Simultaneous reconstruction of 1-D susceptibility and conductivity from electromagnetic data

Transcription

1 GEOPHYSICS, VOL. 64, NO. 1 (JANUARY-FEBRUARY 1999); P , 17 FIGS. Simultaneous reconstruction of 1-D susceptibility and conductivity from electromagnetic data Zhiyi Zhang and Douglas W. Oldenburg ABSTRACT In this paper, we develop an inversion algorithm to simultaneously recover 1-D distributions of electric conductivity and magnetic susceptibility from a single data set. The earth is modeled as a series of homogeneous layers of known thickness with constant but unknown conductivities and susceptibilities. The medium of interest is illuminated by a horizontal circular loop source located above the surface of the earth. The secondary signals from the earth are received by a circular loop receiver located some distance from the source. The model objective function in the inversion, which we refer to as the cost function, is a weighted sum of model objective functions of conductivity and susceptibility. We minimize this cost function subject to the data constraints and show how the choice of weights for the model objective functions of conductivity and susceptibility affects the results of the inversion through 1-D synthetic examples. We also invert 3-D synthetic and field data. From these examples we conclude that simultaneous inversion of electromagnetic (EM) data can provide useful information about the conductivity and susceptibility distributions. INTRODUCTION Both conductivity and susceptibility are important physical parameters. The traditional way to obtain information about susceptibility is through inversion of static magnetic data. Since electromagnetic (EM) surveys are not affected by remanent magnetism and the artificial sources used in EM surveys are highly localized compared to the relatively uniform geomagnetic field in magnetic surveys, EM surveys can provide complementary information about susceptibility. Work has been done to estimate the physical and geometric parameters of some simple models. Ward (1959) describes a method of determining the ratio of magnetic susceptibility of a conducting magnetic sphere to the susceptibility of the background rock. He uses a uniform field for frequencies that span a large range but encompass the critical frequency at which the frequency-independent magnetic field cancels the in-phase component as a result of induced current. Fraser (1973) proposes a way to estimate the amount of magnetite contained in a vertical dike, assuming the body is nonconductive. Fraser (1981) also develops a magnetite mapping technique for the horizontal coplanar coils of a closely coupled multicoil airborne EM system. That technique yields contours of apparent weightpercent magnetite under the assumption that the conductivity of earth is represented by a resistive homogeneous half-space. Although the problem of 1-D inversion of EM data in both the time and frequency domains has been studied extensively in the literature, most of these studies have assumed knowledge of either conductivity or susceptibility. A typical procedure in conductivity inversion is to assume that magnetic susceptibility equals its free-space value. In many cases this assumption is valid because most rocks are nonmagnetic. However, quite often the geological targets are not only conductive but are also permeable, so the data are affected by both conductivity and susceptibility. A common example of the existence of strong magnetization is the negative in-phase coplanar data in airborne EM surveys. Those negative in-phase data result from magnetic polarization, and no pure conductivity model can explain them. If we invert those data under the assumption that µ = µ 0, then the recovered conductivity model will be incorrect and valuable information about susceptibility in the data will also be wasted. On the other hand, when inverting EM data to recover susceptibility, incorrect knowledge about conductivity may also cause severe distortions in the recovered susceptibility models (Zhang and Oldenburg, 1996a). To illustrate this we invert a synthetic data set generated over a 1-D earth with variable conductivity and susceptibility. The data were calculated at 900, 7200, and Hz. The coil separation is 10 m, and the observation height is 30 m. Because of the presence Presented at the 66th Annual Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor May 15, 1997; revised manuscript received June 16, Formerly UBC-Geophysical Inversion Facility, 2219 Main Mall, Vancouver, BC, Canada V6T 1Z4; currently Western Atlas Logging Services, Westheimer, Houston, TX 77042; zhiyiz@sun180.aws.waii.com. Department of Earth & Ocean Sciences, UBC-Geophysical Inversion Facility, 2219 Main Mall, Vancouver, BC, Canada V6T 1Z4; doug@geop.ubc.ca. c 1999 Society of Exploration Geophysicists. All rights reserved. 33

