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 agation is arrested at r=an. 228 T. YAMASHITA constant value υj(j=0,…,n-1) in the range aj 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. In Eqs. (2.1) and (2.2) the following relations are assumed: (2.3) where υ(x) is the crack tip velocity at location x. The semi-infinite two-djmensi onal cracks were assumed to be located along the x-axis in a Cartesian coordinate system xy. In Eqs. (2.1) and (2.2), μ is the rigidity, ω is the angular frequency, i is the imaginary unit, and VP, VS, and VR are velocities of compressional, shear, and Rayleigh waves, respectively. The crack tip velocity was assumed to make sudden changes from υj to υj+1 at x=dj (j=1,…,n), and the crack tip propagation was assumed to be arrested at x=dn, so that υn+1≡0. K2(dj) and K3(dj) are the static stress intensity factors, associated with plane strain and longitudinal shear cracks, respectively, that would be caused if the crack tips stopped to propagate at x= dj. Acceleration spectrum a*c(x,y;ω)(c=x,y or z) is connected with the accelera- tion time history in the form (2.4) It should be noted that the expressions (2.1) and (2.2) were obtained as the leading terms in the asymptotic expansions for large ω. 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 high- frequency 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- sional radiation, replacing the cylindrical geometrical spreading factor rj-1/2, in Eqs. (2.1) and (2.2), by rj-1/2(1+rj/ρ)-1/2, where ρ=aj/sinθ, θ is the angle measured from the z-axis (see Fig. 2, and see Fig. 1 as to aj). Only the far-field radiation will be considered in the present paper, so that rj-1/2(1+rj/ρ)-1/2〓 ρ-1/2R-1, where is the distance from the nucleation point of the rupture. Since all 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 T. YAMASHITA Fig. 2. Coordinate system. (2.5.2) where ν is Poisson's ratio, ψ is the angle measured from the x-axis on the xy-plane (see Fig. 2), and SH and SV denote the contributions from SH and SV waves, respec- tively. 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 the radius aj of modes II and III,respectively. According to KASSIR and SIH (1975,p.24) those are describedin the forms (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 ob- servable frequency range is required (HANKS, 1979). We will treat the accelera- tion in the frequency domain. The radiation at the high-frequency limit was ob- tained in Eqs. (2.5). The radiation at the low-frequency limit will be well approxi- mated by one from a point-source dislocation. Then, in the determination of ac- celeration 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, (2.7) SATOand 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*SHand a*SV,are given by Eqs. (2.5). The expression for (2.8) as the θ- and ψ-components of radiated amplitude spectrum (AKI and RICHARDS, 1979, p.84), where ρ is the density and M0 is the seismic moment. The far-field approximation is employed in deriving Eqs. (2.8), and the displacement-disconti- nuity 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 M0=16a3σ0/7. (2.9) (ESHELBY,1957). If we substitute into Eq. (2.7), we have the expression for (1) Below what frequencycan thelow-frequency asymptotic solution (2.10) (AKI and RICHARDS, 1979, p.805) in the far-field, where Σ is the fault surface, f*(x,ω) is the Fourier transform of the time function for displacement disconti- nuity, x is the location vector of the surface element dΣ measured from the origin, ν is the unit vector pointing to the receiver from the origin and k=ω/Vs. As stated above f(x,t) is assumed in the form (2.11) where υ(s)=υj in the range ja/n ω, the expression for Ω reduces to a rather simple form, and its envelope-amplitude is written as Peak and Root-Mean-Square Accelerations 233 (2.12) This is obtained by the asymptotic evaluation of Fourier integrals and by taking only the leading term; see YAMASHITA (1983). Although the envelope-amplitude of acceleration spectrum radiated by the source model in Fig. 1 is independent of ω at high-frequency limit, it is proportional to when the source time function is 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 Fig. 3. Dependence of spectral amplitude of acceleration on the number of discoritinui- ties in rupture velocity, n; θ is defined in Fig. 2. The source function is described by Eq. (2.11). The broken lines denote the asymptotic solutions valid at ω → ∞. The curve parameters in the figure denote the value of n. 234 T. YAMASHITA Fig. 4. Threshold angular frequency above which the asymptotic solution given by Eq. (2.12) is regarded as valid. n is the number of discontinuities in rupture velocity. Numerals in the figure denote the value of θ. In the numerical calculation, the threshold angular frequency cannot be identified at a single frequency, so that it is shown as a probable range. The solid lines with the letters A and B denote the lines ω=2πVs n/a and ω=10Vs n/a, respectively. Fig. 5. Models for spectral amplitude of acceleration in the intermediate frequency range, ωc<ω<ωn. The high-frequency and low-frequency asymptotic solutions are assumed to be valid for ω>ωn and ω<ωc, respectively. Refbr to the text for ωc and ωn. approximation to the spectral amplitude. This figure suggests that the equation ω=2πVsn/a(≡ ωn) gives a good approximation for the threshold angular frequency, so that the expression quency range (see Fig. 5). Model 1 assumes that and has a discontinuous change at ω=ωn. The results from dislocation approach support Model 2. Although these models are rather arbitrarily chosen, many seismic observations suggest that seismic wave spectra do not show much devia- tion from these two models. 