J. Phys. Earth, 31, 225-249, 1983 PEAK AND ROOT-MEAN-SQUARE ACCELERATIONS RADIATED FROM CIRCULAR CRACKS AND STRESS-DROP ASSOCIATED WITH SEISMIC HIGH-FREQUENCY RADIATION Teruo YAMASHITA Earthquake Research Institute, the University of Tokyo, Tokyo, Japan (Received July 25, 1983) We derive an approximate expression for far-field spectral amplitude of acceleration radiated by circular cracks. The crack tip velocity is assumed to make abrupt changes, which can be the sources of high-frequency radiation, during the propagation of crack tip. This crack model will be usable as a source model for the study of high-frequency radiation. The expression for the spectral amplitude of acceleration is obtained in the following way. In the high-frequency range its expression is derived, with the aid of geometrical theory of diffraction, by extending the two-dimensional results. In the low-frequency range it is derived on the assumption that the source can be regarded as a point. Some plausible assumptions are made for its behavior in the intermediate- frequency range. Theoretical expressions for the root-mean-square and peak accelerations are derived by use of the spectral amplitude of acceleration obtained in the above way. Theoretically calculated accelerations are compared with observed ones. The observations are shown to be well explained by our source model if suitable stress-drop and crack tip velocity are assumed. Using Brune's model as an earthquake source model, Hanks and McGuire showed that the seismic ac- celerations are well predicted by a stress-drop which is higher than the statically determined stress-drop. However, their conclusion seems less reliable since Brune's source model cannot be applied to the study of high-frequency radiation. According to our results, the seismic accelerations can be explained by a lower stress-drop, even by the same value of static stress-drop, if there are more abrupt changes in the crack tip velocity, the magnitude of its change is larger, or the crack tip velocity averaged over the crack surface is higher. Precise information about crack tip velocity is necessary to estimate the stress-drop associated with high-frequency radiation. 1. Introduction It is well known that low-frequency seismic waves provide useful information on earthquake source processes. Low-frequency seismic waves are usually an- alysed with the aid of a simple dislocation source model. Here, the source param- eters, rise time, rupture velocity and final displacement discontinuity, are assumed 225 226 T. YAMASHITA to be uniform on a fault plane. This kind of source analysis explains only the characteristics of earthquake rupture averaged over a fault plane. High-frequency seismic waves, especially acceleration waves, are not well explained by the above simple dislocation source model, since the radiation of high-frequency seismic waves has been shown to be closely related to the localized bursts on a fault plane (e.g., WALLACEet al., 1981; HARTZELLand HELMBERGER, 1982). The construction of a source model to explain high-frequency radiation is useful not only for investigating the fine structure of earthquake rupture, but also for predicting the strong ground motion. It is theoretically known that the radiation of high-frequency elastic waves is sensitive to the singularity at a propagating rupture tip (MADARIAGA,1977; YAMASHITA,1983). Therefore, it is necessary to make a physically plausible treat- ment for a rupture tip in the study of high-frequency radiation. This need suggests that fracture mechanics should be applied to the construction of a source model. Through linear fracture mechanics-influenced theoretical analysis (MADARIAGA, 1977, 1982; YAMASHITA,1983), the following two are known to be sources of high- frequency radiation: (1) Sudden changes in the velocity of rupture tip during rupture propagation; (2) Stress-drophaving a square-rootsingularity of the form, where r is a distance from some point, and A is a parameter independent of r. In these two cases, the highest high-frequency radiation is expected, and the envelope-amplitude of radiated acceleration spectral amplitude is independent of frequency at the high-frequency limit. Crack branching or sudden changes in the distribution of specific fracture energy can be the cause leading to phenomenon (1). Theoretical analysis in linear fracture mechanics shows that abrupt change in the crack tip velocity is caused by change in the spatial distribution of specific fracture energy (e.g., FREUND,1976). Experimental observations have shown that the crack normally branches and momentarily decelerates when a propagating crack in a brittle material reaches a terminal velocity (e.g., KOBAYASHIet al., 1974). Sudden start or dynamic coalescence of preexisting cracks will cause phenomenon (2) (MADARIAGA,1977, 1981). It is necessary to determine which one of (1) and (2) is the main cause of high- frequency radiation in the actual earthquake rupture processes. To do this, it is