1887
Volume 36 Number 1
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

The wavefield in, and at the surface of, a homogeneous, isotropic, perfectly elastic half‐space, excited by a traction distribution at the surface of the medium is investigated. The emitted wavefield is a spatial convolution of the surface tractions and the spatial impulse response. The properties of the wavefield in the far‐field of the medium are derived and it is shown that the far‐field particle velocity is essentially equal to a weighted sum of the time derivative of the integrated surface tractions, that is, of the components of the ‘ground force’. The theory is valid for an arbitrary geometry and orientation of the surface tractions, and is independent of the boundary conditions at the surface of the medium.

The surface tractions are related to a source that consists of a mass distribution with an arbitrary force distribution imposed upon it. A boundary condition is introduced that accounts for the mass load and the forces applied to it but neglects vibrations within the mass. The boundary condition follows from the equation of motion of the surface mass load.

The theory is applied to the Vibroseis configuration, using a P‐wave vibrator model with a uniformly distributed force imposed on top of the baseplate, and assuming that horizontal surface traction components are absent. The distribution of displacement and stress directly underneath the baseplate of a single vibrator and an array of vibrators is investigated. Three different boundary conditions are used: (1) assuming uniform pressure, (2) assuming uniform displacement, (3) using the equation of motion of the baseplate as a boundary condition. The calculations of the distribution of stress and displacement over the plate for different elastic media and several frequencies of operation show that only the results obtained with the mixed boundary condition agree with measurements made in the field.

The accuracy of three different phase‐feedback signals is compared using synthetic data. Baseplate velocity phase‐feedback leads to huge deviations in the determination of the far‐field wavelet; reaction mass acceleration phase‐feedback looks stable but neglects the differentiating earth filter; and phase‐feedback to a weighted sum of baseplate and reaction mass accelerations becomes unstable with increasing frequency. The instability can be overcome using measurements over the whole baseplate.

The model can be extended to a lossy layered earth.

Loading

Article metrics loading...

/content/journals/10.1111/j.1365-2478.1988.tb02149.x
2006-04-27
2024-03-28
Loading full text...

Full text loading...

References

  1. Abramowitz, M. and Stegun, I.A.1970. Handbook of Mathematical Functions. Dover Publications, Inc.
    [Google Scholar]
  2. Aki, K. and Richards, P.G.1980. Quantitative Seismology: Theory and Methods, Vol. 1. W.H. Freeman and Co.
    [Google Scholar]
  3. Awojobi, A.O. and Grootenhuis, P.1965. Vibration of rigid bodies on semi‐infinite elastic media. Proceedings of the Royal Society of London, Series A, 28727–63.
    [Google Scholar]
  4. Båth, M. and Berkhout, A.J.1984. Mathematical Aspects of Seismology, Vol. 17, in the Seismic Exploration Series, section 1, 2nd enlarged edition. Geophysical Press.
    [Google Scholar]
  5. Berg, VanDen, P.M.1984. Iterative computational techniques in scattering based upon the integrated square error criterion. IEEE Transactions on Antennas and Propagation, AP‐32, 1063–1071.
    [Google Scholar]
  6. Bycroft, G.N.1956. Forced vibrations of a rigid circular plate on a semi‐infinite elastic space and on an elastic stratum. Philosophical Transactions of the Royal Society, Series A, 248, 327–368.
    [Google Scholar]
  7. Herman, G.C.1981. Scattering of transient acoustic waves in fluids and solids. Ph.D. thesis, Delft University of Technology, 165–176.
  8. Hoop, De A.T.1970. The surface line source problem in elastodynamics. Nederlands Elektronika en Radiogenootschap.35, 19–21.
    [Google Scholar]
  9. Lamb, H.L.1904. On the propagation of tremors over the surface of an elastic solid. Philosophical Transactions of the Royal Society, Series A, 2031–42.
    [Google Scholar]
  10. Lerwill, W.E.1981. The amplitude and phase response of a seismic vibrator. Geophysical Prospecting29, 503–528 (see also 30, 939–941)
    [Google Scholar]
  11. Lerwill, W.E.1982. Reply to comments by Sallas and Weber on “The amplitude and phase response of a seismic vibrator”. Geophysical Prospecting30, 939–941.
    [Google Scholar]
  12. Miller, G.F. and Pursey, H.1954. The field and radiation impedance of mechanical radiators on the free surface of a semi‐infinite isotropic solid. Proceedings of the Royal Society,Series A, 223521–541.
    [Google Scholar]
  13. Onselen van, C.1980. Analysis of the elastic wave motion in a semi‐infinite solid, excited by a vibrating disk on its stress‐free surface (“Vibroseis problem”). Internal report no. 1980–22, Laboratory of Electromagnetic Research, Department of Electrical Engineering, Delft University of Technology.
    [Google Scholar]
  14. Onselen van, C.1982. Vibroseismic excitation of elastic waves in a semi‐infinite solid, Internal report no. 1982–10, Laboratory of Electromagnetic Research, Department of Electrical Engineering, Delft University of Technology.
    [Google Scholar]
  15. Robertson, I.A.1966. Forced vertical vibration of a rigid circular disk on a semi‐infinite elastic solid. Proceedings of the Cambridge Philosophical Society62, 547–553.
    [Google Scholar]
  16. Sallas, J.J.1984. Seismic vibrator control and the downgoing P‐wave. Geophysics49, 732–740.
    [Google Scholar]
  17. Sallas, J.J. and Weber, R.M.1982. Comments on “The amplitude and phase response of a seismic vibrator” by W.E. Lerwill. Geophysical Prospecting30, 935–938.
    [Google Scholar]
  18. Tan, T.H.1985. The elastodynamic field of N interacting vibrators (two‐dimensional theory). Geophysics50, 1229–1252.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1111/j.1365-2478.1988.tb02149.x
Loading
  • Article Type: Research Article

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error