3. Expression for Root Mean Square Acceleration According to HANKS(1979), the rms-acceleration arms is written as (3.1) where a(t) is the acceleration time history associated with shear waves, recorded on a single horizontal component of accelerograph, a*(ω) is its Fourier spectrum, ωmax is the maximum angular frequency observed on a recording device, and Td is 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) where (3.3) For Td, too, we consider the value averaged over a focal sphere. Upon follow- ing the procedure in Eq. (2.7), we obtain the expected value in the form (3.4) To simplify the analysis we make the following approximation (3.5) where <υ> is the value of υ averaged over the crack surface. If we assume <υ>/Vs= 236 T. YAMASHITA 0.6～0.9, as has been done in our calculation, the result is the above approximation. Studies of earthquake rupture processes suggest that observed values of rupture velocity are in this range (e.g., PURCARUand BERCKHEMER,1982). Upon substituting (3.2) and (3.5) into (3.1), we obtain the theoretical expres- sion for arms. We have, for Model 1, (3.6.1) in the case ωc<ωn<ωmax, and (3.6.2) in the case ωc<ωmax<ωn, where (3.7) The expression for PSH is obtained if |QSV(± θ)| in the expression for PSV is re- placed by |QSH(± θ)|. The above integrals are numerically evaluated. We have, for Model 2, (3.8.1) in the case ωc<ωn<ωmax, and (3.8.2) in the case ωc<ωmax<ωn. Peak and Root-Mean-Square Accelerations 237 4. Comparison between Theoreticaland Observed arms HANKS(1979) and MCGUIREand HANKS(1980) theoretically derived the ex- pression for arms, associated with the far-field SH wave, on the basis of BRUNE'S (1970) earthquake source model. MCGUIRE and HANKS(1980) and HANKSand MCGUIRE(1981) obtained the arms from the analysis of strong-motion records dur- ing 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. In Figs. 6 (a) and 7 (a), the dependence ofρarms/σ0 on the number of disconti- 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 (PURCARUand BERCKHEMER,1982). MCGUIREand HANKS(1980) and HANKSand 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 theoretical calculations we assume, fmax(=ωmax/2π)=25Hz, ρ=2.8g/cm3, and Vs=3.2km/sec. Hanks and McGuire also assumed these values for the analysis 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 (MCGUIREand HANKS,1980). The behavior of spectral amplitude is rather arbitrarily assumed in the inter- mediate frequency range, ωc<ω<ωn, in our study. In order to yield meaningful results the extent of this range should be much smaller than that of the whole integration range, ωmax. If the ratio (ωn-ωc)/ωmax is close to or larger than unity, then the calculated results will depend greatly on the assumption in the frequency range ωc<ω<ωn. Hence we will assume (ωn-ωc)/ωmax≪1 in the following. If the parameters assumed in the calculations of Figs. 6 (a) and 7 (a) are adopted, this inequality is satisfied when n≪90. It is seen in Figs. 6 (a) and 7 (a) that Models 1 and 2 yield nearly the same results in the range 1〓n〓10. Therefore we assume n in this range in the following calculations. 238 T. YAMASHITA The letters A, B, C, and D show the ranges of ρarms/σ0 harmonious with the distribution of arms observed near R=50km in the case of the 1971 San Fernando earthquake. According to MCGUIRE and HANKS (1980), the arms observed near R=50km is distributed in the range from 20 gals to 80 gals (see Fig. 4 in their paper, or Fig. 6 (b) or 7 (b) in the present paper). In the cases of A, B, C, and D, σ0=50, 100, 200, and 400 bars are assumed, respectively. The comparison be- tween our numerical results and the ranges A, B, C, and D suggests that the stress- Fig. 6. rms acceleration expected from Model 1. (a) Dependence of rms acceleration on the number of discontinuities in the crack tip velocity, n, Refer to the text for the ranges A, B, C, and D. Three kinds of distributions are assumed for crack tip velocity, and numerical results are illustrated by the three symbols ●, ○, and △. We denote, by the notation υ/Vs=a-b in the figure, that the crack tip velocity, υ, takes the value aVs in the range a2j tip velocity; Q=500 in the uppermost curve, Q=300 in the middle curve, and Q= 200 in the lowermost curve. (b) Attenuation of rms acceleration. Solid circles denote the rms accelerations observed by MCGUIRE and HANKS (1980) at the 1971 San Fernando earthquake. 