# Source Location in Laterally Varying Media

## Abstract

This paper deals with the interpretation of travel times of seismic waves from earthquakes and explosions. The location of seismic sources comes ultimately from travel time data but the same data are used in evaluating the wave speeds within the Earth itself, so the two problems of source location and velocity determination cannot be separated. In the early work of Jeffreys and Bullen a large set of travel times from earthquakes was used to refine the locations as well as to calculate the travel times of waves as a function of distance. Since that time instruments have improved and more data has become available but the J-B model is still in date. The principal problem with improving travel time models has been that the Earth is not spherically symmetric and it is simply not possible to fit the data with a spherically symmetric model. For example Herrin *et al* ^{(22)} give an improved set of travel time tables that are considered to be a better fit to the oceanic regions of the world. This reflects improved station coverage rather than any new understanding about Earth. It is clear that further developments must take account of lateral variations within the Earth.

Considerable efforts have been made recently towards finding lateral variations immediately beneath arrays of seismometers (e.g. Aki *et al*.^{(1)}). These studies are often restricted to regions where there happens to be an array of seismometers, such as at NORSAR or LASA. The velocity models derived from the data are in many instances rather ambiguous. A more serious problem is that *all* the travel time anomalies are assumed to arise from lateral variations beneath the array whereas we know that the most inhomogeneous parts of the Earth are near sources and so a possible bias in the results will come from source effects. While the idea of finding lateral variations without relocating the sources is an attractive one, there is no sure alternative to finding locations and velocity models simultaneously, as Jeffreys & Bullen did for spherically symmetric models.

