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1 Efficient 3D inversion of magnetic data via octree mesh discretization, space-filling curves, and wavelets Kristofer Davis and Yaoguo Li, Center for Gravity, Electrical, and Magnetics, Colorado School of Mines, Golden, Colorado SUMMARY Inversion of potential-field data involve large and dense coefficient matrices that often exceed the limitations of physical memory in commonly available computers. The repeated multiplications of such matrices to vectors during inversion require an immense amount of CPU power. These two factors pose a significant challenge to solving large-scale problems in practice and can render many realistic problems intractable. To overcome these limitations, we develop a new computational approach for this class of problems by combining an adaptive octree model discretization and wavelet transforms on a re-ordered parameter set. The adaptive mesh discretizes the model region by starting with large cells and splitting the region into smaller cells for localized anomalies to maintain resolution. Hilbert space-filling curves and similar ordering of the reduced parameter set produce higher compression of the coefficient matrix to form its sparse representation in the 1D wavelet domain. This combination can reduce the storage requirement by 1 to 1 times and, therefore also speeds up the computation of the inversion by the same amount. As a result, problems can now be solved that were computationally prohibitive. We present the algorithm and illustrate its effectiveness with synthetic and field examples. BACKGROUND Numerous methods have been developed in order to reduce inversion time or physical memory required. In general, the most computational efficiency has been achieved through integral transforms by storing only significant coefficients of those functions. This not only increases the speed of solving the inverse problem, but also enables larger datasets to be inverted on a computer with limited resources. The Fourier transform (Cordell, 1992; Pilkington, 1997) accelerates problems based on the convolution theorem. The separable wavelet transform (Li and Oldenburg, 1999, 23) has a compression property by winnowing small coefficients that do not significantly change the function after the inverse transform. These two techniques have been the principal approaches for accelerating this class of inverse problems in geophysics. The main limitation of inverting in the Fourier domain is the requirement of a grid of data on a plane. The principal drawback of wavelets is the entire construction of the dense coefficient matrix prior to applying the transform. However, the wavelet transform can be improved upon by reducing the size of the dense coefficient matrix which must be calculated prior to utilizing it. Ridsdill-Smith (2) used wavelets along profiles in order to find edges and reduce the number of parameters. We use a data-adaptive octree-based mesh in order to reduce the number of model parameters. Octree-based methods have been used for 3D inversion of electromagnetic data (Ascher and Haber, 21). Further more, grid refinement (e.g. multi-grid methods) based on hierarchical mesh structures can increase the efficiency of non-linear problems (e.g. Haber et al., 27). We create an octree based mesh prior to inversion and use it throughout because of linearity of the problem. For a quantitative comparison of techniques, we will define the ratio of total coefficients in the dense matrix to the number of stored coefficients (i.e. after the wavelet transform and thresholding) known as the compression ratio. Computation time is inversely proportional to this ratio. In this paper, we first describe the inversion technique. We then discuss a data-adaptive model discretization in order reduce the number of sources prior to inversion. It is desirable to use the wavelet transform for the compression properties. We discuss how to apply the transform to an irregular mesh by implementing the transform to the Green s functions of the model parameters for each respective datum. We also use the ordering the model parameters to increase compression ability. A field example illustrates the different steps of the method. INVERSION ALGORITHM FOR MAGNETIC DATA For computational purpose, we discretize the model into a set of contiguous prisms and assume a constant susceptibility in each. The observed total-field, d, is then given by G κ = d, (1) where κ = (κ 1,κ 2,,κ m ) T that are the magnetic susceptibilities of the model, d = (d 1,d 2,,d n ) T are the total-field data, and G is a dense coefficient matrix, referred to as the sensitivity matrix. It describes the geometry and physics between the sources (prismatic cells) and the points of observation (Li and Oldenburg, 1996; Blakely, 1996). The sensitivity for the i th datum given the j th model susceptibility is G i, j = µ o 4π V j ˆB 1 r i r B o dv, (2) where V j is the volume of the j th model prism, ˆB is the magnetization