2D reconstruction of gravity anomalies using the level set method

Document Type : Research Article


1 M.Sc. Graduated, Department of Earth Physics, Institute of Geophysics, University of Tehran, Tehran, Iran

2 Assistant Professor, Department of Earth Physics, Institute of Geophysics, University of Tehran, Tehran, Iran


In order to properly understand the subsurface structures, the issue of inversion of geophysical data has received much attention from researchers. Since accurate reconstruction of the shape and boundaries of the mass using gravimetric data is very important in some issues, it is important to use an effective and efficient method that has a high ability to draw and reconstruct the boundaries of a mass. In recent years, the level set method introduced by Asher and Stein has been widely used to solve this problem. From the expansion of the level set function in some bases of the problem, the effective number of parameters is greatly reduced and an optimization problem is created which its behavior is better than the least squares problem. As a result, the level set parameterization method will be presented for the reconstruction of inversion models. A common advantage of the parametric level set method is the careful examination of the boundary for optimum sensitivities, which significantly reduces the dimensional problem, and many of the difficulties of traditional level set methods, such as regularization, reconstruction, and basis function. Level set parameterization is performed by radial basis functions (RBF); which causes an optimal problem with an average number of parameters and high flexibility; and the computational and optimization process for Newton's method is more accurate and smooth. The model is described by the zero contour of a level-set function, which in turn is represented by a relatively small number of radial basis functions. This formulation includes some additional parameters such as the width of the radial basis functions and the smoothness of the Heaviside function. The latter is of particular importance as it controls the sensitivity to changes in the model. In this algorithm adaptively chooses the required smoothness parameter and tests the method on a suite of idealized Earth models.
 In this evolutionary approach, the reduction gradient method usually requires many iterations for convergence, and the functions are weakened for low-sensitivity problems. Although the use of Quasi- Newton methods to improve the level set function increases the degree of convergence, they are computationally challenging, and for large problems and relatively finer grids, a system of equations must be solved in each iteration. Moreover, based on the fact that the number of underlying parameters in a parametric approach is usually much less than the number of pixels resulting from the discretization of the level set function, we make a use of a Newton-type method to solve the underlying optimization problem.
In this research, the algorithm is used to investigate its strengths and weaknesses for applying geophysical gravity data, coding and programming, and it is tested using several two-dimensional synthetic models. Finally, the method is tested on gravity data from the Mobrun ore body, north east of Noranda, Quebec, Canada.
The results of this study show that the application of the optimization algorithm of the level set function will lead to a relatively more accurate and realistic detection of mass boundaries. It shows that the tested mass has spread from a depth of 10 meters to a depth of 160 meters.


