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 and 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), 759-780. doi: 10.1006/jcph.2001.6937.

[2]

J. Brandman, A level-set method for computing the eigenvalues of elliptic operators defined on compact hypersurfaces, J. Sci. Comput., 37 (2008), 282-315. doi: 10.1007/s10915-008-9210-z.

[3]

M. Burger, A level set method for inverse problems, Inverse Problems, 17 (2001), 1327-1356. doi: 10.1088/0266-5611/17/5/307.

[4]

M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301. doi: 10.1017/S0956792505006182.

[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), 458-473. doi: 10.1016/j.jcp.2005.08.020.

[6]

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.

[7]

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

[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), 2580-2594. doi: 10.1137/100804139.

[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), 317-388. doi: 10.1016/j.jcp.2004.02.010.

[10]

V. Isakov, "Inverse Source Problems," AMS, Providence, 1990.

[11]

V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry, Commun. Comput. Phys., 10 (2011), 1044-1070. doi: 10.4208/cicp.100710.021210a.

[12]

M. Keldysh, On the solubility and stability of the Dirichlet's problem, Uspekhi Matem. Nauk., 8 (1940), 171-231.

[13]

S. Leung, Eulerian approach for computing the finite time Lyapunov exponent, J. Comput. Phys., 230 (2011), 3500-3524. doi: 10.1016/j.jcp.2011.01.046.

[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), 685-706. doi: 10.1088/0266-5611/14/3/018.

[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), 4330-4350. doi: 10.1137/080740003.

[16]

C. Miranda, "Partial Differential Equations of Elliptic Type," Springer-Verlag, New York-Berlin, 1970.

[17]

P. Novikov, Sur le probleme inverse du potential, Dokl. Akad. Nauk SSSR, 18 (1938), 165-168.

[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), 12-49. doi: 10.1016/0021-9991(88)90002-2.

[19]

A. I. Prilepko, D. G. Orlovskii and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Marcel Dekker, New York, 2000.

[20]

J. Qian, L.-T. Cheng and S. J. Osher, A level set based Eulerian approach for anisotropic wave propagations, Wave Motion, 37 (2003), 365-379. doi: 10.1016/S0165-2125(02)00101-4.

[21]

J. Qian and S. Leung, A level set method for paraxial multivalued traveltimes, J. Comput. Phys., 197 (2004), 711-736. doi: 10.1016/j.jcp.2003.12.017.

[22]

J. Qian and S. Leung, A local level set method for paraxial multivalued geometric optics, SIAM J. Sci. Comp., 28 (2006), 206-223. doi: 10.1137/030601673.

[23]

F. Santosa, A level-set approach for inverse problems involving obstacles, Control, Optimizat. Calculus Variat., 1 (1996), 17-33.

[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), 44-66. doi: 10.1007/s10915-009-9341-x.

[25]

J. Xu and H. K. Zhao, An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput., 19 (2003), 573-594. doi: 10.1023/A:1025336916176.

[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), 179-195. doi: 10.1006/jcph.1996.0167.

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), 759-780. doi: 10.1006/jcph.2001.6937.

[2]

J. Brandman, A level-set method for computing the eigenvalues of elliptic operators defined on compact hypersurfaces, J. Sci. Comput., 37 (2008), 282-315. doi: 10.1007/s10915-008-9210-z.

[3]

M. Burger, A level set method for inverse problems, Inverse Problems, 17 (2001), 1327-1356. doi: 10.1088/0266-5611/17/5/307.

[4]

M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301. doi: 10.1017/S0956792505006182.

[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), 458-473. doi: 10.1016/j.jcp.2005.08.020.

[6]

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.

[7]

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

[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), 2580-2594. doi: 10.1137/100804139.

[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), 317-388. doi: 10.1016/j.jcp.2004.02.010.

[10]

V. Isakov, "Inverse Source Problems," AMS, Providence, 1990.

[11]

V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry, Commun. Comput. Phys., 10 (2011), 1044-1070. doi: 10.4208/cicp.100710.021210a.

[12]

M. Keldysh, On the solubility and stability of the Dirichlet's problem, Uspekhi Matem. Nauk., 8 (1940), 171-231.

[13]

S. Leung, Eulerian approach for computing the finite time Lyapunov exponent, J. Comput. Phys., 230 (2011), 3500-3524. doi: 10.1016/j.jcp.2011.01.046.

[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), 685-706. doi: 10.1088/0266-5611/14/3/018.

[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), 4330-4350. doi: 10.1137/080740003.

[16]

C. Miranda, "Partial Differential Equations of Elliptic Type," Springer-Verlag, New York-Berlin, 1970.

[17]

P. Novikov, Sur le probleme inverse du potential, Dokl. Akad. Nauk SSSR, 18 (1938), 165-168.

[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), 12-49. doi: 10.1016/0021-9991(88)90002-2.

[19]

A. I. Prilepko, D. G. Orlovskii and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Marcel Dekker, New York, 2000.

[20]

J. Qian, L.-T. Cheng and S. J. Osher, A level set based Eulerian approach for anisotropic wave propagations, Wave Motion, 37 (2003), 365-379. doi: 10.1016/S0165-2125(02)00101-4.

[21]

J. Qian and S. Leung, A level set method for paraxial multivalued traveltimes, J. Comput. Phys., 197 (2004), 711-736. doi: 10.1016/j.jcp.2003.12.017.

[22]

J. Qian and S. Leung, A local level set method for paraxial multivalued geometric optics, SIAM J. Sci. Comp., 28 (2006), 206-223. doi: 10.1137/030601673.

[23]

F. Santosa, A level-set approach for inverse problems involving obstacles, Control, Optimizat. Calculus Variat., 1 (1996), 17-33.

[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), 44-66. doi: 10.1007/s10915-009-9341-x.

[25]

J. Xu and H. K. Zhao, An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput., 19 (2003), 573-594. doi: 10.1023/A:1025336916176.

[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), 179-195. doi: 10.1006/jcph.1996.0167.

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