direction of the volume, and B o is the inducing field. This dense matrix is a function of four spatial variables. The elements along the columns correspond to different observation locations (x,y,z), while the elements along the rows correspond to different source locations (x,y,z ). Thus the dense sensitivity matrix is m model parameters by n data and stores the coefficients of the kernel between each datum and respective model parameter. The calculation of the sensitivity matrix, G, for the total field assumes that each cell has a constant susceptibility throughout its volume. We formulate the linear inverse problem to construct magnetic sources by minimizing a global objective function subject to 21 SEG SEG Denver 21 Annual Meeting Downloaded 23 May 211 to Redistribution subject to SEG license or copyright; see Terms of Use at

2 Octree inversion of magnetic data the data constraints in equation 1 (Menke, 1989; Parker, 1994). To achieve this, we construct the magnetic susceptibility distribution through Tikhonov formalism (Tikhonov and Arsenin, 1977). The optimal model solution is found when minimizing the global objective function, Φ(λ), such that minφ(λ) = Φ d + βφ m λ M ln(κ j ), (3) where β is the trade-off (or Tikhonov) parameter, Φ d is the data misfit function, Φ m is the model objective function, and λ M j=1 ln(κ j) is a log-barrier function to impose positivity (Wright, 1997). The global objective function is subject to Φ d = Φ d which, depends upon the noise in the data. The data misfit function defines the fit of the predicted data to the observed data and is given by: j=1 Φ d = Wd (G κ d) 2, (4) Northing (m) e e e e e+6 Depth (m) Easting (m) where W d is a diagonal data weighting matrix normalizing the data its respective standard deviation, σ i. The model objective function quantifies the structural complexity of the model and takes the following form Φ m = W m ( κ κ o ) 2, (5) where κ o is a reference model, W m is a model weighting matrix that includes an empirical depth weighting function, (Li and Oldenburg, 1996). Solving the system of equations to invert for a model that reproduces the data with a dense matrix of size m m, requires at least O(m 3 ) operations. For large datasets, inversion can become computationally prohibitive. One way of increasing the efficiency is minimizing the number of parameters, m, that are required. This is the topic of the next section. OCTREE MESH DISCRETIZATION Octree mesh discretization minimizes the number of model parameters by regrouping finer cells with a coarser ones in regions where high resolution is not needed. The approach keeps finer cells where resolution is desirable to recover a model. The method has been primarily used in geophysics for nonlinear inversion such as electromagnetics (Ascher and Haber, 21; Haber and Heldmann, 27) and large-scale earthquake finite element modeling (Kim et al., 22; Bielak et al., 25). The octree method is particularly useful in large scale problems. Octree design has 2 i 2 j 2 k number of prisms in the mesh as a result of a series of the model splitting to quarters iteratively. There are a few different approaches one can use to incorporate an octree mesh. Information such as the source geometry, for example known geology or a reservoir in timelapse cases, can be used to pre-define large cells and small cells to retain resolution in the desired region. Using the geometry of a source and receiver configuration can also aid in where the mesh should have smaller cells, such as in the case of electromagnetics. In the case of exploration magnetics, the extent of sources are only known laterally to a certain degree. Therefore, we adopt an algorithm dividing the mesh laterally Figure 1: An example of an octree mesh shown in plan view (Top). A slice of the same mesh cut through the main anomalous zone in depth (bottom). The cells retain the aspect ratio assigned from the top view (e.g. Davis and Li, 27), then carry a consistent aspect ratio of the prisms in depth (Figure 1). The adaptive octree mesh architecture places larger model cells where no or little signal change is present in the overlying magnetic data. Smaller cells are assigned to regions requiring increased resolution such as edges of source bodies in order to properly fit overlying, highly-changing magnetic data. The mesh starts with large prismatic cells (i.e. a mesh) and if one of the cells has a property value higher, from known geologic parameters or transformed data, than a given threshold, it is split in half. This process continues until all of the cells are within the property threshold, or to a user-defined width. This may mean if a source is large enough, cells within the source may be larger than the edges of the source. It is important to note that the threshold should automated as it is uneconomical to perform the mesh discretization more than once for the linear problem. Practically, the value with which to choose the mesh are located in the middle of