located every 10m in the interval x = (- 100,100 m). Each of the 21 electrode positions can be activatedas a current site and when it is, electric potentials arerecorded at the remaining electrodes. The observed data setconsists of 420 potential values, each of which has beencontaminated by Gaussian noise having a standard deviationequal to 5 percent of the true potential. The data aregenerated using a finite-difference code (McGillivray, 1992),and the mesh, used both for forward modeling and for theinversion, consists of 1296 elements.

Because of the nonuniqueness of the inverse solution,the character of the final model is heavily influenced by themodel objective function. Our choice for Iji m is guided by thefact that we often wish to find a model that has minimumstructure in the vertical and horizontal directions and at thesame time is close to a base model mo. This model, becauseit is "simple" in some respect, may well be representative ofthe major earth structure; however, other earth modelsmight be closer to reality. Also, even if a geologicallyreasonable model has been found, it is insightful to generatedifferent models that fit the data. This can provide understanding about whether features observed in the constructedmodel are required by the observations or if they are merelythe result of minimizing a particular model objective func-tion. An objective function that has the flexibility to accomplish these goals is

q

(lm = L: aivi == ya.i= I

1330

The inverse problem is posed as a standard optimization:

minimize !jJm(m, mo) = IIWm(m - mo)!!2

subject to ljid(d, do) = IIWd(d - do)ll2 = Iji'd.

In equation (7), mo is a base model and Wm is a generalweighting matrix that is designed so that a model withspecific characteristics is produced. The minimization of themodel objective function !jJ m yields a model that is close tomo with the metric defined by Wm and so the characteristicsof the recovered model are directly controlled by these twoquantities. The choice of Iji m is crucial to the solution of theinverse problem, but we defer the details until later. Wd is adatum weighting matrix. We shall assume that the noisecontaminating the jth observation is an uncorrelatedGaussian random variable having zero mean and standarddeviation Ej' As such, an appropriate form for the N x Nmatrix is w, = diag I l/s , , "', liEN}' Wirh this choice.w,is the random variable distributed as chi-squared with Ndegrees of freedom. Its expected value is approximatelyequal to N and accordingly, Iji~, the target misfit for theinversion, should be about this value. The appropriateobjective function to be minimized is

Iji(m) = ljim(m, mo) + 11(ljid(d, do) -lji'd)' (8)

where 11 is a Lagrange multiplier.The inverse problem is nonlinear and is generally attacked

by linearizing equation (8) about the current model m(n),differentiating with respect to parameters mj and solving theresultant M x M system of equations for a perturbation om.This can be computationally intensive when M becomeslarge and hence the use of the subspace methods. In thesubspace method, the "model" perturbation is restricted tobe a linear combination of search vectors {Vi} i = I, q. Thus

The linearized objective function, obtained by substitutingmin) + ya into equation (8), is

(13)

(12)

+ JJ{axwxC(m a~ mo)f

+ azwz(a(m ~ mo»)2} dx dz:

In equation (12), the functions w" wx ' W z are specified bythe user, and the constants a" ax, a z control the importance of closeness of the constructed model to the basemodel mo and control the roughness of the model in thehorizontal and vertical directions. The discrete form ofequation (12) is

Ijim=(m-mo)T{asW;Ws +axW;Wx + azW;Wz}(m-mo)

11{ljid + 'Y~Ya + ~ aTyTW~Wdya -lji'd}'

(10)

where 'Ym = VmIji m and 'Yd = Vm!jJ d are gradient vectors,and Vm is the operator (alam), ... , alamM)T. In equation(10), ljim is understood to be !jJm(m(n) , mO)'!jJd is !jJd(d(n) ,do), and the sensitivity matrix J has elements Jij =ad,.(m(n»/amj' Differentiating equation (10) with respect toa and setting the resultant equations equal to zero yields

The solution of this system requires that a line search becarried out to find the value of the Lagrange multiplier 11 sothat a specific target value Iji~ is achieved. This involves aninitial guess for 11, solving equation (11) by SVD for thevector a, computing the perturbation om, carrying out aforward modeling to evaluate the true responses and misfit,and adjusting 11. The estimation of an acceptable value of 11typically requires three or four forward modelings.



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Thus the ith datum is

M a<j><j>T] = <!>(a - TJa) = <j>(a) - L - TJjaj + H.O.T.

j = I aaj

This can be written approximately as

We shall develop three procedures for inverting IP data.Method I assumes that the chargeability TJ -e: 1.0.Equation (6) is linearized about the conductivity a b recovered from the inversion of the DC potentials, and a linearinverse problem is solved to recover TJ. In Method II werearrange the mapping defined by equation (6) and computeTJ by manipulating the conductivities obtained by performingtwo DC resistivity inversions. In Method III we present ageneral approach that does not require that chargeability besmall and solves the IP problem as a nonlinear inverseproblem.

INVERSION OF IP DATA

and this defines the matrix Wm in equation (7). For theinversion of the synthetic data, we have set w s' W x' W zequal to unity, and Sv in equation (13) Ws is a diagonalmatrix with elements AxAz, where Ax is the length of thecell and Az is its thickness; Wx has elements ±YAz/ox,where ox is the distance between the centers of horizontallyadjacent cells; and Wz has elements ± YAx/oz, where OZ isthe distance between the centers of vertically adjacent cells.In defining the objective function, we can choose m to be aor In a. Because earth conductivities typically vary overmany orders of magnitude, we choose m = In a. This choicealso ensures that the recovered conductivity will remainpositive. In addition, we specify as = .0002, ax = 1.0,a z = 1.0 and the reference model mo corresponding to abackground conductivity of ao = 5.0 mS/m.

