On geophysical inverse problems and constraints
Article Sidebar
Published:
May 20, 1977
Keywords:
Geophysical inverse problems,
Constraints,
Convex problems
Volumes
Vols. 1-18 (1924-1944), ISSN 0044-2801
Main Article Content
Abstract
Mathematical methods for linearized geophysical inverse problems are reviewed, in cases with and in cases without constraints. The role of constraints receives particular attention, both in linear convex problems and in an exactly solvable non-linear example.
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Article Details
How to Cite
Sabatier, P. (1977). On geophysical inverse problems and constraints. Journal of Geophysics, 43(1), 115-137. Retrieved from https://journal.geophysicsjournal.com/JofG/article/view/178
Section
References
Aki, K., Ware, J.A. (1969) Continuous and discrete inverse-scattering problems in a stratified elastic medium. J. Acoust. Soc. Am. 45:911-921
Anderssen, R.S. (1975a) On the linearisation of geophysical inversion problems. Preprint Computer Centre, Australian National University
Anderssen, R.S. (1975b) On the inversion of global electromagnetic induction data. Phys. Earth Planet. Interiors (Netherlands) 10:292-298
Backus, G.E., Gilbert, F.J. (1967) Numerical applications of a formalism for geophysical Inverse Problems. Geophys. J. R. Astr. Soc. 13:247-276
Backus, G., Gilbert, F. (1968) The resolving power of gross earth data. Geophys. J. R. Astr. Soc. 16:169-205
Backus, G., Gilbert, F. (1970) Uniqueness in the inversion of inaccurate gross earth data. Phil. Trans. Roy. Soc. London 266:123-192
Backus, G. (1970) Inference from inadequate and inaccurate data, I. Proceedings of the National Academy of Sci. 65(1):1-7
Backus, G. (1970b) Inference from inadequate and inaccurate data, IL Proceedings of the National Academy of Sciences 65(2):281-287
Backus, G. (1970c) Inference from inadequate and inaccurate data, III. Proceedings of the National Academy of Sciences 67(1):282-289
Bailey, R.C. (1970) Inversion of the geomagnetic induction problem. Proc. Roy. Soc., London A315:185-194
Barcilon, V. (1976) A discrete model of the inverse Love wave problem. Geophys. J. R. Astr. Soc. 44:61-76
Burkhard, N., Jackson, D.D. (1976) Application of stabilized linear inverse theory to gravity data. J. Geophys. Res. 81(8):1513-1518
Chadan, K., Sabatier, P.C. (1977) Inverse problems of quantum scattering theory. To be published by Springer, Berlin-Heidelberg-New York
Crosson, R.S. (1976) Crustal structure modeling of earthquake data I - Simultaneous least squares estimation of hypocenter and velocity parameters. J. Geophys. Res. 81(17):3036-3046
Ducruix, J., LeMouel, J.L., Courtillot, V. (1974) Continuation of three-dimensional potential fields measured on an uneven surface. Geophys. J. R. Astr. Soc. 38:299-314
Foster, M. (1961) An application of the Wiener-Kolmogorov smoothing theory to matrix inversion. J. Soc. Indust. Appl. Math. 9(3):387-392
Franklin, J.N. (1970) Well-posed stochastic extensions of ill-posed linear problems. J. Math. Anal. and Applications 31:682-716
Gilbert, F., Dziewonski, A., Brune, J. (1973) An informative solution to a seismological inverse problem. Proc. Natl. Acad. Sci. USA 70:1410-1413
Gerver, M., Markushevich, V. (1966) Determination of a seismic wave velocity from the traveltime curve. Geophys. J. R. Astr. Soc. 11:165-173
Gerver, M.L., Markushevich, V.M. (1967) On the characteristic properties of travel-time curves. Geophys. J. R. Astr. Soc. 13:241-146
