Deformation of a homogeneous Earth model by finite dislocations

Abstract

The deformation of a homogeneous, nongravitating, elastic sphere induced by a finite displacement dislocation inside the sphere is obtained in the form of rapidly converging series for arbitrary value of the Poisson ratio. Surface displacements, tilts, strains, and stresses are computed for (1) strike-slip and (2) dip-slip source models with various vertical extents. The numerical results are mapped on an epicentral tangential plane. It is estimated that a strike-slip fault 600 km long extending from the free surface to a depth of 100 km and with a dislocation of 20 meters would yield a strain of the order of 10^≄9 at an equatorial station with respect to the source. The corresponding estimate for a dip-slip source is 10⁻¹¹. However, for deeper sources the difference in the estimated values of the strains decreases. Thus the seismic events such as the Alaskan earthquake of March 28, 1964, should produce strains large enough to be detected by modern extensometers and tiltmeters in the far field. Though the field for a strike-slip source is approximately proportional to the vertical extent of the source, the field for a dip-slip source varies as the square of the vertical extent. The problem of determining the number and position of nodal lines for a sphere is much more complicated than for a half-space. For example, it is known that for a half-space the vertical component of the displacement for strike-slip has one more nodal line than the source in an infinite medium. On the other hand, for a sphere there may be two more nodal lines or none at all, depending on the vertical dimensions of the source. Similarly, though the azimuthal component of the displacement for strike-slip has no nodal line when the source is in a half-space, there is always one nodal line when the source is in a sphere. The finiteness of the source may introduce a strong asymmetry in the field if the pole is not taken at mid-fault. Comparison of results for a sphere and for a half-space reveals two important facts for strike-slip. First, the absolute value of the ratio (sphere/half-space) of the displacements lies between zero and ~4; second, the sign of this ratio for the radial and the azimuthal components of the displacement is negative everywhere in the far field. The negative sign cannot be due to a different sign convention, because, as the point of observation approaches the epicenter, the ratio tends to the value +1. From the stress patterns it is concluded that on the basis of our model the stresses from one earthquake are not likely to trigger another one in the far field.