Comprehensive approaches to 3D inversion of magnetic data affected by remanent magnetization

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1 GEOPHYSICS, VOL. 75, NO. 1 JANUARY-FEBRUARY 1 ; P. L1 L11, 13 FIGS., TABLES / Comprehensive approaches to 3D inversion of magnetic data affected by remanent magnetization Yaoguo Li 1, Sarah E. Shearer, Matthew M. Haney 3, and Neal Dannemiller 4 ABSTRACT Three-dimensional 3D inversion of magnetic data to recover a distribution of magnetic susceptibility has been successfully used for mineral exploration during the last decade. However, the unknown direction of magnetization has limited the use of this technique when significant remanence is present. We have developed a comprehensive methodology for solving this problem by examining two classes of approaches and have formulated a suite of methods of practical utility. The first class focuses on estimating total magnetization direction and then incorporating the resultant direction into an inversion algorithm that assumes a known direction. The second class focuses on direct inversion of the amplitude of the magnetic anomaly vector. Amplitude data depend weakly upon magnetization direction and are amenable to direct inversion for the magnitude of magnetization vector in 3D subsurface. Two sets of high-resolution aeromagnetic data acquired for diamond exploration in the CanadianArctic are used to illustrate the methods usefulness. INTRODUCTION Quantitative interpretation of magnetic data through inversion for general distributions of magnetic susceptibility has played an increasingly important role in mineral exploration in recent years. Such applications range from district-scale to deposit-scale problems. Most of the currently available algorithms require knowledge of magnetization direction, an essential piece of information for carrying out forward modeling e.g., Li and Oldenburg, 1996; Pilkington, In most cases, one can assume there is no remanent magnetization, and the self-demagnetization effect can be neglected. Consequently, the direction of magnetization is assumed to be the same as the current inducing field direction. This is a valid assumption in most cases, as evidenced by many successful applications. However, there are well-documented cases in which such an assumption is inadequate because of the presence of remanent magnetization. The total magnetization direction can be significantly different from that of the inducing field. Without prior knowledge of the direction of resultant total magnetization, current inversion algorithms become ineffective. For instance, simulations by Shearer 5 indicate that the algorithm by Li and Oldenburg 1996 yields erroneous results when the error in specified magnetization direction exceeds 15. This difficulty has limited the application of these algorithms. To address this issue, researchers have taken several different routes. For example, Paine et al. 1 transform magnetic data into quantities that resemble the total-field anomaly but have less dependence on magnetization direction prior to applying 3D inversions. However, this approach suffers from the inconsistency between the transformed data, which are not actual magnetic anomalies, and their inversion using an algorithm designed for magnetic anomalies produced by induced magnetization. Lelièvre and Oldenburg 9 expand the inversion to recover three components of a total magnetization vector. We focus on the magnetization direction as extra parameters in the inversion and have developed two approaches. The first is to estimate the direction of total magnetization and supply it to the inversion algorithm, assuming the magnetization direction does not vary greatly within the target region. Alternatively, we accept the fact that a single direction cannot be estimated for a particular data set and therefore opt to directly invert a quantity that is calculated from mag- Manuscript received by the Editor 8 October 8; revised manuscript received 9April 9; published online 5 January 1. 1 Colorado School of Mines, Center for Gravity, Electrical, and Magnetic Studies CGEM, Department of Geophysics, Golden, Colorado, U.S.A. ygli@mines.edu. Formerly Colorado School of Mines, Center for Gravity, Electrical, and Magnetic Studies CGEM, Department of Geophysics, Golden, Colorado, U.S.A. Presently Ultra Petroleum Corp., Denver, Colorado, U.S.A. sshearer@ultrapetroleum.com. 3 Formerly Colorado School of Mines, Center for Gravity, Electrical, and Magnetic Studies CGEM, Department of Geophysics, Golden, Colorado, U.S.A. Presently United States Geological SurveyAlaska Volcano Observatory, Anchorage, Alaska. mhaney@usgs.gov. 4 Formerly Colorado School of Mines, Center for Gravity, Electrical, and Magnetic Studies CGEM, Department of Geophysics, Golden, Colorado, U.S.A. Presently Pioneer Natural Resources, Denver, Colorado, U.S.A. neal.dannemiller@pxd.com. 1 Society of Exploration Geophysicists. All rights reserved. L1

2 L Li et al. netic data but is weakly dependent upon magnetization direction. Based on these two methods, we have formulated a comprehensive approach to interpret any magnetic data affected by significant remanent magnetization. We first present three methods to estimate magnetization direction for use in subsequent inversions and evaluate the performance in interpreting 3D magnetic data with strong remanent magnetization. Then we present the basics of 3D inversion of amplitude data that are weakly dependent upon magnetization direction. Both approaches are illustrated with synthetic and field data sets from diamond exploration. We conclude by discussing the conditions and limitations of the two approaches and thereby provide explicit guidance on choosing appropriate strategies for interpreting any magnetic data through 3D inversion. ESTIMATING MAGNETIZATION DIRECTION Current 3D magnetic inversion algorithms usually assume a known magnetization direction and construct the 3D distribution of magnetic susceptibility or magnitude of magnetization as a function of 3D position Li and Oldenburg, 1996; Pilkington, Given the critical role of magnetization direction, it is reasonable to attempt to estimate it independently prior to inversion. This is perhaps the simplest and most straightforward modification to the aforementioned algorithms. We develop this approach by first examining direction estimation techniques and