the smallest cell size (a regular, or base mesh), enabling the actual discretization of the mesh to be fast. The observed data is approximately gridded using a nearest neighbor interpolation of the data to the center location of each of the smallest cell size for the top layer. The base mesh is designed to have 2 i 2 j 2 k cells. Though this could mean that the mesh is finer than normally desired, the effect will be absorbed within the octree discretization. The accuracy of the gridded data is not as important as the speed of the algorithm. However, the gridding will allow for an examination of a property for each model cell. It is important to note that this procedure is only used to choose the close to optimal mesh for inversion. In this case, two main issues drive the need for smaller cells to 21 SEG SEG Denver 21 Annual Meeting Downloaded 23 May 211 to Redistribution subject to SEG license or copyright; see Terms of Use at

3 Octree inversion of magnetic data retain accurate results: (1) a long wavelength change in total amplitude over the area (i.e. regional field) or (2) a localized anomaly. We use the curvature of the amplitude of magnetic data normalized by the amplitude. This allows for both the low frequency change of amplitude throughout the examined area and the edges of mid-frequency anomalies to be detected and the mesh adjusted accordingly. One could use the totalgradient, but amplitude data are not as affected by remenant magnetization (Shearer, 25) that can occur often in mining exploration (Li et al., 21). The amplitude, A, of magnetic anomaly is given by A = B 2 x + B 2 y + B 2 z, (6) with B x, B y, and B z the three orthogonal components of the anomalous magnetic field, B a, respectively and are computed in the Fourier domain. The curvature, c, approximated by c = ( 2 A x 2 )2 + ( 2 A y 2 )2, (7) and computed by the second derivatives of the amplitude. This step is straightforward because the data values are already gridded and padded for the Fourier transform. The mesh is then split based on the value q given by dividing equation (7) by equation (6) where q = c A, (8) such that if a change in q is more than twice the standard deviation of q for the entire dataset, the cells in region of interest are split. The threshold prevents smaller cells from forming where small deviations in the data are present by noise from acquisition or gridding. The sensitivity matrix is calculated based on the nodal points of the octree model parameters and the inversion of the data can carried out as it would be for a regular mesh. The octree mesh design has the ability to reduce the model parameters an order of magnitude on average and is highly data dependent. The algorithm is designed to compress the rows of the sensitivity matrix (i.e. model parameters). With the average reduction, the compression ratio for the octree mesh discretization alone averages between 1 and 4. Unfortunately, this is not enough compression to be practical as the desired compression ratio should be a minimum of 5 to advance current methodology. WAVELET TRANSFORMS The wavelet transform expands a function in the bases formed by the translation and dilation of a function called the mother wavelet (Mallet, 1989; Daubechies, 1992; Meyer, 1993). The wavelet transform of a function is based on orthonormal, compactly supported wavelets. Winnowing the coefficients whose magnitudes are below a certain level still allows reconstruction of the original function with a high degree of accuracy. Thus a function can have a sparse representation in the wavelet domain. The compression property of the wavelet transform arises from the presence of a large number of small coefficients in the transformed function. These small wavelet coefficients are produced because the wavelets are localized and orthogonal to a low-degree polynomial. Smoothly decaying portions of the matrix can be represented accurately by only a few wavelets at coarse scales. At low threshold levels, the reconstruction of the function is virtually identical to the function itself, yet only has significant elements within a narrow band near the location of the peaks in the original matrix. As the threshold level increases, small scale distortions begin to appear but the long wavelength features remain. This is the essential property of the wavelet transform that is utilized to achieve high compression ratios. With a regularly gridded mesh, the separable 3D wavelet transform can be applied to each direction of the mesh in the coefficient matrix G. Analogous to a 3D Fourier transform, the 3D wavelet transform is a series of consecutive 1D transforms performed on each respective direction of the 3D mesh. It is this characteristic that enables the wavelet transform to be applied to non-planar data. With a irregularly gridded mesh, such as with a octree discretization, the separable wavelet transform is no longer applicable. Therefore, we apply the 1D wavelet to each row of the sensitivity matrix