### Keywords

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### References

- (1).Aki, K., Christoffersson, A. and Husebye, E.S.: 1976. Three dimensional structure of the lithosphere under Montana LASA.
*Bull. Seism. Soc. Am.**66*, pp. 501–524.Google Scholar - (2).Aki, K., Christoffersson, A. and Husebye, E.S: 1977. Determination of the three dimensional seismic structure of the lithosphere.
*J. Geophys. Res.**82*, pp. 277–296.CrossRefGoogle Scholar - (3).Aki, K. and Lee, W.H.K.: 1976. Determination of three-dimensional velocity anomalies under a seismic array using first P arrival times from local earth-quakes. 1. A homogeneous initial model.
*J. Geophys. Res.**81*, pp. 4381–4399.CrossRefGoogle Scholar - (4).Ansell, J.H. and Smith, E.G.C.: 1975. Detailed structure of a mantle seismic zone using the homogeneous station method.
*Nature**253*, pp. 518–520.CrossRefGoogle Scholar - (5).Backus, G.E. and Gilbert, F.: 1967. Numerical applications of a formalism for geophysical inverse problems.
*Geophys. J. R. Astr. Soc.**13*, pp. 247–276.Google Scholar - (6).Berteussen, K.A., Husebye, E.S., Mereu, R.F. and Ram, A.: 1977. Quantitative assessment of the crust-upper mantle heterogeneities beneath the Gauribi- danur seismic array in Southern India.
*Earth Planet. Sci. Lett.**37*, pp. 326–332.CrossRefGoogle Scholar - (7).Bessonova, E.N., Fishman, V.M., Ryaboyi, V.Z. and Sitnikova, G.A.: 1974. The tau method for the inversion of travel times — I. Deep seismic sounding data.
*Geophys. J. R. Astr. Soc.**36*, pp. 377–398.Google Scholar - (8).Bessonova, E.N., Fishman, V.M., Johnson, L.R., Shnirman, M.G. and Sitnikova, G.A.: 1976. The tau method for the inversion of travel times — II. Earthquake data.
*Geophys. J. R. Astr. Soc.*,*46*, pp. 87–108.Google Scholar - (9).Buland, R.: 1976. The mechanics of locating earthquakes.
*Bull Seism. Soc. Am.**66*, pp. 173–187.Google Scholar - (10).Bullen, K.E.: 1965. An Introduction to the Theory of Seismology. Cambridge Univ. Press., London, 381 pp.Google Scholar
- (11).Chou, C.W. and Booker, J.R.: 1979. A Backus-Gilbert approach to inversion of travel time data for three-dimensional velocity structure.
*Geophys. J. R. Astr. Soc.**59*, pp. 325–344.Google Scholar - (12).Christoffersson, A. and Husebye, E.S.: 1979. On three dimensional inversion of P wave time residuals: Option for geological modelling.
*J. Geophys. Res.**84*, pp. 6168–6176.CrossRefGoogle Scholar - (13).Crosson, R.S.: 1976. Crustal structure modeling of earthquake data. 1. Simultaneous least squares estimation of hypocenter and velocity parameters.
*J. Geophys. Res.**81*, pp. 3036–3046.CrossRefGoogle Scholar - (14).Ellsworth, W.L.: 1977. Three dimensional structure of the crust and mantle beneath the Island of Hawaii. Ph.D. thesis, Massachusetts Institute of Technology.Google Scholar
- (15).Evernden, J.F.: 1969. Precision of epicenters obtained by small numbers of world-wide stations.
*Bull. Seism. Soc. Am.**59*, pp. 1365–1398.Google Scholar - (16).Fitch, T.J.: 1975. Compressional velocity in source regions of deep earthquakes: An application of the master earthquake technique.
*Earth Planet. Sci. Lett.**26*, pp. 156–166.CrossRefGoogle Scholar - (17).Garmany, J.: 1979. On the inversion of travel times.
*Geophys. Res. Lett.**6*, pp. 277–279.CrossRefGoogle Scholar - (18).Garmany, J., Orcutt, J.A. and Parker, R.L.: 1979. Travel time inversion: a geometrical approach.
*J. Geophys. Res.**84*, pp. 3615–3622.CrossRefGoogle Scholar - (19).Gerver, M.L. and Markushevich, V.M.: 1966. Determination of seismic wave velocity from the travel-time curve.
*Geophys. J. R. Astr. Soc.**11*, pp. 165–173.Google Scholar - (20).Gerver, M.L. and Markushevich, V.M.: 1967. On the characteristic properties of travel time curves.
*Geophys. J. R. Astr. Soc.**13*, pp. 241–246.Google Scholar - (21).Haddon, R.A.W. and Husebye, E.S.: 1978. Joint interpretation of P-wave time and amplitude anomalies in terms of lithospheric heterogeneities.
*Geophys. J. R. Astr. Soc.**55*, pp. 19–44.Google Scholar - (22).Herrin, E., Arnold, E.P., Bolt, B.A., Clawson, G.E., Engdahl, E.R., Freedman, H.W., Gordon, D.W., Hales, A.L., Lobdell, J.L., Nuttli, O, Romney, C., Taggart, J. and Tucker, W.: 1978. Seismological tables for P phases.
*Bull Seism. Soc. Am.**58*, pp. 1193–1241.Google Scholar - (23).Hovland, J., Gubbins, D. and Husebye, E.S.: 1980. Upper mantle heterogeneities beneath Central Europe.
*Geophys. J. R. Astr. Soc*., in press.Google Scholar - (24).Husebye, E.S., Christoffersson, A., Aki, K. and Powell, C: 1976. Preliminary results on the 3-dimensional seismic structure of the lithosphere under the USGS central California seismic array.