Green, W.R. (1975) Inversion of gravity profiles by use of a Backus-Gilbert approach. Geophysics 40(5):763-772
Hanson, R.J. (1971) A numerical method for solving Fredholm integral equations of the first kind using singular values. Siam Y. Num. An. 8:616-622
Jackson, D.D. (1972) Interpretation of inaccurate, insufficient and inconsistent data. Geophys. J. R. Astr. Soc. 28:97-110
Jackson, D.D. (1973) Marginal solutions to quasi-linear inverse problems in geophysics: the Edgehog method. Geophys. J. R. Astr. Soc. 35:121-135
Jaynes, E.T. (1957) Information theory and statistical mechanics. Phys. Rev. 106:620-630
Johnson, L.E., Gilbert, F. (1971) Inversion and inference for teleseismic ray data. Institute of Geophysics and Planetary Physics, University of California, San Diego
Jordan, T.H., Franklin, J.N. (1971) Optimal solutions to a linear inverse problem in geophysics. Proceedings of the National Academy of Sciences. 68(2):291-293
Jordan, T.H., Minster, J.B. (1975) Application of a stochastic inverse to the geophysical inverse problem. Contribution no: 2036, Geological and Planetary Sci., California Institute of Technology,
Pasadena, California Ju pp, D.L.B., Vozoff, K.: St.able iterative methods for the inversion of geophysical data. Geophys. J. R. Astr. Soc. 42:957-976
Kennett, B.L.N. (1976) A comparison of travel-time inversions. Geophys. J. R. Astr. Soc. 44:517-536
Knopoff, L. (1961) Green's function for eigenvalue problems and the inversion of Love wave dispersion data. Geophys. J. R. Astr. Soc. 161-173
Knopoff, L. (1962) Higher order Born Approximations for the inversion of Love wave dispersion. Geophys. J. R. Astr. Soc. 7(2):149-157
Lanczos, C. (1961) Linear differential operators. Van Nostrand, New York
Marquardt, D.W. (1963) An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11:431-441
Marquardt, D.W. (1970) Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation. Technometrics 12(3):591-612
Osborne, M.R. (1972) A class of methods for minimizing a sum of squares. Aust. Com. J. 4:164-169
Parker, R.L. (1970) The inverse problem of electrical conductivity in the mantle. Geophys. J. R. Astr. Soc. 22:121-138
Parker, R.L. (1974) Best bounds on density and depth from gravity data. Geophysics 39(5):644-649
Parker, R.L. (1975) The theory of ideal bodies for gravity interpretation. Geophys. J. R. Astr. Soc. 42:315-334
Penrose, R. (1955) A generalized inverse for matrices. Proc. Cambridge Phil. Soc. 51:406-413
Rao, C.R., Mitra, S.K. (1971) Generalized inverse of matrices and its applications. Wiley & Sons, New York
Sabatier, P.C. (1972) Comparative evolution of the inverse problems. Mathematics of Profile Inversion NASA TMX 62
Sabatier, P.C. (1974a) Probleme direct et inverse de diffraction d'une onde elastique par une zone perturbee spherique. I et Complements. C.R. Acad. Sci. Paris 278, 545-547 and 603-606
Sabatier, P.C. (1974b) Remarks on approximate methods in geophysical inverse problems. Proc. Roy. Soc., London A337:49-71
Sabatier, P.C. (1976a) Positivity constraints in linear inverse problems I general theory. Preprint Montpellier PM/75/18, to be published in Geophys. J. R. Astr. Soc. (accepted for publ.)
Sabatier, P.C. (1976b) Positivity constraints in linear inverse problems II. Applications. Preprint Montpellier PM/75/18 to be published in Geophys. J. R. Astr. Soc. (accepted for publ.)
