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Numerical study of diffraction of plane elastic waves by a finite crack with application to location of a magma lens

DEPARTMENT OF EARTH AND PLANETARY SCIENCES MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139

Abstract

Location, shape, and size of near-surface heterogeneities in the Earth may be determined by active or passive seismic diffraction experiments. An array of geophones would be set up to record near-field wave motion resulting from diffraction by the inhomogeneity. As an example of such inhomogeneities, we study the diffraction of plane P and S waves by a finite two-dimensional crack as a function of wavelength and direction of incident wave. We develop a method which can be used to solve for diffraction effects by empty and fluid-filled (zero viscosity) cracks. The magnitude of the effect of the fluid in the crack is governed by a parameter called the crack stiffness factor which is the ratio of bulk modulus of the fluid in the crack to the rigidity of the solid divided by the crack aspect ratio.

The diffraction problem is solved using the finite difference scheme developed by Madariaga (1976) to study dynamic crack propagation problems. It is found that a crack has little effect on wave motion for incident wave with wavelengths greater than 10 times the crack length. For wavelengths less than 5 times the crack length there is a complicated diffraction pattern. The amplitude and phase of both components of motion are affected by the crack. We find that the diffraction effect decreases as the crack stiffness factor increases. For wavelengths less than 2 times the crack length there is a region of reduced amplitude (shadow zone) behind an empty crack. Relative phase of two components of particle motion varies rapidly with position inside the shadow zone.

The results of the theoretical calculation are used to try to locate the edge of a magma lens in Kilauea Iki Crater, Hawaii. It is found that the ratio of vertical-to-horizontal displacement and the direction of rotation of initial particle motion measured by a seismic array located on the floor of the crater agree qualitatively with what the theory predicts. The edge of the magma found by this method shows excellent agreement with the one inferred from the distribution of seismic events originating in the crust of the lava lake (Aki et al., 1977).

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