Inverse Methods and Imaging

  • K. J. Langenberg
  • P. Fellinger
  • R. Marklein
  • P. Zanger
  • K. Mayer
  • T. Kreutter
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 330)


Ultrasonic nondestructive testing (NDT) of solid materials exploits the scattering of elastic waves by defects such as cracks, voids, inclusions, and other inhomogeneities. The scattered waves carry information about location, size, shape, and orientation of these defects which has to be extracted appropriately from measurements [1, 2]. The ultimate goal is to produce threedi-mensional images of the interior of the material. In principal, this can be achieved “inverting” the scattering of ultrasound with the aid of inverse scattering theories. Particularly, in three spatial dimensions this turns out to be a complicated and ill-conditioned task even for the much simpler case of scalar acoustic waves. Therefore, approximations and simplifying assumptions are introduced as a trade-off between complexity of algorithms and proper assessment of the integrity of the material [6]. In addition, for practical applications, data recording and processing has to be fast and cheap, which is only available if the algorithms are simple enough; this essentially means, that they are capable of producing images — even in three dimensions — but, due to the fact, that they are not exact inverse scattering solutions, these images have to be properly assessed and understood. For that reason, apart from making many test experiments, numerical modeling of the radiation, propagation, and scattering of elastic waves for given NDT situations and using the computed data as input for the available imaging schemes seems to be a powerful tool to understand the images better, particularly, if elastic waves are under concern in those imaging algorithms which have been designed for scalar acoustic waves only.


