GEOPHYSICS, VOL. 65, NO. 6 (NOVEMBER-DECEMBER 2000); P. 1931–1945, 16 FIGS., 2 TABLES.
3-D inversion of induced polarization data Yaoguo Li∗ and Douglas W. Oldenburg‡ In application, the technique has matured sufficiently that it is now routinely applied to data sets acquired in mineral exploration projects and in environmental problems. The 2-D IP data are commonly inverted using a linearized approach (LaBrecque, 1991; Oldenburg and Li, 1994), in which the chargeability is assumed to be relatively small and the apparent chargeability data are expressed as a linear functional of the intrinsic chargeability. A linear inverse problem is solved to obtain the chargeability model. In addition to the linearized approach, Oldenburg and Li (1994) also propose two other methods. The second obtains the chargeability by performing two separate dc resistivity inversions and then taking the relative difference of the recovered conductivities. The third method makes no assumption about the magnitude of the chargeability and performs a full nonlinear inversion to construct its distribution. The effectiveness of IP inversions has been documented in several case histories (e.g., Oldenburg et al., 1997; Kowalczyk et al., 1997; Mutton, 1997). When the data set is acquired in a truly 2-D environment, the inversion algorithm has performed well. However, 2-D inversions face difficulties when the basic 2-D assumption is violated because of the use of 3-D acquisition geometry or the presence of a 3-D geoelectrical structure such as severe 3-D topography or 3-D variation of conductivity and chargeability. Under these circumstances, a 3-D algorithm is required. The methods developed in 2-D are general and applicable to 3-D problems. For instance, recovering chargeability by computing the difference between two conductivity inversions is demonstrated by Ellis and Oldenburg (1994) using pole–pole data. The implementation of the linearized approach is also straightforward in principle; however, numerical and computational challenges require specific treatment. The foremost challenge is the computational complexity related to generating background conductivity in three dimensions and the solution of the large-scale constrained minimization problem to construct the 3-D chargeability model. This paper concentrates on these associated computational issues. We assume that the
ABSTRACT
We present an algorithm for inverting induced polarization (IP) data acquired in a 3-D environment. The algorithm is based upon the linearized equation for the IP response, and the inverse problem is solved by minimizing an objective function of the chargeability model subject to data and bound constraints. The minimization is carried out using an interior-point method in which the bounds are incorporated by using a logarithmic barrier and the solution of the linear equations is accelerated using wavelet transforms. Inversion of IP data requires knowledge of the background conductivity. We study the effect of different approximations to the background conductivity by comparing IP inversions performed using different conductivity models, including a uniform half-space and conductivities recovered from one-pass 3-D inversions, composite 2-D inversions, limited AIM updates, and full 3-D nonlinear inversions of the dc resistivity data. We demonstrate that, when the background conductivity is simple, reasonable IP results are obtainable without using the best conductivity estimate derived from full 3-D inversion of the dc resistivity data. As a final area of investigation, we study the joint use of surface and borehole data to improve the resolution of the recovered chargeability models. We demonstrate that the joint inversion of surface and crosshole data produces chargeability models superior to those obtained from inversions of individual data sets.
INTRODUCTION
In recent years, there has been much progress in rigorous inversion of induced polarization (IP) data assuming a 2-D earth structure. Published work on 2-D inversions has demonstrated that inversion can help extract information that is otherwise unavailable from direct interpretation of the pseudosections.
Manuscript received by the Editor February 23, 1999; revised manuscript received June 2, 2000. ∗ Formerly University of British Columbia, Department of Earth and Ocean Sciences; presently Colorado School of Mines, Department of Geophysics,1500 Illinois St., Golden, Colorado 80401. E-mail:
[email protected]. ‡University of British Columbia, Department of Earth and Ocean Sciences, 2219 Main Mall, Vancouver, B.C. V6T1Z4, Canada. E-mail: doug@eos. ubc.ca. c 2000 Society of Exploration Geophysicists. All rights reserved. 1931
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chargeability is small and that the data are not affected by EM coupling effect. Therefore, we adopt the linearized representation of the IP response and develop the inversion methodology applicable for general electrode configurations, including surface arrays, downhole arrays, and crosshole electrode configurations. We present a detailed algorithm that solves large-scale problems. Our paper begins with a summary of the basics of IP inversion and the formulation of the inverse solution. The use of approximate conductivity models in the 3-D IP inversion is discussed next to demonstrate how an efficient IP solution can be obtained in practice. We then study the joint inversion of surface and crosshole data and its improvement in model resolution. We conclude with an application to a field data set and a discussion. BACKGROUND
The commonly used electrode configurations in most exploration work include the pole–pole, pole–dipole, dipole–dipole, and gradient arrays. These arrays are usually arranged in a colinear configuration, and the source and potential electrodes are generally aligned parallel to the traverse direction. However, to image a 3-D structure, truly 3-D data are often needed. This requires that off-line or cross-line data be acquired and that the orientation of the current electrodes be varied. In addition, high-resolution surveys carried out in ore delineation and geotechnical investigations often acquire surface-to-borehole and crosshole data in three dimensions. Thus, a generally applicable inversion algorithm must be able to work with arbitrary electrode configurations. In this paper, we assume that the time-domain IP measurements are acquired using an arbitrary electrode geometry over a 3-D structure. The current source can be a single pole, dipole, or widely separated bipole either on the earth’s surface or in boreholes. The resulting potential or potential difference can be measured as data anywhere on the surface or in the borehole. The commonly used pole–pole, pole–dipole, and dipole–dipole arrays on the surface or in the borehole constitute only a small number of possible configurations. Let σ (r) be the conductivity as a function of position in three dimensions beneath the earth’s surface and η(r) be the chargeability as defined by Seigel (1959). The dc potential produced by a current of unit strength placed at rs is governed by the partial differential equation
∇ · (σ ∇φσ ) = −δ(r − rs ),
(1)
where φσ denotes the potential in the absence of IP effect. When the chargeability is nonzero, it effectively decreases the electrical conductivity of the media by a factor of (1−η)(Seigel, 1959). The corresponding total potential φη is given by
∇ · (σ (1 − η)∇φη ) = −δ(r − rs ).