then evaluating its utility in 3D inversions. Many workers recognize the importance of magnetization direction in interpreting magnetic data. For example, Zietz and Andreasen 1967 examine the relationship between position and intensity of the maximum and minimum produced by a simple causative body. Roest and Pilkington 1993 correlate the amplitude of the D total gradient of the magnetic field and the absolute value of the horizontal gradient of the pseudogravity produced by D sources. Lourenco and Morrison 1973 develop a method based upon the integral relationships of magnetic moments derived by Helbig 196. More recently, Haney and Li develop a wavelet-based method for determining magnetization direction in D data sets. Dannemiller and Li 6 introduce an improvement to Roest and Pilkington s 1993 method and extend it to 3D cases. For this paper, we present Helbig s moment method, wavelet method, and cross-correlation method. The first two methods directly explore the relation between the anomaly and magnetization direction and compute magnetization direction from the data; whereas, the third method estimates magnetization direction using the symmetry property of the reduced-to-pole RTP field. In all three methods, we assume magnetization direction does not vary drastically for sources within the volume under examination. Once the direction is estimated, it can be incorporated into a commonly used inversion algorithm that assumes a known magnetization direction. The inversion proceeds with the estimated magnetization direction and attempts to recover an effective susceptibility defined as the ratio of the magnitude of magnetization over the strength of the inducing magnetic field H. In the following, we present the salient features of the estimation methods. We then illustrate the inversion using such estimates through application to a synthetic example. Helbig s moment method Helbig s method Lourenco and Morrison, 1973; Phillips, 5 is based on the integral relations between the moments of a magnetic anomaly and the magnetic dipole moment developed by Helbig 196 : xb z x,y dxdy m x, yb z x,y dxdy m y, xb x x,y dxdy m z, where B x, B y, and B z are, respectively, the x-, y-, and z-components of the magnetic anomaly and where m x, m y, and m z are the three components of the magnetic moment of the source. Once the magnetic moment is estimated, it can be used to calculate the inclination and declination of the magnetization, assuming they are constant within the source body. Although the integral relationships in equation 1 do not assume any specific source geometry, we have observed that the method is best applied to data sets produced by compact source bodies. Two scenarios arise in practical applications. First, we usually have only the total-field anomaly data and need to calculate the three components from the total-field anomaly by the corresponding wavenumber-domain operators e.g., Pedersen, 1978; Blakely, 1996; Schmidt and Clark, Difficulties may arise when the data are acquired in low magnetic latitudes because the conversion involves a half-reduction to the pole. Therefore, additional efforts are required near the magnetic equator. Alternatively, vector magnetic surveys are now becoming available e.g., Dransfield et al., 3, and the observed three-component data can be used directly in the estimation. Wavelet multiscale edge method Haney and Li develop a method for estimating the magnetization direction in two domensions using multiscale edges of a magnetic anomaly derived by a continuous wavelet transform. The multiscale edges correspond to the trajectories of the extrema of the wavelet transform of the anomaly profile, and their positions in the x-z-plane are dependent upon the inclination of magnetization. Tracking the multiscale edges allows one to determine the inclination of the magnetization in D sources. Given a profile of magnetic data collected at a height z in an ambient magnetic field with inclination I, a continuous wavelet transform CWT can be performed using a set of natural wavelets that is equivalent to the magnetic field produced by a line dipole in a particular direction Hornby et al., To estimate magnetization direction, we use a wavelet whose corresponding dipole orientation has an inclination of I. Such a wavelet leads to a CWT that is dependent upon magnetization direction only. Carrying out the calculation for multiple dilation factors yields the complete wavelet transform. 1

3 Magnetic inversion with remanence L3 The magnitude of the multiscale edges depends on the source geometry, whereas the location depends primarily on the intrinsic properties of the source, i.e., the magnetization direction I m. For example, for a line of dipoles, there are four trajectories, given by s x cot I m, x cot I 3 m 4, 3 3 where s is the dilation factor and x is the location along the magnetic profile. The dipole line is assumed to be directly below x. Given a profile of magnetic data, the problem of estimating the direction of magnetization becomes one of tracking the trajectories of the multiscale edges and calculating I m by regression according to equation. Although the above approach is developed for D problems, it can be applied to 3D data sets with isolated anomalies. In such cases, we can integrate the data map in each of the two horizontal directions to synthesize two profiles. Integrating in the easting direction simulates a profile along a north-south traverse above a D source that strikes east-west. The translation invariant property of magnetic anomalies means that integrating the anomalous field is equivalent to adding 3D bodies along the strike direction to build up a D causative body. The corresponding magnetization is given by the projection of the 3D magnetization vector in the north-south cross section. Applying the wavelet estimation to this profile produces the apparent inclination of the magnetization within the north-south section. Performing similar operations in the perpendicular direction yields the apparent inclination in the east-west section. The true inclination and declination of magnetization in the original 3D source can then be reconstructed from these two apparent inclinations in the easting and northing cross sections, respectively. Crosscorrelation method The RTP anomaly theoretically has the least asymmetry of all magnetic anomalies produced by a given