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

Victor Isakov; Shingyu Leung; Jianliang Qian

doi:10.3934/ipi.2013.7.523

Article Contents

2013, Volume 7, Issue 2: 523-544. Doi: 10.3934/ipi.2013.7.523

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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

Received: August 31, 2012

Revised: January 31, 2013

Published: April 2013

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Abstract

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.

Keywords:

Inverse gravimetry,
level set method,
alternating minimization,
weighted essentially non-oscillatory schemes,
Green function.

Mathematics Subject Classification: 65N21, 65N06, 65N80.

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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.