(2)
Thus, the secondary potential measured in an IP survey is given by the difference
φs = φη − φσ ,
(3)
while the apparent chargeability is defined
ηa =
φη − φσ . φη
(4)
The apparent chargeability is the preferred form of IP data, and it is well defined in some surface and downhole surveys.
However, in crosshole experiments using dipole sources or receivers, the electric field often reverses direction along the borehole, and the measured total potential differences can approach zero in the vicinity of the zero crossing. The zero crossing can also occur with noncolinear arrays on the surface. These near-zero potentials cause the apparent chargeability to be undefined. It is therefore necessary to use the secondary potential as data when these conditions occur. When the magnitude of the chargeability is moderate, the secondary potential φs measured in an IP experiment is well approximated by a linear relationship with the intrinsic chargeability. Applying a Taylor expansion to equation (3), neglecting higher order terms, and discretizing the earth into cells of constant conductivity σ j and chargeability η j results in the following equation (e.g., Seigel, 1959; Oldenburg and Li, 1994):
φsi =
M
−η j
j=1
M ∂φηi φ ≡ η j Ji j , ∂ ln σ j j=1
(5)
where Jiφj is the sensitivity of the secondary potential φsi and φηi is the corresponding total potential. If the total potentials do not approach zero, the linearized equation for apparent chargeability ηa is given by
ηai =
M j=1
−η j
M ∂ ln φηi η ≡ η j Ji j , ∂ ln σ j j=1
(6)
where Jiηj is the corresponding sensitivity. Note that Jiηj is undefined when φηi approaches zero. Given a set of measured IP data, inversion of either equation (5) or (6) allows the recovery of the intrinsic chargeability model. Since the true conductivity structure is unknown in practical applications, an approximation to it is substituted in calculating the sensitivities. This approximation is usually obtained by inverting the accompanying dc potential data. Thus, the IP inverse problem is a two-stage process. In the first stage, an inverse problem is solved to recover a background conductivity from the dc resistivity data. This conductivity is then used to generate the sensitivity for the IP inversion, and a linear inverse problem is solved to obtain the chargeability. FORMULATING THE INVERSION
Assume we have a set of N IP data, which can be apparent chargeabilities or secondary potentials. Further assume that a dc resistivity inversion has been performed (see next section) to obtain a reasonable approximation to the true conductivity; the IP sensitivity is calculated from it. To invert these IP data for a 3-D model of chargeability, we first use the same mesh as in the dc resistivity inversion to divide the model region into M cells and assume a constant chargeability value in each cell. The data are formally related to the chargeabilities in the cells by the relation in equations (5) and (6),
d = Jη ,
(7)
where the data vector d = (d1 , . . . , d N )T and the model vector η = (η1 , . . . , η M )T . J is the sensitivity matrix corresponding to the data, whose elements Ji j are calculated from the assumed approximation to the background conductivity by using an adjoint equation approach (McGillivary and Oldenburg, 1990). For this calculation and all the numerical simulations in this
3-D Inversion of IP Data
paper, we use the finite-volume method (Dey and Morrison, 1979) to solve equation (1) to obtain the electrical potentials. The number of model cells is generally far greater than the number of data available; thus, an underdetermined problem is solved. To obtain a particular solution, we minimize a model objective function, subject to the data constraints in equation (7). We used a model objective function that is similar to that for the 2-D case but that has an extra derivative term in the third dimension. Let m = η generically denote the model. The objective function is given by
∂(m − m 0 ) 2 dv ∂x V V ∂(m − m 0 ) 2 ∂(m − m 0 ) 2 dv + αz dv, + αy ∂y ∂z V V (8)
ψm = α s
(m − m 0 )2 dv + αx
where m 0 is a reference model. The positive scalars αs , αx , α y , and αz are coefficients that affect the relative importance of the different components. We usually choose αs to be much smaller than the other three coefficients, so the recovered model becomes smoother as the ratios αx /αs , α y /αs , and αz /αs increase. For numerical solutions, equation (8) is discretized using the finite-difference approximation. The resulting matrix equation has the following form;
ψm = (m − m0 )T αs WsT Ws + αx WTx Wx + α y WTy W y + αz WzT Wz m − m0 ) ≡ Wm (m − m0 )2 .
(9)
The data constraints are satisfied by requiring that the total misfit between the observed and predicted data be equal to a target value. We measure the data misfit using the function
2 ψd = Wd d pr e − dobs ,
(10)
where d pr e and dobs are, respectively, predicted and observed data and where Wd is a diagonal matrix whose elements are the inverse of the standard deviation of the estimated error of each datum: Wd = diag{1/1 , . . . , 1/ N }. If we assume that the contaminating noise is independent Gaussian noise with zero mean, then ψd has X 2 distribution with N degrees of freedom, and its expected value is equal to N . Thus, a reasonable target value is ψd = N . In addition to the data constraints, we also need to impose a lower and an upper bound on the recovered chargeability. The bounds are required because the chargeability is defined in the range [0,1). The bound constraints ensure that the recovered model is physically plausible. For numerical implementation, the lower bound must be zero since the chargeability of the general background is zero. The upper bound, denoted by u, can take on the theoretical value of unity or can be smaller if a better estimate of the upper bound is known. Having defined the model objective function, the data misfit and its expected value, and the appropriate bounds, we now solve the inverse problem of constructing the 3-D chargeability model by the Tikhonov regularization method (Tikhonov and Arsenin, 1977) with additional bound constraints:
minimize ψ = ψd + µψm subject to 0 ≤ m < u,
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where µ is the regularization parameter that controls the tradeoff between the model norm and misfit. Ultimately, we want to choose µ such that the data misfit function is equal to a prescribed target value ψd . The minimization is solved when a minimizer m is found whose elements are all within the bounds. This is a quadratic programming problem, and the main difficulties arise from the presence of the bound constraints. We use an interior-point method to perform the minimization. The original problem in equation (11) is solved by a sequence of nonlinear minimizations in which the bound constraints are implemented by including a logarithmic barrier term in the objective function (e.g., Gill et al., 1991; Saunders, 1995):
mj B(m, λ) = ψd + µψm − 2λ ln u j=1 M
mj ln 1 − + u j=1
,
(12)
where λ is the barrier parameter and the regularization parameter µ is fixed during the minimization. The minimization starts with a large λ and an initial model whose elements are well within the lower and upper bounds. It then iterates to the final solution as λ is decreased toward zero. As λ approaches zero, the sequence of solutions approaches the model that minimizes the original total objective function ψ in equation (11). Since we are only interested in the final solution, we do not carry out the minimization completely for each value of λ in the decreasing sequence. Instead, we take only one Newton step and limit the step length during the model update to keep the model within the bounds throughout the minimization. The steps of the algorithm are as follows: 1) Set the initial model m and the µ, and calculate the starting value of the barrier parameter by
λ= −2
M j=1
ψd + µψm
.