causative body. It follows that the vertical derivative of the RTP anomaly is also least asymmetrical. It has been shown that the total gradient amplitude of the gradient vector in three dimensions of the RTPanomaly is the envelope of the vertical derivative of the anomaly produced under arbitrary inducing-field and magnetization directions Haney et al., 3. The envelope, by definition, is the most symmetric form. Using these properties, Dannemiller and Li 6 develop a method to estimate magnetization by examining the symmetry of various RTP fields. This method is an extension of the D crosscorrelation method developed by Roest and Pilkington 1993 to three dimensions and improves upon the latter by using two quantities that have the same decay rate with distance to the source. As a result, Dannemiller and Li s 6 approach also extends the crosscorrelation method to anomalies produced by dipping causative bodies. The method searches for the particular magnetization direction that yields the maximum symmetry in the resultant RTP field. The symmetry is measured by the crosscorrelation between the vertical derivative and total gradient of the RTP anomaly that is calculated using an assumed magnetization direction. These two quantities achieve the maximum correlation near the correct magnetization direction. The key operation of the method is the RTP process. Consequently, we will encounter difficulties at low magnetic latitudes as before. Fortunately, several stable RTP methods are available for low latitudes e.g, Hansen and Pawlowski, 1989; Mendonca and Silva, 1993; Li and Oldenburg, 1. There is also a trade-off between the accuracy of estimated inclination and the accuracy of estimated declination. The accuracy of inclination improves as it approaches 9 or 9, but the accuracy of the corresponding declination decreases. However, this does not pose a major problem because the influence of the declination becomes less important at high magnetic latitudes. Synthetic example To illustrate the direction estimation algorithms and the utility of the resultant magnetization direction, we apply them to a synthetic example consisting of a dipping dike buried in a nonmagnetic background. Figure 1 shows the model and the total-field anomalies with and without the influence of remanent magnetization. For simplicity, the effective susceptibility for both cases is set to.5 SI. The inducing-field direction for this model has an inclination of 65, a declination of 5, and a strength of B 5, nt. The total magnetization with remanence has an inclination of 45 and a declination of 75. The negative anomaly toward the northeast of the positive peak in the data map Figure 1b is primarily related to the altered magnetization direction. We first apply Helbig s method to the data in Figure 1b to estimate the magnetization direction. To achieve this, we transform the totalfield anomaly in Figure 1b into three orthogonal components of the anomalous magnetic field shown in Figure. Applying equation 1 to these components yields an estimate of the magnetic source dipole moment. The corresponding magnetization direction is I m,d m 46,61. To apply the wavelet estimation, we first carry out integration in northing and easting directions, respectively, to simulate two totalfield profiles in the easting and northing directions Figure 3. The corresponding apparent inclination values of total magnetization are 51 and 71 in easting and northing sections, respectively. The final estimate for the magnetization direction in three dimensions is I m,d m 49,68. Last, we apply the crosscorrelation method. The vertical and total gradients of computed RTP field achieve maximum correlation at I m,d m 45,8, which yields our third estimate for the magnetization. Figure 4 displays the crosscorrelation map with the maximum point shown by the plus sign. Table 1 compares these three estimates with the true values. All three approaches provide reasonable estimates of the direction. The largest deviation between true and estimated directions in three dimensions is 1 Helbig s method. This is well within the error tolerance of 15 for the assumed magnetization established by Shearer 5. We then use these estimated magnetization directions as input in a 3D inversion algorithm Li and Oldenburg, 3 to invert the data shown in Figure 1b. All three inversions produce consistent models that represent the true dipping slab. For brevity, we only display the recovered effective susceptibility from the inversion using the magnetization direction from Helbig s estimate Figure 5. We can see that the dip of the anomalous source body is clearly visible, and its horizontal and vertical extents are well defined. Overall, the recovered anomalous body is a good representation of the true model. For this synthetic example, the following observations can be made. All three estimation methods produce consistent and reliable estimates of the magnetization direction. Using these estimates, subsequent inversions produce magnetic models that are excellent representations of the true model. Thus, the results amply demonstrate

4 L4 Li et al. the validity of the two-step approach. It is also noteworthy, however, that the estimation methods all rely on the global property of the magnetic data set. The estimated direction effectively summarizes the bulk direction of the magnetization in the causative bodies. The validity of the estimated direction relies on the assumption that actual direction does not vary significantly. In practice, the applicability of the approach is therefore limited to anomalies produced by single sources of compact geometry. For more complex scenarios, and when the magnetization direction is expected to vary greatly within the region of interest, an alternative approach is needed. 1 8 B x Depth (m) c) T B y B z B Figure 1. A synthetic example with strong remanent magnetization. a Dipping body with an effective susceptibility of.5 SI. Totalfield data b with and c without remanent magnetization. The inducing field has I,D 65, 5, and the total magnetization direction in the presence of remanence is I m,d m 45,75. Figure. Three orthogonal components of the magnetic anomaly vector computed from the total-field anomaly in Figure 1. The firstorder moments of the x- and z-components yield the three components of the source magnetic dipole and therefore the magnetization direction.