Our search vectors for the subspace inversion are obtained by partitioning IjJ d into data sets associated withindividual current electrodes. Steepest descent vectors associated with these 21 data objective functions are combinedwith the steepest descent vector for the model gradient anda constant vector to form a basis for the subspace. Theinversion begins with a halfspace of conductivity 5.0 mS/m.At every iteration we ask for a 50 percent decrease in themisfit objective function until a final misfit IjJd = N isachieved, where N is the number of data. A line search usingforward modeling ensures that this is achieved, or in caseswhere it is not achievable, the line search is used to find thatvalue of f.l which provides the greatest decrease in the misfit.Once the target misfit has been obtained, the line searchensures that the misfit remains at the target value, and hencesubsequent iterations alter only IjJ m' The desired misfitIjJd = 420 is achieved by iteration 13, but a few moreiterations are carried out until no further decrease in themodel objective function is obtained. The model obtained atiteration 20 is shown in Figure 3c. It compares favorablywith the true model in Figure 3a. The surface variation iswell defined and so is the conductive anomaly in the centerof the figure. There is no manifestation of the resistive ledgeat the bottom left of the picture, but this might have beenexpected since its depth is 67 m and the electrodes span theregion (-100, 100) m.

Method I: Linearization of the data equations2 .56 3 .4 5 4 .6 5 6 . 25 8 .41 11.3 15 .2 20. 5 27 .8

Let the earth model be partitioned into M cells and let TJiand a i denote the chargeability and electrical conductivity ofthe ith cell. Linearizing the potential <j>T] about the conductivity model a yields

FIG. 3. The true and apparent conductivities are plottedrespectively in (a) and (b). Inversion of the DC potentialsyields the recovered model in (c). The grey scales indicateconductivity in mS/m.


l1J = IIW1)(TJ - TJo)ll2 + f.L(IIWd(.[TJ - TJ~bS)112 -$'d),

(19)

where $'d is a target misfit. In the following inversions, W1)is identical to the model weight matrix used for inverting theDC resistivity data, and the reference chargeability model TJohas been set to zero. The inversion is carried out with the

is ijth element of the sensitivity matrix. In practice, the trueconductivity (J' is not known, and hence the backgroundconductivity (J' b recovered in the DC resistivity inversion isused to compute the sensitivities in equation (18).

In the inversion of the DC potentials, we ensured positivity of the conductivity by working with m = In (J' in themodel objective function. That was a natural choice sinceconductivity varies over many orders of magnitude, and it isrelative change in conductivity that is often geologicallymeaningful rather than absolute change. Minimizing anobjective function that penalizes variation of the logarithm ofconductivity can therefore yield a geologically interpretablecross-section. Intrinsic chargeability, although positive, isconfined to the region [0, 1). We are usually interested in theabsolute deviation of chargeability from zero, and this rangeis rather small. Generally, we are not interested in thevariation of chargeability in the range between zero andsome small number (e.g., 0.01), and working with logarithmscan put undue emphasis on these small values, especially ifthere is a smallest model component in the objective function, that is, if $m has a component with as 'i' O. The basicquestion is really whether the final model is more easilyinterpretable as a logarithmic quantity or as a linear quantity.Our belief is that linear chargeability is likely to be moremeaningful and hence $ m should be evaluated with m = TJ.We adopt this throughout most of the paper; however, thereis a degree of subjectivity in our choice, and we thereforepresent an example using m = In TJ.

If m = TJ, then invoking positivity requires that an extramapping be introduced since variables in the subspacesolution are intrinsically positive or negative. The inclusionof this mapping is simply made in the subspace algorithm anddetails are relegated to the Appendix.

As a test example, we consider the synthetic chargeabilitymodel shown in Figure le. It consists of a chargeable layer ofTJ = 0.05 at the surface and a chargeable block of TJ = 0.15 atdepth. The position of the chargeable block has been offsetlaterally from the position of the conductive block shown inFigure la. Offsetting the chargeable block from the conductive block is physically realistic and it will also be used toemphasize the ability of our inversion to work in suchsituations.

The 420 IP data collected in the survey are contaminatedwith Gaussian noise having a standard deviation equal to.002. This corresponds to about 5 percent of the averagevalue of the apparent chargeabilities. After setting m = TJand writing Wm as W1)' the inverse problem is solved byminimizing the objective function

(18)

Oldenburg and Li

same 23 subspace vectors used in the DC inversion. Theresultant chargeability model, which has a desired misfit of420 and was obtained after 18 iterations, is shown inFigure 4c. There has been good recovery of both thechargeable surface layer and the subsurface block.

