# Linearized Inversion of (Teleseismic) Data

• Guust Nolet
Chapter
Part of the Ettore Majorana International Science Series book series (EMISS, volume 11)

## Summary

This paper consists of two parts. First, I show how the inverse problem for teleseismic waves can be linearized. The second part deals with linear inversion methods, and is applicable to geophysical data in general.

The two principles of stationarity in seismology, Fermat’s Principle and Rayleigh’s Principle, can be used to establish a linearized relationship between perturbations in model parameters on the one hand, and the resulting perturbations in travel times or dispersion data on the other hand. Tables of formulae are presented for the calculations of the direct problem for Love and Rayleigh dispersion and for travel times, with flat and spherical Earth models and different interpolation rules between model points. Integral kernels for inversion of these data are also catalogued.

Backus-Gilbert theory for the inversion of linearized relationships between model and data perturbations in general is briefly reviewed and extended to the case of discretized models. Interpolation functions may be used to span a subspace of the Hilbert space of all possible Earth models. This leads to a discrete set of equations of much smaller size than the original Backus-Gilbert formulation, thus allowing for the simultaneous inversion of much larger sets of data. Orthonormalization of the basis of interpolation functions leads to simplifications in the theory, a more logical approach to discrete inversion, and allows the resolution calculations of Backus and Gilbert to be carried over from the domain of piecewise continuous models to be discretized case. The method presented here has considerable computational advantages over earlier methods.

## Keywords

Travel Time Rayleigh Wave Love Wave Linear Inversion Simultaneous Inversion

## References

1. Aki, K., 1977, Three dimensional seismic velocity anomalies in the lithosphere. Method and summary of results, J. Geophysics, 43:235.Google Scholar
2. Aki, K., A. Christoffersson, and Husebye, E.S., 1977, Determination of three-dimensional seismic structure of the lithosphere, J. Geophys. Res., 82:277.
3. Backus, G., and Gilbert, F., 1967, Numerical application of a formalism for geophysical inverse problems, Geophys. J.R. astr. Soc, 13:247.
4. Backus, G., and Gilbert, F., 1968. The resolving power of gross Earth data, Geophys. J.R. astr. Soc., 16:169.
5. Backus, G., and Gilbert, F., 1970, Uniqueness in the inversion of inaccurate gross Earth data, Phil. Trans. Roy. Soc. Lond., A266:123.Google Scholar
6. Cervený, V., and Ravindra, R., 1971, Theory of seismic head waves, Univ. of Toronto Press.Google Scholar
7. Cloetingh, S.A.P.L., Nolet, G., and Wortel, M.J.R., 1980, Standard graphs and tables for the interpretation of Rayleigh wave group velocities in crustal structures, Proc. Roy. Neth. Ac. Sci., B83(1): 101.Google Scholar
8. Fischer, A.G., and Judson, S. (eds.), 1975, Petroleum and Global Tectonics, Princeton University Press.Google Scholar
9. Garmany, J., 1979, On the inversion of travel times, Geophys. Res. Lett., 6:277.
10. Gilbert, F., 1971, Ranking and winnowing gross Earth data for inversion and resolution, Geophys. J.R. astr. Soc., 23:125.
11. Jackson, D.D., 1972, Interpretation of inaccurate, insufficient and inconsistent data, Geophys. J.R. astr. Soc., 28:97.
12. Jackson, D., 1976, Most squares inversion, J. Geophys. Res., 81:1027.
13. Jackson, D.D., 1979, The use of a priori data to resolve non-uniqueness in linear inversion, Geophys. J.R. astr. Soc., 57: 137.
14. Jeffreys, H., 1961, Small corrections in the theory of surface waves, Geophys. J.R, astr. Soc., 6: 115.
15. Johnson, R.E., and Gilbert, F., 1972, Inversion and inference for teleseismic ray data, Meth. Comput. Phys., 12:231.
16. Julian, B.R., and Anderson, D.L., 1968, Travel times, apparent velocities and amplitudes of body waves, Bull. Seism. Soc. Am., 58:339.Google Scholar
17. Julian, B.R., and Gubbins, D., 1977, Three dimensional seismic ray tracing, J. Geophysics, 43:95.Google Scholar
18. Kennett, B.L.N., and Nolet, G., 1978, Resolution analysis for discrete systems, Geophys. J.R. astr. Soc, 53:413.
19. Matthews, J., and Walker, R.L., 1973, Mathematical methods of physics, 2nd. ed., W.A. Benjamin Inc., Menlo Park, CA.Google Scholar
20. Mendiguren, J.A., 1977, Inversion of surface wave data in source mechanism studies, J. Geophys. Res., 82:889.
21. Neigauz, M.G., and Shkadinskaya, G.V., 1972, Method for calculating surface Rayleigh waves in a vertically inhomogeneous half-space, in: Computational Seismology, ed. V.I. Keilis-Borok, Consultants Bureau, N.Y.Google Scholar
22. Nolet, G., 1978, Simultaneous inversion of seismic data, Geophys. J. R. astr. Soc., 55:679.
23. Nolet, G., 1980, Backus-Gilbert theory for models with arbitrary parametrization, submitted for publication.Google Scholar
24. Parker, R.L., 1977a, Understanding inverse theory, Ann. Rev. Earth Plan. Sci., 5:35.
25. Parker, R.L., 1977b, Linear inference and underparametrized models, Rev. Geophys. Space Phys., 15: 446.
26. Rodi, W.L., Glover, P., Li, T.M.C., and Alexander, S.S., 1975, A fast, accurate method for computing group-velocity partial derivatives for Rayleigh and Love modes, Bull. Seism. Soc Am., 65:1105.Google Scholar
27. Sabatier, P.C., 1979, Comment on “The use of a priori data to resolve non-uniqueness in linear inversion” by D.D. Jackson, Geophys. J.R. astr. Soc, 58:523.
28. Strong, D.F. (ed.), 1974, Metallogeny and Plate Tectonics, The Geological Association of Canada, Special paper 14.Google Scholar
29. Takeuchi, H., and Saito, M., 1972. Seismic surface waves, Meth. Comp. Phys., 11: 217.Google Scholar
30. Vlaar, N.J., 1976, On the excitation of the Earth’s seismic normal modes, Pure Appl. Geophys., 114:863.
31. Vlaar, N.J., and Nolet, G., 1978, Seismic surface waves, in: Modern problems in elastic wave propagation, ed. J. Miklowitz and J.D. Achenbach, J. Wiley and Sons.Google Scholar
32. 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.
33. Woodhouse, J.H., 1976, On Rayleigh’s principle, Geophys. J.R. astr. Soc., 46:11.
34. Woodhouse, J.H., and Dahlen, F.A., 1978, The effect of a general aspherical perturbation on the free oscillations of the Earth, Geophys. J.R. astr. Soc., 53:335.