Safon, C., Vasseur, G., Cuer, M. (1976) Some applications of linear programming to the inverse gravity problem. (submitted for publication in Geophysics)
Smith, M.L., Franklin, J.N. (1969) Geophysical application of generalized inverse theory. J. Geophys. Res. 74(10):2783-2785
Strand, O.N., Westwater, E.R. (1968a) Minimum rms estimation of the numerical solution of a Fredholm integral equation of the 1st kind. Siam Y. Num. An. 5:287-295
Strand, O.N., Westwater, E.R. (1968b) Statistical estimation of the numerical solution of a Fredholm integral equation of the 1st kind. J. Ass. Comp. Mech. 15:100-114
Twomey, S. (1965) The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements. J. Franklin Inst. 279:95-109
Varah, J.H. (1973) On the numerical solution of ill-conditionned linear systems with applications to illposed problems. Siam J. Num. Anal. 10:257-287
Weidelt, P. (1972) The inverse problem of geomagnetic induction. Z. Geophys 38:257-289
Wiggins, R.A. (1972) The general linear inverse problems: implication of surface waves and free oscillations for earth structure. Rev. Geophys. Space Phys. 10:251-285
Wiggins, R.A., Larner, K.L., Wisecup, R.D. (1976) Residual statics analysis as a general linear inverse problem. Preprint Western Geoph. C0 Houston
Anderssen, R.S. (1975a) On the linearisation of geophysical inversion problems. Preprint Computer Centre, Australian National University
Anderssen, R.S. (1975b) On the inversion of global electromagnetic induction data. Phys. Earth Planet. Interiors (Netherlands) 10:292-298
Backus, G.E., Gilbert, F.J. (1967) Numerical applications of a formalism for geophysical Inverse Problems. Geophys. J. R. Astr. Soc. 13:247-276
Backus, G., Gilbert, F. (1968) The resolving power of gross earth data. Geophys. J. R. Astr. Soc. 16:169-205
Backus, G., Gilbert, F. (1970) Uniqueness in the inversion of inaccurate gross earth data. Phil. Trans. Roy. Soc. London 266:123-192
Backus, G. (1970) Inference from inadequate and inaccurate data, I. Proceedings of the National Academy of Sci. 65(1):1-7
Backus, G. (1970b) Inference from inadequate and inaccurate data, IL Proceedings of the National Academy of Sciences 65(2):281-287
Backus, G. (1970c) Inference from inadequate and inaccurate data, III. Proceedings of the National Academy of Sciences 67(1):282-289
Bailey, R.C. (1970) Inversion of the geomagnetic induction problem. Proc. Roy. Soc., London A315:185-194
Barcilon, V. (1976) A discrete model of the inverse Love wave problem. Geophys. J. R. Astr. Soc. 44:61-76
Burkhard, N., Jackson, D.D. (1976) Application of stabilized linear inverse theory to gravity data. J. Geophys. Res. 81(8):1513-1518
Chadan, K., Sabatier, P.C. (1977) Inverse problems of quantum scattering theory. To be published by Springer, Berlin-Heidelberg-New York
Crosson, R.S. (1976) Crustal structure modeling of earthquake data I - Simultaneous least squares estimation of hypocenter and velocity parameters. J. Geophys. Res. 81(17):3036-3046
Ducruix, J., LeMouel, J.L., Courtillot, V. (1974) Continuation of three-dimensional potential fields measured on an uneven surface. Geophys. J. R. Astr. Soc. 38:299-314
Foster, M. (1961) An application of the Wiener-Kolmogorov smoothing theory to matrix inversion. J. Soc. Indust. Appl. Math. 9(3):387-392
Franklin, J.N. (1970) Well-posed stochastic extensions of ill-posed linear problems. J. Math. Anal. and Applications 31:682-716
Gilbert, F., Dziewonski, A., Brune, J. (1973) An informative solution to a seismological inverse problem. Proc. Natl. Acad. Sci. USA 70:1410-1413
Gerver, M., Markushevich, V. (1966) Determination of a seismic wave velocity from the traveltime curve. Geophys. J. R. Astr. Soc. 11:165-173
Gerver, M.L., Markushevich, V.M. (1967) On the characteristic properties of travel-time curves. Geophys. J. R. Astr. Soc. 13:241-146
Green, W.R. (1975) Inversion of gravity profiles by use of a Backus-Gilbert approach. Geophysics 40(5):763-772
Hanson, R.J. (1971) A numerical method for solving Fredholm integral equations of the first kind using singular values. Siam Y. Num. An. 8:616-622
Jackson, D.D. (1972) Interpretation of inaccurate, insufficient and inconsistent data. Geophys. J. R. Astr. Soc. 28:97-110