The inversion of IP data is a multistep process andconsequently there are numerous reasons why our inversionresult is not identical to the true model. In particular, thediscrepancies between Figures 4c and 4a can be caused by:(1) the availability of limited inaccurate data, (2) the choiceof model objective function, (3) choice of subspace vectorsor other aspects of the subspace inversion algorithm, (4)incorrect sensitivities Jij resulting from the fact that theestimated conductivity differs from the true conductivity, (5)incorrect sensitivities resulting from numerical calculation,and (6) errors caused by the neglect of second-order terms inthe linearized expansion that led to equation (17). It ispossible to gain some insight as to which of these factorsmight be the most important. To do so, we perform threemore inversions. The purpose of the first inversion is toevaluate the differences between the true and recoveredmodel that are attributable to specifics of the inversionalgorithm. This includes the effect of limited inaccurate data,the form of the objective function to be minimized, and allaspects of the subspace inversion. The test is performed bygenerating a new set of data TJa = .[TJt where J is thesensitivity used in the inversion that produced the result inFigure 4c and TJt is the true chargeability. The same realization of error has been added to the data and the inversion hasbeen carried out with the same parameters. The model isshown in Figure 4d. There are differences betweenFigures 4c and 4d, but they are slight. The amplitude of theburied anomalous chargeable body in Figure 4c is slightlyhigher than in Figure 4d and there is slightly more roughnessin the surface zone. This may have been caused by the slightroughness of the recovered conductivity in Figure 3c. Thesimilarity in the inversions indicates that the primary difference between the true and recovered models in Figures 4aand 4c is a result of the model objective function and thelimited inaccurate data. The algorithm is designed to produce a chargeability that has minimal structure in the horizontal and vertical directions subject to fitting erroneousdata. As a result, the buried chargeable body does notachieve amplitudes as high as those of the true body and itsboundaries are smoothed.

The next two inversions provide some indication about theeffect of inaccurate estimation of the background conductivity used to generate J. We first compute J from the trueconductivity in Figure la. The inversion result is given inFigure 4e. The result is similar to that in Figure 4c eventhough the sensitivities have been developed from twodifferent conductivities. It would seem that for purposes ofinverting IP data, the conductivity obtained from invertingthe DC data is an adequate approximation to the trueconductivity. Last, we develop the sensitivities for a constant conductivity half-space. The sensitivities are independent of the value of the half-space, and the results from theinversion are shown in Figure 4f. There are significantdifferences between the models in Figures 4c and 4f. Nevertheless, the approximate location of the buried chargeablebody is quite evident, particularly in the region that has

a In <\li[IT]J .. =-

tj a In ITj

1332

where



lower conductivity, and there is no question that this imageis more easily interpretable than the pseudosection inFigure 4b. This is encouraging in that it suggests that arelatively crude approximation may serve as an adequatebackground conductivity for the IP inversion. This is an areaof future research.

As a last investigation, we carry out the inversion withm = In TJ. It is not possible to incorporate a reference modelof TJo = 0 and so the objective function is modifiedby settingas = O. Otherwise IjIm is identical to that used in theprevious chargeability inversions. The In TJ model is shownin Figure 5a. The chargeable body is clearly observed but itsdepth is somewhat greater than the true depth and thatshown in Figure 4c. The chargeable region on the surface isalso visible. Plotting the results in Figure 5a on a linear scaleproduces Figure 5b. The chargeable body has been contracted to a circular feature whose central value is about40 percent greater than the true chargeability. Nevertheless,the essential features of the true chargeability model areclearly visible. To be precise with comparisons, we havererun the inversion in Figure 4c with an objective function in

which Us = O. The result is given in Figure 5c, and it differsimperceptably from the result in Figure 4c which usedas = .0002. Comparison of Figures 4a, 5a, and 5c showsthat the center of the chargeable body in Figure 5c coincidesmore closely with that of the true model, but its value isabout 40 percent less than the true chargeability. The contours in Figure 5c exhibit some elongation in the horizontaldirection, indicating that the body might be wider than it isthick. At a first level of interpretation, however, all threemodels in Figure 5 provide consistent information aboutearth chargeability, and certainly anyone is preferable tomaking an interpretation from the pseudosection. We shallcontinue to use m = TJ and make our inversions andinterpretations directly with linear chargeability.

Method II: An exact formalism

Our second method for inverting IP data follows directlyfrom the definitions in equations (I) and (3) and the formalintroduction of an inverse mapping operator ?F icl . Applyingthis operator to equations (I) and (3) respectively yields

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X (m) X (m)

0 .... I. eo 1 .'70 3. 10 .... &.40 e.30 ' .20 8 .10

FIG. 4. Inversion results using Method I. The true and apparent chargeability models are shown in (a) and (b). An inversion thatuses sensitivities obtained from the inverted conductivity in Figure 3(c) produces the chargeability model in (c). Discrepanciesbetween (a) and (c) that are related to aspects of the subspace algorithm are inferred from (d). That panel shows the results ofinverting synthetic data obtained by multiplying the sensitivity matrix J by the true chargeability model in (a) and inverting theresultant data. The chargeability models in (e) and (f) are the inversion results by respectively using the sensitivity matrixdeveloped from the true conductivity model and the sensitivity matrix from a half-space.



shown in Figures 6a and 6b. The differences between them

0 .0'" 0 .141 a .2M e.see 0 .131 1.'77 3 .32 I .U 11.7 are subtle and not readily visible to the eye . The differencebetween the conductivity models is shown in Figure 6c, and

.. the chargeability recovered by applying equation (22) isgiven in Figure 6d. It is a rather good representation of the

2• • true chargeability.

!...

Method In: Nonlinear inversionN ...

