# American Institute of Mathematical Sciences

June  2021, 15(3): 387-413. doi: 10.3934/ipi.2020073

## Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods

 1 School of Science, Harbin Institute of Technology, Shenzhen, Shenzhen 518055, China 2 Departments of Mathematics and CMSE, Michigan State University, East Lansing, MI 48824, USA

* Corresponding author: Jianliang Qian

Received  April 2020 Revised  October 2020 Published  June 2021 Early access  November 2020

We develop new efficient algorithms for a class of inverse problems of gravimetry to recover an anomalous volume mass distribution (measure) in the sense that we design fast local level-set methods to simultaneously reconstruct both unknown domain and varying density of the anomalous measure from modulus of gravity force rather than from gravity force itself. The equivalent-source principle of gravitational potential forces us to consider only measures of the form $\mu = f\,\chi_{D}$, where $f$ is a density function and $D$ is a domain inside a closed set in $\bf{R}^n$. Accordingly, various constraints are imposed upon both the density function and the domain so that well-posedness theories can be developed for the corresponding inverse problems, such as the domain inverse problem, the density inverse problem, and the domain-density inverse problem. Starting from uniqueness theorems for the domain-density inverse problem, we derive a new gradient from the misfit functional to enforce the directional-independence constraint of the density function and we further introduce a new labeling function into the level-set method to enforce the geometrical constraint of the corresponding domain; consequently, we are able to recover simultaneously both unknown domain and varying density from given modulus of gravity force. Our fast level-set method is built upon localizing the level-set evolution around a narrow band near the zero level-set and upon accelerating numerical modeling by novel low-rank matrix multiplication. Numerical results demonstrate that uniqueness theorems are crucial for solving the inverse problem of gravimetry and will be impactful on gravity prospecting. To the best of our knowledge, our inversion algorithm is the first of such for the domain-density inverse problem since it is based upon the conditional well-posedness theory of the inverse problem.