necessary to make a theoretical study of the properties of high-frequency elastic waves radiated by both source models. In the present paper, we study the far- field radiation from a three-dimensional crack model where the crack tip velocity undergoes sudden changes. With the aid of KELLER'S(1957, 1962) geometrical theory of diffraction, we derive the expression for high-frequency radiation from the three-dimensional crack model, extending the two-dimensional results obtained by YAMASHITA(1983). We specifically assume a circular crack, the simplest among the various three-dimensional fault models. We compare high-frequency radiation expected theoretically and actual ob- servation of seismic acceleration waves. As the observed data, we use the rms Peak and Root-Mean-Square Accelerations 227 (root mean square) acceleration radiated by the 1971 San Fernando earthquake, observed by MCGUIREand HANKS(1980), and the peak acceleration data compiled by CAMPBELL(1981) and JOYNERand BOORE(1981). The attenuations of both data associated with seismic acceleration waves prove to be fairly accurately predicted by our source model. HANKSand MCGUIRE(1981) showed, using BRUNE'S(1970) source model, that the rms accelerations were better predicted by stress-drop nearly equal to 100 bars than by static stress-drop. However the applicability of Brune's source model for high-frequency radiation is questionable from the viewpoint of fracture mechanics, and their estimates of stress-drop are less believable. In Brune's model, the sources of high-frequency radiation stated above are not taken into account. It will be shown in the present paper that precise rupture velocity information is necessary to estimate the stress-drop from the observation of seismic accelera- tion waves. 2. Radiation from a Circular Crack 2.1 Outline of source mode' A planar circular crack is assumed to be an earthquake source model (Fig. 1). The crack is located in a homogeneous, isotropic, linearly elastic solid, and the rupture is nucleated at the origin of coordinates. The velocity of crack tip is as- sumed to make sudden changes, which are regarded as the sources of high-frequency radiation, during its propagation. The rupture front is assumed to form a circle at all times, and the crack tip velocity makes a discontinuous change simultaneously over the rupture front at r=aj (j=1,…,n), where an≡a (final radius of crack surface) and r is the radial distance, on the crack surface, from the nucleation point of rupture. For the sake of simplicity, the crack tip velocity is assumed to take a Fig. 1. Earthquake source model. Rupture surface is located on the xy-plane. The crack tip velocity makes abrupt changes at (j=1,…,n), and takes constant values υj in the range aj<r<aj+1 (j=0,…,n-1). The crack tip prop- agation is arrested at r=an. 228 T. YAMASHITA constant value υj(j=0,…,n-1) in the range aj<r<aj+1, and aj+1-aj to be con- stant, where a0≡0. The stress component pzx is assumed to be released, by the amount σ0, at the passage of the rupture front. We will consider only the shear wave radiation, which plays a primary role in the strong ground motion. If the analytical expression for the displacement discontinuity at the source is known, then the radiation of elastic waves is fairly easily calculated with the aid of the representation theorem (DE HOOP, 1958). However when we assume the dynamic stress-drop on the crack surface, it will be impossible to obtain the expres- sion for displacement discontinuity in a simple form if the crack tip propagation is arrested at some time; the dynamic stress-drop is usually assumed instead of dis- placement discontinuity in a fracture-mechanics approach. Thus numerical studies are often made in the problem of propagation of three-dimensional finite crack (e.g., MADARIAGA,1976). Numerical work is, however, expensive and it has inherent inaccuracies due to discretization, so that it is inadequate for the study of high- frequency radiation. In the present paper we will obtain the expression for high- frequency acceleration radiated by the crack, illustrated in Fig. 1, extending the two- dimensional results derived by YAMASHITA(1983). KELLER'S(1957, 1962)geometri- cal theory of diffraction is employed in the extension. MADARIAGA(1977) and ACHENBACHand HARRIS(1978) also used this approach to derive the expressions for high-frequency radiation from three-dimensional sources. 2.2 High-frequency radiation from the two-dimensional crack YAMASHITA(1983) derived the expressions for near-source high-frequency radi- ation, associated with sudden changes in crack tip velocity, from two-dimensional semi-infinite cracks. The expressions for shear-wave radiations are reproduced here. The longitudinal shear crack radiates the high-frequency acceleration spec- trum given by (2.1) The x- and y-components of high-frequency acceleration spectrum radiated by the plane strain shear crack are (2.2.1) and (2.2.2) Peak and Root-Mean-Square Accelerations 229 respectively.