May  2013, 7(2): 523-544. doi: 10.3934/ipi.2013.7.523

A three-dimensional inverse gravimetry problem for ice with snow caps

1. 

Wichita State University, 1845 Fairmount, Wichita, KS 67260-0033

2. 

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China, China

Received  September 2012 Revised  February 2013 Published  May 2013

We propose a model for the gravitational field of a floating iceberg $D$ with snow on its top. The inverse problem of interest in geophysics is to find $D$ and snow thickness $g$ on its known (visible) top from remote measurements of derivatives of the gravitational potential. By modifying the Novikov's orthogonality method we prove uniqueness of recovering $D$ and $g$ for the inverse problem. We design and test two algorithms for finding $D$ and $g$. One is based on a standard regularized minimization of a misfit functional. The second one applies the level set method to our problem. Numerical examples validate the theory and demonstrate effectiveness of the proposed algorithms.
Citation: Victor Isakov, Shingyu Leung, Jianliang Qian. A three-dimensional inverse gravimetry problem for ice with snow caps. Inverse Problems & Imaging, 2013, 7 (2) : 523-544. doi: 10.3934/ipi.2013.7.523
References:
[1]

M. Bertalmio, L.-T. Cheng, S. Osher and G. Sapiro, Variational problems and partial differential equations on implicit surfaces,, J. Comput. Phys., 174 (2001), 759.  doi: 10.1006/jcph.2001.6937.  Google Scholar

[2]

J. Brandman, A level-set method for computing the eigenvalues of elliptic operators defined on compact hypersurfaces,, J. Sci. Comput., 37 (2008), 282.  doi: 10.1007/s10915-008-9210-z.  Google Scholar

[3]

M. Burger, A level set method for inverse problems,, Inverse Problems, 17 (2001), 1327.  doi: 10.1088/0266-5611/17/5/307.  Google Scholar

[4]

M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design,, European J. Appl. Math., 16 (2005), 263.  doi: 10.1017/S0956792505006182.  Google Scholar

[5]

T. Cecil, S. J. Osher and J. Qian, Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension,, J. Comput. Phys., 213 (2006), 458.  doi: 10.1016/j.jcp.2005.08.020.  Google Scholar

[6]

O. Dorn and D. Lesselier, Level set methods for inverse scattering,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/4/R01.  Google Scholar

[7]

A. Elcrat, V. Isakov, E. Kropf and D. Stewart, A stability analysis of the harmonic continuation,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075016.  Google Scholar

[8]

N. Halko, P. G. Martinsson, Y. Shkolnisky and M Tygert, An algorithm for the principal component analysis of large data sets,, SIAM J. Sci. Comput., 33 (2010), 2580.  doi: 10.1137/100804139.  Google Scholar

[9]

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.  doi: 10.1016/j.jcp.2004.02.010.  Google Scholar

[10]

V. Isakov, "Inverse Source Problems,", AMS, (1990).   Google Scholar

[11]

V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry,, Commun. Comput. Phys., 10 (2011), 1044.  doi: 10.4208/cicp.100710.021210a.  Google Scholar

[12]

M. Keldysh, On the solubility and stability of the Dirichlet's problem,, Uspekhi Matem. Nauk., 8 (1940), 171.   Google Scholar

[13]

S. Leung, Eulerian approach for computing the finite time Lyapunov exponent,, J. Comput. Phys., 230 (2011), 3500.  doi: 10.1016/j.jcp.2011.01.046.  Google Scholar

[14]

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.  doi: 10.1088/0266-5611/14/3/018.  Google Scholar

[15]

C. B. Macdonald and S. J. Ruuth, The implicit closest point method for the numerical solution of partial differential equations on surfaces,, SIAM J. Sci. Comput., 31 (2009), 4330.  doi: 10.1137/080740003.  Google Scholar

[16]

C. Miranda, "Partial Differential Equations of Elliptic Type,", Springer-Verlag, (1970).   Google Scholar

[17]

P. Novikov, Sur le probleme inverse du potential,, Dokl. Akad. Nauk SSSR, 18 (1938), 165.   Google Scholar

[18]

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.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[19]

A. I. Prilepko, D. G. Orlovskii and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics,", Marcel Dekker, (2000).   Google Scholar

[20]

J. Qian, L.-T. Cheng and S. J. Osher, A level set based Eulerian approach for anisotropic wave propagations,, Wave Motion, 37 (2003), 365.  doi: 10.1016/S0165-2125(02)00101-4.  Google Scholar

[21]

J. Qian and S. Leung, A level set method for paraxial multivalued traveltimes,, J. Comput. Phys., 197 (2004), 711.  doi: 10.1016/j.jcp.2003.12.017.  Google Scholar

[22]

J. Qian and S. Leung, A local level set method for paraxial multivalued geometric optics,, SIAM J. Sci. Comp., 28 (2006), 206.  doi: 10.1137/030601673.  Google Scholar

[23]

F. Santosa, A level-set approach for inverse problems involving obstacles,, Control, 1 (1996), 17.   Google Scholar

[24]

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.  doi: 10.1007/s10915-009-9341-x.  Google Scholar

[25]

J. Xu and H. K. Zhao, An Eulerian formulation for solving partial differential equations along a moving interface,, J. Sci. Comput., 19 (2003), 573.  doi: 10.1023/A:1025336916176.  Google Scholar

[26]

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.  doi: 10.1006/jcph.1996.0167.  Google Scholar

show all references

References:
[1]

M. Bertalmio, L.-T. Cheng, S. Osher and G. Sapiro, Variational problems and partial differential equations on implicit surfaces,, J. Comput. Phys., 174 (2001), 759.  doi: 10.1006/jcph.2001.6937.  Google Scholar