mj mj + ln 1 − ln u u
(13)
2) Take one Newton step for each value of λ by solving the following equation for a model perturbation m:
JT WdT Wd J + µWmT Wm + λX−2 + λY−2 m
= −JT WdT Wd (d − dobs ) − µWmT Wm (m − m0 ) + λ(X−1 − Y−1 )e,
(14)
where X = diag{m 1 , . . . , m M }, Y = uI − X, and e = (1, . . . , 1)T . 3) Determine the maximum step length of the model update that satisfies the bounds:
mj , m j 0
(11)
M
u −mj , m j
ρ = min(ρ− , ρ+ ).
(15)
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4) Update the model and barrier parameter by the limited step length:
m ← m + γρ m, λ ← [1 − min(γ , ρ)]λ.
(16)
5) Return to step 2 and iterate until convergence according to the criteria that ψd + µψm has reached a plateau and the barrier term is much smaller than this quantity. The parameter γ is usually prescribed to be a value close to unity. Its role is to prevent model elements from reaching the bounds exactly so that the logarithmic barrier iteration can continue. Values between (0.99, 0.999) have been commonly used in literature (e.g., Gill et al., 1991). Our experience with IP inversions suggests that a slightly smaller value works better, so we have typically used 0.925 in our algorithm. The central task of the algorithm is solving the linear system in equation (14). We obtain the solution by using the conjugate gradient (CG) technique. The reasons for using a CG solver are twofold. First, any practical application will require a large number of cells (at least on the order of 104 ) to represent the geology reasonably. As a result, the linear system in equation (14) is large, and explicit formation of JT WdT Wd J is impractical. This precludes the use of any direct solver. The CG technique is the obvious choice for an iterative solver since the matrix (JT WdT Wd J + µWmT Wm ) is symmetric. Also, each subproblem at a given value of barrier parameter only generates one step among a sequence that leads to the final solution. It is unnecessary to solve equation (14) precisely. Instead, it is common to solve the central equation approximately to produce a partial solution. The resulting update is called a truncated Newton step. This is designed to reduce the required amount of computation without compromising the quality of the final solution. The CG technique can be terminated at an early stage by supplying it with a relaxed stopping criterion. We have typically used a criterion that the ratio of the norm of the residual and the norm of the right-hand side in equation (14) be less than 10−2 . This has led to large computational savings. CG iterations require the repeated multiplication of the matrix JT WdT Wd J and WmT Wm to vectors. The matrix WmT Wm is extremely sparse, and the matrix–vector multiplication is easily obtained. However, applying JT WdT Wd J is computationally intensive since J is dense. We perform a fast matrix–vector multiplication by using wavelet transforms (Li and Oldenburg, 1999a), in which a sparse representation of the dense matrix is formed in the wavelet domain and matrix–vector multiplication is carried out by sparse multiplications. The remaining issue is how to determine the optimal value µ so that the data constraint ψd = ψd is satisfied. There are two situations that require different treatments. In the first, a reliable estimate is available for the standard deviation of the errors that have contaminated the data and, therefore, the value ofψd is known. We then need to find the value of µ that yields this target misfit. This is achieved by an efficient linesearch technique that uses a number of approximate solutions to the minimization problem. In the second case, the standard deviations of errors are unknown; hence, the optimal value of µ must be estimated independently. We achieve this with the generalized cross-validation technique (Golub et al., 1979;
Wahba, 1990; Haber and Oldenburg, 2000). The use of these techniques in large-scale 3-D inversions is detailed in Li and Oldenburg (1999a). We now illustrate our algorithm using a test model composed of five anomalous rectangular prisms embedded in a uniform half-space (Figure 1). Three surface prisms simulate nearsurface distortions, and two buried prisms simulate deeper targets. The conductivity and chargeability of the prisms are listed in Table 1. The dc resistivity and IP data from both surface and crosshole experiments have been computed. The surface experiment is carried out using a pole–dipole array with a = 50 m and n = 1 to 6. There are seven traverses spaced 100 m apart in both east–west and north–south directions. There are 1384 observations, and these have been contaminated with uncorrelated Gaussian noise whose standard deviation is equal to 2% of the datum value. Figures 2 and 3 show apparent conductivity pseudosections and apparent chargeability pseudosections at three selected east–west traverses. The pseudosections are dominated by the responses to the near-surface prisms, and there are only subtle indications of the buried conductive prism. We first inverted the dc resistivity data using a Gauss-Newton approach that constructs a minimum structure model using a model objective function similar to that in equation (8) but applied to the logarithmic conductivity as the model. We set the coefficients to αs = 0.0001 and αx = α y = αz = 1 and used a reference conductivity model of 1 mS/m. The recovered conductivity model is shown by two plan sections and one cross section in Figure 4. It is a reasonably good representation of the true conductivity model. All three surface prisms and the buried conductive prism are clearly imaged, and there is indication of a resistive prism at depth. This conductivity model is then used to calculate the sensitivity for the subsequent IP inversion. The inverted chargeability model is shown in Figure 5 in the same plan and cross-sections. The surface prisms are clearly imaged, and the chargeability at depth is concentrated at the location of the two buried targets. The separation of these bodies is not clearly defined, but this decrease of anomaly definition with increasing depth is expected when surface data are inverted. Overall, the model is a good representation of the true anomalous chargeability zone. The contrast between the pseudosections shown in Figure 3 and the cross-section of the recovered model in Figure 5 illustrates the improvement gained by performing the 3-D inversion. CONSTRUCTION OF APPROXIMATE CONDUCTIVITIES
As discussed in the preceding section, the inversion of IP data requires a background conductivity model for calculating the sensitivity. IP inversion is therefore a two-stage process, and its success depends upon the availability of a conductivity model that is a reasonable approximation to the true conductivity. The usual approach to generating such a conductivity model is to invert the dc resistivity data that accompany the IP data. Numerous papers have been published on 3-D dc resistivity inversions, and different approaches have been proposed. For example, Park and Van (1991), Ellis and Oldenburg (1994a), Sasaki (1994), Zhang et al. (1995), and LaBrecque and Morelli (1996) all perform linearized inversion to construct a conductivity model from the dc resistivity data, although details of their algorithms and implementations may vary greatly.