Jackson, D.D. (1973) Marginal solutions to quasi-linear inverse problems in geophysics: the Edgehog method. Geophys. J. R. Astr. Soc. 35:121-135
Jaynes, E.T. (1957) Information theory and statistical mechanics. Phys. Rev. 106:620-630
Johnson, L.E., Gilbert, F. (1971) Inversion and inference for teleseismic ray data. Institute of Geophysics and Planetary Physics, University of California, San Diego
Jordan, T.H., Franklin, J.N. (1971) Optimal solutions to a linear inverse problem in geophysics. Proceedings of the National Academy of Sciences. 68(2):291-293
Jordan, T.H., Minster, J.B. (1975) Application of a stochastic inverse to the geophysical inverse problem. Contribution no: 2036, Geological and Planetary Sci., California Institute of Technology,
Pasadena, California Ju pp, D.L.B., Vozoff, K.: St.able iterative methods for the inversion of geophysical data. Geophys. J. R. Astr. Soc. 42:957-976
Kennett, B.L.N. (1976) A comparison of travel-time inversions. Geophys. J. R. Astr. Soc. 44:517-536
Knopoff, L. (1961) Green's function for eigenvalue problems and the inversion of Love wave dispersion data. Geophys. J. R. Astr. Soc. 161-173
Knopoff, L. (1962) Higher order Born Approximations for the inversion of Love wave dispersion. Geophys. J. R. Astr. Soc. 7(2):149-157
Lanczos, C. (1961) Linear differential operators. Van Nostrand, New York
Marquardt, D.W. (1963) An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11:431-441
Marquardt, D.W. (1970) Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation. Technometrics 12(3):591-612
Osborne, M.R. (1972) A class of methods for minimizing a sum of squares. Aust. Com. J. 4:164-169
Parker, R.L. (1970) The inverse problem of electrical conductivity in the mantle. Geophys. J. R. Astr. Soc. 22:121-138
Parker, R.L. (1974) Best bounds on density and depth from gravity data. Geophysics 39(5):644-649
Parker, R.L. (1975) The theory of ideal bodies for gravity interpretation. Geophys. J. R. Astr. Soc. 42:315-334
Penrose, R. (1955) A generalized inverse for matrices. Proc. Cambridge Phil. Soc. 51:406-413
Rao, C.R., Mitra, S.K. (1971) Generalized inverse of matrices and its applications. Wiley & Sons, New York
Sabatier, P.C. (1972) Comparative evolution of the inverse problems. Mathematics of Profile Inversion NASA TMX 62
Sabatier, P.C. (1974a) Probleme direct et inverse de diffraction d'une onde elastique par une zone perturbee spherique. I et Complements. C.R. Acad. Sci. Paris 278, 545-547 and 603-606
Sabatier, P.C. (1974b) Remarks on approximate methods in geophysical inverse problems. Proc. Roy. Soc., London A337:49-71
Sabatier, P.C. (1976a) Positivity constraints in linear inverse problems I general theory. Preprint Montpellier PM/75/18, to be published in Geophys. J. R. Astr. Soc. (accepted for publ.)
Sabatier, P.C. (1976b) Positivity constraints in linear inverse problems II. Applications. Preprint Montpellier PM/75/18 to be published in Geophys. J. R. Astr. Soc. (accepted for publ.)
Safon, C., Vasseur, G., Cuer, M. (1976) Some applications of linear programming to the inverse gravity problem. (submitted for publication in Geophysics)
Smith, M.L., Franklin, J.N. (1969) Geophysical application of generalized inverse theory. J. Geophys. Res. 74(10):2783-2785
Strand, O.N., Westwater, E.R. (1968a) Minimum rms estimation of the numerical solution of a Fredholm integral equation of the 1st kind. Siam Y. Num. An. 5:287-295
Strand, O.N., Westwater, E.R. (1968b) Statistical estimation of the numerical solution of a Fredholm integral equation of the 1st kind. J. Ass. Comp. Mech. 15:100-114
Twomey, S. (1965) The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements. J. Franklin Inst. 279:95-109
Varah, J.H. (1973) On the numerical solution of ill-conditionned linear systems with applications to illposed problems. Siam J. Num. Anal. 10:257-287
Weidelt, P. (1972) The inverse problem of geomagnetic induction. Z. Geophys 38:257-289
Wiggins, R.A. (1972) The general linear inverse problems: implication of surface waves and free oscillations for earth structure. Rev. Geophys. Space Phys. 10:251-285
Wiggins, R.A., Larner, K.L., Wisecup, R.D. (1976) Residual statics analysis as a general linear inverse problem. Preprint Western Geoph. C0 Houston