8• .The third method for inverting IP data is the most preci se

a 18 .8 in that it solves a nonlinear inverse problem. With this"I. approach, there is no necessity for the chargeability to be

- 111 . - ' I , -21 . 21. 61 . 11' . 17.'small. The inverse problem is stated asx (m).. 1&.4

2• •minimize IjJm (TJ , TJo ) = IIWT] (TJ - TJ o)11 2

13 .2 (23)

!... 11 .0

subject to IjJd(d, do) = IIWd(d - do)112 = 1jJ'd ,

N ... where the data d are now the apparent chargeabilities, and8 . 8

81 .the equation for producing the predicted data from any

b 8 . 8 chargeability model TJ is given by equation (6) rather than'U. ... equation (18) as it was with Method I.

·1'., -b' . ...2• . 2' . 61 . ", , The inversion proceeds by linearizing the data equationsx (m )2 .2.. about the current model TJ (n) and computing the sensitivities

2• • J ;j = odj /oTJj ' As in all of our methods, it is understood that

... the background conductivity model has been recovered by a

! previous inversion of the DC resistivit y data. Writing theN 61. data equations as

8• .<l> ~ - <l> ~c

d;= (24)'". <l>~-,... -61 . ...21 . 21. ... 11• .X (m)

and differentiating with respect to the parameter TJj yields

(26)

(25)

0 .800 ..ao ."'0 3 .10 4 .&0 ' .40 ' .30 7 .20 1 .10

FIG. 5. Inversion results from Method I using m = In TJ as thevariable in the objective function . The recovered logarithmicchargeability is shown in (a) and it has been replotted inlinear form in (b). For comparison, the inversion carried outusing m = TJ and setting U s = 0 is shown in (c). Thus (a) and(c) are a direct comparison of the results of using the sameobjective function form but applying it to m = In 1] and tom = TJ, respectively.

(20)

ad ; <l> ~ o<l> ~

oTJj = (<l>~) 2 0TJj .

We therefore need only evaluate o <l> ~/ o TJj ' Writing a =

abO - TJ) as the conductivity that produces the potential<l>T]' then

o<l>~ o<l>~o1]j = -abj oa j == -abj Gij

is simply a scaled value of the sensitivity for a DC resistivityproblem. The final sensitivity is

od j <l> ~J ij = OTJj = -abj ( <l> ~)2 G ij . (27)

To solve the inverse problem, we appeal to the subspacemethodology and carry out an identical procedure used toinvert the potentials <l> rr (see section: Inversion of DCPotentials). In fact , because of the relationship between theIP sensitivities and the sensitivities for the DC problem givenby equation (27), the algorithm to invert DC data requiresonly minor changes so that IP data can be inverted. Wheninverting the IP data we use 23 search vectors: 21 steepestdescent vectors associated with data groupings from eachcurrent electrode, the steepest descent vector associatedwith the model objective function, and a constant vector. Weuse the same model objective function that produced theresults in Figure 4. The positivity constraint in the subspaceformulation has been implemented by the transformation in

(21)

(22)

and

?Ji ic'( <l> rr) - ?Ji dcl(<l> 1])

TJ = ?Ji dcl(<l>rr)

Thus the chargeability model is produced by manipulatingthe conductivities obtained by inverting two sets of potentialdata taken in a DC/IP survey. To implement equation (22),we have inverted accurate <l>" and <l>1] data and have used thesame model objective function that was used to recover themodel in Figure 3c. It is important to minimize the same IjJm;otherwise the nonuniqueness inherent in the inversion canallow radically different conductivities to be obtained andhence produce a poor estimate of TJ. The two inversions are

rr(] - TJ) = ?Ji dcl (<l> 1] ) '

from which it follows that



the Appendix. The result of inverting the test data set,achieved after 23 iterations, is shown in Figure 7b. There hasbeen a good recovery of the chargeability model. A comparison between Figures 4c and 7b quantifies the differencebetween solving a linearized problem and solving the complete nonlinear inverse problem for this example. The differences are small but this might be expected since the maximum chargeability in the model is only 0.15.

NONUNIQUENESS AND HYPOTHESIS TESTING IN IPINVERSIONS

Minimization of a particular objective function provides asingle model from which to make geologic inferences. Without further investigation it cannot be determined whichfeatures of the constructed model are demanded by theobservations and which are the result of the objectivefunction that has been minimized. Some insight can beachieved by altering the objective function and carrying outother inversions. We present two examples that illustrate thepotential usefulness of the flexible model objective functiongiven in equation (12).

In some geologic environments it might be expected thatthe earth has greater continuity in the lateral directioncompared to the vertical direction. To see the effect of thison the current example we carry out the inversion withax = 1.0 and a z = 0.1. The result is shown in Figure 8a.Comparison with Figure 4c shows that the surface structureis smoothed in the horizontal direction and that the buriedanomaly has been elongated. The transition from the bottomof the buried anomaly to the background is more rapid andthe depth extent of the anomaly is reduced. The chargeabil-

ity low between the surface layer and the buried prism isbetter defined. Redoing the inversion with a z = 0.01, asshown in Figure 8b, further enhances these effects. It isnoted that the horizontal extent and thickness of the surfacechargeable layer, and the central position of the buriedanomaly, remain unchanged from those recovered in theprevious inversions. These results suggest that in this example, the locations of the recovered anomalies are primarilycontrolled by the data and not altered by different objectivefunctions. This is further illustrated by the next example.