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 crack tip velocity distribution υ/Vs=0.7-0.9 are assumed, then even a stress-drop 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 stress- drop of about 50 bars is correct. In Figs. 6 (b) and 7 (b) we plot the attenuation of arms observed in the case of the 1971 San Fernando earthquake, which is reproduced from Fig. 4 in MCGUIRE and HANKS(1980). On the observed data we superpose the theoretical attenua- tion 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 im- possible 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 T. YAMASHITA 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 HANKSand MCGUIRE(1981): the relation amaxand arms, derived by VANMARCKEand LAI (1980) using a stochastic approach, is employed. Accord- ing 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 amaxis easily obtained. We will use the data set on peak acceleration compiled by CAMPBELL(1981) and JOYNERand 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 ac- celeration. 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 logM0=1.5M+16.1, (5.2) so that the magnitude M is essentially the moment magnitude (HANKSand KANA- MORI,1979). However we also know the following empirical relations: logM0=1.5ML+16.0 (THATCHER and HANKS, 1973), (5.3) logM0=1.5MS+(16.1±0.1) (PURCARU and BERCKHEMER, 1978), 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 great- er than or equal to 6.0, and assumed ML when both magnitudes were below this value. JOYNERand 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 cal- culation 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. When υ/Vs=0.8 and 0.9, the stress-drops 100-200 bars and 50-100 bars well 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- ity as in the preceding section. The stress drop σ0 is assumed to be 50 and 100 bars Fig. 8. Attenuation and magnitude-dependence of peak acceleration. We assume n=1 and υ/Vs=0.8 in the theoretical calculations. The solid circles denote the observed peak acceleration compiled by CAMPBELL (1981) and JOYNER and BOORE (1981). The curve parameter denotes the value of σ0. Q=500 (the uppermost broken curve), Q=300 (the middle solid curve), and Q=200 (the lowermost bro- ken curve) are assumed in the cases σ0=50 bars and 400 bars. Only Q=300 is assumed in the cases σ0=100 bars and 200 bars. 242 T. YAMASHITA in one example υ/Vs=0.6-0.8, and is fixed at 50 bars in the other examples. Since 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 ac- celeration. It is seen in these figures that the trend of our observed data distribu- tion 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 discontinui- ties 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. Fig. 9. Attenuation and magnitude-dependence of peak acceleration. We assume n=1, υ/Vs=0.9, and Q=300 in the theoretical calculations. See the caption of Fig. 8 for the other notations. Peak and Root-Mean-Square Accelerations 243 244 T. YAMASKITA Peak and Root-Mean-Square Accelerations 245 246 T. YAMASHITA Peak and Root-Mean-Square Accelerations 247 6. Discussion and Conclusions We obtained the approximate expression for spectral amplitude of accelera- tion radiated by the discontinuously propagating circular crack. The expressions for root-mean-square and peak accelerations are obtained on the basis of the ex- pression 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 HANKSand 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 seis- mic high-frequency radiation. We assumed that the threshold angular frequency, ωn, above which the high- frequency asymptotic solution can be regarded as valid is given by the equation ωn=2πVsn/a. This equation has a physical meaning. However it will be shown that a slight change of coefficient of n does not cause much difference in the cal- culated accelerations if n is not very large. Figure 4 suggests that even ωn=10Vs n/a may be in harmony with the numerical results. The rms accelerations are calculated for both cases, ωn=2πVs n/a and ωn=10Vs n/a, and shown in Fig. 14. If n is smaller than 10, there is nearly no difference in the calculated value of ρarms/σ0. The threshold angular frequency, ωc, below which the low-frequency asympto- tic solution can be regarded as valid, is assumed to be given by the corner frequency of circular crack with constant crack tip velocity. Since ωc is generally far smaller than ωmax and the value of Fig. 14. Dependence of rms accelerationon the assumption of ωn. The upper and lower curves are expected from the assumptions ωn=2πVs n/a and ωn=10Vs n/a, respectively, for each example of the distribution of crack tip velocity. The distri- butions of crack tip velocity are the same as in Figs. 6 and 7, and Q=300 is as- sumed here. 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 rup- ture 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. 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