2 34 Zhang and Oldenburg of the susceptibility structure, the in-phase datum at 90 Hz has a negative value of 13.0 ppm. Gaussian noise with standard deviation of about 1% of the data strength is added to the data. In the first inversion we attempt to recover the susceptibility distribution while specifying that the earth s conductivity is a half-space of S/m. The inversion is carried out following the work of Zhang and Oldenburg (1997). The recovered susceptibility in Figure 1a is not a good representation of the true three-layer model. In the next inversion, where we use correct information about the conductivity, the recovered susceptibility model represents the true model very well (Figure 1b). We carry out the same tests on the conductivity inversion. Figure 1c shows the recovered conductivity model under the assumption that µ = µ 0. The recovered model is distorted severely at depth and overshoots the true model. However, when true information about susceptibility is used, the recovered conductivity model in Figure 1d delineates the true model quite successfully. Those results show that in an individual inversion we need accurate information about either susceptibility or conductivity to recover the other. Potentially, DC resistivity data or static magnetic field data could be inverted to provide information about conductivity or susceptibility, respectively. The conductivity could subsequently be used to carry out inversion of frequency- or time-domain EM data to recover susceptibility structure, or the susceptibility could be used when inverting the EM data for conductivity structure. The practical difficulty is that we do not always have DC resistivity or static magnetic data along with frequency- or time-domain EM data. Therefore, the ideal way to attack this problem is to recover both conductivity and susceptibility at the same time through a simultaneous inversion. Simultaneous inversion of EM data has been investigated by only a few authors. Habashy et al. (1986) presents a method of simultaneous reconstruction of dielectric permittivity and conductivity profiles in a cylindrically stratified geometry. They assume that both permittivity and conductivity vary only in the radial direction. The unknown region is an annulus of specified thickness. In their inversion they recover a single complex model parameter m(ɛ, σ). After m(ɛ, σ) is recovered, the permittivity and conductivity are obtained by taking the real and imaginary parts of this complex model. Sena and Toksöz (1990) discuss a similar technique to recover conductivity and permittivity from the inversion of crosshole EM data. Zhang and Oldenburg (1996a) present a nonlinear inverse scheme to recover conductivity and susceptibility through 1-D simultaneous inversions. Qian et al. (1996) propose a method to calculate apparent resistivity and susceptibility based upon a half-space model. Beard and Nyquist (1996) calculate apparent resistivities and susceptibilities from a half-space and two-layer models. FIG. 1. Individual inversions with accurate and inaccurate information about conductivity or susceptibility. Solid lines denote the recovered models; dashed lines denote the true models. (a) The recovered susceptibility from the inversion with inaccurate information about conductivity. The conductivity model used in the inversion was a s/m half-space. (b) Recovered susceptibility from the inversion with accurate information about conductivity. (c) Recovered conductivity from the inversion under the assumption that the susceptibility is equal to its free-space value. (d) The recovered conductivity when accurate information about susceptibility was used in the inversion.

