PEAK AND ROOT-MEAN-SQUARE ACCELERATIONS RADIATED FROM CIRCULAR CRACKS AND STRESS-DROP ASSOCIATED WITH SEISMIC HIGH-FREQUENCY RADIATION

PEAK AND ROOT-MEAN-SQUARE ACCELERATIONS RADIATED FROM CIRCULAR CRACKS AND STRESS-DROP ASSOCIATED WITH SEISMIC HIGH-FREQUENCY RADIATION 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 intermediatefrequency 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 accelerations 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. It is well known that low-frequency seismic waves provide useful information on earthquake source processes. Low-frequency seismic waves are usually analysed with the aid of a simple dislocation source model. Here, the source parameters, rise time, rupture velocity and final displacement discontinuity, are assumed 225

226 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., WALLACE et al., 1981; HARTZELL and 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 treatment 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 highfrequency radiation: (1) Sudden changes in the velocity of rupture tip during rupture propagation; 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., KOBAYASHI et 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 highfrequency 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 farfield 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 observation 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 MCGUIRE and HANKS (1980), and the peak acceleration data compiled by CAMPBELL (1981) and JOYNER and BOORE (1981). The attenuations of both data associated with seismic acceleration waves prove to be fairly accurately predicted by our source model. HANKS and 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 acceleration waves. 2. Radiation from a Circular Crack 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 assumed 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 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

228 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 expression 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 displacement 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 highfrequency radiation. In the present paper we will obtain the expression for highfrequency acceleration radiated by the crack, illustrated in Fig. 1, extending the twodimensional results derived by YAMASHITA (1983). KELLER'S (1957, 1962) geometrical theory of diffraction is employed in the extension. MADARIAGA (1977) and ACHENBACH and 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 radiation, 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 spectrum given by (2.1) The x- and y-components of high-frequency acceleration spectrum radiated by the plane strain shear crack are and (2.2.1) (2.2.2)

Peak and Root-Mean-Square Accelerations 229 respectively. In Eqs. (2.1) and (2.2) the following relations are assumed: (2.3) (2.4) 2.3 High-frequency acceleration radiated by the circular crack Let us consider wavelengths which are much shorter than the radius of the curvature of rupture front. In this case the elastodynamic field in the vicinity of rupture front can be regarded to be two-dimensional with components of both plane strain and longitudinal shear slips. The expression for the three-dimensional highfrequency radiation is easily derived with the aid of Eqs. (2.1) and (2.2) and KELLER'S (1957, 1962) results. As shown by Keller, we obtain the three-dimen- the other dynamic properties of two-dimensional waves remain valid, we may write the envelope-amplitude of spectral amplitude of high-frequency acceleration radiated by the circular crack in the form

230 Fig. 2. Coordinate system. where (2.5.2) Fig. 2), and SH and SV denote the contributions from SH and SV waves, respectively. The SH waves have motion parallel to the edge of the fault and SV waves have motion in a plane perpendicular to the edge. Poisson's ratio is assumed to be 0.25 in the present paper. In the derivation of Eqs. (2.5), K2(aj) and K3(aj) are regarded as the stress-intensity factors associated with a static circular crack with (2.6) 2.4 Envelope-amplitude of spectral amplitude of acceleration As stated in the introduction we will specifically study the rms acceleration and

Peak and Root-Mean-Square Accelerations 231 the peak acceleration. When we calculate the rms acceleration, knowledge of acceleration time history or of spectral amplitude of acceleration in the whole observable frequency range is required (HANKS, 1979). We will treat the acceleration in the frequency domain. The radiation at the high-frequency limit was obtained in Eqs. (2.5). The radiation at the low-frequency limit will be well approximated by one from a point-source dislocation. Then, in the determination of acceleration spectrum, there remains arbitrariness only in the intermediate frequency range below and above which the low- and high-frequency asymptotic solutions are valid, respectively. Some simple assumptions will be made for the acceleration spectrum in the intermediate frequency range. We will consider the envelope-amplitude of acceleration spectral amplitude instead of spectral amplitude itself, which will greatly simplify the analysis. In this subsection the vectorial sum of two perpendicular shear-wave components averaged over a focal sphere, <a*s>, is investigated. If the stations are randomly distributed on a focal sphere, the spectral amplitude of acceleration is, on the average, written as (2.7) SATO and HIRASAWA (1973) employed this averaging procedure to obtain the expected value of corner-frequency. In the high-frequency range, where Eqs. (2.5) are valid, we have the equality, where a*sh and a*sv, are given by Eqs. (2.5). The expression for <a*s>, thus obtained, is denoted as <a*s>h. At the low-frequency limit a seismic source can be well approximated by a point-source dislocation. Thus we have (2.8) approximation is employed in deriving Eqs. (2.8), and the displacement-discontinuity is assumed to have a non-zero component only in the x-direction (see Fig. 2). In our circular seismic source model the seismic moment should have the value (ESHELBY, 1957). If we substitute into Eq. (2.7), we have the expression for <a*s> in the low-frequency range. The expression, thus obtained, is In this way we obtain the expressions for the envelope-amplitude of spectral amplitude of acceleration both at low- and high-frequency limits. The following problems remain.