5 Magnetic inversion with remanence L5 INVERSION OF DATA WEAKLY DEPENDENT ON MAGNETIZATION DIRECTION Integrated mag Integrated mag Figure 3. Two synthesized profiles obtained from the data map shown in Figure 1b. The east-west profile top is obtained by integrating the map in the north-south direction; hence, it corresponds to the anomaly produced by a D causative body with a north-south strike. The north-south profile in the lower panel is obtained by integration in the east-west direction and corresponds to a D body with a strike in that direction. These profiles are used in wavelet estimation of the total magnetization direction. Inclination ( o ) Declination ( o ) Figure 4. Crosscorrelation map between the total gradient of and vertical gradient of the RTP field computed from assumed inclination and declination of the magnetization. The crosscorrelation achieves maximum near the correct magnetization direction We now examine the more complex scenarios where a single estimated magnetization direction is no longer applicable. Such cases may arise if a geologic unit has undergone deformation because of tectonic activities so that the magnetization direction changes significantly within the source bodies, or when multiple source bodies acquire remanent magnetization at different times and have significantly different magnetization directions. The challenge faced in this case requires a completely new approach that does not rely on knowledge of magnetization direction. The inversion of data associated with magnetic anomalies but weakly dependent upon magnetization direction offers the needed alternative. The amplitude of the anomalous magnetic field vector and the total gradient of the magnetic anomaly are independent of the magnetization direction in D problems Nabighian, 197 because the amplitude of the magnetic anomaly vector is the envelope of each component of the vector. The same holds true for the amplitude of the gradient vector of a D field component and the derivative of the same field component in any direction. Although such a property does not extend exactly to 3D problems, both quantities are only weakly dependent on magnetization direction. This is especially so when the anomaly has been transformed to the vertical component i.e., half RTP. This property provides the opportunity for direct inversion of the anomaly amplitude or total gradient to recover the magnitude of magnetization without knowing its direction. Shearer and Li 4 develop such an algorithm by formulating a generalized inversion using Tikhonov regularization and imposing a Table 1. Magnetization direction estimated using three different methods for the dipping-slab data set shown in Figure 1c. The deviation is defined as the angle between the true and estimated directions in three dimensions. Method Inclination Declination Deviation True value Helbig s method Wavelet method Crosscorrelation Depth (m) k (SI) Figure 5. Effective susceptibility obtained through the inversion of the synthetic data in Figure 1b with estimated direction of total magnetization from Helbig s method shown in Table 1. The inversion is performed using the algorithm of Li and Oldenburg a Cross section at 5 m north. b Plan section at a depth of 15 m. The true position of the source body is shown by the black outline. The resultant model is a good representation of the true model. The peak value of effective susceptibility is consistent with the true value of.5 SI.

6 L6 positivity constraint on the amplitude of magnetization. Shearer 5 carries out a detailed investigation of the approach and demonstrates that the amplitude data are far less dependent on magnetization direction than the total gradient data. Furthermore, the amplitude data preserve the low-wavenumber content in the data and therefore retain the signal from deeper causative bodies that is present in the total-field magnetic anomaly. Consequently, inversion of amplitude data offers a better alternative than does inversion of total gradient data. We describe the amplitude inversion below, but readers are referred to Shearer 5 for more details. The inversion of total gradient data is exactly parallel. Basic algorithm for amplitude inversion The algorithm starts by calculating the amplitude of the anomalous magnetic field from the observed total-field anomaly. This is accomplished by first transforming the total-field anomaly into the three orthogonal components in the x-, y-, and z-directions. The amplitude data are given by B a B a Bx B y B z, where B a are the amplitude and B x,b y,b z are the transformed magnetic anomaly vectors. A common approach to obtain the three orthogonal components is to use the wavenumber-domain expressions e.g., Pedersen, 1978 when the data are located on a plane. Alternatively, equivalent-source techniques Dampney, 1969 can be used to carry out the transformation when ground data are acquired in areas with high topographic relief. In such cases, the wavenumber-domain approach, which assumes that all observations lie on a planar surface, is inappropriate. The amplitude data are then treated as the input data and inverted to recover the distribution of magnetization as a function of 3D position in the subsurface. One advantage of the approach is that it is not limited to a single anomaly nor does it require that adjacent anomalies have the same magnetization direction. Therefore, the approach is generally applicable to a wide range of problems where the source distribution is more complicated. The basic inversion algorithm follows that of Li and Oldenburg 1996, 3 in which the Tikhonov formalism is used to trade off between the data misfit and the structural complexity of the recovered model. The data misfit is defined as N d i 1 B obs pre ai B ai i 3, 4 where B obs ai and B pre ai are observed and predicted amplitude data, respectively, and i are the standard deviation of the amplitude data. Although we commonly assume a Gaussian distribution for errors in magnetic field component data, the corresponding errors in the computed amplitude data no longer follow such a distribution. The model objective function is chosen as m s w z dv V w z x V y V x w z y dv dv Li et al. z V w z z dv, where is the effective susceptibility, defined as the ratio of magnitude of magnetization over the strength of the inducing field H, is a reference model, and w z is a depth-weighting function. The inverse solution is given by the minimization