Citation: Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems and Imaging, 2021, 15 (3) : 387-413. doi: 10.3934/ipi.2020073
##### References:
 [1] J. E. Bain, T. R. Horscroft, J. Weyand, A. H. Saad and D. N. Bulling, Complex salt features resolved by integrating seismic, gravity and magnetics, EAEG/EAPG Annual Meeting, Expanded Abstracts. [2] M. K. Ben Hadj Miled and E. L. Miller, A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography, Inverse Problem, 23 (2007), 2375-2400. doi: 10.1088/0266-5611/23/6/007. [3] M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301.  doi: 10.1017/S0956792505006182. [4] O. Dorn and D. Lesselier, Level set methods for inverse scattering, Inverse Problems, 22 (2006), R67-R131. doi: 10.1088/0266-5611/22/4/R01. [5] W. Freeden and M. Zuhair Nashed, Handbook of Mathematical Geodesy: Functional Analytic and Potential Theoretic Methods, Springer Nature, 2018. doi: 10.1007/978-3-319-57181-2. [6] S. Hou, K. Solna and H.-K. Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys., 199 (2004), 317-338.  doi: 10.1016/j.jcp.2004.02.010. [7] V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, Rhode Island, 1990. doi: 10.1090/surv/034. [8] V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry,, Comm. in Computational Physics, 10 (2011), 1044-1070. doi: 10.4208/cicp.100710.021210a. [9] V. Isakov, S. Leung and J. Qian, A three-dimensional inverse gravimetry problem for ice with snow caps, Inverse Problems and Imaging, 7 (2013), 523-544.  doi: 10.3934/ipi.2013.7.523. [10] G. J. Jorgensen and J. L. Kisabeth, Joint 3-D inversion of gravity magnetic and tensor gravity fields for imaging salt formations in the deepwater gulf of mexico, in Expanded Abstracts, Soc. Expl. Geophys., Tulsa, OK, 2000, 424-426. doi: 10.1190/1.1816085. [11] B. Kirkendall, Y. Li and D. Oldenburg, Imaging cargo containers using gravity gradiometry, IEEE Transactions on Geoscience and Remote Sensing, 45 (2007), 1786-1797.  doi: 10.1117/12.660979. [12] R. A. Krahenbuhl and Y. Li, Inversion of gravity data using a binary formulation, Geophysical Journal International, 167 (2006), 543-556.  doi: 10.1111/j.1365-246X.2006.03179.x. [13] K. Kunisch and X. Pan, Estimation of interfaces from boundary measurements, SIAM J. Control Optm., 32 (1994), 1643-1674.  doi: 10.1137/S0363012992226338. [14] S. Leung and J. Qian, Transmission traveltime tomography based on paraxial Liouville equations and level set formulations, Inverse Problems, 23 (2007), 799-821.  doi: 10.1088/0266-5611/23/2/019. [15] J. P. Leveille, I. F. Jones, Z. Z. Zhou, B. Wang and F. Liu, Subsalt imaging for exploration, production, and development: A review, Geophysics, 76 (2011), WB3-WB20. doi: 10.1190/geo2011-0156.1. [16] W. Li, W. Lu and J. Qian, A level set method for imaging salt structures using gravity data, Geophysics, 81 (2016), G35-G51. doi: 10.1190/geo2015-0295.1. [17] W. Li, W. Lu, J. Qian and Y. Li, A multiple level set method for three-dimensional inversion of magnetic data, Geophysics, 82 (2017), J61-J81. doi: 10.1190/segam2017-17729331.1. [18] W. Li, J. Qian and Y. Li, Joint inversion of surface and borehole magnetic data: A level-set approach, Geophysics, 85 (2020), J15-J32. doi: 10.1190/segam2018-2996911.1. [19] Y. Li and D. Oldenburg, 3D inversion of gravity data, Geophysics, 63 (1998), 109-119. [20] A. Litman, D. Lesselier and F. Santosa, Reconstruction of a 2-D binary obstacle by controlled evolution of a level-set, Inverse Problems, 14 (1998), 685-706.  doi: 10.1088/0266-5611/14/3/018. [21] W. Lu, S. Leung and J. Qian, An improved fast local level set method for three-dimensional inverse gravimetry, Inverse Problems and Imaging, 9 (2015), 479-509.  doi: 10.3934/ipi.2015.9.479. [22] W. Lu and J. Qian, A local level set method for three-dimensional inversion of gravity gradiometry data, Geophysics, 80 (2015), G35-G51. [23] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503. [24] S. J. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2. [25] D. Peng, B. Merriman, S. Osher, H. K. Zhao and M. Kang, A pde-based fast local level set method, J. Comput. Phys., 155 (1999), 410-438.  doi: 10.1006/jcph.1999.6345. [26] F. Santosa, A level-set approach for inverse problems involving obstacles, Control, Optimizat. Calculus Variat., 1 (1996), 17-33.  doi: 10.1051/cocv:1996101. [27] M. Sussman, P. Smereka and S. J. Osher, A level set approach for computing solutions to incompressible two-phase flows, J. Comput. Phys., 114 (1994), 146-159. [28] K. van den Doel, U. Ascher and A. Leitao, Multiple level sets for piecewise constant surface reconstruction in highly ill-posed problems, J. Sci. Comput., 43 (2010), 44-66.  doi: 10.1007/s10915-009-9341-x. [29] H.-K. Zhao, T. Chan, B. Merriman and S. J. Osher, A variational level set approach for multiphase motion, J. Comput. Phys., 127 (1996), 179-195.  doi: 10.1006/jcph.1996.0167.