3-D Inversion of IP Data
Li and Oldenburg (1994), on the other hand, apply approximate inverse mapping (AIM) formalism to construct a model that reproduces the data. For the current study, we implement a regularized inversion and use Gauss-Newton minimization to accomplish this (Li and Oldenburg, 1999b). The basics of that algorithm are summarized here. Let m = ln σ define the model used in the conductivity inversion, and let dobs be the dc potential data. As in the IP inver-
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sion, we minimize the model objective function in equation (8) subject to fitting the data to the degree determined by the estimated error. Thus, the desired conductivity model solves the following minimization problem:
minimize ψ = ψd + µψm subject to ψd = ψd∗ ,
(17)
where ψd and ψm are the same as those defined in equations (8) and (10), µ is the regularization parameter, and ψd∗ is the target misfit value for the dc resistivity problem. The potential data depend nonlinearly upon the conductivity; hence, minimization (17) must be solved iteratively. Let m be the current model, d its predicted data, and m a model perturbation. Performing a first-order Taylor series expansion of the predicted data as a functional of the new model m+ m and substituting into the total objective function in equation (17), we obtain
2 ψ(m + m) ≈ Wd (d + J m − dobs )
+ µWm (m + m − m0 )2 .
(18)
Table 1. Conductivity and chargeability of the prisms. The half-space has a conductivity of 1 mS/m and zero chargeability. Prism FIG. 1. Perspective view of the five-prism model. Seven surface traverses in the east–west direction and four boreholes are also shown. For clarity, seven traverses in the north–south direction are not shown. The physical property values of the prisms are listed in Table 1.
FIG. 2. Examples of the apparent conductivity pseudosections at three east–west traverses. The data are simulated for a pole–dipole array, and they have been contaminated by independent Gaussian noise with a standard deviation equal to 2% of the accurate datum magnitude. The pseudosections are dominated by the surface responses. The grayscale shows the apparent conductivity in mS/m.
S1 S2 S3 B1 B2
Conductivity (mS/m)
Chargeability (%)
10 5 0.5 0.5 10
5 5 5 15 15
FIG. 3. Examples of the apparent chargeability pseudosections along three east–west traverses. The data have been contaminated by independent Gaussian noise with a standard deviation equal to 2% of the accurate datum magnitude plus a minimum of 0.001. The same masking effect of near-surface prisms observed in apparent-conductivity pseudosections is also present here. The grayscale shows the apparent chargeability multiplied by 100.
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The value J is the sensitivity matrix of the potential data. Its elements are given by
= −JT WdT Wd (d − dobs ) − µWmT Wm (m − m0 ). (20)
This is the basic equation solved for a Gauss-Newton step. The new model is then formed by updating the current model: m ← m + m. This process is repeated iteratively until the minimization converges and an optimal value of µ is found to produce the desired data misfit in equation (17). The nonlinear inversion of 3-D dc resistivity data provides the best approximation to the actual conductivity distribution, but it is a costly undertaking. One may not always want to expend that amount of computation, especially when the recovery of the conductivity model is but an intermediate step toward the end goal of constructing a chargeability model. More importantly, good IP inversion results are often obtained by using less rigorous approximations to the conductivity. Our
FIG. 4. The conductivity model recovered from inversion of surface data using a Gauss-Newton method. The model is shown in one cross-section and two plan sections. The positions of the true prisms are indicated by the white lines.
FIG. 5. The chargeability model recovered from inversion of surface data. The conductivity from full 3-D dc inversion is used to calculate sensitivities. The positions of the true prisms are indicated by the white lines.
