Skip to main content
SpringerLink
Log in

Inverse Theory, Linear

  • Living reference work entry
  • First Online:
Encyclopedia of Solid Earth Geophysics

Part of the book series: Encyclopedia of Earth Sciences Series ((EESS))

  • 230 Accesses

Introduction

The activity of solving inverse problems is all pervading. We perform this activity throughout our life right since birth. As a new born baby the first inverse problem we solve, albeit through pattern recognition, is identification of mother in a group of persons. In this vein, the geophysical inverse problems are just a member of the class of inverse problems encountered and studied in science. Formally, the inverse problem can be defined as an estimation of system parameters together with their uncertainties from the observed system response to a given source excitation. In contrast, the forward problem is defined as computation of response of the system for a given model of its properties and for a given source excitation. Most of the geophysical inverse problems are nonlinear by virtue of the system response being a nonlinear function of system parameters. However, a widely employed class of methodologies solves the nonlinear inverse problems through...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Bibliography

  • Aki K, Richards PG (2002) Quantitative seismology. University Science Book, Sausalito

    Google Scholar 

  • Backus GE (1970) Inference from inadequate and inaccurate data. Proc Natl Acad Sci 65:1–7, 281–287; 67:282–289

    Article  Google Scholar 

  • Backus GE (1996) Trimming and procrastination as inversion techniques. Phys Earth Planet Inter 98:101–142

    Article  Google Scholar 

  • Backus GE, Gilbert JF (1967) Numerical applications of a formalism for geophysical inverse problems. Geophys J R Astron Soc 13:247–273

    Article  Google Scholar 

  • Backus G, Gilbert JF (1968) Numerical applications of a formalism for geophysical inverse problems. Geophys J R Astron Soc 16:169–205

    Article  Google Scholar 

  • Backus G, Gilbert JF (1970) Uniqueness in the inversion of inaccurate gross earth data. Philos Trans R Soc Lond A 266:123–192

    Article  Google Scholar 

  • Bjorck A (1996) Numerical methods for least squares problems. SIAM, Philadelphia

    Book  Google Scholar 

  • Borcea L (2002) Electrical impedance tomography – topical review. Inverse Prob 18(6):R99–R136

    Article  Google Scholar 

  • Calvetti D, Morigi S, Reichel L, Sgallari F (2000) Tikhonov regularization and the L-curve for large discrete ill-posed problems. J Comput Appl Math 123:423–446

    Article  Google Scholar 

  • Ghosh DP (1971a) The application of linear filter theory to the direct interpretation of geoelectrical resistivity sounding measurements. Geophys Prospect 19:192–217

    Article  Google Scholar 

  • Ghosh DP (1971b) Inverse filter coefficients for the computation of resistivity standard curves for a horizontally stratified earth. Geophys Prospect 19:769–775

    Article  Google Scholar 

  • Gupta PK (1998) Chapter 7: The Backus Gilbert method. In: Indira NK, Gupta PK (eds) Inverse methods: general principles and applications in earth system sciences. Narosa Publishing House, New Delhi, pp 60–68

    Google Scholar 

  • Gupta PK, Niwas S, Gaur VK (1996) Straightforward inversion scheme (SIS) for one-dimensional magnetotelluric data. Proc Indian Acad Sci (EPS) 105(4):413–429

    Google Scholar 

  • Gupta PK, Niwas S, Gaur VK (1997) Straightforward inversion of vertical electrical sounding data. Geophysics 62(3):775–785

    Article  Google Scholar 

  • Habashy TM, Abubakar A (2004) A general framework for constraint minimization for the inversion of electromagnetic measurements. Prog Electromagn Res 46:265–312

    Article  Google Scholar 

  • Hadamard J (1902) Sur les problemes aux derives partielles et leur signification physique. Princeton Univ Bull 13:49–52. Reprinted in his Oeuvres, Vol. III, Centre Nat. Recherche Sci., Paris, 1968, 1099–1105

    Google Scholar 

  • Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34:561–580

    Article  Google Scholar 

  • Indira NK, Gupta PK (eds) (1998) Inverse methods: general principles and applications to earth system sciences. Narosa, New Delhi

    Google Scholar 

  • Jackson DD (1972) Interpretation of inaccurate, insufficient, and inconsistent data. Geophys J R Astron Soc 28:97–109

