3D inversion of magnetic data using Lanczos bidiagonalization and unstructured element

Document Type : Research Paper


School of Mining Engineering, College of Engineering, University of Tehran, Tehran, Iran



This work presents an algorithm to construct a 3D magnetic susceptibility property from magnetic geophysical data. Physical model discretization has substantial impact on accurate inverse modeling of the sought sources in potential field geophysics, where structural meshing suffers from edge preserving of complex-shaped geological sources. In potential field geophysics, a finite-element (FE) methodology is usually employed to discretize the desired physical model domain through an unstructured mesh. The forward operator is calculated through a Gauss-Legendre quadrature technique rather than an analytic equation. To stabilize mathematical procedure of inverse modeling and cope with the intrinsic non-uniqueness arising from magnetometry data modeling, regularization is often implemented by utilizing a norm-based Tikhonov cost function. A so-called fast technique, “Lanczos Bidiagonalization (LB) algorithm”, can be utilized to solve the central system of equations derived from optimizing the function, where it decreases the execution time of the inverse problem by replacing the forward matrix with a lower dimension one. In addition, to obtain best regularization parameter, a weighted generalized cross-validation (WGCV) curve is plotted, that makes a balance between misfit norm and model norm introduced in the cost function. In order to tackle the normal propensity of physical structures to focus at the shallow depth, an expression of depth weighting is used. This procedure is applied to a synthetic scenario presenting a complex-shaped geometry along with a real set of magnetic data in central part of Iran. So the capability of the proposed algorithm for inversion indicates the accuracy of the inversion algorithm. Additionally, the modeling results pertaining to a field case study are in good agreement with the drilling data.