as if it were simply a 1D function. REORDERING MODEL PARAMETERS To increase the compression of the rows, we attempt to make each row of the sensitivity matrix smoother by re-ordering the model parameters. Indexing the model cells appropriately results in a large number of small model responses consecutively which require few coefficients in the wavelet domain. The idea is similar to the one used by Lamarque and Robert (1996) for edge detection of images. The approach of re-ordering the model is to optimally map the 3D structure of the model mesh onto the 1D row of the sensitivity matrix for a given datum. Therefore we examine general means of mapping an n- dimensional space onto a one dimensional space: space-filling curves (Butz, 1971; Sagan, 1994; Jin and Mellor-Crummey, 25). Hilbert (1891) presented a space-filling curve to efficiently and consecutively map every node for a closed 2 n cube for which became the base for the entire class of space-filling curves (Moon et al., 21). Three dimensional space-filling curves are utilized in multi-grid methods for up-scaling and down-scaling mesh (Aftosmis et al., 24; Griebel et al., 1998; Behrens and Zimmermann, 2). The nature of the curve allows for the minimization of large spatial jumps of cells in the octree mesh. Simple examples of 3D Hilbert curves of orders 1 and 2 onto a base mesh are shown in Figures 2(a) and 2(b). respectively. For our application, the Hilbert curve starts in the south-west portion of the mesh. Since the base mesh is 2 i 2 j 2 k, multiple Hilbert curves are easily incorporated. If i = j = k is not true, multiple Hilbert curves are merged throughout the mesh. The curve searches throughout the base mesh in order get the next prism in the octree mesh to ensure spatial continuity. 21 SEG SEG Denver 21 Annual Meeting Downloaded 23 May 211 to Redistribution subject to SEG license or copyright; see Terms of Use at

4 Octree inversion of magnetic data 1 Amplitude e+6 1.5e+6 2e+6 Parameter number m y (a) m x (a) Amplitude Parameter number (b) m y 2 4 m x (b) Figure 2: The Hilbert space-filling curve is used for ordering model parameters with the 1D wavelet. The first order curve (a) for a 2 1 cube. The second order curve (b) is for 2 2 cube. The Hilbert curve allows for model cells to be spatially continuous during the adaptive mesh parameter ordering. The ordering starts in the south-west top corner and concludes in the north-west top corner. EXAMPLE As an illustration, we show the response for a datum of a base mesh for a total of 2,97,152 parameters. Using 3D wavelet transforms, the sensitivity matrix row requires 3,489 non-zeros for a compression ratio of 61. We show the row in Figure 3(a) prior to wavelet compression. The same row after octree discretization and Hilbert-curve indexing of the model parameters is shown in Figure 3(b). The octree-based method reduced the model to 59,76 parameters and compresses the row 1,186 times for a storage requirement of 1,768 non-zeros. For a comparison of the separable and 1D wavelet, we show the histograms of 1D wavelet (gray) and 3D wavelet (black) coefficient amplitudes Figure 3(c). The multidimensional wavelets require many more small amplitude coefficients than the one-dimensional wavelet which decreases the storage and increases computation time. These small coefficients are required between the many peaks that are constant throughout the row. The Hilbert-curve re-ordering of the model produces two main peaks, requiring larger, but fewer coefficients. Frequency Coefficient amplitude (c) Figure 3: (a) Model response for a datum (e.g. row of the kernel matrix) for a regular mesh prior to wavelet transform. (b) The same row after octree discretization. The compression achieved with the separable wavelet transform was 61, where as with the octree approach was 1,186. (c) Histograms of the 1D (gray) and separable wavelet (black) show the large number of small-amplitude coefficients decreasing the efficiency. CONCLUSIONS Inversion of potential-field methods involve large and dense coefficient matrices that often exceed the limitations of physical memory in commonly available computers. The adaptive octree approach is useful in decreasing the number of model parameters, effectively reducing the m m system of equations needed for inversion. Model ordering before applying the 1D wavelet transform for the compression each row of the kernel matrix creates a three-step approach towards achieving maximum compression potential. This combination can consistently reduce the storage requirement by 1s to 1s times and, therefore also speeds up the computation of the inversion process by the same amount. A field example will further demonstrate the effectiveness of the approach. ACKNOWLEDGMENTS The authors thank Robert Eso for helpful discussions involving mesh generation. We would also Dave Hale for insightful discussions on octree methodology and Hilbert space-filling curves. This research was supported financially through the companies of Gravity and Magnetics Research Consortium: Anadarko, BGP, BP, ConocoPhillips, and Valé. 21 SEG SEG Denver 21 Annual Meeting Downloaded 23 May 211 to Redistribution subject to SEG license or copyright; see Terms of Use at