[2]

J. Brandman, A level-set method for computing the eigenvalues of elliptic operators defined on compact hypersurfaces,, J. Sci. Comput., 37 (2008), 282.  doi: 10.1007/s10915-008-9210-z.  Google Scholar

[3]

M. Burger, A level set method for inverse problems,, Inverse Problems, 17 (2001), 1327.  doi: 10.1088/0266-5611/17/5/307.  Google Scholar

[4]

M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design,, European J. Appl. Math., 16 (2005), 263.  doi: 10.1017/S0956792505006182.  Google Scholar

[5]

T. Cecil, S. J. Osher and J. Qian, Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension,, J. Comput. Phys., 213 (2006), 458.  doi: 10.1016/j.jcp.2005.08.020.  Google Scholar

[6]

O. Dorn and D. Lesselier, Level set methods for inverse scattering,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/4/R01.  Google Scholar

[7]

A. Elcrat, V. Isakov, E. Kropf and D. Stewart, A stability analysis of the harmonic continuation,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075016.  Google Scholar

[8]

N. Halko, P. G. Martinsson, Y. Shkolnisky and M Tygert, An algorithm for the principal component analysis of large data sets,, SIAM J. Sci. Comput., 33 (2010), 2580.  doi: 10.1137/100804139.  Google Scholar

[9]

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.  doi: 10.1016/j.jcp.2004.02.010.  Google Scholar

[10]

V. Isakov, "Inverse Source Problems,", AMS, (1990).   Google Scholar

[11]

V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry,, Commun. Comput. Phys., 10 (2011), 1044.  doi: 10.4208/cicp.100710.021210a.  Google Scholar

[12]

M. Keldysh, On the solubility and stability of the Dirichlet's problem,, Uspekhi Matem. Nauk., 8 (1940), 171.   Google Scholar

[13]

S. Leung, Eulerian approach for computing the finite time Lyapunov exponent,, J. Comput. Phys., 230 (2011), 3500.  doi: 10.1016/j.jcp.2011.01.046.  Google Scholar

[14]

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.  doi: 10.1088/0266-5611/14/3/018.  Google Scholar

[15]

C. B. Macdonald and S. J. Ruuth, The implicit closest point method for the numerical solution of partial differential equations on surfaces,, SIAM J. Sci. Comput., 31 (2009), 4330.  doi: 10.1137/080740003.  Google Scholar

[16]

C. Miranda, "Partial Differential Equations of Elliptic Type,", Springer-Verlag, (1970).   Google Scholar

[17]

P. Novikov, Sur le probleme inverse du potential,, Dokl. Akad. Nauk SSSR, 18 (1938), 165.   Google Scholar

[18]

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.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[19]

A. I. Prilepko, D. G. Orlovskii and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics,", Marcel Dekker, (2000).   Google Scholar

[20]

J. Qian, L.-T. Cheng and S. J. Osher, A level set based Eulerian approach for anisotropic wave propagations,, Wave Motion, 37 (2003), 365.  doi: 10.1016/S0165-2125(02)00101-4.  Google Scholar

[21]

J. Qian and S. Leung, A level set method for paraxial multivalued traveltimes,, J. Comput. Phys., 197 (2004), 711.  doi: 10.1016/j.jcp.2003.12.017.  Google Scholar

[22]

J. Qian and S. Leung, A local level set method for paraxial multivalued geometric optics,, SIAM J. Sci. Comp., 28 (2006), 206.  doi: 10.1137/030601673.  Google Scholar

[23]

F. Santosa, A level-set approach for inverse problems involving obstacles,, Control, 1 (1996), 17.   Google Scholar

[24]

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.  doi: 10.1007/s10915-009-9341-x.  Google Scholar

[25]

J. Xu and H. K. Zhao, An Eulerian formulation for solving partial differential equations along a moving interface,, J. Sci. Comput., 19 (2003), 573.  doi: 10.1023/A:1025336916176.  Google Scholar

[26]

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.  doi: 10.1006/jcph.1996.0167.  Google Scholar

[1]

Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems & Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479

[2]

Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431

[3]

Yonggui Zhu, Yuying Shi, Bin Zhang, Xinyan Yu. Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing. Inverse Problems & Imaging, 2014, 8 (3) : 925-937. doi: 10.3934/ipi.2014.8.925

[4]

Yuan Shen, Xin Liu. An alternating minimization method for matrix completion problems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020103

[5]

Feng Ma, Jiansheng Shu, Yaxiong Li, Jian Wu. The dual step size of the alternating direction method can be larger than 1.618 when one function is strongly convex. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020016

[6]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[7]

Yue Lu, Ying-En Ge, Li-Wei Zhang. An alternating direction method for solving a class of inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 317-336. doi: 10.3934/jimo.2016.12.317

[8]

Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897

[9]

Sandro Zagatti. Minimization of non quasiconvex functionals by integro-extremization method. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 625-641. doi: 10.3934/dcds.2008.21.625

[10]

Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems & Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459

[11]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487

[12]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[13]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[14]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

[15]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307

[16]

Kohei Ueno. Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 985-996. doi: 10.3934/dcds.2011.31.985

[17]

Kohei Ueno. Weighted Green functions of polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2283-2305. doi: 10.3934/dcds.2014.34.2283

[18]

Russell E. Warren, Stanley J. Osher. Hyperspectral unmixing by the alternating direction method of multipliers. Inverse Problems & Imaging, 2015, 9 (3) : 917-933. doi: 10.3934/ipi.2015.9.917

[19]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767

[20]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]