Ji j =
∂φi ∂ ln σ j
(19)
and are evaluated at the current model. Differentiating with respect to m and setting the derivative to zero yields the equation for the model perturbation:
JT WdT Wd J + µWm Wm m
3-D Inversion of IP Data
experience with 2-D inversions (Oldenburg and Li, 1994) has shown that good first-order results concerning the chargeability distribution can often be obtained by approximating the earth using a homogeneous conductive half-space. This suggests that a reasonable recovery of a 3-D chargeability model might be achieved by using intermediate approximations between the two end members corresponding to a uniform half-space and the conductivity model recovered from a full nonlinear 3-D dc inversion. To explore this, we compare five options for generating a conductivity model to be used in the IP inversion. The first four require much less computation than does the full 3-D inversion: 1) A uniform half-space: This is the simplest approximation, and no inversion of dc data is involved. When inverting apparent chargeability data, the actual value of the half-space conductivity is arbitrary since the sensitivity is independent of it. However, when the secondary potentials are inverted, the best fitting half-space from the dc resistivity data should be used. 2) One-pass approximate 3-D inversion: This conductivity model is obtained from a linear inversion of the dc data assuming that the actual conductivity consists of weak perturbations of a uniform half-space (e.g., Li and Oldenburg, 1994; Mø/ller et al., 1996). Such a model captures the gross features in the conductivity structure and demands the least amount of computation. We have implemented the approximate inversion in the spatial domain, in which the model objective function in equation (8) is minimized explicitly so that a minimum structure model is obtained. This is identical to performing the first iteration of the Gauss-Newton inversion with both the initial and reference model being equal to the chosen background, mb . The equation to be solved is
JT WdT Wd J + µWm Wm m = −JT WdT Wd (d − dobs ),
(21)
where d is the predicted data. The approximate conductivity is given by m = mb + m. 3) Composite 2-D inversions: When surface data along parallel traverses are available, independent 2-D inversions can be carried out along each line so that a 2-D model that reproduces the observations is generated (Oldenburg et al., 1993; Loke and Barker, 1996). The series of 2-D models are then combined to form a 3-D representation of the true conductivity structure. Such a model should perform well when there are strong 2-D features in the data. 4) Limited 3-D AIM updates: Using the one-pass 3-D inversion as an AIM, we can iteratively update the conductivity model by the AIM algorithm (Oldenburg and Ellis, 1991)
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such that a final model reproducing the 3-D observations is constructed. The greatest misfit reduction is achieved within the first two or three iterations (Li and Oldenburg, 1994). Thus, by performing only a limited number of AIM updates, we can obtain a conductivity model for the IP inversion. Let F −1 denote the one-pass approximate inversion and m be the current model. The model perturbation is defined by the difference between models generated by applying the approximate inverse mapping to the observed and predicted data, respectively:
m ← m + F −1 [dobs ] − F −1 [d],
(22)
where d is the predicted data from the current model. The iteration starts with an initial model which can be supplied by m = F −1 [dobs ]. 5) Full 3-D nonlinear inversion: We use the Gauss-Newton inversion discussed at the beginning of this section. This approach provides the best approximation to the conductivity, but it is the most computationally intensive. Each iteration requires calculation of the sensitivity and several additional forward modelings. The relative merits of these five methods will probably depend upon the complexity of the actual conductivity distribution. A general statement may therefore be difficult to make, but insight can be obtained from applications to specific data sets. We have applied these five methods to the inversion of our synthetic test data set shown in Figures 2 and 3. We first focus upon generating the approximate conductivities. The composite 2-D conductivity was obtained by inverting the data from the seven east–west lines using a 2-D algorithm and stitching together the resulting 2-D conductivities to form a 3-D model. The one-pass approximate 3-D inversion was carried out using a uniform background of 1 mS/m and a model objective function with αs = 0.0001 and αx = α y = αz = 1; we chose an optimal regularization parameter by the L-curve criterion (Hansen, 1992) to account for both the linearization error and the added random errors. The selected regularization parameter, together with the objective function, also defined the approximate inverse mapping. We performed two iterations of AIM updates to produce the AIM approximation of the conductivity. Last, we had the conductivity model from a full 3-D inversion (Figure 4). For comparison, we have listed in Table 2 the data misfit between the observed dc potential data and the predicted data obtained by applying 3-D forward modeling to each of the five approximate conductivity models. A comparison of these models with the true conductivity model is shown in Figure 6. We selected the cross-section at northing = 475 m, which passes through four of the five prisms in the model. The four models from different inversions show different levels of detail about the conductivity anomaly, and they present a general progression toward better representations of the true model. However, the improvement diminishes
Table 2. List of the dc and IP misfit for different approximations to the background conductivity. The dc misfit is calculated between the observed dc resistivity data and the predicted data obtained from 3-D forward modeling of the approximate conductivities. The IP misfit is the value achieved by the IP inversion when an approximate conductivity is used to calculate the sensitivity. Misfit
Half-space
One-pass 3-D approximation
2-D composite
Limited AIM updates
Nonlinear inversion
DC IP
2.96 × 10 2184
6.44 × 10 1720
1.22 × 10 1637
6.96 × 10 1634
1.35 × 103 1510
6
4
5
3
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as the approximation approaches the best model that is obtained from the full 3-D inversion. Although the inverted conductivity models are similar, there is a substantial difference between the true model and any one of these approximations. Using these five approximations to calculate the sensitivities, we performed five different inversions of the IP data. Since some of the conductivity models are poorer approximations, the corresponding IP inversions are not expected to achieve the expected data misfit. Instead, we chose an optimal regularization parameter for each inversion according to a generalized cross-validation criterion. The result is that different inversions misfit the observed IP data by different amounts (Table 2). The resulting chargeability models are compared with the true model in Figure 7. Each panel in that figure is the cross-section of the recovered chargeability model at northing 475 m. All five models recover the essential features of the true model, and they present a general trend of improvement as the approximation to the background conductivity improves. However, the improvement in the recovered chargeability model is not proportional to the increased computational cost involved in constructing a better conductivity approximation. Less rigorous approximations of conductivity which require much less computation have produced good representations of the true chargeability model.
JOINT INVERSION OF SURFACE AND CROSSHOLE DATA
Crosshole data have been used to achieve higher resolution image of the subsurface structure obtained from dc resistivity and IP experiments (e.g., Spies and Ellis, 1995; LaBrecque and Morelli, 1996). However, although crosshole data are sensitive to the vertical variation of conductivity and chargeability, they have rather poor sensitivity to the lateral variation because the data have limited spatial distribution and the array separation is restricted to a small range. Surface data, however, usually have good areal coverage and therefore possess better resolving power for determining lateral variations in the subsurface structure. Surface data can provide good complementary information to the crosshole data if the targets are within the depth of penetration of the surface arrays. Joint inversion of these two data sets was expected to improve the resolution of the recovered chargeability model. We placed four vertical boreholes around the anomalous region in the test model. The locations are shown in Figure 1. We simulated crosshole data from a pole–dipole tomographic experiment. Current sources were placed along the source hole from 0 to 400 m depth at an interval of 25 m. For each current location, potentials in another borehole (receiver hole) were measured with a 50-m dipole at an interval of 25 m between z = 0 and 400 m. Figure 8 illustrates the electrode configuration
FIG. 6. Comparison between the five approximate conductivity models with the true conductivity. All sections are at northing = 475 m, which passes through four of the five prisms. The positions of the true prisms are outlined by the white boxes. As the approximation improves, the inverted conductivity model is a better representation of the true model.