3 Simultaneous Inversion 35 In this paper we present a method to solve the simultaneous inverse problem in a layered earth for a horizontal coplanar EM system. The number of layers chosen is large enough to adequately represent possible conductivity and susceptibility structures. The thickness of each layer is fixed and increases with depth to compensate for the associated loss of resolution of the data because of the attenuation of the EM fields. In each of these homogeneous layers, a pair of model parameters for the conductivity and susceptibility needs to be recovered. Methods similar to the one presented in this paper were previously used in the inversion of either the susceptibility (Zhang and Oldenburg, 1995, 1997) or conductivity structure (Fullagar and Oldenburg, 1984) in a 1-D environment. Here, however, we simultaneously invert for two model parameters, σ and κ. We use a weighted sum of model objective functions of conductivity and susceptibility to construct the cost function. The technique is applied to both synthetic and field data examples. FORWARD MODELING AND CALCULATION OF SENSITIVITIES For a horizontal loop system sitting at a height h 0 above a layered half-space (Figure 2), the vertical component of the secondary magnetic field is given by Ryu et al. (1970): H z (r,ω,z obs ) = 1 jωµ 0 0 E(λ, ω, z obs )λ 2 J 0 (λr) dλ, (1) where J 0 is the Bessel function of zero order and the electric field is given by E(λ, ω, z obs ) = A Z 1 Z 0 Z 1 + Z 0 e 2u 0 h 0 +u 0 z obs, A = jωµ 0aIJ 1 (λa) 2u 0, (2) where j = 1, z obs is the vertical distance between the source and receiver planes, λ is the Hankel transformation parameter, ω is the angular frequency, u 2 0 = λ2 ω 2 ɛ 0 µ 0, a is the radius of the source loop, and I is the amplitude of the current in the source. The input impedance of the first layer, Z 1, can be found by a recursive procedure outlined by Morrison et al. (1969): Z i Z i+1 + Z i tanh(u i h i ) = Z i Z i + Z i+1 tanh(u i h i ), (3) where h i is the thickness of the ith layer, the intrinsic impedance Z i is given by and Z i = jωµ i u i, (4) u 2 i = λ 2 ω 2 ɛ 0 µ i + jωσ i µ i. (5) In the bottom layer, which is actually a half-space, there is no upgoing wave; hence, the input impedance is equal to the intrinsic impedance, Z M = Z M. (6) If the induced voltage is measured, then it is related to the magnetic field by FIG. 2. Geometry of the horizontal coplanar coil system. A horizontal loop of radius a is located at h 0 above the surface of a 1-D earth. The source carries a current Ie iωt. The receiver is separated from the source by a radial distance r and a vertical distance z obs. V (r,ω,z obs ) = jω (r,ω,z obs ) = jωµ 0 H z (r,ω,z obs ) ds, (7) where D is the effective area of the receiver, is the magnetic flux through the receiver, and the time derivative has been replaced by jω since we assume harmonic field of the form e jωt. Usually the raw data are normalized by the corresponding primary field and the final data are given in parts per million (ppm). Since the permeability µ is connected to susceptibility κ via µ = µ 0 (1 + κ), we can work with permeability when computing the sensitivities. The sensitivities for conductivity and permeability are given by Zhang and Oldenburg (1994, 1997): E(λ, ω, z obs ) zi+1 = jωµ i G(λ, ω, z)e(λ, ω, z) dz σ i z i (8) and E(λ, ω, z obs ) µ i [ ] zi+1 = 2u2 i µ i z i + jω( jωɛ 0 + σ i ) D G(λ, ω, z)e(λ, ω, z) dz 2A i b i h i zi+1 z i G(λ, ω, z)e(λ, ω, z) dz, (9)