Main Subjects

[1] Li, Y., Oldenburg, D.W., 1998. 3-D inversion of gravity data. Geophysics 63, 109–119.
[2] Blakely, R. J. (1996). Potential theory in gravity and magnetic applications. Cambridge university press.
[3] Menke, W., 1989. Geophysical Data Analysis: Discrete Inverse Theory. Academic Press, Inc.
[4] Cella, F., & Fedi, M. (2012). Inversion of potential field data using the structural index as weighting function rate decay. Geophysical Prospecting60(2), 313-336.
[5] Abedi, M., Gholami, A., Norouzi, G.H., 2014. 3D inversion of magnetic data seeking sharp boundaries: a case study for a porphyry copper deposit from now Chun in Central Iran. Near Surf. Geophys. 2, 657–666.
[6] Abedi, M., 2019. AIRRLS: an augmented iteratively re-weighted and refined least squares algorithm for inverse modeling of magnetometry data. J. Geol. Res. 3 (1), 16–27.
[7] Essa, K.S., Elhussein, M., 2017. A new approach for the interpretation of magnetic data by a 2-D dipping dike. J. Appl. Geophys. 136, 431–443.
[8] Biswas, A., 2020. Interpretation of gravity anomaly over 2D vertical and horizontal thin sheet with finite length and width. Acta Geophys. 68, 1083–1096.
[9] Zhang, Y., Yan, J., Li, F., Chen, C., Mei, B., Jin, S., Dohm, J.H., 2015. A new bound constraints method for 3D potential field data inversion using Lagrangian multipliers. Geophys. J. Int. 201, 267–275.
[10] Yang, M., Wang, W., Kim Welford, J., Farquharson, C.G., 2019. 3D gravity inversion with optimized mesh based on edge and center anomaly detection. Geophysics 84 (3), G13–G23.
[11] Li, Y., Oldenburg, D.W., 2003. Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method. Geophys. J. Int. 152, 251–265.
[12] Leli`evre, P.G., Oldenburg, D.W., Williams, N., 2009. Integrating geological and geophysical data through advanced constrained inversions. Explor. Geophys. 40 (4), 334–341.
[13] Fournier, D., Oldenburg, D.W., 2019. Inversion using spatially variable mixed ℓp norms. Geophys. J. Int. 218 (1), 268–282.
[14] Portniaguine, O., Zhdanov, M.S., 2002. 3D magnetic inversion with data compression and image focusing. Geophysics 67, 1532–1541.
[15] Jahandari, H., & Farquharson, C. G. (2015). Finite-volume modelling of geophysical electromagnetic data on unstructured grids using potentials. Geophysical Journal International,
 202(3), 1859-1876.
[16] Jahandari, H., Ansari, S.M., Farquharson, C.G., 2017. Comparison between staggered grid finite–volume and edge–based finite–element modelling of geophysical electromagnetic data on unstructured grids. J. Appl. Geophys. 138, 185–197.
[17] Liu, S., Hu, X., Xi, Y., Liu, T., 2015a. 2D inverse modeling for potential fields on rugged observation surface using constrained Delaunay triangulation. Comput. Geosci. 76, 18–30.
[18] Singh, A., 2020. Triangular grid-based fuzzy cross-update inversion of gravity data: case studies from mineral exploration. Nat. Resour. Res. 29, 459–471.
[19] Baranwal, V.C., Franke, A., Borner, R.U., Spitzer, K., 2011. Unstructured grid based 2-D inversion of VLF data for models including topography. J. Appl. Geophys. 75, 363–372.
[20] Darijani, M., Farquharson, C.G., Leli`evre, P.G., 2020. Clustering and constrained inversion of seismic refraction and gravity data for overburden stripping: application to uranium exploration in the Athabasca Basin, Canada. Geophysics 85 (4), B133–B146.
[21] Cai, Y., & Wang, C. Y. (2005). Fast finite-element calculation of gravity anomaly in complex geological regions. Geophysical Journal International, 162(3), 696-708.
[22] Key, K., & Ovall, J. (2011). A parallel goal-oriented adaptive finite element method for 2.5-D electromagnetic modelling. Geophysical Journal International, 186(1), 137-154.
[23] Jahandari, H., & Farquharson, C. G. (2013). Forward modeling of gravity data using finite-volume and finite-element methods on unstructured grids. Geophysics, 78(3), G69-G80.
[24] Kim, G. S., Ryu, J. C., Sin, O. C., Han, J. S., & Kim, S. G. (2014). Body-growth inversion of magnetic data with the use of non-rectangular grid. Journal of Applied Geophysics, 102, 47-61.
[25] Gross, L., Altinay, C., & Shaw, S. (2015). Inversion of potential field data using the finite element method on parallel computers. Computers & geosciences, 84, 61-71.
[26] Roussel, C., Verdun, J., Cali, J., & Masson, F. (2015). Complete gravity field of an ellipsoidal prism by Gauss–Legendre quadrature. Geophysical Supplements to the Monthly Notices of the Royal Astronomical Society, 203(3), 2220-2236.