of the total objective function, consisting of a weighted sum of d and m, subject to the constraint that the effective susceptibility must be nonnegative: 5 minimize d m subject to, 6 where is the regularization parameter. The positivity is implemented by using a primal logarithmic barrier method Wright, 1997; Li and Oldenburg, 3. The solution is obtained iteratively because nonlinearity is introduced by the positivity constraint and the nonlinear relationship between amplitude data and effective susceptibility. We discuss the basics of this aspect next, but readers are referred to Shearer 5 for more details. We adopt a commonly used model representation that discretizes the model region in three dimensions into a set of contiguous rectangular prisms by an orthogonal mesh, and we assume a constant effective susceptibility value within each prism. Under this assumption, each component of the magnetic anomaly vector is given by a matrix-vector product: d x G x, d y G y, d z G z, where d x B x1,,b xn T is an algebraic vector holding the x-components of the anomalous magnetic field; d y and d z are similarly defined; 1,, M T is the vector of unknown effective susceptibility to be recovered; and G x, G y, and G z are the sensitivity matrices relating the respective components of anomalous magnetic field to effective susceptibilities. The elements of the sensitivity matrices quantify the field produced at the ith observation location by a unit effective susceptibility in the jth prism. Assuming the susceptibility model is n at the nth iteration, substituting equation 7 into equation 3 and differentiating B ai with respect to j yields n B ai B ai n j b ij, B ai where B n ai is the predicted anomalous magnetic vector at the ith observation location by the model n at the nth iteration and where b ij is the magnetic vector produced at the same location by a unit susceptibility in the jth prisms. Thus, the sensitivity has a simple and elegant form given by the inner product of the unit vector of the predicted magnetic field at the nth iteration and the magnetic field produced by a unit effective susceptibility in a prism. For computational purposes, this means we only need to compute and store the three sensitivity matrices corresponding to the three components of the field in equation 7 and calculate the sensitivity for the amplitude data by a sequence of matrix-vector multiplications when solving the minimization in equation 6. When calculating the three individual sensitivity matrices, we 7 8

7 Magnetic inversion with remanence L7 must formally specify a magnetization direction. Given that we are utilizing the weak dependence of amplitude data on magnetization direction, it is sufficient to use the direction of the current-inducing field. Thus, the sensitivity is computed as if the magnetization vector were given by the product of the effective susceptibility and the inducing field. It is important to note, however, that the interpretation of the inversion result is independent of the direction of the inducing field. The strength of the inducing field, on the other hand, defines the magnitude of the total magnetization if desired. Revisiting the synthetic example We now return to the synthetic example shown in Figure 1 and illustrate the approach of amplitude-data inversion. To calculate the amplitude data, we must first convert the total-field anomaly into three orthogonal components in x-, y-, and z-directions as shown in Figure. The desired amplitude data shown in Figure 6a are obtained by equation 3. For comparison, Figure 6b displays the amplitude data for the same model when the magnetization is aligned with the inducing field. For consistency, we have added the same amount of noise to total-field anomalies in both cases prior to conversion to amplitude data. The high degree of similarity between the two maps demonstrates the direction insensitivity of the amplitude data. Using the same mesh as in the previous inversion, we invert the amplitude data in Figure 6a. 1 The result is shown in Figure 7, which can be compared with the model recovered using an estimated magnetization direction Figure 5. The location and spatial extent of the anomalous body is recovered well, but the dip is less visible.also, the peak value of the recovered effective susceptibility is much higher. Overall, however, the inversion result is much improved, compared to the inability to invert the original data without the magnetization direction when using existing algorithms. The lack of recovered dip is to be expected in this case. The reasons are twofold. First, much of the dip information is encoded in the phase of magnetic anomaly, and that information is greatly diminished in the amplitude data. Second, the particular model does not have a pronounced elongation in dip direction, and the remaining phase information in amplitude data is not enough to constrain the dip. In general, the ability to recover the dip depends on the orientation of the magnetization relative to the geometry of the causative body and on its aspect ratio. Although not reproduced here for brevity, numerical tests have shown that the dip can be recovered when the depth extent of a causative body is much longer than its width. The peak value of the recovered effective susceptibility in the amplitude inversion is much higher than that in the inversion based on estimated magnetization direction. This difference has been observed consistently in all synthetic and field data examples that we have studied. This difference appears to be related to the following two factors. First, because of the partial lack of the phase information in amplitude data, the recovered effective susceptibility tends to spread out B a Figure 6. Comparison of amplitude data computed from the totalfield anomalies shown in Figure 1. a The amplitude data computed from the total-field anomaly in Figure 1b when the magnetization is affected by remanence. b Data computed from the total-field anomaly in Figure 1c when the magnetization is purely induced Depth (m) k (SI) Figure 7. Inversion of the amplitude data in Figure 6a using an amplitude-inversion algorithm. The model is shown as effective susceptibility, which is defined by the magnitude of the magnetization vector divided by the strength of the inducing magnetic field. a Cross section at 5 m north. b Plan section at 15 m depth. The outline of the true source body is shown by the solid black line.