4 36 Zhang and Oldenburg where A i and b i are the upgoing and downgoing coefficients for the primary field E(λ, ω, z) and the auxiliary field G(λ, ω, z)in the ith layer and ui 2 = λ 2 ω 2 ɛ 0 µ i + jωσ i µ i. The sensitivities for H z are connected to the sensitivities for the electric field through H z (r,ω,z obs ) = 1 E(λ, ω, z obs ) J 0 (λr)λ 2 dλ. m i jωµ i 0 m i (10) In equation (10), m i stands for either σ i or κ i. INVERSION ALGORITHM The goal of the inversion is to find a model that reproduces the data and exhibits desired characteristics. Our choice for the objective function is guided by the desire to find a model that has minimum structure in the vertical direction and at the same time is close to a reference model. To accomplish this we set up the model objective functions for conductivity and susceptibility as [ ( )] σ 2 φ σ = α σ ln dz σ 0 [ ] (ln σ ln σ0 ) 2 + (1 α σ ) dz (11) z and φ κ = α κ [m(κ) m(κ 0 )] 2 dz { } [m(κ) m(κ0 )] 2 + (1 α κ ) dz, (12) z where σ 0 and κ 0 are the reference models for conductivity and susceptibility. The parameters α σ and α κ control the relative importance of smallest and flattest components in the model objective functions. The use of ln (σ ) as the model parameter ensures the recovered conductivity is positive and also accommodates the wide range of conductivity variations. A nonlinear mapping is used to project the susceptibility into m(κ): m b κ<κ b [ ( ) ] κ m(κ) = κ 1 ln + 1 κ b κ κ 1 (13) κ 1 κ κ > κ 1 and where m m m 1 ( mk1 κ(m) = 1) κ 1 e m b < m < m 1, (14) κ b m m b m b = κ 1 [ ln ( κb κ 1 ) ] + 1. (15) In above equations, m 1 = κ 1 and κ 1 and κ b are mapping parameters. Details about this mapping are given in Zhang and Oldenburg (1997). This mapping can provide positivity constraints on the recovered susceptibility model and prevents small values of susceptibility from carrying too much weight in the inversion. For the discrete 1-D inversion, these two model objective functions can be rewritten as ( ) φ σ = σ 2 W σ ln (16) σ 0 and φ κ = Wκ [m(κ) m(κ 0 )] 2, (17) where W σ and W κ are M M weighting matrices. One of the most distinguishing aspects of a simultaneous inversion scheme is that two objective functions, φ σ and φ κ, are to be minimized. We let our final cost function be Coefficients ϱ and γ are given by φ m = ϱφ σ + γφ κ, (18) ϱ = s, γ = s 1 + s, (19) where 0 s is the desired magnifying factor. When s 0,φ m φ σ ; when s,φ m φ κ. Now we solve the simultaneous inverse problem by minimizing equation (18) subject to the constraint that the data are adequately reproduced: minimize φ = (ϱφ σ + γφ κ ) + β 1( φ d φ tar), (20) where β 1 is a Lagrange multiplier and φ tar is the target misfit level. In equation (2), the data objective function φ d is φ d = W d (D obs D) ( ) 2 N 2 d obs l d l =, (21) ξ l=1 l where D obs and D are the observed and predicted data, respectively, and ξ l is the standard deviation of the error associated with the lth datum. We assume that the noise in the data is Gaussian and independent. Let m σ = ln(σ ) and m κ = m(κ) be vectors with M components, and let δm σ and δm κ denote perturbations at the nth iteration. The predicted data can be approximated as D [ m (n) σ + δm σ, m (n) ] κ + δm κ F [ m (n) σ, ] m(n) κ + Jσ δm σ + J κ δm κ, (22) where F is the forward modeling operator and J σ and J κ are sensitivities whose elements are (J σ ) li = d l / m σi and (J κ ) li = d l / m κi. Let J = (J σ, J κ ) be a global sensitivity matrix, m = (m σ, m κ ) be a global model parameter vector, and ( ϱwσ ) 0 W m = (23) 0 γ Wκ be the global weighting matrix. Our linearized problem becomes the minimization of φ = β W m [ δm + m (n) m 0 ] 2 + { W d { D obs F [ m (n)] + Jδm } 2 φ tar(n+1)}, (24) where φ tar(n+1) is the target level for data misfit at the (n + 1)th iteration. Usually we reduce the target misfit level from one iteration to the next by a factor between two and ten. Setting