3-D Inversion of IP Data
between two holes. Only one borehole in any pair of boreholes was used as the source hole, and the reverse configuration of switching the source and receiver holes was not used. This resulted in six independent pairs of source–receiver holes. A total of 1530 observations were generated for both dc and IP experiments using this configuration. Because of the presence of zero crossings in the measured total potentials, we used the secondary potential, instead of apparent chargeability, as the IP data. The data were contaminated with independent Gaussian noise. The standard deviation for dc potentials was equal to 2% of each accurate datum; for secondary potentials it was equal to 5% of each accurate datum plus a minimum of 0.1 mV. (All the potentials were normalized to unit current strength.) Figure 9 displays the crosshole dc data as the apparent conductivity between two pairs of boreholes. The vertical axis of the plot indicates the position of the current electrode in the source hole, and the horizontal axis indicates the midpoint of the potential dipole in the receiver hole. The data plots are remarkably featureless, and identification of individual prisms in the true model is impossible. Figure 10 displays the secondary potentials in the same two pairs of boreholes. Again, there is no distinct feature in the secondary potential plots. The lack of distinct features in the borehole data is a direct indication of the data’s poor sensitivities to the lateral variations in the conductivity and chargeability distributions.
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Next, we compared the chargeability models obtained from inverting the crosshole data alone with the model obtained by jointly inverting the surface and crosshole data. For this study, we inverted the dc resistivity data using the full nonlinear inversion so that the best conductivity approximation at our disposal was used for the sensitivity calculation. In both dc and IP inversions, we chose a model objective function by setting the coefficients to αs = 0.0001 and αx = α y = αz = 1. A uniform half-space of 1 mS/m was used as the reference model for dc resistivity inversion, and a zero reference model was used for the IP inversion. In the inversions, the known values of the error standard deviations were used, and the target misfit value was set to the number of data points being inverted. All inversions converged to the expected misfit value. Figure 11 shows the conductivity model obtained from inverting the crosshole dc resistivity data, displayed in two plan sections and one cross-section. This model is a crude representation of the true conductivity. Only the large surface conductor and the buried conductor are identified, and the recovered anomaly amplitude is very low. We used this conductivity in the inversion of crosshole IP data and the recovered chargeability model (Figure 12). This model is a poor representation of the true chargeability. Anomalies are recovered near the surface, but they do not correspond to the locations of the true prisms. The two deep prisms are marginally identified. The vertical
FIG. 7. Comparison of chargeability models recovered from the 3-D inversion of surface IP data using five different approximations to the background conductivity. The process by which each conductivity approximation is obtained is shown in each panel. The lower-right panel is the true chargeability model.
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FIG. 8. Crosshole electrode configuration for collecting tomographic data. The current source A in hole B moves at an interval of 25 m from z = 0 m to z = 400 m. For each current location, the potential electrodes M and N , separated by 50 m, measure the potential differences in hole D. The midpoint of the potential dipole moves at an interval of 25 m from z = 25 m to z = 375 m. For a given pair of holes, no data are collected by interchanging the current and potential holes.
extent is well imaged, as would be expected from borehole data, but the orientations and horizontal boundaries of the recovered anomalies differ from those of the true model. In addition, there is excessive structure in the region immediately surrounding the boreholes. This is typical when crosshole data are inverted unless special weighting in the objective function is included to counter it. We next jointly inverted the surface data in Figures 2 and 3 and the crosshole data in Figures 8 and 9. Figure 13 displays the recovered conductivity model from the joint inversion of surface and borehole data. This model is dominated by the features recovered in the surface data inversion. The minor improvements are the increased amplitude and the slightly better definition of the depth extent of the conductivity prism. Using this model we calculated the sensitivity and then performed the joint inversion of the two IP data sets to recover the chargeability model shown in Figure 14. It shows dramatic improvement compared with the models from individual inversions in Figures 5 and 12. All five prisms are well resolved, and artifacts surrounding the boreholes are minimized. The most noticeable improvement is the clear image of the two separate buried targets. The recovered amplitudes, positions, and orientations of the two anomalies all correspond well with the true model.
FIG. 9. The crosshole plot of apparent-conductivity data. The vertical axis is the location of the current source in the source hole, and the horizontal axis is the midpoint of the potential dipole in the receiver hole. The left panel is for data between holes C and A, and the right panel is for data between holes B and D, as shown in Figure 1.
FIG. 10. The crosshole plot of secondary potential data in the same format as the crosshole apparent conductivity plots in Figure 8. The potentials are normalized to unit current strength, and the grayscale indicates the value in millivolts.
3-D Inversion of IP Data FIELD EXAMPLE
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As our last example, we illustrate the 3-D inversion algorithm using a set of pole–dipole data from the Mt. Milligan copper–gold porphyry deposit in central British Columbia, Canada. These data were first analyzed by Oldenburg et al. (1997) using a 2-D algorithm. We invert them using the 3-D algorithm, which illustrates the 3-D inversion in a mineral exploration setting and provides a comparison with the result from a series of 2-D inversions. The Mt. Milligan deposit lies within the Early Mesozoic Quesnel terrane, which hosts a number of Cu-Au porphyry deposits, and it occurs within porphyritic monzonite stocks and adjacent volcanic rocks. The initial deposit model consists of a vertical monzonitic stock, known as the MBX stock, intruded
into volcanic host rocks. Dykes extend from the stock and cut through the porous trachytic units in the host. Emplacement of the monzonite intrusive is accompanied by intensive hydrothermal alteration primarily near the boundaries of the stock and in and around the porous trachytic units cross-cut by monzonite dykes. Potassic alteration, which produced chalcopyrite, occurs in a region surrounding the initial stock, and its intensity decreases away from the boundary. Propylitic alteration, which produces pyrite, exists outward from the potassic alteration zone. Strong IP effects are produced by these alteration products, and the IP survey is well suited for mapping the alteration zones. The pole–dipole dc resistivity and IP surveys over Mt. Milligan were carried out along east–west lines spaced 100 m apart. The dipole length was 50 m, and n-spacing was from 1 to 4. This yielded 946 data points along 11 lines in
FIG. 11. Conductivity model recovered from the crosshole data alone. This model shows an elongated conductor on the surface and a broad conductor at depth. Both conductors are surrounded by resistive halos, and the amplitude is small.