    Article  Google Scholar 

  • Jackson DD (1979) The use of a-priori data to resolve nonuniqueness in linear inversion. Geophys J R Astron Soc 57:137–158

    Article  Google Scholar 

  • Kunetz G (1972) Processing and interpretation of magnetotelluric soundings. Geophysics 37:1005–1021

    Article  Google Scholar 

  • Lawson CL, Hanson RJ (1995) Solving least squares problems. SIAM, Philadelphia

    Book  Google Scholar 

  • Lines LR, Treitel S (1984) Tutorial: a review of least-squares inversion and its application to geophysical problems. Geophys Prospect 32:159–186

    Article  Google Scholar 

  • Mallat S (2009) A wavelet tour of signal processing: the sparse way. Elsevier, Amsterdam

    Google Scholar 

  • Menke W (1984) Geophysical data analysis: discrete inverse theory. Academic, Orlando

    Google Scholar 

  • Moore EH (1920) On the reciprocal of the general algebraic matrices. Bull Am Math Soc 26:394–395

    Google Scholar 

  • Oldenburg DW (1976) Calculation of Fourier transform by the Backus-Gilbert method. Geophys J R Astron Soc 44(2):413–431

    Article  Google Scholar 

  • Oldenburg DW, Li Y (2004) Inversion for applied geophysics – a tutorial, presented at EMI workshop held at Hyderabad

    Google Scholar 

  • Penrose R (1955) A generalized inverse for matrices. Proc Camb Philos Soc 51:406–413

    Article  Google Scholar 

  • Rao AR, Bhimasankaram P (1992) Linear algebra. Tata McGraw-Hill, New Delhi

    Google Scholar 

  • Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York

    Google Scholar 

  • Rawlinson N, Pozgay S, Fishwick S (2010) Seismic tomography: a window into deep Earth – review. Phys Earth Planet Inter 178:101–135

    Article  Google Scholar 

  • Reichel L, Sadok H (2008) A new L-curve for ill-posed problems. J Comput Appl Math 219:493–508

    Article  Google Scholar 

  • Rezghi M, Hosseini SM (2009) A new variant of L-curve for Tikhonov regularization. J Comput Appl Math 231:914–924

    Article  Google Scholar 

  • Rojas M, Sorensen DC (2002) A trust region approach to the regularization of large scale discrete forms of ill-posed problems. SIAM J Sci Comput 23(6):1843–1861

    Article  Google Scholar 

  • Saltelli A, Ratto M, Tarantola S, Campolongo F (2006) Sensitivity analysis practices: strategies for model based inferences – review. Reliab Eng Saf 91:1109–1125

    Article  Google Scholar 

  • Scales JA, Smith ML, Treitel S (2001) Introductory geophysical inverse theory. Smizdat, Golden, White River Junction

    Google Scholar 

  • Sneider R, Trampert J (1999) Inverse problems in geophysics. In: Wirgin A (ed) Wavefield inversion. Springer, New York, pp 119–190

    Google Scholar 

  • Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. SIAM, Philadelphia

    Book  Google Scholar 

  • Tikhanov AN, Arsenin VY (1977) Solutions ill-posed problems. Wiley, New York

    Google Scholar 

  • Treitel S, Lines L (2001) Past, present, and future of geophysical inversion – a new millennium analysis. Geophysics 66(1):21–24

    Article  Google Scholar 

  • Twomey S (1977) Introduction to the mathematics of inversion in remote sensing and indirect measurements. Elsevier Scientific, Amsterdam

    Google Scholar 

  • Zhdanov MS (2002) Geophysical inverse theory and regularization problems. Elsevier, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Earth Sciences, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Pravin K. Gupta

Authors
  1. Pravin K. Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pravin K. Gupta .

Editor information

Editors and Affiliations

  1. Seismology, National Geophysical Research Insti, Hyderabad, India

    Harsh K. Gupta

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Gupta, P.K. (2020). Inverse Theory, Linear. In: Gupta, H. (eds) Encyclopedia of Solid Earth Geophysics. Encyclopedia of Earth Sciences Series. Springer, Cham. https://doi.org/10.1007/978-3-030-10475-7_151-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-10475-7_151-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10475-7

  • Online ISBN: 978-3-030-10475-7

  • eBook Packages: Springer Reference Earth and Environm. ScienceReference Module Physical and Materials ScienceReference Module Earth and Environmental Sciences

Publish with us

Policies and ethics

Search

Navigation