show all references

##### References:
 [1] J. E. Bain, T. R. Horscroft, J. Weyand, A. H. Saad and D. N. Bulling, Complex salt features resolved by integrating seismic, gravity and magnetics, EAEG/EAPG Annual Meeting, Expanded Abstracts. [2] M. K. Ben Hadj Miled and E. L. Miller, A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography, Inverse Problem, 23 (2007), 2375-2400. doi: 10.1088/0266-5611/23/6/007. [3] M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301.  doi: 10.1017/S0956792505006182. [4] O. Dorn and D. Lesselier, Level set methods for inverse scattering, Inverse Problems, 22 (2006), R67-R131. doi: 10.1088/0266-5611/22/4/R01. [5] W. Freeden and M. Zuhair Nashed, Handbook of Mathematical Geodesy: Functional Analytic and Potential Theoretic Methods, Springer Nature, 2018. doi: 10.1007/978-3-319-57181-2. [6] S. Hou, K. Solna and H.-K. Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys., 199 (2004), 317-338.  doi: 10.1016/j.jcp.2004.02.010. [7] V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, Rhode Island, 1990. doi: 10.1090/surv/034. [8] V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry,, Comm. in Computational Physics, 10 (2011), 1044-1070. doi: 10.4208/cicp.100710.021210a. [9] V. Isakov, S. Leung and J. Qian, A three-dimensional inverse gravimetry problem for ice with snow caps, Inverse Problems and Imaging, 7 (2013), 523-544.  doi: 10.3934/ipi.2013.7.523. [10] G. J. Jorgensen and J. L. Kisabeth, Joint 3-D inversion of gravity magnetic and tensor gravity fields for imaging salt formations in the deepwater gulf of mexico, in Expanded Abstracts, Soc. Expl. Geophys., Tulsa, OK, 2000, 424-426. doi: 10.1190/1.1816085. [11] B. Kirkendall, Y. Li and D. Oldenburg, Imaging cargo containers using gravity gradiometry, IEEE Transactions on Geoscience and Remote Sensing, 45 (2007), 1786-1797.  doi: 10.1117/12.660979. [12] R. A. Krahenbuhl and Y. Li, Inversion of gravity data using a binary formulation, Geophysical Journal International, 167 (2006), 543-556.  doi: 10.1111/j.1365-246X.2006.03179.x. [13] K. Kunisch and X. Pan, Estimation of interfaces from boundary measurements, SIAM J. Control Optm., 32 (1994), 1643-1674.  doi: 10.1137/S0363012992226338. [14] S. Leung and J. Qian, Transmission traveltime tomography based on paraxial Liouville equations and level set formulations, Inverse Problems, 23 (2007), 799-821.  doi: 10.1088/0266-5611/23/2/019. [15] J. P. Leveille, I. F. Jones, Z. Z. Zhou, B. Wang and F. Liu, Subsalt imaging for exploration, production, and development: A review, Geophysics, 76 (2011), WB3-WB20. doi: 10.1190/geo2011-0156.1. [16] W. Li, W. Lu and J. Qian, A level set method for imaging salt structures using gravity data, Geophysics, 81 (2016), G35-G51. doi: 10.1190/geo2015-0295.1. [17] W. Li, W. Lu, J. Qian and Y. Li, A multiple level set method for three-dimensional inversion of magnetic data, Geophysics, 82 (2017), J61-J81. doi: 10.1190/segam2017-17729331.1. [18] W. Li, J. Qian and Y. Li, Joint inversion of surface and borehole magnetic data: A level-set approach, Geophysics, 85 (2020), J15-J32. doi: 10.1190/segam2018-2996911.1. [19] Y. Li and D. Oldenburg, 3D inversion of gravity data, Geophysics, 63 (1998), 109-119. [20] A. Litman, D. Lesselier and F. Santosa, Reconstruction of a 2-D binary obstacle by controlled evolution of a level-set, Inverse Problems, 14 (1998), 685-706.  doi: 10.1088/0266-5611/14/3/018. [21] W. Lu, S. Leung and J. Qian, An improved fast local level set method for three-dimensional inverse gravimetry, Inverse Problems and Imaging, 9 (2015), 479-509.  doi: 10.3934/ipi.2015.9.479. [22] W. Lu and J. Qian, A local level set method for three-dimensional inversion of gravity gradiometry data, Geophysics, 80 (2015), G35-G51. [23] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503. [24] S. J. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2. [25] D. Peng, B. Merriman, S. Osher, H. K. Zhao and M. Kang, A pde-based fast local level set method, J. Comput. Phys., 155 (1999), 410-438.  doi: 10.1006/jcph.1999.6345. [26] F. Santosa, A level-set approach for inverse problems involving obstacles, Control, Optimizat. Calculus Variat., 1 (1996), 17-33.  doi: 10.1051/cocv:1996101. [27] M. Sussman, P. Smereka and S. J. Osher, A level set approach for computing solutions to incompressible two-phase flows, J. Comput. Phys., 114 (1994), 146-159. [28] K. van den Doel, U. Ascher and A. Leitao, Multiple level sets for piecewise constant surface reconstruction in highly ill-posed problems, J. Sci. Comput., 43 (2010), 44-66.  doi: 10.1007/s10915-009-9341-x. [29] H.-K. Zhao, T. Chan, B. Merriman and S. J. Osher, A variational level set approach for multiphase motion, J. Comput. Phys., 127 (1996), 179-195.  doi: 10.1006/jcph.1996.0167.