232 (3) In what form is the envelope-amplitude described in the intermediate frequency range? It is plausible to assume that the expression <a*s>l is valid below the frequency at which the low- and high-frequency asymptotes intersect, where n=1 is assumed for <a*s>h. This frequency is nothing but the corner frequency of crack with constant crack tip velocity. The angular frequency at this intersection point is here- who solved the radiation from a circular crack with constant crack tip velocity by the finite difference method. If we observe the source process through very lowfrequency waves, the rupture velocity will be seen to be constant over the fault plane. It will take the value averaged over the fault plane. Hence the crack tip It is physically plausible to assume that the expressions (2.5) are valid for wavelengths smaller than a/n. Let this be confirmed in the investigation of radiation from a circular dislocation source which has the same rupture velocity as in Fig. 1. A step function is assumed for its source time function and constant displacement discontinuity is assumed over the fault plane. If the displacement discontinuity is known, the far-field radiation is obtained in a simple form. Although the property of this dislocation source is considerably different from our crack model in Fig. 1, it may be possible to get insight into a problem like the present one. The acceleration spectrum of radiated shear waves is known to be proportional to (2.10) (2.11)

Peak and Root-Mean-Square Accelerations 233 (2.12) described by Eq. (2.11). This is due to the difference in the singularity at the tip of displacement discontinuity (YAMASHITA, 1983). We compare, in Fig. 3, the high-frequency envelope-amplitude given by Eq. (2.12) and the spectral amplitude given by Eq. (2.10). The integral in Eq. (2.10) is numerically evaluated. We show in Fig. 4 the threshold frequency above which Eq. (2.12) becomes a good

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Peak and Root-Mean-Square Accelerations 235 support Model 2. Although these models are rather arbitrarily chosen, many seismic observations suggest that seismic wave spectra do not show much deviation from these two models. 3. Expression for Root Mean Square Acceleration According to HANKS (1979), the rms-acceleration arms is written as (3.1) the duration of shear-wave strong motion at the station. We will compare the theoretical arms-value expected from our crack model with the observed one. We expect the acceleration radiated from our source model to be observed on a single horizontal component of accelerograph in the form (3.2) (3.2) accounts for approximate free-surface amplification of shear waves, and the factor vector partition into two horizontal components of equal magnitude. These are the same assumptions made by HANKS and MCGUIRE (1981). If the source model in Fig. 1 is assumed and the far-field approximation is made, then Td is written in the form (3.3) For Td, too, we consider the value averaged over a focal sphere. Upon following the procedure in Eq. (2.7), we obtain the expected value in the form To simplify the analysis we make the following approximation (3.4) (3.5)

236 Studies of earthquake rupture processes suggest that observed values of rupture velocity are in this range (e.g., PURCARU and BERCKHEMER, 1982). Upon substituting (3.2) and (3.5) into (3.1), we obtain the theoretical expression for arms. We have, for Model 1, (3.6.1) (3.6.2) (3.7) (3.8.1) (3.8.2)

