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3D Inversion of magnetic total gradient data in the presence of remanent magnetization Sarah Shearer* and Yaoguo Li Gravity and Magnetics Research Consortium, Department of Geophysics, Colorado School of Mines Summary Inversion of magnetic data has long been hampered by the need to specify the direction of magnetization. We present a general approach that utilizes the minimal dependence on magnetization direction of amplitude and total gradient data and thereby overcome the difficulty associated with inversion when unknown remanent magnetization is present. To construct the inversion algorithm for the magnitude of magnetization, we solve a nonlinear minimization problem formulated using Tikhonov regularization. A positivity constraint is also incorporated to improve the solution. The algorithm will be illustrated with both synthetic and field data sets. Introduction The presence of strong remanent magnetization can pose severe challenges to the quantitative interpretation of magnetic data. Problems stem from the fact that the direction of total magnetization is unknown and can be significantly different from that of the current inducing field. Thus, a vital piece of a priori information is missing and the forward modeling required in interpretation can no longer be reliably carried out. Although this problem has existed since the beginning of the magnetic method, it has not received much attention for two reasons. First, the remanence is either very weak or aligned with the current field direction for the majority of applications and therefore not of major consequence. Secondly, many earlier interpretation techniques, such as depth estimation methods (e.g., Hartman et al., 1971; Reid et al., 1990), seek semi-quantitative information and are not affected by the lack of unknown magnetization direction. However, in applications such as crustal scale studies, mining exploration, and archaeological investigations, the remanence is often very strong and cannot be ignored. Inversion techniques require recovered causative bodies to reproduce the observed data, and this dictates that we must know the magnetization direction. Unrealistic distribution of magnetization and susceptibilities can result from inversion if incorrect direction is specified. Faced with this difficulty, one can take several different approaches. The first is to physically measure the remanent magnetization by oriented sample. This can be an expensive proposition and a small number of samples may not necessarily give good characterization of the bulk magnetization direction. One can also attempt to estimate the total magnetization direction by utilizing the properties of the magnetic anomaly (e.g., Roest and Pilkington, 1993) and use it in the interpretation. An alternative approach is to work with quantities that are computed from the observed magnetic data and have little or no dependence on the magnetization direction. The wellknown analytic-signal approach by Nabighian (1972) belongs to this class, in which it is shown that the amplitude of the gradient (referred to as total gradient hence forth) in 2D is independent of magnetization direction and its interpretation yields source locations and geometry. In an effort to invert for 3D magnetic source distribution in the presence of remanence, Paine et al. (2001) examined two related quantities. The first is the total gradient of the vertically integrated magnetic anomaly (termed ASVI by Paine et al.) and the second is the vertical integral of the total gradient (VIAS). Because of the weak dependence and the fact both quantities have the dimension of magnetic field, nT, Paine et al. (2001) treated the quantities as RTP fields and applied the 3D inversion algorithm by Li and Oldenburg (1996). Compared to inverting total field anomaly by assuming induced magnetization only, this approach has produced somewhat better results. However, there are a number of theoretical difficulties associated with this approach and results are not always as expected. This paper improves upon the work by Paine et al. (2001) by developing an algorithm that directly inverts quantities having a minimal dependence on the direction of the magnetization. Such quantities include the magnitude of the anomalous magnetic field vector and the magnitude of the gradient vector of a magnetic anomaly. Our algorithm fully incorporates the nonlinear relationship between these quantities and the subsurface magnetization. Theory As discussed above, the horizontal and vertical derivatives of the magnetic field can be combined into a two-dimensional quantity known as the analytic signal, introduced by Nabighian (1972). The amplitude of the analytic signal, which is the amplitude of the gradient vector, is independent of magnetization direction and allows for depth estimation along the profile of a magnetic source without assumed knowledge of magnetization. Inversion of magnetic total gradient Many authors have tried to extend this approach to 3D problems, but to date no one has found a quantity that can be calculated directly from the observed data and is entirely independent of magnetization direction. However, it has been demonstrated that the magnitude of the gradient vector (total gradient), often erroneously termed 3D analytic signal, has a minimal dependence on magnetization direction. In particular, if the anomaly is half reduced to the pole, the corresponding total gradient has an even weaker dependence (Haney et al., 2003). Because of this property, total gradient has been widely used for interpreting magnetic data acquired in both exploration and environmental problems. We can make use of this property and invert it directly to recover the distribution of the magnitude of the magnetization without knowing its direction. Figure 1 illustrates this property by displaying the total field anomalies and the corresponding total