5 Simultaneous Inversion 37 the gradient δm φ = 0 we obtain the perturbation on the model at the nth iteration: δm = [ βw T m W m + J T W T d W d J ] 1 { J T W T d W dδd n + βw T m W m[ m0 m (n)]}, (25) where δd n = D obs F[m (n) ]. At each iteration a nonlinear line search is required to find the value of β that generates the desired target misfit. If the target misfit level cannot be met, then the β that leads to the minimum misfit level is adopted. At each iteration, the perturbation of the data is caused by the change in conductivity and susceptibility, and the trade-off between these two model parameters is an important issue. Zhang (1997) has attempted to address this trade-off issue. Physically, the secondary field results from both induced eddy currents and magnetic polarization. Eddy currents are related to the term jωµσ. As long as the product of µ and σ in each layer remains the same, the forward response is not affected. Independent information about the susceptibility appears only in the boundary conditions used to calculate the input impedances. The simultaneous inversion is useful only when the secondary fields caused by magnetic polarization become sufficiently large. NUMERICAL RESULTS In the following synthetic examples, the earth is divided into 44 layers to a depth of 500 m. All inversions use the mapped parameter m(κ) as the model parameter, and the mapping parameters κ 1 and κ b are set to 10 3 and The parameters α σ and α κ are chosen as 0.02 for all of the synthetic examples. The starting and reference models for susceptibility are 0.0 and 10 6 SI unit half-spaces, respectively. In the following 1-D examples, the starting and reference models for conductivity are 1 ms/m half-spaces. The same data set for the example in Figure 1 is reinverted to simultaneously recover conductivity and susceptibility. In this inversion we set s = 6. Figure 3 shows the results. The recovered conductivity model in Figure 3a is a good representation of the true model. The recovered susceptibility model slightly undershoots the true model, but it recovers the susceptibility high at the right depth (Figure 3b). Figures 3c,d show the data misfit curve and the data, respectively. The appropriate choice of weighting parameters in equation (18) is important. By increasing the parameter s we give conductivity more freedom to vary; by reducing s, we give susceptibility more room to vary. When s is too large, the susceptibility is overdepressed. On the other hand, if s is set too small, a large susceptibility is generated that may invalidate the linearization and convergence difficulties may occur. We illustrate this by repeating the inversion with s = 20 and s = 0.1. Figure 4 shows the results of inversion with s = 20. The recovered conductivity in Figure 4a is a good representation of the true model. But the susceptibility in Figure 4b is overdepressed into a near-surface layer, and it is not a good representation of the true model. The inversion converged after six iterations and fit the data to the desired misfit level. Figures 4c,d present the misfit curve and the data. When s = 0.1 is used in the inversion, the conductivity is recovered successfully (Figure 5a), but the recovered susceptibility in Figure 5b overshoots the true model significantly. Starting from the ninth iteration, the inversion cannot reduce the misfit any further (Figure 5c). The real component of the data at 900 Hz is fit to about 7% instead of the desired level of 1% of the amplitude of the data (Figure 5d). This is because too little weighting has been given to susceptibility; hence, the susceptibility changes too much at the final stage of the inversion. The 1-D inversions above, and other synthetic modeling, indicate that conductivity, compared to susceptibility, is relatively insensitive to the choice of the value of parameter s. Because the data are more sensitive to the change of susceptibility, more weight should be applied to the model objective function for susceptibility. The appropriate choice of s is model dependent, and several test runs may be needed to decide the right weighting. From our experience and for the type of data considered here, a value between 2 and 20 is a good starting point. Geological targets usually are 3-D, yet full 3-D inversion is still computationally prohibitive. As a first step toward interpretation, we can apply a 1-D algorithm to 3-D data. In the following example, we investigate if 1-D inversion of a 3-D data set is justified. This 3-D data set was generated from the model shown in Figure 6 by Newman and Alumbaugh with a staggered finite-difference method (Newman and Alumbaugh, 1996). The background conductivity and susceptibility are 0.01 s/m and 0 SI units, respectively. The conductivity in the upper prism is 0.1 s/m, and susceptibility is 0.1 SI units. In the lower prism, the conductivity is 0.5 s/m and the susceptibility is 0.2 SI units. The observation height is 30 m, and the coil separation is 10 m. Data were calculated at ten frequencies, ranging from 110 to Hz, with equal logarithmic spacing. Line spacing is 50 m, and station interval is 25 m. The weighting parameter s was set to 3. The standard deviations were 5 ppm plus 10% of the data. Other parameters were kept as in previous 1-D examples. Figures 7 and 8 plot the real and imaginary components of the predicted and observed data at 110, 7040, and Hz. The data clearly indicate the presence of two anomaly bodies. But it is difficult to determine whether those two bodies are conductive, resistive, or permeable, and it is even more difficult to estimate the depths. In the inversion, the data were all fit to the desired misfit level (including the negative in-phase data at 110 Hz). The positions of the two prisms are denoted by white rectangles. The simultaneous inversion recovered two conductive and permeable bodies at depth. Figure 9 shows the crosssections of the conductivity at y = 250, 350, and 450 m, and Figure 10 presents the corresponding cross-sections of susceptibility. The recovered conductivity represents the true model reasonably well, even though it is shallower and wider than the true model. The recovered susceptibility is also a reasonable representation of the true model. Those two prisms are clearly separated from each other in the recovered susceptibility model. In contrast to conductivity, the recovered susceptibility body extends deeper than the true model. Figures 11a c show the x-y plan views of the recovered conductivity at 20, 40, and 60 m depth, and Figures 11d f plot the corresponding x-y plane views of the recovered susceptibility. At all corresponding depths, the susceptibility is more localized than the conductivity. As a whole, the 1-D simultaneous inversion of the 3-D data set has generated conductivity and susceptibility models that represent the true model reasonably well. Two conductivity