In the synthetic example, the chargeable prism has beenoffset from the conductive prism. The inverted chargeabilities reflected this shift. For some geologic settings, however,it might be expected that locations of high conductivity andchargeability coincide. To see if this hypothesis is compatible with the data example analyzed here, we have rerun theinversion but have introduced a weighting function thatattempts to make the chargeability coincide with the conductive prism. This is done by introducing weighting functions, derived from the recovered conductivity anomaly,that attempt to force the recovered chargeability anomaly tocoincide with those regions that differ from the backgroundconductivity. To form the weight functions, we first convertthe conductivity model ab to r IT = IloglO(ablaO)I, whereao = 0.01 Sim for this example. This quantity is then scaledto the range of (0.02, 1) and its reciprocal is generated toform w s : The weighting functions w x and w z for the derivative terms are generated in a similar manner from thegradient of rIT so that the low values for wx and wzcorrespond to regions of high gradient in the recoveredconductivity model. As an example, the function w s gener-

2 .88 3 .83 4 .80 8 .81 8 .81 \2 .0 \8 .2 21.8 28 .8 0 .100 0 .300 0 .&00 0 .'700 0 .800 1..0 1.30 1.&0 1.70

..2•.

] ...N 6' .

8• •

eI I ' .

10' . - 181. -68 . -2'. 2• . 6• • '8•.X (m)

..28 .

] ...N 68 .

88 .

d,,,..

nl. -tie . -61 , -211. 2'. U . 118 ,X (m)

0 . 200 0 .100 I." • •30 3 .00 3 .70 .. ..0 lS. l0 ....-21 . 28 . 61 .

X (m)

-2' . 21 . 6• .X (m)

..21 .

] U .

N 6'.

se.

ue .-1"". -6' .

..21 .

] U .

N U .

81 .

UI ,

-,,,". -61 .

2. 88 3.83 4 .80 8 .8\ 8 .81 \2. 0 \8 .2 21. 8 28.8

FIG. 6. Inversion results from Method II: The recovered conductivity structures obtained by inverting <!>IT and <!>Tj are shown in(a) and (b), respectively. The difference is shown in (c), and the recovered chargeability is shown in (d). The gray scales indicateconductivity in mSlm and chargeability in percent.


1336 Oldenburg and LI

ated from the conductivity model in Figure 3c is shown inFigure 9a. The function has two zones of reduced weightscorresponding to the surface resistive zone and the buriedconductive prism. The latter would attempt to center theburied chargeability anomaly at x = O. The model obtainedby carrying out the weighted inversion is shown in Figure 9band this can be compared with the unweighted inversionreplotted in Figure 9c. The attempt to make the chargeabilityhigh coincide with the conductive high has not been successful. There has been some distortion of the contours towardthe conductor, but the center of chargeability high remainsclose to its original and true position of x = -25 m. Thisprovides greater confidence that the location of the chargeable high is demanded by the data and is not an artifact of theobjective function that is being minimized.

PRACTICAL CONSIDERATIONS WITH RESPECT TO FIELDDATA

In developing the IP inversion algorithms we have madethe assumption that the IP response is the apparent chargeability obtained from an idealized time domain experiment.Using Figure 2, the apparent chargeability is defined as theratio of the secondary potential s to the total potentialmeasured <1>1'] that is measured just before the current is cutoff. This is a dimensionless number and usually has a range(0, 0.5). In reality, sand <1>1'] are not recorded directly withthe field acquisition system. In time-domain systems, <1>1'] ismeasured and then either a single sampling of s (t) at sometime beyond the current shutoff, or an integration of s< t)over a time window, is recorded and used as an IP datum. In

a frequency-domain system, sinusoidal current waveformsare input to the ground at a low-frequency We and at a highfrequency W h » and a percent frequency effect (PFE) is usedfor the IP response. In a phase IP system, the phase lagbetween a transmitted sinusoidal current and a receivedvoltage can be measured directly. Although each of thesesystems measures something related to IP potentials, theydo not measure the same quantity nor do any of themmeasure the quantity that is required for our inversions. Infact, the various data, still generally called "apparentchargeability," can be dimensionless or have units of ms ormrad.

To use the algorithms presented here, it is necessary toconvert collected IP data into the form defined byequation (5). The only exception is when relative chargeabilities are required as output; that is, if the final distribution ofchargeabilities can be in error by a constant factor. In suchcases, Method I, which solves a linear problem, can still beused. With that algorithm, if IP data have been scaled by aconstant factor F, then the recovered model will also bescaled by F and the output model has the same units as thedata.

If the output chargeability model is to be calibrated, or ifMethods II or III are used, then the data collected from anyIP experiment must be converted to the response used here.This can be done. A study of the relationships between theseparameters is given in Van Voorhis et al. (1973) and Nelsonand Drake (1973).

Another practical aspect about data collection is that thepotential <1>" is not generally measured. Yet it is this potentialthat should be inverted to recover the background conductivity. Usually the potential <1>1'] is measured and inverted torecover the background conductivity, but the inversion of

1.50 3 .00 4 .50 s.ec 7 . 50 s.oe 10.5 12 .0 13 .5

I .

21 .

!... a.lo

N bI . 7 .20

el . a S.30

nl . 0 .40-1'1 . -61 . -21 . 21. bl . ,,,..

X (m)4 .~

I. 3 .10

21. 2 .70

! '1 . I.SO

N 61. 0 .80

SI . bIII .

-nl. - 61 . -21 . 21 . bI . UI .X em)nl .

118 .

b

a

bl .

bl .