Peak and Root-Mean-Square Accelerations 237 HANKS (1979) and MCGUIRE and HANKS (1980) theoretically derived the expression for arms, associated with the far-field SH wave, on the basis of BRUNE'S (1970) earthquake source model. MCGUIRE and HANKS (1980) and HANKS and MCGUIRE (1981) obtained the arms from the analysis of strong-motion records during California earthquakes. They compared the model arms estimate with those obtained from the recorded accelerograms, and determined the stress-drop related to the generation of seismic high-frequency radiation. They concluded that all the earthquakes they analysed have a stress-drop very nearly equal to 100 bars, which are rather higher than the usual static stress-drop. However, their conclusion is doubtful since the sources of high-frequency radiation are not included in Brune's source model as stated in the introduction of the present paper. In our source model, this effect is explicitly introduced so that our source model will yield more reliable results. Let us estimate the stress-drop related to the generation of seismic high-frequency radiation on the basis of our source model, and compare it with the result of Hanks and McGuire. nuities in the distribution of crack tip velocity, n, is shown, which is expected from our source model. Models 1 and 2 are assumed in Figs. 6 (a) and 7 (a), respectively and R=50km is assumed in each figure. Three kinds of distributions are assumed for the crack tip velocity, and the values 200, 300, and 500 are assumed for Q. Each of the assumed distributions of crack tip velocity has an average value in the observed range of earthquake rupture velocity (PURCARU and BERCKHEMER, 1982). MCGUIRE and HANKS (1980) and HANKS and MCGUIRE (1981) assumed Q=300 for California earthquakes. Since acceleration radiated by California earthquakes is also investigated in the present paper, Q=300 will be used hereafter. In the of California earthquakes. The frequency 25Hz corresponds to the nominal natural frequency of SMA-1 recording devices. We assume a=11.9km in Figs. 6 and 7, which seems to be appropriate as the source-radius of the 1971 San Fernando earthquake (MCGUIRE and HANKS, 1980). The behavior of spectral amplitude is rather arbitrarily assumed in the inter-

238

Peak and Root-Mean-Square Accelerations 239 drop is not easily determined from the observation of arms: information about the distribution of crack tip velocity is required. The observed arms can be explained by a rather high stress-drop of about 200 bars if n=1. However if n=10 and a of about 50 bars, which is nearly equal to the static one, determined by the usual method (MIKUMO, 1973) can satisfy the observation for Models 1 and 2. Then it may be stated that the rupture velocity made abrupt changes during the rupture propagation in the case of the 1971 San Fernando earthquake if the static stressdrop of about 50 bars is correct. In Figs. 6 (b) and 7 (b) we plot the attenuation of arms observed in the case of and HANKS (1980). On the observed data we superpose the theoretical attenuation curves expected from our source model. All theoretical curves in the figures seem to explain the observed attenuation. Thus it can be concluded that it is impossible to estimate the stress drop only from the information on observed arms. Fig. 7. rms acceleration expected from Model 2. See the caption of Fig. 6 for the other notations.

240 5. Peak Accelerations In the preceding section the observed arms-attenuation was compared with our theoretical results. However we now have a greater observed-data accumulation for peak acceleration than for root-mean-square acceleration. Therefore, more insight may be obtained from the comparison with peak accelerations. We will obtain the theoretical expression for peak acceleration, amax, using the same method as HANKS and MCGUIRE (1981): the relation amax and arms, derived by VANMARCKE and LAI (1980) using a stochastic approach, is employed. According to Vanmarcke and Lai, its relation is written as (5.1) where T0 is the predominant period of earthquake motion, and s0 is the duration of strong motion. We will assume T0=1/fmax and s0=td following Hanks and MCGUIRE. Since we already know the expression for arms, that for amax is easily obtained. We will use the data set on peak acceleration compiled by CAMPBELL (1981) and JOYNER and BOORE (1981). Data associated with California earthquakes are purposely used; by doing so, we will be able to assume the same parameters as adopted in the preceding section. Let us investigate whether our source model well predict the attenuation and earthquake-magnitude dependence of peak acceleration. In the calculation of radiation from our source model, the earthquake magnitude, M, is assumed to be determined, from the seismic moment, through the relation so that the magnitude M is essentially the moment magnitude (HANKS and KANA- MORI, 1979). However we also know the following empirical relations: (5.3) where ML is the local magnitude and Ms is the surface-wave magnitude. Then no matter which magnitude scale among the above M, ML, and Ms is assumed, there is not a large difference. CAMPBELL (1981), in the compilation of data, assumed Ms as the scale of earthquake magnitude when both Ms and ML were greater than or equal to 6.0, and assumed ML when both magnitudes were below this value. JOYNER and BOORE (1981) assumed the moment magnitude. We used the peak-acceleration data observed in the distance-range R=28-50km in the investigation of its dependence on the magnitude, and R=40km was assumed for comparison with the observed data, in the theoretical calculation. In the investigation of attenuation of peak acceleration we use the observed data whose

