GROUND MOTIONS DUE TO BURIED SOURCES

Abstract

The problem of determining the response of the earth to buried sources of elastic waves is of primary importance in seismology and seismic prospecting. The problem was first considered by H.Lamb in a classic paper entitled 'on the Propagation of Tremors over the Surface of an Elastic Solid'. Since then a number of investigators have focussed their attention on various aspects of this problem, and have tried to obtain analyti cal as well as numerical solutions thereof. The present work deals with the response of an elastic solid half space to buried point sources of strike slip and dip slip type. The treatment has two novel features. Firstly, generalized elastic wave potentials have been constructed for representing multipolar seismic sources including those which simulate an earthquake source in an infinite elastic medium, in a manner simpler than hitherto used. These potentials are then used to obtain the formal solution to the boundary value-initial value problem for a point source of strike slip and dip slip type in a solid elastic half space using the method of images. The second novelty appears in the method of evaluating the integrals of the formal solution. An application of this method of evaluation to a problem of interest in seismic prospecting is also discussed.

From the seismological point of view, the elastic wave sources of practical interest are explosions and earthquakes. The former has generally been represented as a point source of spherical waves of the compressional type. However, in the case of an earthquake source there is no such simple solution. An equivalence has been shown to exist between an elastic dislocation representing an earthquake and a double couple of forces without moment in so far as their effects are concerned, both for static and dynamic cases. Thus two alternative methods of tackling buried source problems have been followed-: one based entirely on the theory of dislocations and the other making use of body force equivalents of elastic dislocations. In the present work a given source is conceived as a suitable set of discontinuities in components of stress or displacement across a suitable surface through the source. For example, a single point force acting in an infinite, homogeneous, isotropic and perfectly elastic medium is representable as a discontinuity in the normal stress component across a plane perpendicular to the force and passing through its point of action. Mathematical representations of such discontinuities are built up from P,SH and SV potentials satisfying the homogeneous equations of motions for such a medium. Formal solutions for displacement on the surface of an elastic solid half space in response to buried point sources of strike slip and dip slip type have been obtained in the form of integrals. These integrals have been evaluated approximately by the method of contour integration. The poles of the integrands give rise to surface wave terms in the solutions. For evaluating each branch line integral an approximation has been made through which square roots of the sum or difference of two terms have been represented by the first two terms of their binomial expansions. These branch line integrals yield waves which have travelled most of the distance between the source and receiving point as P or S waves. In the present case four kinds of body wave arrivals have been obtained- (i) a direct P wave, (ii) a direct S wave, an SP wave, starting as an S wave from the source, meeting the free surface at a critical angle i = sin (j}/a) > where a and p are velocities of P and S waves in the half space, and then travelling just below the free surface to the receiving point as a P wave, (iv) an inhomogeneous wave travelling with the S-wave velocity. Displacements in the time domain have been obtained for an impulsive source. Theoretical seismograms have been computed for a few representative cases. The problem of obtaining the response of a layered half space to near surface sources of elastic waves has much wider application to practical problems, especially in seismic prospecting. In the present work a simple problem of this kind has been taken up. The branch line integrals occurring in the solution to the problem of propagation of explosive sound in a two layered half space have been evaluated employing the same method of contour integration as used earlier. This yields characteristics of head wave arrivals at distances large compared with the thickness of the first layer. The results of this study are the same as obtained by earlier workers.