Regularizations by solving equation (37) with $\lambda_f = 0.01$. (a) Using Dirichlet boundary condition (38); (b) using Neumann boundary condition (41)
Example 1. (a) True anomalous mass distribution; (b) initial guess; (c) clean data; (d) data contaminated with $5\%$ Gaussian noise
Example 1. Inversion results. (a) Using $(g_1,g_2)$ with clean measurements; (b) using $(g_1,g_2)$ contaminated with $5\%$ Gaussian noise; (c) using $d$ with clean measurement; (d) using $d$ contaminated with $5\%$ Gaussian noise
Example 2. (a) True anomalous mass distribution; (b) gravity data contaminated with $5\%$ Gaussian noise; (c) recovered solution using $(g_1,g_2)$; (d) recovered solution using $d$
Example 3. (a) True mass distribution 1; (b) true mass distribution 2; (c) error of the recovered solution $(\rho-\rho_{exact})$ for distribution 1; (d) error $(\rho-\rho_{exact})$ for distribution 2
Example 4: first case. (a) Anomalous mass distribution with $f = 2-x_2$; (b) recovered solution; (c) cross section at $x_1 = 0.5$
Example 4: second case. (a) Anomalous mass distribution with $f = 1+2\sin(2\pi x_2)$; (b) recovered solution; (c) cross section at $x_1 = 0.5$
Example 5. (a) Anomalous mass distribution with $f = 2-2\,x_2^2$; (b) modulus data $d = |\nabla U|$ with $5\%$ Gaussian noises
Example 5. (a) Initial guess of $\phi$; (b) initial structure of the anomalous mass distribution; (c) labeling function $F(x)$; (d) recovered solution; (e) error $\rho-\rho_{exact}$; (f) cross section of $\rho$ at $x_1 = 0.25$
Example 6. (a) Anomalous mass distribution with $f = 1+\sin x_2$; (b) modulus data $d = |\nabla U|$ with $5\%$ Gaussian noise
Example 6. (a) Initial guess of $\phi$; (b) initial structure of the anomalous mass distribution; (c) labeling function $F(x)$; (d) recovered solution; (e) error $\rho-\rho_{exact}$; (f) cross section of $\rho$ at $x_1 = 0.25$
 [1] Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029 [2] Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems and Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019 [3] Victor Isakov, Shingyu Leung, Jianliang Qian. A three-dimensional inverse gravimetry problem for ice with snow caps. Inverse Problems and Imaging, 2013, 7 (2) : 523-544. doi: 10.3934/ipi.2013.7.523 [4] Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems and Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709 [5] Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems and Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163 [6] Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems and Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465 [7] Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems and Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479 [8] Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems and Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 [9] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [10] Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations and Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 [11] Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405 [12] Mikko Orispää, Markku Lehtinen. Fortran linear inverse problem solver. Inverse Problems and Imaging, 2010, 4 (3) : 485-503. doi: 10.3934/ipi.2010.4.485 [13] A. Doubov, Enrique Fernández-Cara, Manuel González-Burgos, J. H. Ortega. A geometric inverse problem for the Boussinesq system. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1213-1238. doi: 10.3934/dcdsb.2006.6.1213 [14] Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3473-3489. doi: 10.3934/dcdss.2020235 [15] Lu Zhao, Heping Dong, Fuming Ma. Inverse obstacle scattering for acoustic waves in the time domain. Inverse Problems and Imaging, 2021, 15 (5) : 1269-1286. doi: 10.3934/ipi.2021037 [16] Frank Hettlich. The domain derivative for semilinear elliptic inverse obstacle problems. Inverse Problems and Imaging, 2022, 16 (4) : 691-702. doi: 10.3934/ipi.2021071 [17] Hermann Gross, Sebastian Heidenreich, Mark-Alexander Henn, Markus Bär, Andreas Rathsfeld. Modeling aspects to improve the solution of the inverse problem in scatterometry. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 497-519. doi: 10.3934/dcdss.2015.8.497 [18] Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems and Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297 [19] Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014 [20] Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems and Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551

2021 Impact Factor: 1.483