[27] Schaa, R., Gross, L., & Du Plessis, J. (2016). PDE-based geophysical modelling using finite elements: examples from 3D resistivity and 2D magnetotellurics. Journal of Geophysics and Engineering, 13(2), S59-S73.
[28] Li, J., Lu, X., Farquharson, C. G., & Hu, X. (2018). A finite-element time-domain forward solver for electromagnetic methods with complex-shaped loop sources. Geophysics, 83(3), E117-E132.
[29] Uwiduhaye, J. D. A., Mizunaga, H., & Saibi, H. (2019). A case history: 3-D gravity modeling using hexahedral element in Kinigi geothermal field, Rwanda. Arabian Journal of Geosciences, 12(3), 86.
[30] Codd, A. L., Gross, L., & Aitken, A. (2021). Fast multi-resolution 3D inversion of potential fields with application to high-resolution gravity and magnetic anomaly data from the Eastern Goldfields in Western Australia. Computers & Geosciences, 157, 104941.
[31] Toushmalani, R., & Saibi, H. (2015). Fast 3D inversion of gravity data using Lanczos bidiagonalization method. Arabian Journal of Geosciences, 8(7), 4969-4981.
[32] Rezaie, M., Moradzadeh, A., & Kalateh, A. N. (2017). Fast 3D inversion of gravity data using solution space priorconditioned lanczos bidiagonalization. Journal of Applied Geophysics, 136, 42-50.
[33] Tikhonov, A. N., & Arsenin, V. I. (1977). Solutions of ill-posed problems (Vol. 14).
[34] Martin, R., Monteiller, V., Komatitsch, D., Perrouty, S., Jessell, M., Bonvalot, S., & Lindsay, M. (2013). Gravity inversion using wavelet-based compression on parallel hybrid CPU/GPU systems: application to southwest Ghana. Geophysical Journal International, 195(3), 1594-1619.
[35] Kim, Kang-sop, Hu, Xiang-yun, et al., 2009. Study on isoparametric finite-element integral algorithm of gravity and magnetic anomaly for body with complex shape. Oil Geophys. Prospect. 44 (2), 231–239 (in Chinese).
[36] Pilkington, M. (1997). 3-D magnetic imaging using conjugate gradients. Geophysics, 62(4), 1132-1142.
[37] Meng, Z., Li, F., Xu, X., Huang, D., & Zhang, D. (2017). Fast inversion of gravity data using the symmetric successive over-relaxation (SSOR) preconditioned conjugate gradient algorithm. Exploration Geophysics, 48(3), 294-304.
[38] Moradzadeh, A. (1998). Electrical imaging of the Adelaide geosyncline using magnetotellurics (MT) (Doctoral dissertation, Flinders University of South Australia).
[39] Zhdanov, M. S. (2002). Geophysical inverse theory and regularization problems (Vol. 36). Elsevier.
[40] Abedi, M. (2022). Cooperative fuzzy‑guided focused inversion for unstructured mesh modeling of potential feld geophysics, a case study for imaging an oil‑trapping structure. Acta Geophysica, doi.org/10.1007/s11600-022-00857-w.
[41] Danaei, K., Moradzadeh, A., Norouzi, G. H., Smith, R., Abedi, M., & Fam, H. J. A. (2022). 3D inversion of gravity data with unstructured mesh and least-squares QR-factorization (LSQR). Journal of Applied Geophysics, 206, 104781.
[42] Abedi, M., Gholami, A., Norouzi, G. H., & Fathianpour, N. (2013). Fast inversion of magnetic data using Lanczos bidiagonalization method. Journal of Applied Geophysics, 90, 126-137.
[43] Daliran, F., Stosch, H.G., Williams, P., 2010. Lower Cambrian iron oxide-apatite-REE (U) deposits of the Bafq district, east-Central Iran. In: Corriveau, L., Mumin, A.H. (Eds.), Exploring for Iron Oxide Copper-Gold Deposits. Canada and Global Analogues. Geological Society of Canada Short Course Notes 20, St. John’s, Newfoundland Canada, pp. 147–159.
[44] Nabatian, Gh., Rastad, E., Neubauer, F., Honarmand, M., Ghaderi, M., 2015. Iron and FeMn mineralisation in Iran: implications for Tethyan metallogeny. Aust. J. Earth Sci.
62, 211–241.
[45] Alamdar, K., 2016a. Interpretation of the magnetic data from Shavaz iron ore using
enhanced local wavenumber (ELW) and comparison with Euler deconvolution method. Arab. J. Geosci. 9, 597.
[46] Alamdar, K., 2016b. Development of the gradient ratio method for depth estimation of the subsurface bodies using Bouguer gravity map data. J. Res. Appl. Geophys. 1 (2), 131–141.
[47] Rahimi, E. (2018),  Depth estimation of the potential field data using wavelet transform, (case study: gravity and magnetic data from Shavaz iron ore mine in Yazd). M.Sc. thesis (in Persian), Department of Mining & Metallurgical Engineering, Yazd University, Iran.
 [48] Abedi, M. (2020). A focused and constrained 2D inversion of potential field geophysical data through Delaunay triangulation, a case study for iron-bearing targeting at the Shavaz deposit in Iran. Physics of the Earth and Planetary Interiors 309, 106604.