Peak and Root-Mean-Square Accelerations 241 magnitudes were from 6.0 to 6.9, and M=6.5 was assumed in the theoretical calculation of radiation. The above distance- and magnitude-ranges were assumed mainly because the observed data appeared to be concentrated in those ranges. First, let us assume the crack tip velocity constant on the crack surface, and examine whether our source model explains the observed peak acceleration. The comparison between theoretical results and observed data is made in Figs. 8 and 9. explain the data distribution, respectively. Next the crack tip velocity is assumed to make discontinuous changes during the crack tip propagation. The same distributions are assumed for crack tip veloc-

242 the acceleration is directly proportional to stress-drop in our source model, the effect of stress-drop is easily inferred from these calculated examples. In Figs. 10-13, we make comparisons between the theoretical results and the observed peak acceleration. It is seen in these figures that the trend of our observed data distribution is well predicted by our source model if suitable distribution is assumed for crack tip velocity and suitable value is assumed for stress-drop. Figures 10-13 suggest that a lower stress-drop can explain the observed data if more discontinuities exist in the crack tip velocity, or if the magnitude of discontinuity is larger, or the average value of crack tip velocity is higher. It will be impossible to estimate the stress-drop accurately by observating peak acceleration if we have no knowledge of the distribution of crack tip velocity.

Peak and Root-Mean-Square Accelerations 243

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Peak and Root-Mean-Square Accelerations 245

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Peak and Root-Mean-Square Accelerations 247 6. Discussion and Conclusions We obtained the approximate expression for spectral amplitude of acceleration radiated by the discontinuously propagating circular crack. The expressions for root-mean-square and peak accelerations are obtained on the basis of the expression for the spectal amplitude. The theoretical accelerations thus obtained were compared with the observed seismic accelerations. It was shown that our source model predicts the observations rather well if suitable stress-drop and the distribution of crack tip velocity are assumed. The main object of the present paper is to compare our results with those obtained by HANKS and MCGUIRE (1981). They concluded, using BRUNE'S (1970) source model, that the observed accelerations are better predicted by stress-drop higher than the static stress-drop. The difference between their treatment and ours lies in the assumption of earthquake source model, and all the other assumptions will be the same or similar. Although Brune's model is simple enough and very tractable, it is not employable in the study of high-frequency radiation. According to our source model, a lower stress-drop can explain the observed acceleration if a suitable distribution is assumed for crack tip velocity. Accurate information on crack tip velocity is unavoidable in estimating the stress-drop associated with seismic high-frequency radiation. culation of the integral (3.1). It is generally believed that the value of Q has the most crucial effect on the high-frequency radiation. We assumed three kinds of Q, that is Q=200, 300, and 500, in the calculations of rms accelerations in Figs. 6 (a) and 7 (a) and of peak accelerations in Fig. 8. These figures show that the variations of accelerations due to variations in the value of Q are fairly small in comparison to the scatter of observed data. Hence our result, obtained on the assumption of Q=300, will be approximately valid for any Q if Q is in the range 200<Q<500. In the present paper we assumed the discontinuity in crack tip velocity as the

248 source of high-frequency radiation. Although our results appear to be in harmony with the observations, it is not our intention to say that the abrupt change in rupture tip velocity is the main cause of seismic high-frequency radiation. What we want to emphasize is that the observed acceleration is not necessarily related to a high stress-drop. In order to study the cause of seismic high-frequency radiation, the property of high-frequency radiation associated with the square-root singularity of stress-drop should also be studied. The author is grateful to Prof. Ryosuke Sato, of the University of Tokyo, for his valuable suggestions and advice. This research was supported in part by the Science Research Fund of the Ministry of Education of Japan REFERENCES ACHENBACH, J.D. and J.G. HARRIS, Ray method for elastodynamic radiation from a slip zone of arbitrary shape, J. Geophys. Res., 83, 2283-2291, 1978. AKI, K. and P.G. RICHARDS, Quantitative Seismology, Vol. 1 and 2, W.H. Freeman and Company, San Francisco, 1979. BRUNE, J.N., Tectonic stress and spectra of seismic shear waves from earthquakes, J. Geophys. Res., 75, 4997-5009, 1970.