gradients due to two identical prismatic sources with different magnetization directions. Although the total field anomalies are very different under two different magnetization directions, the corresponding total gradients appear to be similar. Since total gradient data has minimal dependence on the magnetization direction, an assumption of such will not drastically influence our results. We note that the magnitude of the anomalous magnetic vector itself has the similar property. The magnitude of the anomalous field vector in 2D is also independent from the magnetization direction. This arises from the fact that the horizontal and vertical derivatives of magnetic potential forms a Hilbert transform pair and the amplitude therefore forms an envelope of either component. The envelope is, by definition, independent of the direction. Although it is not entirely direction independent in 3D, the amplitude can be demonstrated as showing much less sensitivity to it. Therefore, we can also invert it in the similar manner as we do total gradient data. It is interesting that the quantity ASVI used by Paine et al. (2001) is equal to the amplitude of the anomalous field if the observed magnetic data is converted to vertical component by applying a half reduction to the pole. This is easily shown by the fact that vertically integrating the vertical magnetic anomaly yields the magnetic potential and the three partial derivatives of the potential, which define the three components of the anomalous field. For brevity, we develop the inversion methodology below by focusing on the total gradient, but note that it is exactly parallel for the amplitude of the anomaly vector. Therefore, the following algorithm is general and applicable to both. The total gradient g is a calculated quantity defined using partial derivatives of the anomalous field B in three dimensions: g = ∇B = (∂B / ∂x ) 2 + (∂B / ∂y ) + (∂B / ∂z ) 2 2 , (1) where B is a given component of the anomalous field such as the total field anomaly or the vertical anomaly. Given a set of field data, the total gradient can be derived in different ways. The horizontal derivatives can be calculated using a stable algorithm in either the space or Fourier domain. The vertical derivative is then obtained by applying the generalized 3D Hilbert transform. Forward modeling Given a subsurface magnetization model, we can take a more direct approach to calculate its total gradient. Assume the subsurface volume of interest is discretized into a set of rectangular prisms with constant magnetization. Let the values of the magnetization amplitude be given by the product of an effective susceptibility κ and the strength of the current inducing field. Denote the effective v susceptibility model by κ = (κ 1 , L , κ M ) T , where M is the number of cells in the model. Each component of the gradient is linearly related to the magnetization by a matrix system, M M M j =1 j =1 j =1 g xi = ∑ κ j G xij , g yi = ∑ κ j G yij , g zi = ∑ κ j G zij , (2) where Gxij is the ∂B / ∂x produced at the i’th observation location by unit susceptibility in the j’th cell. We have implicitly assumed an anomaly projection direction and a nominal magnetization direction, with the understanding that the latter will not drastically affect the total gradient given by, g ipred = g xi2 + g 2yi + g zi2 . (3) This process provides the mechanism to forward model the predicted total gradient data from a given magnetization model. It also is clear that the total gradient is nonlinearly dependent upon the magnetization magnitude in each model cell. Inversion Given the above relationship between the subsurface magnetization and the total gradient data, we can perform a nonlinear inversion to recover the amplitude of magnetization without knowing its exact direction. In general, magnetic anomaly is a combined response of a distribution of subsurface sources. Thus, we formulated the inversion as an underdetermined problem. In addition, observed magnetic data, and therefore the computed total gradient, is contaminated with noise. To deal with these two aspects, we use Tikhonov regularization (Tikhonov and Arsenin, 1977) to formulate the inversion as an optimization problem that minimizes the following objective function: φ = φ d + βφ m , (4) Inversion of magnetic total gradient where φd is the data misfit, φm is the model objective function, and β is the regularization parameter. The regularization parameter is chosen to prohibit the model from fitting noise-contaminated data exactly as well as to produce a geologic level of features in the model. The data misfit function is defined as r r 2 φd = Wd (d obs − d pre ) , v d obs = ( g1obs , L , g Nobs )T where are the (5) total gradient v calculated from the observed magnetic data and d pre are the predicted total gradient data, and Wd is a data weighting matrix. The model objective function is designed to minimize the structural complexity of the recovered effective susceptibility, r r 2 φ m = Wm (κ − κo) , r (6) where κ = (κ1 , L, κ M )T is the effective susceptibility model to be recovered, Wm is the model weighting matrix, and is the reference susceptibility model. r κo d + βφ m − 2 λ ∑ ln κ j , (7) j where λ is a barrier parameter. The logarithmic barrier term ensures that feasible solutions of the minimization problem are all positive. Since the total gradient data are nonlinearly related to the susceptibility model, and the logarithmic barrier term introduces additional nonlinearity, the minimization problem in eq. (7) must be carried out iteratively. We use a GaussNewton method. At each iteration, the model perturbation is give by the following equation, r ( J T WdT Wd J + βwTz wz + λΧ −2 )∆κ , r r v = − J T WdT Wd δd n − βWmT Wmδκ n − λΧ −1e The updated model is given by v (8) v r r r v r r r where δd n = d n − d obs , δκ n = κ n − κ o , and, κ n and d n are the model and predicted data at the n’th iteration. X is a