6 38 Zhang and Oldenburg FIG. 3. The result of the simultaneous inversion with s = 6. (a) The true and recovered conductivity models. The solid line denotes the recovered model, and the dashed line denotes the true model. (b) The true (dashed line) and reconstructed (solid line) susceptibility models; (c) The data-misfit curve. (d) The real (solid line) and imaginary (dashed line) components of the predicted and observed data. Lines denote predicted data; dots denote observed data. FIG. 4. The results from the inversion with s = 20. (a) The true and recovered conductivity models. The solid line denotes the recovered model, and the dashed line denotes the true model. (b) The true (dashed line) and reconstructed (solid line) susceptibility models. (c) Data misfit as a function of iteration. (d) The predicted and observed data. The solid line denotes the real component of the predicted data, and the dashed line denotes the imaginary component of the predicted data. Observed data are denoted by dots.

7 highs and two separated susceptibility highs over the tops of the two prisms are recovered. The recovered susceptibility has greater depth extent than the true model, while the recovered conductivity is shallower and thinner than the true model. The recovered susceptibility model also has better horizontal separation than the one for recovered conductivity. Simultaneous Inversion 39 FIELD DATA EXAMPLE In the following example, we invert field data collected over the Stratmat Main Zone, located 40 km southwest of the city of Bathurst and 2 km north of the Heath Steele Mine site in northern New Brunswick. The area is underlain by felsic to mafic and metasedimentary rocks of the Ordovician-age Tetagouche Group and is host to several major polymetallic base metal sulfide deposits. A large metagabbro intrusion is adjacent to, and in some places assimilates, the sulfide deposits. The volcanic rocks are very resistive, while the massive sulfide deposits are very conductive. Therefore, EM surveys register a strong anomaly over the deposits. The magnetic minerals in the gabbroic dykes are expected to affect the EM data, too. This Main Zone was discovered in 1957 as the result of follow-up surveys of an airborne EM anomaly. More recently, an Aerodat airborne EM system was flown over this region. The coaxial data were collected at frequencies of 935 and 4600 Hz, and the coplanar data were measured at 4175 Hz, with the same coil separation of 7 m. The line interval is about 200 m, and the station spacing is about 8 m. There are 696 stations total. The flight height of the bird is between 20 and 50 m. Since the coplanar data were measured at only one frequency, we need to include the coaxial data in the inversion to obtain better depth resolution. Zhang et al. (1996b) propose a simple method to construct inverse algorithms for the coaxial, perpendicular, and vertical coplanar data, based upon FIG. 6. The x-y plan view and the x-z cross-section of the 3-D model. The background conductivity and susceptibility σ 0 and κ 0 are 0.01 s/m and 0 SI units, respectively. The conductivity and susceptibility in the upper prism are 0.1 s/m and 0.1 SI units. For the lower prism, σ 2 and κ 2 are 0.5 s/m and 0.2 SI units, respectively. Data are calculated at 110, 220, 440, 880, 1760, 3520, 7040, , and Hz. The coil separation is 10 m, and the flight height is 30 m. The station interval is 25 m, and the line spacing is 50 m. Flight lines are in the x-direction. FIG. 5. The inversion with s = 0.1. In (a) and (b), the solid lines denote the recovered model, and the dashed lines denote the true model. (a) The true and recovered conductivity models. (b) The true and recovered susceptibility models. (c) The data-misfit curve. (d) The real component of the predicted data (solid line) and the imaginary component of the predicated data (dashed line). Dots denote the observed data.

8 40 Zhang and Oldenburg FIG. 7. The real components of the predicted and observed data for the 3-D model. Panels a, b, and c show the predicted data at 110, 7040, and Hz. Panels d, e, and f plot the observed data at the same frequencies. FIG. 8. The imaginary components of the predicated and observed data at 110, 7040, and Hz for the 3-D model. Panels a, b, and c show the predicted data. Panels d, e, and f plot the observed data.

9 Simultaneous Inversion 41 FIG. 9. The x-z cross-sections of the recovered conductivity from the inversion of the 3-D data. White rectangles denote the positions of those two prisms. Panels a, b, and c show the x-z cross-section at y = 450, 350, and 250 m, respectively. FIG. 10. The cross-sections in the x-direction of the recovered susceptibility. The positions of the prisms are denoted with white rectangles. Panels a, b, and c show the cross-sections at y = 450, 350, and 250 m, respectively.

10 42 Zhang and Oldenburg existing inverse algorithms for the coplanar data. Following their work, we adapted our simultaneous inverse algorithm to invert the coplanar and coaxial data jointly. In carrying out the inversions, the earth was divided into 50 layers to a depth of 300 m. The model objective function was that given in equation (20). Parameters α σ and α κ were set to The reference model was a conductive and magnetic half-space whose conductivity and susceptibility were 0.1 ms/m and 10 6 SI units, respectively. The starting model was a nonmagnetic half-space with 1 ms/m conductivity. The error assigned to the data was 1 ppm plus 10% of the data strength. The parameter s was set to three and was fixed throughout the inversion. After ten iterations at each station, the total chi-squared misfit level at all the stations was reduced to 4265 from the accumulative initial misfit of Figures 12 and 13 show the real and imaginary components of the predicted and observed data from the inversion. Both components of the data show a distinct anomalous high at the center of line 12.8 km, where the Main Zone resides. The negative real component of the coplanar data is a manifestation of the existence of magnetization in this region and justifies the necessity for a simultaneous inversion. The recovered models at each station are assembled to form a 3-D model. Figure 14a shows the x-y plan view of the recovered conductivity at 30 m. A conductivity high is recovered between stations at m and m of line m. Figure 14b shows the y-z cross-section of the recovered conductivity at line m. The white line in Figure 14a indicates where the section in Figure 14b is taken. The recovered model suggests that the conductive body has a depth extent of about 150 m. While this depth extent may be true, this is also about the maximum limit of depth penetration of the airborne EM system in use, and termination of the conductor might be a consequence of the objective function being minimized. The data from an aeromagnetic survey have been inverted to recover a 3-D model of susceptibility (Li et al., 1996). Figure 15 FIG. 11. The x-y plan views of the recovered conductivity and susceptibility. Panels a, b, and c show the recovered conductivity at z = 20, 40, and 60 m. Panels d, e, and f present the susceptibility models at the corresponding depths.