*Geophys. J. R. Astr. Soc.**46*, pp. 319–340.Google Scholar - (25).Husebye, E.S., Haddon, R.A.W. and King, D.W.: 1977. Precursors to
*P’P’*and upper mantle discontinuities.*J. Geophys.**43*, pp. 535–543.Google Scholar - (26).ISC Bulletin of the International Seismological Centre. July 1977. ISC, Newbury Berkshire, UK.Google Scholar
- (27).Iyer, H.M.: 1974. Teleseismic evidence for the existence of low velocity material deep into the upper mantle under the Yellowstone Caldera. (abstract)
*EOS Trans. A. Geophys. Union**56*, pp. 1190.Google Scholar - (28).Jackson, D.D.: 1972. Interpretation of inaccurate, insufficient and inconsistent data.
*Geophys. J. R. Astr. Soc.**28*, pp. 97–109.Google Scholar - (29).Jackson, J.: 1980 Errors in focal depth determination and the depth of seismicity in Iran and Turkey.
*Geophys. J. R. Astr. Soc.**61*, pp. 285–301.Google Scholar - (30).Jeffreys, H.: 1961.
*Theory of Probability*. Oxford University Press, Oxford, England, 447 pp.Google Scholar - (31).Jeffreys, H.A. and Bullen, K.E.: 1970. Seismological Tables. British Association for the Advancement of Science, Gray Miln Trust, London.Google Scholar
- (32).Johnson, L.E. and Gilbert, F.: 1972a. A new datum for use in the body wave travel time inverse problem
*Geophys. J. R. Astr. Soc*.*30*, pp. 373–380.Google Scholar - (33).Johnson, L.E. and Gilbert, F.: 1972b. Inversion and inference for teleseismic ray data.
*Methods in Comp. Phys.**12*, pp. 231–266.Google Scholar - (34).Julian, B.R.: 1970. Ray tracing in arbitrarily heterogeneous media.
*Lincoln Lab. Tech. Note*,*1970-45*.Google Scholar - (35).Julian, B.R. and Gubbins, D.: 1977. Three dimensional seismic ray tracing.
*J. Geophys.**43*, pp. 95–113.Google Scholar - (36).Kennett, B.L.N.: 1976. A comparison of travel time inversions.
*Geophys. J. Astr. Soc.**44*, pp. 517–536.Google Scholar - (37).Kennett, B.L.N.: 1978. Ray theoretical inverse methods in geophysics. In Applied Inverse Problems, ed. P.C. Sabatier, Springer-Verlag, Berlin.Google Scholar
- (38).Lanczos, C: 1961.
*Linear Differential Operators*. Van Nostrand, London, 564 PP.Google Scholar - (39).Lawson, C.L. and Hanson, R.J.: 1973.
*Solving Least Squares Problems*. Prentice-Hall Inc., Englewood Cliffs, New Jersey, 340 pp.Google Scholar - (40).Lillwall, R.C. and Douglas, A.: 1970. Estimation of P-wave travel times using the joint epicentre method.
*Geophys. J. R. Astr. Soc.*19, pp. 165–181.Google Scholar - (41).Masters, T.G.: 1979. Observational constraints on the chemical and thermal structure of the Earth’s deep interior.
*Geophys. J. R. Astr. Soc.**57*, pp. 507–534.Google Scholar - (42).Menke, W.H.: 1977. Lateral inhomogeneities in P velocity under the Tarbela array of the Lesser Himalayas of Pakistan.
*Bull Seism. Soc. Am.**67*, pp. 725–734.Google Scholar - (43).Mitchell, B.J., Cheng, C.C. and Stauder, W.: 1977. A three dimensional velocity model of the lithosphere beneath the new Madrid seismic zone.
*Bull. Seism. Soc. Am.**62*, pp. 1061–1074.Google Scholar - (44).Orcutt, J.A.: 1980. Joint linear, extremal inversion of seismic kinematic data.
*J. Geophys. Res.**85*, pp. 2649–2660.CrossRefGoogle Scholar - (45).Parker, R.L.: 1977. Linear inference and underparameterized models.
*Rev. Geophys. Space Phys.**15*, pp. 446–456.CrossRefGoogle Scholar - (46).Pavlis, G.L. and Booker, J.R.: 1980. The mixed discrete-continuous inverse problem: Application to the simultaneous determination of earthquake hypocenters and velocity structure.
*J. Geophys. Res*., in press.Google Scholar - (47).Romanowicz, B.: 1979. Seismic structure of the upper mantle beneath the United States by three dimensional inversion of body wave arrival times.
*Geophys. J. R. Astr. Soc.**57*, pp. 479–506.Google Scholar - (48).Romanowitz, B.A.: 1980. Large scale lateral variations of P velocity in the upper mantle beneath Western Europe. Geophys.
*J. R. Astr. Soc*., in press.Google Scholar - (49).Smith, M.L., Julian, B.R., Engdahl, E.R., Gubbins, D. and Gross, R.: 1979. Linearized inversion of travel times for three dimensional Earth structure (Abstract).
*EOS Trans. Amer. Geophys. Union**59*, p. 12.Google Scholar - (50).Spencer, C. and Gubbins, D.: 1980. Travel time inversion for simultaneous earthquake location and velocity structure determination in laterally varying media.
*Geophys. J. R. Astr. Soc*.*62*., in press.Google Scholar - (51).Wiggins, R.: 1972. The general linear inverse problem: Implications of surface waves and free oscillations on Earth structure.
*Rev. Geophys. Space Phys.**10*, pp. 251–258.CrossRefGoogle Scholar - (52).Wu, J.C.: 1977. Inversion of travel time data for seismic velocity structure in three dimensions. Ph.D. Thesis, University of Washington.Google Scholar