diagonal matrix containing the values of the current susceptibility model and it stems from the presence of the v v κ ( n+1) = κ ( n ) + α∆κ , (9) where α is a step length determined by a linear search. This v process starts with an initial model κ ( 0) and a large value for λ. The iteration terminates when the objective function has stopped decreasing and the barrier parameter is sufficiently small. The majority of the computational cost for the inversion lies in the solution of the Gauss-Newton equation in eq. (8) and the calculation of the required sensitivity matrix J. For this reason, we discuss the calculation of sensitivity next. Sensitivity calculation The sensitivity matrix, J, characterizes the change of the total gradient data with respect to susceptibility values in the model. Substituting in eq. (2) into eq. (1) and carrying out differentiation with respect to κ j yields, J ij = In addition, we also impose a positivity constraint on the recovered susceptibility model, for the magnitude of magnetization is positive by definition. This is accomplished by using a logarithmic barrier method, in which the constraint is incorporated into the objective function as an additional nonlinear term, φ =φ logarithmic barrier term. The matrix J is the sensitivity of total gradient with respect to the effective susceptibility. ∂g i 1 = ∇Bipred ⋅ ∇bij , ∂κ j ∇Bipred (9) where ∇Bipred is the gradient vector of the anomalous field predicted by the current magnetization at the i’th observation location and ∇bij is the magnetic gradient at the i’th location due to unit susceptibility in the j’th cell. That is, the sensitivity of the total gradient with respect to a particular cell has a simple form: it is the inner product of the unit directional vector of the gradient produced by the entire model and the gradient produced by the unit susceptibility in that cell. We note that the components of ∇bij are the elements of component sensitivity matrices Gx, Gy, and Gz in eq. (2). These three matrices are therefore v required in order to calculate both the predicted data d ( n ) = ( g 1( n ) , L , g N( n ) ) T as well as the sensitivity matrix J at each iteration. In our algorithm, we first compute the three component sensitivity matrices and store them for subsequent use since they model independent. The calculation of sensitivity for total gradient at each iteration is therefore a trivial operation that makes use of the stored components. As a result, although the problem is nonlinear, the computational cost to obtain a solution is similar to that for solving the 3D magnetic inversion when the magnetization direction is known. Discussion To overcome the restriction of requiring the direction of magnetization in the presence of unknown remanent Inversion of magnetic total gradient magnetization, we have developed a 3D magnetic inversion algorithm that utilizes the minimal dependence on magnetization direction by quantities such as total gradient. We first compute total gradient of observed total field magnetic anomaly data and then invert the new data to recover 3D variation of the amplitude of magnetization in the subsurface. Although the forward modeling still requires an assumed magnetization direction, it has little effect on the inversion other than to facilitate the forward modeling. The ability to invert magnetic data with little information about the magnetization direction gives us is great flexibility in practical applications. It is now possible to take any magnetic data and carry out a 3D inversion to some extent. This ability opens the door to applying 3D magnetic inversion in a number of areas including the depth estimation of basement in petroleum exploration, imaging kimberlites in diamond exploration, and crustal studies. Acknowledgments The authors would like to thank Misac Nabighian for his discussion. This work was partially supported by the Gravity and Magnetics Research Consortium at the Colorado School of Mines. One of the authors (SS) was supported by a National Science Foundation Grant award # DGE0231611. Hartman, R.R., Teskey, D.J., and Friedbery, J.L., 1971, A system for rapid digital aeromagnetic interpretation: Geophysics, 36, 891-918. Li, Y., and Oldenburg, D.W., 1996, 3-D inversion of magnetic data: Geophysics, 61, 394-408. Nabighian, M., 1972, The analytic signal of two-dimensional magnetic bodies with polygonal cross-section: its properties and use for automated anomaly interpretation: Geophysics, 37, 507-517. Paine, J., Haederle, M., and Flis, M., 2001, Using transformed TMI data to invert for remanently magnetized bodies: Exploration Geophysics, vol. 32, no. 3 & 4, 238-242. Reid, A. B., Allsop, J. M., Granser, H., Millett, A. J., and Somerton, I. W., 1990, Magnetic interpretation in three dimensions using Euler deconvolution: Geophysics, 55, 80– 91. Roest, W. R. and Pilkington, M., 1993, Identifying remanent magnetization effects in magnetic data: Geophysics, 58, 653659. Tikhonov, A. and Arsenin, V., 1977, Solutions of ill-posed problems, V.H. Winston & Sons. References Haney, M., Johnston, C., Li, Y., and Nabighian, M., 2003, Envelopes of 2D and 3D magnetic data and their relationship to the analytic signal: 73rd Ann. Internat. Mtg., Soc, Expl. Geophys., Expanded Abstracts. Im=65º Dm=25º Im=25º Dm=-65º Figure 1: An illustration of the weak dependence on magnetization direction by total gradient data (the amplitude of the gradient vector). The assumed inducing field is in the direction of I=45º and D=45º. The total field anomalies produced by two identical prismatic sources with different magnetization directions (Im and Dm) appear very different, whereas the corresponding total gradients are much more similar. Thus, the assumption of total gradient’s minimal dependence on the magnetization direction will not drastically influence inversion results. This is especially true after the total field data are converted to vertical anomaly.