11 Simultaneous Inversion 43 FIG. 12. The real component of the predicted and the observed data at Heath Steele Stratmat (HSS). (a, b) Predicted coaxial data; c) Predicated coplanar data. (d f) Corresponding observed data. FIG. 13. The imaginary component of the predicted and the observed data at HSS. (a, b) Predicted coaxial data. (c) Predicted coplanar data. (d f) Corresponding obser ved data.

12 44 Zhang and Oldenburg shows the aeromagnetic data. The average sampling rate and line interval are 8 and 200 m, respectively. The data show a general pattern of magnetic anomaly that is consistent with the negative in-phase data shown in Figure 7. We plot the recovered susceptibility model from our 1-D simultaneous inversion along with that obtained from the 3-D inversion of the aeromagnetic data. Figure 16 shows the plan sections of the recovered 3-D susceptibility at 20, 30, and 100 m. Figures 16d f plots the susceptibility recovered from the simultaneous inversion. It is unclear which recovered susceptibility model is more correct, but the major feature of the susceptibility recovered from the simultaneous inversion is generally consistent with the result from the inversion of aeromagnetic data in Figures 16a c. The anomalies Figures 16d f appear as isolated peaks because of the sparse line spacing. Two major susceptibility highs one in the north and one in the south are registered by the simultaneous inversion. But the anomaly centered at station of line m in Figures 16a c does not appear in the recovered model from the simultaneous inversion. This is because the signal level needed to resolve this missing susceptibility structure is completely overwhelmed by the signal related to the eddy currents. Based upon forward modeling, we estimate that the susceptibility causes a response of 1 ppm for the real component of the coaxial data at both frequencies and 9 ppm for the real component of the coplanar data. Yet the maximum values for the real components of the coaxial data at 935 and 4600 Hz are 45 and 62 ppm, respectively, and for the coplanar data the peak value is 287 ppm. In order not to over-fit the data contaminated by the 3-D effects, we used 10% of the data strength FIG. 14. The recovered conductivity model from the simultaneous inversion. (a) Plan section of the model at 30 m. (b) Cross-section of the model at line m. The white line in (a) denotes the position of the section in (b).

13 Simultaneous Inversion 45 FIG. 15. The aeromagnetic data at HSS, measured in nanoteslas. as the error in the inversion, so the signal resulting from the susceptibility is smaller than the assigned noise level. Figure 17 shows the cross-sections of the recovered susceptibility along the survey line m. Figure 17a presents the results from the simultaneous inversion; Figure 17b plots the corresponding section of the recovered susceptibility from the inversion of the aeromagnetic data. The two susceptibility highs around stations and m in Figure 17a correspond to the anomalous highs in Figure 17b, even though the left susceptibility high in Figure 17b shifts leftward slightly. The recovered model from the simultaneous inversion indicates the anomalous bodies extend to depths of about 150 m. Another anomalous high, which does not show in Figure 17b, is recovered at the right end of the section in Figure 17a. The 1-D simultaneous inversion has generated 3-D images of susceptibility and conductivity. The recovered conductivity model not only clearly indicates the existence of a conductor at the location of the Main Zone but also suggests that the anomaly is very conductive and has a depth extent of about 150 to 200 m. This depth extent is about the maximum depth of FIG. 16. The plan views of the recovered susceptibility in msi units from the inversions of aeromagnetic data and airborne EM data. (a c) Recovered model from the inversion of the aeromagnetic data. (b f) Corresponding sections of the recovered model from the simultaneous inversion.