Main Subjects

Allaire, G., Jouve, F. and Toader, A.-M., 2002, A level-set method for shape optimization. C R Acad Sci., Paris Ser I, 334, 1–6.
Allaire, G., Jouve, F. and Toader, A.-M., 2004, Structural optimization using sensitivity analysis and a level-set method. J Comput Phys; 194, 363–93.
Aghasi, A., Kilmer, M. and Miller, E. L., 2011 Parametric level set methods for inverse problems. SIAM Journal on Imaging Sciences, 4(2), 618–650.
Belytschko, T., Xiao, SP. and Parimi, C., 2003, Topology optimization with implicitly function and regularization. Int J Numer Method Eng; 57, 1177–96.
Ben Hadj Miled, M. K. and Miller, E.L., 2007, A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography, Inverse Problems, 23, 2375–2400.
Berger, M.S., 1977, Nonlinearity and Functional Analysis, Academic Press, New York, London.
Bernard, O., Friboulet, D., Th´ evenaz, P. and Unser, M., 2009, Variational B-spline level-set: A linear filtering approach for fast deformable model evolution, IEEE Trans. Image Process., 18, 1179–1191.
Blakely, R. J., 1996, Potential theory in gravity and magnetic applications. Cambridge university press.
Chan, T.F. and Vese, L.A., 2001, Active contours without edges, IEEE Trans. Image Process. 10, 266-277.
Charles, A., 1984, Micchelli. Interpolation of scattered data: distance matrices and conditionally positive definite functions. In Approximation theory and spline functions, pages 143–145.
Dorn, O. and Lesselier, D., 2006, Level set methods for inverse scattering, Inverse Problems, 22, R67–R131.
Feng, H., Karl, W.C. and Castanon, D., 2003, A curve evolution approach to object-based tomographic reconstruction, IEEE Trans. Image Process., 12, 44–57.
Gage, M. E., 1983, An isoperimetric inequality with applications to curve shortening. Duke Mathematical Journal, 50(4), 1225-1229.
Grant, F. S. and West, G. F., 1965, Interpretation theory in applied geophysics, McGraw, 70, 39-43.
Isakov, V., Leung, S. and Qian, J., 2011, A fast local level set method for inverse gravimetry: Communications in Computational Physics, 10, 1044–1070.
Isakov, V., Leung, S. and Qian, J. 2013, A three-dimensional inverse gravimetry problem for ice with snow caps: Inverse Problems and Imaging, 7, 523–544, doi: 10.3934/ipi.
Kadu, A., Van Leeuwen, T. and Mulder, W., 2017, Parametric level-set full-waveform inversion in the presence of salt bodies. In SEG Technical Program Expanded Abstracts 2017 (pp. 1518-1522). Society of Exploration Geophysicists.
Li, W. and Leung, S., 2013, A fast local level set adjoint state method for first arrival transmission traveltime tomography with discontinuous slowness: Geophysical Journal International, 195, 582–596, doi: 10.1093/gji/ ggt244.
Li, W., Leung, S. and Qian, J., 2014, A level-set adjoint-state method for crosswell transmission-reflection traveltime tomography: Geophysical Journal International, 199, 348–367, doi: 10.1093/gji/ggu262.
Micchelli Charles, A., 1986, Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constructive approximation 2.1, 11-22.
Mulder, W. A. , Osher, S. and Sethian, J. A., 1992, Computing interface motion in compressible gas dynamics. Journal of Computational Physics, 100(2), 209–228.
Ngo, T. A., Lu, Z. and Carneiro, G., 2017, Combining deep learning and level set for the automated segmentation of the left ventricle of the heart from cardiac cine magnetic resonance. Medical image analysis, 35, 159-171.
Osher, S. and Fedkiw, R. P. 2001, Level set methods: an overview and some recent results. Journal of Computational Physics, 169(2), 463–502.
Osher, S.J. and Santosa, F., 2001, Level set methods for optimization problems involving geometry and constraints. I. frequencies of a two-density inhomogeneous drum. J Comput Phys; 171, 272–88.
Paragios, N., Faugeras, O., Chan, T. and Schnoerr, C., (Eds.), 2005, Variational, Geometric, and Level Set Methods in Computer Vision: Third International Workshop, VLSM 2005, Beijing, China, October 16, 2005, Proceedings (Vol. 3752). Springer.
Theillard, M., Djodom, L. F., Vié, J. L. and Gibou, F., 2013, A second-order sharp numerical method for solving the linear elasticity equations on irregular domains and adaptive grids–application to shape optimization. Journal of Computational Physics, 233, 430-448.
Tinós, R. and Júnior, L. O. M., 2009, Use of the q-Gaussian function in radial basis function networks. In Foundations of Computational Intelligence Volume 5 (pp. 127-145). Springer, Berlin, Heidelberg.
Villegas, R., Dorn, O., Moscoso, M., Kindelan, M. and Mustieles, F., 2006 Simultaneous characterization of geological shapes and permeability distributions in reservoirs using the level set method, in Proceedings of the SPE Europec/EAGE Annual Conference and Exhibition.
Wang, M.Y. and Wang, X.M., 2004, PDE-driven level sets, shape sensitivity, and curvature flow for structural topology optimization. Comput Model Eng Sci; 6, 373–95.
Wendland, H., 1995, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, 4(1), 389–396.
Zhang, K., Zhang, L., Lam, K. M. and Zhang, D., 2015, A level set approach to image segmentation with intensity inhomogeneity. IEEE transactions on cybernetics, 46(2), 546-557.
Zhao, H.K., Chan, T., Merriman, B. and Osher, S., 1996, A variational level set approach to multiphase motion, J. Comput. Phys., 127, 179–195.
Zhdanov, M.S., 2002, Geophysical Inverse Theory and Regularization Problems, Elsevier Science, Amsterdam.