Peak and Root-Mean-Square Accelerations 249 CAMPBELL, K.W., Near-source attenuation of peak horizontal acceleration, Bull. Seismal. Soc. Am., 71, 2039-2070, 1981. DE HOOP, A.T., Representation theorems for the displacement in an elastic solid and their application to elastodynamic diffraction theory, D. Sci. Thesis, Technishe Hogeschool, Delft, 1958. ESHELBY, J.D., The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. R. Soc., A241, 376-396, 1957. FREUND, L.B., Dynamic crack propagation, in The Mechanics of Fracture, ed. F. Erdogan, pp. 105-134, the American Society of Mechanical Engineers, New York, 1976. along active crustal fault zones and the estimation of high-frequency strong ground motion, J. Geophys. Res., 84, 2235-2242, 1979. HANKS, T.C. and K. KANAMORI., A moment magnitude scale, J. Geophys. Res., 84, 2348-2350, 1979. HANKS, T.C. and R.K. MCGUIRE, The character of high-frequency strong ground motion, Bull. Seismol. Soc. Am., 71, 2075-2095, 1981. HARTZELL, S. and D.V. HELMBERGER, Strong-motion modeling of the Imperial Valley earthquakes of 1979, Bull. Seismol. Soc. Am., 72, 571-596, 1982. JOYNER, W.B. and D.M. BOORE, Peak horizontal acceleration and velocity from strong motion records including records from the 1979 Imperial Valley, California, earthquake, Bull. Seismol. Soc. Am., 71, 2011-2038, 1981. KASSIR, M.K. and G.C. SIH, Three Dimensional Crack Problems, Noordhoff, Leyden, 1975. KELLER, J.B., Diffraction by an aperture, J. Appl. Phys., 28, 426-444, 1957. KELLER, J.B., Geometrical theory of diffraction, J. Opt. Soc. Am., 52, 116-130, 1962. KOBAYASHI, A.S., B.G. WADE, W.B. BRADLEY, and S.T. CHIU, Crack branching in holomite- 100 sheets, Eng. Fract. Mech., 6, 81-92, 1974. MADARIAGA, R., Dynamics of expanding circular fault, Bull. Seismol. Soc. Am., 66, 639-666, 1976. MADARIAGA, R., High-frequency radiation from crack (stress drop) models of earthquake faulting, Geophys. J.R. Astron. Soc., 51, 625-651, 1977. MADARIAGA, R., A string model for the high frequency radiation from earthquake faulting, Proceedings of the USGS-NRC Workshop on Strong Motion, Lake Tahoe, October, 1981. MCGUIRE, R.K. and T.C. HANKS, RMS accelerations and spectral amplitudes of strong ground motion during the San Fernando, California, earthquake, Bull. Seismol. Soc. Am., 70, 1907-1919, 1980. MIKUMO, T., Faulting process of the San Fernando earthquake of February 9, 1971 inferred from static and dynamic near-field displacements, Bull. Seismol. Soc. Am., 63, 249-269, 1973. PURCARU, G. and H. BERCKHEMER, A magnitude scale for very large earthquakes, Tectonophysics, 49, 189-198, 1978. PURCARU, G. and BERCKHEMER, H., Quantitative relations of seismic source parameters and a classification of earthquakes, Tectonophysics, 84, 57-128, 1982. SATO, T. and T. HIRASAWA, Body wave spectra from propagating shear cracks, J. Phys. Earth, 21, 415-431, 1973. THATCHER, W. and T.C. HANKS, Source parameters of southern California earthquakes, J. Geophys. Res., 78, 8547-8576, 1973. VANMARCKE, E.H. and S.S.P. LAI, Strong-motion duration and rms amplitude of earthquake records, Bull. Seismol. Soc. Am., 70, 1293-1307, 1980. WALLACE, T.C., D.V. HELMBERGER, and J.E. EBEL, A broadband study of the 13 August 1978 Santa Barbara earthquake, Bull. Seismol. Soc. Am., 71, 1701-1718, 1981 YAMASHITA, T., High-frequency acceleration radiated by unsteadily propagating cracks and its near-source geometrical attenuation, J. Phys. Earth, 31, 1-32, 1983.