14 46 Zhang and Oldenburg FIG. 17. The cross-section of the recovered susceptibility model at line m from the inversion of the aeromagnetic data and the simultaneous inversion of the airborne EM data. (top) Recovered model from the simultaneous inversion. (bottom) Recovered model from the inversion of the aeromagnetic data. investigation for most airborne EM systems. The recovered susceptibility from the simultaneous inversion bears similarities to that reconstructed from the inversion of the aeromagnetic data. The agreement in the susceptibility models recovered from airborne EM and aeromagnetic data indicates the target does not have strong remanence. SUMMARY Formulation of the simultaneous inverse problem requires minimization of an objective function, including both conductivity and susceptibility φ = ϱφ σ + γφ κ. By choosing different weighting, we can obtain different solutions. Choice of the final relative weighting between φ σ and φ κ requires additional input or knowledge on the part of the user. Compared to susceptibility, conductivity is more robust with respect to the change of weighting parameter. Tests on both 1-D and 3-D synthetic data sets suggest that simultaneous inversion is a useful tool in providing information about conductivity and susceptibility distributions. The recovered susceptibility model from the inversion of EM data is not affected by magnetic remanence. The results from the inversion of field data at Heath Steele Stratmat are encouraging. By inverting a single EM data set, we recovered not only the conductive deposit at the Main Zone but also the susceptibility structure associated with the gabbroic dyke. The negative inphase data are no longer a source of contamination; they have become an important source of information about susceptibility distribution. ACKNOWLEDGMENTS We sincerely thank Gregory A. Newman and David L. Alumbaugh from Sandia National Laboratories for providing 3-D synthetic data sets. This research was supported by an NSERC IOR grant and an industry consortium, Joint and Cooperative Inversion of Geophysical and Geological Data. The following companies take part in this consortium: Placer Dome, BHP Minerals, Noranda Exploration, Cominco Exploration, Falconbridge, INCO Exploration & Technical Services, Hudson Bay Exploration & Development, Kennecott Exploration Company, Newmount Gold Company, Western Mining Corporation, and CRA Exploration Pty. REFERENCES Beard, L. P., and Nyquist, J. E., 1996, Inversion of airborne electromagnetic data for magnetic permeability: the 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Fraser, D. C., 1973, Magnetite ore tonnage estimates from an aerial electromagnetic survey: Geoexploration, 11, , Magnetite with a multicoil airborne electromagnetic system: Geophysics, 46, Fullagar, P. K., and Oldenburg, D. W., 1984, Inversion of horizontal loop electromagnetic frequency soundings: Geophysics, 49, Habashy, T. M., Chew, W. C., and Chow, E. Y., 1986, Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab: Radio Sci., 21, Li, Y., Oldenburg, D. W., Farquharson, C., and Shekhtman, R., 1996, Inversion of geophysical data sets at Heath Steele Stratmat: Annual report for 1995, prepared for JACI Consortium, Appendix A. Morrison, F. H., Phillips, R. J., and O Brien, D. P., 1969, Quantitative interpretation of transient electromagnetic fields over a layered halfspace: Geophys. Prosp., 17, Newman, G. A., and Alumbaugh, D. L., 1995, Frequency-domain modelling of airborne electromagnetic responses using staggered finite differences: Geophys. Prosp., 43, Qian, W., Gamey, J., Lo, B., and Holladay, J. S., 1996, AEM apparent resistivity and magnetic susceptibility calculation: the 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Ryu, J., Morrison, F. H., and Ward, S. H., 1970, Electromagnetic fields about a loop source of current: Geophysics, 35, Sena, A. G., and Toksöz, M. N., 1990, Simultaneous reconstruction of permittivity and conductivity for crosshole geometry: Geophysics, 55,

15 Simultaneous Inversion 47 Ward, S. H., 1959, Unique determination of conductivity, susceptibility, size, and depth in multifrequency electromagnetic exploration: Geophysics, 24, Zhang, Z., 1997, Reconstruction of conductivity and susceptibility from the inversion of EM data: Ph.D Thesis, Univ. of British Columbia. Zhang, Z., and Oldenburg, D. W., 1994, Inversion of airborne data over a 1-D earth: Annual report for 1993, JACI consortium meeting, Appendix H. 1995, The inversion of AEM data to recover magnetic susceptibility structure in a 1-D environment: at the 66th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, a, Simultaneous reconstruction of 1-D susceptibility and conductivity from EM data: Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, b, Reconstruction of 1-D conductivity from coaxial, coplanar, vertical coplanar and perpendicular EM data: 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, , Recovering magnetic susceptibility from electromagnetic data over a 1-D earth: Geophys. J. Internat., 130,