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L2 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, 1997. 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 2D total gradient of the magnetic field and the absolute value of the horizontal gradient of the pseudogravity produced by 2D sources. Lourenco and Morrison 1973 develop a method based upon the integral relationships of magnetic moments derived by Helbig 1962. More recently, Haney and Li 2002 develop a wavelet-based method for determining magnetization direction in 2D data sets. Dannemiller and Li 2006 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 0 . 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, 2005 is based on the integral relations between the moments of a magnetic anomaly and the magnetic dipole moment developed by Helbig 1962: xB z x,ydxdy2m x , yB z x,ydxdy2m y , xB x x,ydxdy2m 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, 1998. 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., 2003, and the observed three-component data can be used directly in the estimation. Wavelet multiscale edge method Haney and Li 2002 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 2D sources. Given a profile of magnetic data collected at a height z 0 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., 1999. 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
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 sx cot I m 2 3 , x cot I m 3 4 3 , 2 where s is the dilation factor and x is the location along the magnetic profile. The dipole line is assumed to be directly below x0. 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 2. Although the above approach is developed for 2D 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 2D 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 2D 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 RTPanomaly 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., 2003. The envelope, by definition, is the most symmetric form. Using these properties, Dannemiller and Li 2006 develop a method to estimate magnetization by examining the symmetry of various RTP fields. This method is an extension of the 2D 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 2006 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, 2001. 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 90° or 90°, 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 0.05 SI. The inducing-field direction for this model has an inclination of 65°, a declination of 25°, and a strength of B 0 50,000 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 2.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 RTPfield achieve maximum correlation at I m ,D m 45°,80°, 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 10° Helbig’s method. This is well within the error tolerance of 15° for the assumed magnetization established by Shearer 2005. We then use these estimated magnetization directions as input in a 3D inversion algorithm Li and Oldenburg, 2003 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
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