8 L8 Li et al. more. There is essentially more susceptibility distributed at larger depth. Correspondingly, a higher peak value is required to reproduce the data. For example, the depths of the center of mass of the susceptibility are and 8 m, respectively, in the models from inversion using estimated direction Figure 5 and from amplitude inversion Figure 7. The cube of the ratio of the two depths is.35. Given that the amplitude data decay approximately with inverse distance cubed, this ratio is consistent with the peak susceptibility ratio of.45. Second, every small anomalous feature in the amplitude data can be reproduced easily, so the data tend to be overfit when we use standard techniques for estimating the optimal data fit determined by the regularization parameter. This also leads to a greater peak susceptibility value in recovered models. However, the geometry of the recovered source body is similar for the two inversions despite the difference in peak susceptibility values. Both models can be used to carry out the final interpretation. The ability to invert amplitude data has effectively circumvented the need for magnetization direction as a crucial piece of information. As a result, we have bypassed the limitation of the first method that requires a constant magnetization direction within a source body. This opens the door to applying 3D inversion to interpret a wide range of data sets, especially those from areas with complex geology and multiple source regions with variable magnetization directions. APPLICATION TO FIELD DATA SETS We now invert two sets of high-resolution aeromagnetic data and illustrate the application of our methods in their respective scenarios. The data were acquired by TeckCominco and Diamonds North over kimberlites on Victoria Island in Northwest Territory, Canada. The geology in the area is characterized by an Archean granitic basement overlain by a Proterozoic sedimentary sequence with minor volcanics, capped by flat-lying Cambrian-to-Devonian carbonate rocks. The host rocks are largely nonmagnetic, and the kimberlite bodies stand out in the magnetic surveys as distinct, sharp anomalies indicative of shallow bodies, compared with the more rounded anomalies caused by deep features in the basement or within the sedimentary rocks. Positive and negative anomalies are associated with kimberlite intrusions, whereas the negative anomalies are produced by hypabyssal dikes. The ages of the kimberlite intrusions range from 5 to 3 million years J. Lajoie, personal communication, 4. The distinct magnetic signature of these kimberlite dikes above a quiet magnetic background makes high-resolution magnetic surveys an ideal exploration tool in this area. However, the highly variable orientation of these anomalies is produced by magnetization that is dominated by strong remanence with variable direction. As a result, the quantitative interpretation of the anomalies is difficult to achieve through current 3D inversion algorithms. For this reason, the data are ideal for testing our new approaches. We investigate data from two different areas with variable complexity in anomaly patterns. The first data set contains a single anomaly associated with a kimberlite dike; the second contains several different anomalies with large variations in magnetization direction. The data set containing a single anomaly is shown in Figure 8a. The inducing field has an inclination of 86.7 and declination of 6.3. Given the high magnetic latitude and dominant negative anomaly, it is clear that the kimberlite has strong remanence, and the total magnetization is nearly in the opposite direction to the inducing field. Figure 8b displays the amplitude data computed from these data. Given the single anomaly in this data set, we can estimate the direction first and then invert the total-field data or we can directly invert the amplitude data. We present both for comparison. The results of estimation are listed in Table. The estimated values for inclination are similar, but the declination varies greatly. This is expected, given the inclination is close to 9. When used in an inversion, the error in declination does not strongly affect the final result either. Using the direction estimated from multiscale edges, inversion of total-field data Figure 8a is shown in one cross section at 3 m north and one plan section at 5 m depth in Figure 9. The result from inverting the amplitude data is shown in the same format in Figure 1. The two inversions recover models with similar geometry but differing peak susceptibility, as noted in the preceding section. Both effectively image a compact magnetic body. It has a northwest strike and a strike length of approximately 5 m, and it is located at 5 m depth to the center. This result is consistent with the presence of a kimberlite dike. The second data set is shown in Figure 11a. A number of dipolar total-field anomalies with differing orientations occur throughout the survey area. The map has been rotated clockwise by 34 ; the inducing field has an inclination of 86.7 and a nominal declination of 7.7. Two types of anomalies dominate the data set: several smaller, more compact, and high-frequency anomalies surrounding areas T B a Figure 8. a Total-field magnetic anomaly T over a kimberlite dike. The inducing field is in the direction of I 86.7 and D 6.3. Judging from the negative anomaly in the center, the presence of strong remanent magnetization is apparent. b The corresponding amplitude, B a, of the anomalous field vector.