FIG. 12. Chargeability model recovered from the crosshole data alone. This model shows little resolution near the surface where the anomalies are confined to small volumes. The deeper anomalies are identified but not well resolved. As expected, the depth extent of the buried chargeable bodies is well delineated.
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our study area of 1.2 × 1.0 km. This area, directly above the MBX stoc, has a gentle surface topography, and the total relief is about 100 m. Figure 15 displays the apparent chargeability data in plan maps of constant n-spacings. For brevity, we have not shown the dc resistivity data here. The apparent chargeability data show large anomalies toward the western and southern regions. The north-central region of low apparent chargeability is related to the intrusive stock that has significantly less sulfide from the alteration processes. To invert these data, we used a mesh that consisted of cells 25 m wide in both horizontal directions and 12.5 m thick in the region of interest. The mesh was extended horizontally and downward by cells of increasing sizes. The total number of cells in the inversion was about 72 000. We first performed the full nonlinear inversion of the DC resistivity data and then used it to
FIG. 13. Conductivity model recovered from the joint inversion of surface and crosshole data. This model is similar to that obtained from surface data alone, but it has a slightly higher amplitude for the buried conductive anomaly. The depth extent of the anomaly is also slightly better defined.
carry out the IP inversion. The resulting model is shown in Figure 16. For comparison, we also plotted the chargeability model created by combining the 2-D sections obtained from inverting the 11 lines of data separately using a 2-D algorithm. The recovered 3-D chargeability models from these two approaches were consistent, and they both imaged the large-scale anomalies reasonably well. This was not surprising since the limited array length meant there was little redundant information in the data from adjacent lines. The model recovered from the 3D inversion was somewhat smoother and showed less spurious structure than the composite 2-D model. It also showed a welldefined central zone of low chargeability at depth. This was a clearer image of the monzonite stock than what was imaged in the 2-D inversions.
FIG. 14. Chargeability model recovered from the joint inversion of surface and crosshole data. This model shows the great improvement achieved by joint inversion of the two complementary data sets. All five anomalies are well resolved. Especially noticeable is that the boundaries of the two buried chargeable bodies are well delineated.
3-D Inversion of IP Data DISCUSSION
We developed a 3-D IP inversion algorithm that applies to data acquired using arbitrary electrode configurations on a topographically variable earth surface or in boreholes. We assumed that the chargeability is small and formulated the inversion as a two-step process. First, the dc resistivity data are inverted to generate a background conductivity. That conductivity is used to generate the sensitivity matrix for the IP equations. The 3-D chargeability model is then generated by solving the system of equations, subject to a restriction that the chargeability is everywhere positive and smaller than an upper bound. The analysis of large IP data sets often takes place in stages. The first goal is to obtain an image that reveals the major subsurface structures, answers questions about the existence of buried targets, and supplies approximate details about size and location. Major components affecting this image are the choice of model objective function, the amount and type of errors on the data, and the degree to which the data are fit. For our twostep process, we must also pay attention to how valid the linearization process is and how close the recovered conductivity is to the true conductivity. The question of how close the recovered conductivity needs to be to the true conductivity so that sensitivity J is a good estimate of the true sensitivities is not addressed quantitatively in this paper. We have, however, carried out an empirical test in a single example. Four approximate conductivities were used to generate the sensitivities. In general, higher quality dc in-
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versions yielded better IP results, with the half-space conductivity, a one-pass linearized inversion, a few passes of an AIM approach, and the Gauss-Newton inversion giving progressive improvement. However, the differences in the final IP inversions from these various approximations were fairly subtle (see Figure 7). In fact, these differences were smaller than changes in the image obtained by adjusting the degree to which the data are misfit or by slightly altering the model objective function being minimized. Yet the various approximations to the conductivity can be produced with substantially fewer computations than the full Gauss-Newton solution. This allows the user to carry out a number of first-pass inversions with a data set to achieve insight about the gross distribution of earth chargeability. If the conductivity structure is not overly complicated, then this result may be satisfactory for final interpretation. The question of how well the conductivity must be known is a potential area for further research. Another approximate conductivity model is that generated by combining results from 2-D inversions. The prevalence of 2-D inversion algorithms means that this information is generally available when data have been collected along parallel lines. We know that off-line anomalies and 3-D topography will cause distortions in the recovered 2-D conductivity models, so some degree of caution is required. In the synthetic modeling presented here and in the Mt. Milligan example, the 2-D analysis for conductivity worked satisfactorily. Further research is required to provide more detailed rules about when 2-D is applicable.
FIG. 15. The IP data from an area above the MBX stock of the Mt. Milligan copper–gold porphyry deposit in central British Columbia. The data were acquired using a pole–dipole array with a dipole length of 50 m and n-spacing from 1 to 4. The four panels are plan maps of the data corresponding to different n-spacings.