Shear Wave Wave Front Elastic Wave Rayleigh Wave Nondestructive Test 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Lorenz, L.F. van der Wal, A.J. Berkhout: Ultrasonic Imaging with Multi-SAFT; Nondestructive Characterization of Defects in Steel Com-ponents, Nondestr. Test. Eval. 6 (1991) 149CrossRefGoogle Scholar
  2. [2]
    L. Caineri, H.G. Tattersall, J.A.G. Temple, M.G. Silk: Time-of-Flight Diffraction Tomography for NDT Applications, Ultrasonics 30 (1992) 275CrossRefGoogle Scholar
  3. [3]
    J.D. Achenbach: Wave Propagation in Elastic Solids, North-Holland, Amsterdam 1973Google Scholar
  4. [4]
    A. Ben-Menahem, S.J. Singh: Seismic Waves and Sources, Springer-Verlag, New York 1981CrossRefGoogle Scholar
  5. [5]
    M. Spies, P. Fellinger, K.J. Langenberg: Elastic Waves in Homogeneous and Layered Transversely Isotropic Media: Gaussian Wave Packets and Green Functions, in: Review of Progress in Quantitative Nondestructive Evaluation, Eds.: D.O. Thompson, D.E. Chimenti, Plenum Press, New York 1992Google Scholar
  6. [6]
    G.T. Herman, H.K. Tuy, K.J. Langenberg. P. Sabatier: Basic Methods of Tomography and Inverse Problems., Adam Hilger, Bristol 1987Google Scholar
  7. [7]
    K.J. Langenberg, U. Aulenbacher, G. Bollig, P. Fellinger, H. Morbitzer, G. Weinfurter, P. Zanger, V. Schmitz: Numerical Modeling of Ultrasonic Scattering, in: Mathematical Modelling in Nondestructive Testing Eds.: M. Blakemore, G.A. Georgiou, Clarendon Press, Oxford 1988Google Scholar
  8. [8]
    Z.S. Alterman: Finite Difference Solutions to Geophysical Problems, Journal of Physics of the Earth 16 (1968)Google Scholar
  9. [9]
    A. Bayliss, K.E. Jordan, B.J. Le Mesurier, E. Türkei: A Forth-Order Accurate Finite-Difference Scheme for the Computation of Elastic Waves, Bulletin Seism. Soc. Am. 75 (1986)Google Scholar
  10. [10]
    J. Virieux: P-SV Wave Propagation in Heterogeneous Media: Velocity-Stress Finite-Difference Method, Geophysics 51 (1986) 889CrossRefGoogle Scholar
  11. [11]
    L.J. Bond, M. Punjani, N. Saffari: Ultrasonic Wave Propagation and Scattering Using Explicit Finite Difference Methods, in: Mathematical Modelling in Nondestructive Testing, Eds.: M. Blakemore, G.A. Georgiou, Clarendon Press, Oxford 1988Google Scholar
  12. [12]
    R. Ludwig, W. Lord: A Finite-Element Formulation for the Study of Ultrasonic NDT Systems, IEEE Trans. Ultrasonics, Ferroel., and Frequ. Contr. 35 (1988) 809CrossRefGoogle Scholar
  13. [13]
    T. Weiland: On the Numerical Solution of Maxwell’s Equations and Applications in the Field of Accelerator Physics, Particle Accelerators 15 (1984)Google Scholar
  14. [14]
    K.S. Yee: Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media, IEEE Trans. Ant. Prop. AP-14 (1966) 302CrossRefGoogle Scholar
  15. [15]
    P. Fellinger, K.J. Langenberg: Numerical Techniques for Elastic Wave Propagagtion and Scattering, in: Elastic Waves and Ultrasonic Non-destructive Evaluation, Eds.: S.K. Datta, J.D. Achenbach, Y.S. Raja-pakse, North-Holland, Amsterdam 1990Google Scholar
  16. [16]
    P. Fellinger: Ein Verfahren zur numerischen Behandlung elastischer Wellenausbreitungsprobleme im Zeitbereich durch direkte Diskretisie-rung der elastodynamischen Grundgleichungen, Ph.D. Thesis, University of Kassel, Kassel/Germany 1991Google Scholar
  17. [17]
    Y.H. Pao, V. Varatharajulu: Huygens’ Principle, Radiation Conditions, and Integral Formulas for the Scattering of Elastic Waves, J. Acoust. Soc. Am. 59 (1976)Google Scholar
  18. [18]
    H.J. Salzburger, W. Schmidt: Automatische wiederkehrende Ul-traschall-Prüfung der Laufflächen von Hochgeschwindigkeitsschienen-fahrzeugen, in: Mit vernetzten, intelligenten Komponenten zu leistungsfähigeren Meß- und Automatisierungssystemen, Eds.: G. Schmidt, H. SteuflofF, Oldenburg, München 1989Google Scholar
  19. [19]
    K.J. Langenberg, P. Fellinger, R. Marklein: On the Nature of the So-Called Subsurface Longitudinal Wave and/or the Surface Longitudinal Creeping Wave, Res. Nondestr. Eval. 2 (1990) 59Google Scholar
  20. [20]
    B.A. Auld: General Electromechanical Reciprocity Relations Applied to the Calculation of Elastic Wave Scattering Coefficients, Wave Motion 1 (1979) 3CrossRefGoogle Scholar
  21. [21]
    F. Lakestani: Validation of Mathematical Models of the Ultrasonic In-spection of Steel Components, Report PISC DOC (90)12, NDE Lab., JRC-Ispra/Italy 1990Google Scholar
  22. [22]
    J.D. Achenbach, A.K. Gautesen, H. McMaken: Ray Methods for Waves in Elastic Solids, Pitman, Boston 1982Google Scholar
  23. [23]
    R. Marklein: Die Akustische Finite Integrationstechnik (AFIT)Ein numerisches Verfahren zur Lösung von Problemen der Abstrahlung, Ausbreitung und Streuung von Akustischen Wellen im Zeitbereich, Master Thesis, University of Kassel/Germany, Kassel 1992Google Scholar
  24. [24]
    V. Schmitz, W. Müller, G. Schäfer: Practical Experiences with L-SAFT, in: Review of Progress in Quantitative Nondestructive Evaluation, Eds.: D.O. Thompson, D.E. Chimenti, Plenum Press, New York 1986Google Scholar
  25. [25]
    K. J. Langenberg, M. Brandfaß, K. Mayer, T. Kreutter, A. Brüll, P. Fel-linger, D. Huo: Principles of Microwave Imaging and Inverse Scattering, Advances in Remote Sensing (1992) (to be published)Google Scholar
  26. [26]
    K. Mayer, R. Marklein, K.J. Langenberg, T. Kreutter: Threedimen-sional Imaging System based on Fourier Transform Synthetic Aperture Focussing Technique, Ultrasonics 28 (1990) 241–255CrossRefGoogle Scholar
  27. [27]
    K.J. Langenberg: Introduction to the Special Issue on Inverse Problems, Wave Motion 11 (1989) 99–112CrossRefGoogle Scholar
  28. [28]
    T. Kreutter, S. Klaholz, A. Brüll, J. Sahm, A. Hecht: Optimierung und Anwendung eines schnellen Abbildungsalgorithmus für die Schmiedewel-lenprüfung, Seminar “Modelle und Theorien für die Ultraschallprüfung” der Deutschen Gesellschaft für zerstörungsfreie Prüfung, Berlin 1990Google Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • K. J. Langenberg
    • 1
  • P. Fellinger
    • 1
  • R. Marklein
    • 1
  • P. Zanger
    • 1
  • K. Mayer
    • 1
  • T. Kreutter
    • 1
  1. 1.University of KasselKasselGermany

Personalised recommendations