- 21 . 21 .X em)

-21 . 21 .X (m)

-

-se.

-bl .

21 .

I .

I.

SI .

21 .

SI .

111 . L-J... ......... ...L-.wL..............<L..>"-'--'-U...L-"-'-..J....I...L-"-'-..J....I....1-J

-nl .

'"e.-1el .

- .."..sN 61 .

- u ..sN 61.

1. 2 2 . 4 3 .S 4 .8 8 .0 7 .2 8 .4 s.e 10 . 8

FIG. 7. Inversion results from Method III. The true andrecovered chargeabilities are shown in (a) and (b),respectively.

FIG. 8. Exploring nonuniqueness in IP inversions by alteringthe objective function. The model in (a) was obtained bysetting as = 0.0, ax = 1.0, and a z = 0.1. This has anincreased penalty against variation in the horizontal direction and hence the chargeability is elongated horizontally.The effect is increased in (b) where a z = 0.01.



The inversions presented here have concentrated uponusing pole-pole data. The techniques can, without alteration,be used to invert IP data from any electrode geometry . Thefield example shown next inverts dipole-dipole data from agold region in the Abitibi clay belt in Quebec. The geologictarget is a shear-zone-controlled gold mineralization associated with silicification and sulphides. The geophysical interpretation is complicated by the presence of a conductiveoverburden of variable thickness. The overburden maycontain chargeable clays that can dilute or even maskcompletely the IP response from the subcrop. It is also likelythat the effect upon the resistivity caused by silicification inthe subcrop will be confused with the variable thickness ofthe conductive overburden.

It was felt that the geologic structure might be reasonably2-D and hence amenable to inversion with our algorithms. Adipole-dipole survey with 25 m spacings and n = 1 to n = 5was carried out. Figures lOa and lla show the originalpseudosections of apparent conductivity and apparentphase. The main feature in the apparent conductivity sectionis the change in conductivity as one proceeds from theresistive region to the south into the conducting overburdentowards the north. Considering that the apparent conductivity pseudosection is rather smooth, we assume a 5 percenterror in each of the 140 potential data. The 2-D earthstructure is divided into 48 x 27 cells, and so the inverseproblem is to recover the unknown conductivity for each ofthe 1296 cells. The reference model (To is chosen to be auniform half-space of I mS/m. This is based upon the priorknowledge about the deep geological structure from PlacerDome, which indicates resistive country rock. In developingthe objective function [equation (12)], we set the functionswsand w z to unity throughout the model but choose aspecial weighting W x = (Az/8x)2 near the surface. This

FIELD DATA EXAMPLE

We have assumed that data errors are Gaussian and independent. This is surely violated in practice. Also, the " errors" ascribed to the data must reflect not only measurementerrors (additive noise, incomplete removal of electromagnetic coupling, electrode effects, etc .) but also geologic"noise" caused by 3-D bodies and mathematical cellularizations which are too coarse to emulate true geologic variability even if the earth is 2-D. As a consequence , it is notpossible to assign precise standard deviations for the observations. Our approach is pragmatic. For the first inversion,we assign to each datum a standard deviation that is apercentage of the average apparent chargeability . If a modelcan be found that is in acceptable agreement with theseassigned errors, then we adopt this model and carry outanother inversion in which we assign somewhat lower standard deviations to see if more structure can be extractedfrom the data. Alternatively, if the inversion fails to find amodel that adequately reproduces the observations, then weincrease the assigned errors. Contoured maps of misfitsbetween the observed and predicted data are invaluable inassessing appropriateness of assigned standard deviationsand in detecting outliers . In summary, the question of errorassignment is problematic and dictates that the inversion willbe carried out a number of times.0 .030 0 .044 0.01& O.OM 0 .141 0 .201 0 .301 0 .457 0 .1"

I .

21.

!...

N " .II.

a 12.1

111 ,

-" , . - e.l . -2 1 . ~. " . UI . 11.2X (m)

I .' .a

2• • a .4

!.a . 7 .0

N ... a .a

II.

b 4 .2

11• . 2 .a.,... ·.a . -21 . 21. 68. lIa .X (m)

1. 4I .

21 .

!...

N ...II.

C"1 .

.,... .... - 2• . 21• ... 11• •X (m)

O.llOO 1.80 • •70 ' .80 •.se 6 . 40 1 .30 7.20 ' . 10

FIG. 9. Hypothesis testing in IP inversions. A weighingfunction that attempts to move the chargeable zone so that itcoincides with the high conductivity has been incorporatedinto the inversion. The weighting function is shown in (a) andthe recovered model is given in (b). That can be comparedwith the model in (c) which has no preferential weighting.

<1>1] yields the conductivity (T(1 - 'T]) . If'T] is small, then thisdifference may not be important; however, if 11 is large, thena correction may be demanded . Two possible routes exist.The first is to estimate the potentials <1>0' by computing<1>1] - s from the field measurements. Although thisprocedure may enhance the error on the data, it doesproduce a datum that is ideally unaffected by the polarizationof the subsurface. The second method is an iterative correction that could be performed one or more times and consistsof steps: (1) perform a DC inversion on <1>1] to yield (T b1]' (2)use (Tb1] to carry out the IP inversion to recover 'T], (3) forma new background conductivity (Tb = (Tb1]/(l - 'T]), and (4)solve the IP inverse problem again. We have not implemented the above correction procedures on our field dataexample presented in the next section because the chargeabilities are quite small. We do however, recognize itspotential importance for some field data.