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An important aspect of our IP inversion is the incorporation of upper and lower bounds on the chargeability. The lower bound is physical since chargeability is positive. The upper bound might be (1) assignable from a priori knowledge about the nature of the mineralization or (2) assigned to generate a model consistent with the linearized formulation of the equations. Linearization requires that the chargeability be small. The positivity and upper bounds are implemented through a primal logarithmic barrier method. This increases the complexity of the algorithm, but the method is well established in the literature and we provide an explicit algorithm for its implementation. Another major component of our algorithm is the introduction of the wavelet transform to perform the matrix–vector multiplications. The sensitivity matrix can be compressed by a factor of at least 10. This leads to substantial savings in both re-
quired memory and CPU time. This has made the algorithm at least ten times faster than a direct approach and consequently has allowed us to routinely handle problems that have a few thousand data and a hundred thousand cells with relative efficiency. Last, the application of our algorithm to joint surface and crosshole data has demonstrated that the inversion of these two complementary data sets can greatly improve the resolution of the inverted chargeability model. The noticeable gains are in the enhanced definition of both horizontal boundary and vertical extent of buried chargeable zones. ACKNOWLEDGMENTS
We thank Roman Shekhtman for his valuable assistance in programming the code and in running numerical examples. This work has been supported by an NSERC IOR grant
FIG. 16. Comparison of the chargeability models obtained from 2-D and 3-D inversions of the data from Mt. Milligan shown in Figure 15. The column on the left shows one cross-section and two plan sections of the 3-D model obtained by combining eleven 2-D sections recovered from 2-D inversions. The column on the right shows the model obtained by performing a single 3-D inversion of all the data. The two results are generally consistent. However, less spurious structure is present in the model from the 3-D inversion, and the central zone of low chargeability corresponding to the MBX stock is imaged better.
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and an industry consortium, 3-D Inversion of DC resistivity and Induced Polarization Data (INDI). Participating companies are Placer Dome, BHP Minerals, Cominco Exploration, Falconbridge, INCO Exploration & Technical Services, Newmont Gold Company, and Rio Tinto Exploration. REFERENCES Dey, A., and Morrison, H. F., 1979, Resistivity modelling for arbitrarily shaped three-dimensional structures: Geophysics, 44, 753–780. Ellis, R. G., and Oldenburg, D. W., 1994a, The pole-pole 3-D DCresistivity inverse problem: A conjugate-gradient approach: Geophys. J. Internat., 119, 187–194. ——— 1994b, 3-D induced polarization inversion using conjugate gradients: Presented at the John Sumner Memorial Internat. Workshop on Induced Polarization (IP) in Mining and the Environment. Gill, P. E., Murray, W., Ponceleon, D. B., and Saunders, M., 1991, Solving reduced KKT systems in barrier methods for linear and quadratic programming: Stanford Univ. Technical Report SOL 91–7. Golub, G. H., Heath, M., and Wahba, G., 1979, Generalized crossvalidation as a method for choosing a good ridge parameter: Technometrics, 21, 215–223. Haber, E., and Oldenburg, D. W., 2000, A GCV-based method for nonlinear ill-posed problems: Comp. Geosci., in press. Hansen, P. C., 1992, Analysis of discrete ill-posed problems by means of the L-curve: SIAM Review, 34, 561–580. Kowalczyk, P. L., Logan, K. J., and Bradshaw, P. M. D., 1997, New methods in geophysics to visualize geology in tropical terrains: 4th Decennial Internat. Conf. Min. Expl., Proceedings, 829–834. LaBrecque, D. J., 1991, IP tomography: 61st Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 413–416. LaBrecque, D. J., and Morelli, G., 1996, 3-D electrical resistivity tomography for environmental monitoring: Symp. on Appl. Geophys. to Engin. and Environ. Problems, Proceedings. Li, Y., and Oldenburg, D. W., 1994, Inversion of 3-D DC resistivity data using an approximate inverse mapping: Geophys. J. Internat., 116, 527–537. ——— 1999a, Fast inversion of large scale magnetic data using wavelet transforms: Geophysical J. Internat., accepted for publication. ——— 1999b, 3-D inversion of DC resistivity data using an L-curve criterion: 69th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded
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Abstracts, 251–254. Loke, M. H., and Barker, R. D., 1996, Rapid least-squares inversion of apparent conductivity pseudosection using a quasi-Newton method: Geophys. Prosp., 44, 131–152. McGillivary, P. R., and Oldenburg, D. W., 1990, Methods for calculating Frechet derivatives and sensitivities for nonlinear inverse problem: A comparative study: Geophys. Prosp., 38, 499–524. Møller, I., Christensen, N. B., and Jacobsen, B. H., 1996, 2-D inversion of resistivity profile data: Symp. on Appli. Geophys. to Engin. and Environ. Problems, Proceedings. Mutton, A. J., 1997, The application of geophysics during evaluation of the Century zinc deposit: 4th Decennial Internat. Conf. Min. Expl., Proceedings, 599–614. Oldenburg, D. W., and Ellis, R. G., 1991, Inversion of geophysical data using an approximate inverse mapping: Geophys. J. Internat., 105, 325–353. Oldenburg, D. W., and Li, Y., 1994, Inversion of induced polarization data: Geophysics, 59, 1327–1341. Oldenburg, D. W., McGillivary, P. R., and Ellis, R. G., 1993, Generalized subspace method for large scale inverse problems: Geophys. J. Internat., 114, 12–20. Oldenburg, D. W., Li, Y., and Ellis, R. G., 1997, Inversion of geophysical data over a copper-gold porphyry deposit: A case history for Mt. Milligan: Geophysics, 62, 1419–1431. Park, S. K., and Van, G. P., 1991, Inversion of pole-pole data for 3-D resistivity structure beneath arrays of electrodes: Geophysics, 56, 951–960. Sasaki, Y., 1994, 3-D resistivity inversion using the finite-element method: Geophysics, 59, 1839–1848. Saunders, M., 1995, Cholesky-based methods for sparse least squares: The benefits of regularization: Stanford Univ. Technical Report SOL 95–1. Seigel, H. O., 1959, Mathematical formulation and type curves for induced polarization: Geophysics, 24, 547–565. Spies, B. R., and Ellis, R. G., 1995, Cross-borehole resistivity tomography of a pilot-scale, in-situ vitrification test: Geophysics, 60, 886–898. Tikhonov, A. V., and Arsenin, V. Y., 1977, Solution of ill-posed problems, ed. J. Fritz: John Wiley & Sons. Wahba, G., 1990, Spline models for observational data: Soc. Ind. Appl. Math. Zhang, J., MacKie, R. D., and Madden, T. R., 1995, 3-D resistivity forward modelling and inversion using conjugate gradients: Geophysics, 60, 1313–1325.