9 Magnetic inversion with remanence L9 of broad, lower-frequency anomalies. The orientation of the anomalies indicates the total magnetization direction varies greatly from anomaly to anomaly. Thus, it is unlikely that we can invert this data set with a single magnetization direction. Overlapping anomalies also mean that separately inverting each anomaly by first estimating a magnetization direction is not feasible. We resort to the second approach, i.e., we invert the amplitude of the anomalous magnetic field and recover the magnitude of magnetization in the form of an effective susceptibility. The computed amplitude data are shown in Figure 11b. The effective susceptibility recovered from the inversion of the amplitude data in Figure 11b is shown in Figure 1 as a volume-rendered image with an overlain translucent color display of the amplitude data. There are five main anomalous bodies of high susceptibility in the recovered model. The two broad, elongated bodies resemble kimberlite dikes known in this area, whereas the more compact bodies oriented vertically resemble kimberlite pipes. DISCUSSION Table. Magnetization direction estimated using three different methods for the field data set shown in Figure 8a. Method Inclination Declination Helbig s method Wavelet method Crosscorrelation The methodology developed in this paper for inverting magnetic data in the presence of remanent magnetization consists of two approaches. The first approach directly addresses the issue of unknown magnetization direction and estimates it using several existing and newly developed algorithms. The data are then inverted using existing magnetic inversion algorithms. The second approach circumvents the need for reliable knowledge of magnetization direction and, instead, inverts directly the amplitude of the anomalous field to recover the magnitude of the magnetization. Figure 13 summarizes the three routes to the inversion of magnetic data. For a given data set, the first question to be answered is whether the data are affected by strong remanent magnetization that is not aligned with the current inducing field. If the answer is no, then any standard inversion algorithms for 3D magnetic inversion can be applied. If the answer is yes, the data set should be inverted by using one of two approaches depending on the complexity of the magnetic anomaly. The criterion for choosing which method to use is whether a single magnetization direction is a valid assumption and can be estimated. If the answer is yes, then the method based on direction estimation should be used. In practice, this means that only a single compact anomaly is present, although rare cases of multiple anomalies with the same magnetization direction may exist. If the answer is no, then the amplitude-data inversion method should be used. Such cases include a single anomaly produced by a complex source body or multiple anomalies with different orientations. Both methods can effectively construct the source distribution for a compact source body that meets the assumption of a constant magnetization direction. However, when multiple source bodies are present with varying magnetization directions, amplitude-data inversion proves to be much more versatile. The price we pay for that ability is, of course, the partially missing phase information in the data.as a result, the dip of the recovered source distribution may not be clearly imaged if the causative body does not have a pronounced Depth (m) k (SI) Figure 9. Effective susceptibility recovered by inverting the totalfield magnetic anomaly in Figure 8a. The inversion uses the magnetization direction estimated by the multiscale edge method. a Cross section at 3 m north. b Plan section at 5 m depth. Depth (m) k (SI) Figure 1. Effective susceptibility recovered by inverting the amplitude anomaly in Figure 8b. a Cross section at 3 m north. b Plan section at 5 m depth.

10 L1 Li et al T B a Figure 11. a The total-field anomaly data over a group of kimberlites on Victoria Island, Northwest Territories, Canada. The 6 1-m data have been rotated 34 clockwise and regridded at a 1-m spacing. The rotated data map has an inducing field with an inclination of 86.7 and a nominal declination of 7.7. b The corresponding amplitude of anomalous magnetic vector Ba Figure 1. Volume-rendered inversion results of the recovered effective susceptibility with a translucent color display of the amplitude data shown in Figure 11b. The view is from the southeast. The color scale indicates the amplitude data in nt; the effective susceptibility is cut off at.5 SI. Five major magnetic sources are recovered. The two broad, elongated sources resemble kimberlite dikes known in this area; the more compact, vertically oriented bodies resemble kimberlite pipes in the area. Invert total field anomaly Magnetic susceptibility No Yes Estimate magnetization direction Invert total field anomaly Total-field anomaly Remanent magetization? Single anomaly? Magnitude of magnetization elongation along its dip. Computationally, the cost of estimating a magnetization direction is negligible compared to the subsequent inversion that uses the estimation. Therefore, this method incurs a total computational cost similar to 3D inversion of purely induced magnetic data. The amplitude inversion must generate and use three sensitivity matrices; hence, the total cost is approximately three times that of the first method. Our approach deals with the difficulty caused by unknown magnetization direction. In exploration problems, one such occurrence may be the result