A matter of practical importance with respect to any of ourinversion methods is the issue of ascribing errors to the dataand deciding upon the degree to which the data are to be fit.



tends to suppress abrupt horizontal changes that might resultfrom the elongated cells near the surface. The invertedconductivity, achieved after 20 iterations, is shown inFigure lOb. Several features are observed in the model. Themajor conductive overburden begins at 1525, extends northward, and attains a thickness of over 40 m. There is also aconductive layer south of 1525, but its thickness is muchsmaller and variable. The model shows two distinct resistiveregions whose resistivities are almost twice that of thegeneral background. The first is centered at 1525 and thesecond at 1300.

The IP pseudosection in Figure lla displays considerablestructure. There is a well-defined anomaly near 1825, and alarger region from 1300 to 1600 that contains high andvariable chargeabilities. The data provided are apparentphases in milliradians. The IP inversion requires apparentchargeabilities (as per Siegel's definition) as data and invertsfor the intrinsic chargeability model. Since to first orderapparent phase is proportional to apparent chargeability, wehave chosen to use apparent phases as data for the inversion.We use Method I for the inversion and hence the recoveredmodel will also have units of mrad. The apparent phase dataappear to be more noisy than the apparent resistivities, andthis has made the assignment of an appropriate error moredifficult. We have assigned a constant error equal to0.13 mrad to each datum. This corresponds to 5 percent ofthe middle value of the range of the apparent phase. This"estimate" of the error is used in the inversion but, as weshall illustrate, it is likely too small.

The chargeable earth model is discretized into 48 by 27cells so that 1296 unknown intrinsic chargeabilities areestimated from 140 data. The model objective function isidentical to that in the DC inversion except the referencemodel TJo is taken as zero. The target misfit is 140, but thesubspace inversion code was not able to find a model thatreproduced the data to that degree. The model shown inFigure l ld has the smallest misfit obtainable by the algorithm. The misfit is X2 = 1330, and this translates into anaverage misfit of 0.4 mrad rather than 0.13 mrad. There isalways uncertainty about the data errors and equivalently,the degree to which the data should be fit. In general, thefeatures observed in the inversion will decrease in amplitudeas the misfit is allowed to increase. To obtain some confidence about the features observed, we have carried out twomore inversions. The respective misfits were 1400and 1500(i.e., 0.41, and 0.42 mrad), and the models are shown inFigures l lc and l lb. The anomalies become shallower andhave reduced amplitude as the misfit is increased, but thethree major IP bodies are evident in all inversions and thisprovides confidence that the bodies are not artifacts ofoverfitting the data.

Close inspection of Figures lOb and l ld shows that the IPanomalies are offset from the conductivity anomalies. Weemphasize that these results illustrate the need for an algorithm that does not coarsely parameterize the models, butrather allows many cells and finds a solution by minimizingan objective function of the model.

2 .00 2 .51 3 . 16 3 .98 5 .01 6 .31 7 .94 10 .0 12 .6

iNo-0;:lCl)III

0..

25.

35 .

45 .

55 .

65 .

75 .

1175 . 1350 . 1525.X (m)

1700 . 1875 . 2050 .

0 .

40 .

i 80 .

N 120 .

160 .

200 .

1175 . 1350 . 1525 .X (m)

1700 . 1875 . 2050 .

o.70 8 I. 00 1. 41 2 . 00 2 . 82 3 . 98 5 . 62 7 . 94 11. 2

FIG. 10. A dipole-dipole apparent conductivity pseudosection is shown in (a). The inverted conductivity ispresented in (b). Gray scales indicate the conductivity in mS/m.



DISCUSSION

We have presented three methods by which IP data can beinverted. The methods have been illustrated in two dimensions by inverting pole-pole and dipole-dipole data , but noalteration is required to invert any configuration of electrodedata over I-D or 3-D structures . All three methods require aprior inversion of DC potentials to recover a backgroundconductivity ITb : The following summarizes our thoughtsabout the three methods.

Method 1.- The equations are linearized about the background conductivity. An advantage of this approach is that

only a linear inverse problem needs to be solved andpositivity is easily incorporated. The inversion is rapid andcan be carried out many times with different model objectivefunctions or with different estimates of data error and targetmisfits. Additionally, the inversion results are insensitive toscale errors of the data. These are very positive attributes.There is, however , an explicit need to compute and store thesensitivity matrix J. Also, nonlinear effects are unaccountedfor, and these can become important when chargeabilitiesare large.

Method II.- The appeal of this method is that only onecomputing algorithm to invert DC potentials is required . A

4 .67

4 .04

1N

o"tl:l<lltilc,

25.

35 .

45 .

55 .

b5 .

75 .

1175 . 1350 . 1525 .X (rn)

1700. 1875. 2050 .

3 .6\

2 .g8

2 . 46

I. g2

1. 3g

0 .8 80

0 .

40 .

80 .

N 120 .

lb0 .

200 .

0 .330

9 .00

1175 . 1350 . 1525 . 1700. 1875 . 2050 .6.00

0 .

40 .

1 80 .

N 120 .

1b0 .

200 .

1175 . 1350 . 1525 . 1700 . 1875 . 2050 .

7.00

6 .00

5 .00

4 .00

N

0 .

40 .

80 .

120 .

1b0 .

200 .