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 21 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Aftosmis, M. J., M. J. Berger, and S. M. Murman, 24, Applications of space-filling curves to cartesian methods for CFD: Presented at the 42nd Aerospace Sciences Meeting and Exhibit. Ascher, U. M., and E. Haber, 21, Grid refinement and scaling for distributed parameter estimation problems: Inverse Problems, 17, no. 3, , doi:1.188/ /17/3/314. Behrens, J., and J. Zimmermann, 2, Parallelizing an unstructured grid generator with a space-filling curve approach: Presented at the Euro-Par 2 Parallel Processing, 6th International Euro-Par Conference. Bielak, J., O. Ghattas, and E. J. Kim, 25, Parallel octree-based finite element method for large-scale earthquake ground motion simulation: CMES, 1, Blakely, R., 1996, Potential theory in gravity and magnetic applications: Cambridge University Press. Butz, A. R., 1971, Alternative algorithm for hilbert s space-filling curve: IEEE Transactions on Computers, C-2, no. 4, , doi:1.119/t-c Cordell, L., 1992, A scattered equivalent-source method for interpolation and gridding of potential field data in three dimensions: Geophysics, 57, , doi:1.119/ Daubechies, I., 1992, Ten lectures on wavelets: SIAM, Philadelphia, PA. Davis, K., and Y. Li, 27, A fast approach to magnetic equivalent source processing using an adaptive quadtree mesh discretization: Presented at the 19th Geophysical Conf. and Exhibition, ASEG. Griebel, M., N. Tilman, and H. Regler, 1998, Algebraic multigrid methods for the solution of the navierstokes equations in complicated geometries: International Journal of Numerical Methods for Heat & Fluid Flow, 26, Haber, E., and S. Heldmann, 27, An octree multigrid method for quasi-static Maxwells equations with highly discontinuous coefficients: Journal of Computational Physics, 223, no. 2, , doi:1.116/j.jcp Haber, E., S. Heldmann, and U. Ascher, 27, Adaptive finite volume method for distributed non-smooth parameter identification: Inverse Problems, 23, no. 4, Hilbert, D., 1891, Ueber die stetige abbildung einer line auf ein flachenstuck: Mathematische Annalen, 38, no. 3, , doi:1.17/bf Jin, G., and J. Mellor-Crummey, 25, Using space-filling curves for computation reordering: Presented at the Los Alamos National Laboratory, Los Alamos Computer Science Institute Sixth Annual Symposium. Kim, E., J. Bielak, and J. Wang, 22, Octree-based finite element method for large-scale earthquake ground motion modeling in heterogeneous basins: Presented at the American Geophysical Union Fall Meeting, Abstract S12B Lamarque, C. H., and F. Robert, 1996, Image analysis using space-filling curves and 1d wavelet bases: Pattern Recognition, 29, no. 8, , doi:1.116/31-323(95) SEG SEG Denver 21 Annual Meeting Downloaded 23 May 211 to Redistribution subject to SEG license or copyright; see Terms of Use at

6 Li, Y., and D. W. Oldenburg, 1996, 3-D inversion of magnetic data: Geophysics, 61, no. 2, , doi:1.119/ , 1999, Rapid construction of equivalent sources using wavelets: Expanded Abstracts, , 69th Annual International Meeting, SEG. Li, Y., and D. W. Oldenburg, 23, Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method: Geophysical Journal International, 152, no. 2, , doi:1.146/j x x. Li, Y., S. E. Shearer, M. M. Haney, and N. Dannemiller, 21, Comprehensive approaches to 3d inversion of magnetic data affected by remanent magnetization: Geophysics, 75, no. 1, L1 L11, doi:1.119/ Mallet, S., 1989, A theory for multiresolution signal decomposition: The wavelet representation: IEEE Trans. PAMI, 11, Menke, W., 1989, Geophysical data analysis: Discrete inverse theory: Academic Press. Meyer, Y., 1993, Wavelets: Algorithms and applications: SIAM, Philadelphia, PA. Moon, B., H. V. Jagadish, C. Faloutsos, and J. H. Saltz, 21, Analysis of the clustering properties of hilbert space-filling curve: IEEE Transactions on Knowledge and Data Engineering, 13, no. 1, , doi:1.119/ Parker, R. L., 1994, Geophysical inverse theory: Princeton University Press. Pilkington, M., 1997, 3-D magnetic imaging using conjugate gradients: Geophysics, 62, , doi:1.119/ Ridsdill-Smith, T. A., 2, The application of the wavelet transform to the processing of aeromagnetic data: PhD thesis, University of Western Australia. Sagan, H., ed., 1994, Space-filling curves: Springer-Verlag. Shearer, S., 25, Three-dimensional inversion of magnetic data in the presence of remanent magnetization: Master s thesis, Colorado School of Mines. Tikhonov, A. N., and V. Y. Arsenin, 1977, Solution of ill-posed problems: Winston Press. Wright, S. J., 1997, Primal-dual interior-point methods: SIAM. 21 SEG SEG Denver 21 Annual Meeting Downloaded 23 May 211 to Redistribution subject to SEG license or copyright; see Terms of Use at