of the presence of remanent magnetization. Although our approach enables the inversion of data to obtain a 3D distribution of magnitude of magnetization, it does not separate induced and remanent components. Another leading cause of unknown magnetization direction is the self-demagnetization effect in highly magnetic environments such as banded iron formations. The methods developed in this paper may apply to interpretation of data affected by self-demagnetization effect. This aspect is under investigation. CONCLUSION Yes We have developed a comprehensive set of methods to tackle the problem of inverting magnetic data in the presence of remanent magnetization that alters the direction of the total magnetization. Given these methods, we have a set of tools at our disposal for interpreting magnetic data in the presence of significant remanent magnetization. Coupled with existing 3D inversion algorithms, we can confidently state that most of the magnetic data acquired in exploration problems can be interpreted quantitatively by constructing 3D distributions of either magnetic susceptibility of the magnitude of magnetization. This will further enhance the applicability and effectiveness of 3D magnetic inversion. ACKNOWLEDGMENTS Compute amplitude data Invert amplitude data Figure 13. The three routes to the inversion of magnetic data. If the data are not affected by remanent magnetization, any standard inversion algorithms for 3D magnetic inversion can be applied. Otherwise, the data should be inverted by using one of the two approaches developed here. We thank Jules Lajoie, TeckCominco, and Diamond North for providing the data used in the study. We also thank Misac Nabighian and Jeff Phillips for many discussions. Finally, we thank Richard Lane, Peter Lelièvre, and an anonymous reviewer for detailed comments that improved the clarity of the presentation. This work was No

11 Magnetic inversion with remanence L11 partly supported by the Gravity and Magnetics Research Consortium, sponsored by Anadarko, BGP, BP, Chevron, ConocoPhillips, and Vale. Partial funding support was provided by KORES. REFERENCES Blakely, R. J., 1996, Potential theory in gravity and magnetic applications: Cambridge University Press. Dampney, C. N. G., 1969, The equivalent source technique: Geophysics, 34, Dannemiller, N., and Y. Li, 6, Anew method for estimation of magnetization direction: Geophysics, 71, no. 6, L69 L73. Dransfield, M., A. Christensen, and G. Liu, 3, Airborne vector magnetics mapping of remanently magnetized banded iron formations at Rocklea, WesternAustralia: Exploration Geophysics, 34, Haney, M., C. Johnston, Y. Li, and M. Nabighian, 3, Envelopes of D and 3D magnetic data and their relationship to the analytic signal: Preliminary results: 73rd Annual International Meeting, SEG, Expanded Abstracts, Haney, M., and Y. Li,, Total magnetization direction and dip from multiscale edges: 7nd Annual International Meeting, SEG, Expanded Abstracts, Hansen, R. O., and R. S. Pawlowski, 1989, Reduction to the pole at low latitude by Wiener filtering: Geophysics, 54, Helbig, K., 196, Some integrals of magnetic anomalies and their relationship to the parameters of the disturbing body: Zeitschrift für Geophysik, 9, Hornby, P., F. Boschetti, and F. G. Horowitz, 1999, Analysis of potential field data in the wavelet domain: Geophysical Journal International, 137, Lelièvre, P. G., and D. W. Oldenburg, 9, A 3D total magnetization inversion applicable when significant, complicated remanence is present: Geophysics, 74, no. 3, L1 L3. Li, Y., and D. W. Oldenburg, 1996, 3-D inversion of magnetic data: Geophysics, 61, , 1, Stable reduction to the pole at the magnetic equator: Geophysics, 66, , 3, Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method: Geophysical Journal International, 15, Lourenco, J. S., and H. F. Morrison, 1973, Vector magnetic anomalies derived from measurements of a single component of the field: Geophysics, 38, Mendonca, C. A., and J. B. C. Silva, 1993, A stable truncated series approximation of the reduction-to-the-pole operator: Geophysics, 58, Nabighian, M., 197, The analytic signal of two-dimensional magnetic bodies with polygonal cross-section: Its properties and use for automated anomaly interpretation: Geophysics, 37, Paine, J., M. Haederle, and M. Flis, 1, Using transformed TMI data to invert for remanently magnetised bodies: Exploration Geophysics, 3, Pedersen, L. B., 1978, Wavenumber domain expressions for potential fields from arbitrary -, 1 -, and 3-dimensional bodies: Geophysics, 43, Phillips, J. D., 5, Can we estimate total magnetization directions from aeromagnetic data using Helbig s formulas: Earth, Planets, and Space, 57, Pilkington, M., 1997, 3-D magnetic imaging using conjugate gradients: Geophysics, 6, Roest, W., and M. Pilkington, 1993, Identifying remanent magnetization effects in magnetic data: Geophysics, 58, Schmidt, P. W., and D. A. Clark, 1998, The calculation of magnetic components and moments from TMI: A case study from the Tuckers igneous complex, Queensland: Exploration Geophysics, 9, Shearer, S., 5, Three-dimensional inversion of magnetic data in the presence of remanent magnetization: M.S. thesis, Colorado School of Mines. Shearer, S., and Y. Li., 4, 3D Inversion of magnetic total-gradient data in the presence of remanent magnetization: 74th Annual International Meeting, SEG, ExpandedAbstracts, Wright, S. J., 1997, Primal-dual interior-point methods: SIAM. Zietz, I., and G. E. Andreasen, 1967, Remanent magnetization and aeromagnetic interpretation: Mining Geophysics,,