3 .00

2 .00

1.00

1175 . 1350 . 1525 .X (m)

1700 . 1875 . 2050 .

FIG. II. The dipole-dipole apparent chargeability phase pseudosection ~s gi,:en in (a). Va~ues are in f!lrad.Three inversions carried out at successively smaller misfit levels are grven 10 (b)-(d). Chi-squared misfits,assuming an initi~ standard deviation of each datum of 0.13 mrad, are, respectively, 1500, 1400,and 1330forthe three inversions.



difficulty, however, is a potential lack of robustness arisingbecause the desired chargeability is a small quantity that isobtained by subtracting two large quantities. This requiresthat the inversion algorithm is stable, that is, ~ del must besuch that small changes in the data result in small changes inthe conductivity model. The DC inversion algorithm usedhere is robust in this regard, but we still have had difficultyusing this procedure with error contaminated data. Anotherdifficulty with this method is that the two data sets cj>cr and cj>T)are not generally recorded directly. Rather, cj>T) and a numberrelated to cj>s are acquired in field surveys.

Method III.- This has the advantage that it can handlelarge chargeabilities and that essentially the same algorithmcan be used to invert both the DC and the IP data. This isespecially true if positivity is implemented by using In TJ asthe variable in the inversion code. On the other hand, fieldobservations must be converted to chargeabilities defined bycj>s/cj>T) , and the inversion results are sensitive to any errorsresulting from that transformation. Also, a complete nonlinear inversion must be carried out to compute the chargeability. This is a disadvantage compared to Method I if manyinversions are to be completed with different model objective functions and/or solved according to different misfitlevels.

We have no absolute preference for methodology to invertIP data. Both Methods I and III are viable. From a theoretical viewpoint, Method III is likely the best. However, formost of our inversions, we have used Method I, principallybecause it allows us to bypass the step of converting fielddata to true responses and because we can easily carry out anumber of inversions at different misfit levels and withdifferent objective functions. A similar lack of conclusiveness exists with the choice of whether TJ or In TJ is used as thevariable in the model objective function. For reasons alreadystated in the text, we prefer to use TJ, but images ofln TJ maybe just as geologically interpretable.

The important aspect of our inversion methodology is thatthe generic model objective function that is minimizedprovides great flexibility to generate different models. With aproperly designed objective function it is possible to incorporate additional information about the distribution of conductivity or chargeability and to generate a model that is inaccordance with geologic constraints. Such a model may beregarded as a best estimate for the true earth structure andcan be used in a final interpretation. However, altering theobjective function and carrying out additional inversionsallows exploration of model space and provides an indicationof which features are demanded by the data. These twoaspects of constructing a most-likely model, and exploringthe range of acceptable models, form the foundation of aresponsible interpretation.

ACKNOWLEDGMENTS

We thank Peter Kowalczyk and Placer Dome for supplying the field data set and for discussions concerning its

interpretations. This work was supported by an NSERC lORgrant 5-81215 and an industry consortium "Joint and Cooperative Inversion of Geophysical and Geological Data."Participating companies are Placer Dome, BHP Minerals,Noranda Exploration, Cominco Exploration, FalconbridgeLimited (Exploration), INCO Exploration and TechnicalServices, Hudson Bay Exploration and Development, Kennecott Exploration, Newmont Gold Company, WMC, andCRA Exploration. We thank the University of Waterloo formaking available the MATI Sparse Matrix Solver(D'Azevedo et al., 1991,D'Azevedo, E. F., Kightley, J. R.,and Forsyth, P. A., 1991, MATI iterative sparse matrixsolver: User's guide).

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Inversion of Induced Polarization Data

APPENDIX

INVOKING POSITIVITY IN INVERSE PROBLEMS

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Positivity in subspace solutions is effected in the followingmanner. The reader is referred to Oldenburg and Li (1994)for more details. In an attempt to keep a uniform notation,let the symbol m denote the model for the inverse problem,so m, = TJi in what follows. To invoke positivity, letm, = f(Pi) and Pi = f -l(mi) be mappings connecting thechargeability with new parameters Pi' Let p(n) denote themodel at the nth iteration. The value of the model objectivefunction \)J m evaluated at an updated model p (n) + 8p is

\)Jm(p(n) + 8p) = IIWm[f(p(n) + 8p)] - mo11 2•(A-I)

By performing a Taylor expansion onfat the point p(n) then

The minimization of equation (A-4) subject to \)Jd = \)J'd issolved using the subspace formulation. The perturbation 8pis represented as Y.. and the objective function \)J = \)J m +f1(\)Jd - \)J'd) is solved by minimizing with respect to thecoefficients 0:. This produces equations that are the same asthose in equation (11) with the exception that the matrix yhas been replaced by the matrix Yf.

The mapping used here is a two-segment mapping consisting of an exponential and a linear region

(A-2)

where F is a diagonal matrix with elements {

O'

m = e'' - mb,

(P-Pl + l)e P I -mb,

(A-5)

(A-3)afi ami

F u = -Ip(n) = -Ip(n).Bp, api

A similar Taylor expansion applied to the misfit objectivefunctional yields

where P = P I is the transition point between the twosegments, and m i, is selected to be small enough such thatmodel values smaller than mb are not significantly differentfrom zero.

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Inversion of induced polarization data · Inversion of induced polarization data Douglas W. Oldenburg* and Yaoguo Li* ABSTRACT We develop three methods to invert induced polar ization